introduction to quantitative analysis€¦ · introduction to quantitative analysis bus-221-qm...

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Introduction to Quantitative Analysis

Bus-221-QM Lecture 14

Course review

To accompany Quantitative Analysis for Management, Eleventh Edition, by Render, Stair, and Hanna

Course Overview

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Lecture TITLE 1 Break Even Analysis (Chapter 1) 2 Bayes Theorem

Regression Analysis (Chapter 2, not English) 3, 4, 5, 6 Decision Analysis (Chapter 3) 7, 8 Forecasting (Chapter 5) 9 Inventory Control (Chapter 6) 10, 11 Linear Programming (Chapter 7) 12 Project Management (Chapter 13) 13 Waiting Lines (Chapter 14)

The Quantitative Analysis Approach

1-3 Implementing the Results

Analyzing the Results

Testing the Solution

Developing a Solution

Acquiring Input Data

Developing a Model

Defining the Problem

Figure 1.1

Developing a Model

Quantitative analysis models are realistic, solvable, and understandable mathematical representations of a situation.

1-4

There are different types of models: $ Advertising

$ Sa

les

Y = b0 + b1X

Schematic models

Scale models

Acquiring Input Data

Input data must be accurate

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Data may come from a variety of sources such as company reports, company documents, interviews, on-site direct measurement, or statistical sampling.

Garbage In

Process Garbage

Out

How To Develop a Quantitative Analysis Model

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A mathematical model of profit:

Profit = Revenue – Expenses

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1-7

Expenses can be represented as the sum of fixed and variable costs. Variable costs are the product of unit costs times the number of units.

Profit = Revenue – (Fixed cost + Variable cost) Profit = (Selling price per unit)(number of units

sold) – [Fixed cost + (Variable costs per unit)(Number of units sold)]

Profit = sX – [f + vX] Profit = sX – f – vX

where s = selling price per unit v = variable cost per unit f = fixed cost X = number of units sold


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Expenses can be represented as the sum of fixed and variable costs and variable costs are the product of unit costs times the number of units

Profit = Revenue – (Fixed cost + Variable cost) Profit = (Selling price per unit)(number of units

sold) – [Fixed cost + (Variable costs per unit)(Number of units sold)]

Profit = sX – [f + vX] Profit = sX – f – vX

where s = selling price per unit v = variable cost per unit f = fixed cost X = number of units sold

The parameters of this model are f, v, and s as these are the inputs inherent in the model The decision variable of interest is X

Break Even Point

0 = sX – f – vX, or 0 = (s – v)X – f

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Companies are often interested in the break-even point (BEP). The BEP is the number of units sold that will result in $0 profit.

Solving for X, we have f = (s – v)X

X = f

s – v

BEP = Fixed cost

(Selling price per unit) – (Variable cost per unit)

Types of Probability Determining!objec've!probability):)! Rela.ve)frequency)

!  Typically)based)on)historical)data)

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P (event) = Number of occurrences of the event Total number of trials or outcomes

!  Classical or logical method !  Logically determine probabilities without

trials

P (head) = 1 2

Number of ways of getting a head Number of possible outcomes (head or tail)

Types of Probability

Subjec've!probability)is)based)on)the)experience)and)judgment)of)the)person)making)the)es.mate.)

! Opinion)polls)! Judgment)of)experts)! Delphi)method)

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Mutually Exclusive Events

Events)are)said)to)be)mutually!exclusive)if)only)one)of)the)events)can)occur)on)any)one)trial.)

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!  Tossing a coin will result in either a head or a tail.

!  Rolling a die will result in only one of six possible outcomes.

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Collectively Exhaustive Events

Events)are)said)to)be)collec.vely)exhaus.ve)if)the)list)of)outcomes)includes)every)possible)outcome.)

! Both)heads)and))tails)as)possible))outcomes)of))coin)flips.)

! All)six)possible))outcomes))of)the)roll))of)a)die.)

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OUTCOME OF ROLL PROBABILITY

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

Total 1

Three Types of Probabilities !  Marginal)(or)simple))probability)is)just)the)probability)of)a)

single)event)occurring.)P)(A))

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!  Joint probability is the probability of two or more events occurring and is equal to the product of their marginal probabilities for independent events.

P (AB) = P (A) x P (B) !  Conditional probability is the probability of event

B given that event A has occurred. P (B | A) = P (B)

!  Or the probability of event A given that event B has occurred

P (A | B) = P (A)

Revising Probabilities with Bayes� Theorem

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Posterior Probabilities

Bayes� Process

Bayes� theorem is used to incorporate additional information and help create posterior probabilities.

Prior Probabilities

New Information

Figure 2.4

General Form of Bayes’ Theorem

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)()|()()|()()|()|(

APABPAPABPAPABPBAP

!!+=

We can compute revised probabilities more directly by using:

where the complement of the event ; for example, if is the event “fair die”, then is “loaded die”.

AA

A!

= ! A

Random Variables

Discrete!random!variables)can)assume)only)a)finite)or)limited)set)of)values.))

Con'nuous!random!variables)can)assume)any)one)of)an)infinite)set)of)values.)

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A random variable assigns a real number to every possible outcome or event in an experiment.

X = number of refrigerators sold during the day

Probability Distribution of a Discrete Random Variable

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The students in Pat Shannon’s statistics class have just completed a quiz of five algebra problems. The distribution of correct scores is given in the following table:

For discrete random variables a probability is assigned to each event.

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Probability Distribution of a Discrete Random Variable

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RANDOM VARIABLE (X – Score)

NUMBER RESPONDING

PROBABILITY P (X)

5 10 0.1 = 10/100 4 20 0.2 = 20/100 3 30 0.3 = 30/100 2 30 0.3 = 30/100 1 10 0.1 = 10/100

Total 100 1.0 = 100/100

The Probability Distribution follows all three rules: 1. Events are mutually exclusive and collectively exhaustive. 2. Individual probability values are between 0 and 1. 3. Total of all probability values equals 1.

Table 2.6

Expected Value of a Discrete Probability Distribution

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( ) ( )∑=

=n

iii XPXXE

1

( ) )(...)( 2211 nn XPXXPXXPX +++=

The expected value is a measure of the central tendency of the distribution and is a weighted average of the values of the random variable.

where iX)( iXP

∑=

n

i 1)(XE

= random variable’s possible values = probability of each of the random variable’s

possible values = summation sign indicating we are adding all n

possible values = expected value or mean of the random sample

Variance of a Discrete Probability Distribution

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For a discrete probability distribution the variance can be computed by

)()]([∑=

−==n

iii XPXEX

1

22 Varianceσ

where iX)(XE

)( iXP

= random variable’s possible values = expected value of the random variable = difference between each value of the random

variable and the expected mean = probability of each possible value of the

random variable

)]([ XEXi −

Variance of a Discrete Probability Distribution

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A related measure of dispersion is the standard deviation.

2σVarianceσ ==where

σ= square root = standard deviation

Probability Distribution of a Continuous Random Variable

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Since random variables can take on an infinite number of values, the fundamental rules for continuous random variables must be modified.

!  The sum of the probability values must still equal 1.

!  The probability of each individual value of the random variable occurring must equal 0 or the sum would be infinitely large.

The probability distribution is defined by a continuous mathematical function called the probability density function or just the probability function.

!  This is represented by f (X).

Probability Distribution of a Continuous Random Variable

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Prob

abili

ty

| | | | | | | 5.06 5.10 5.14 5.18 5.22 5.26 5.30

Weight (grams)

Figure 2.6

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The Normal Distribution

The)normal!distribu'on)is)the)one)of)the)most)popular)and)useful)con.nuous)probability)distribu.ons.)))

! The)formula)for)the)probability)density)func.on)is)rather)complex:)

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2

2

2

21 σ

µ

πσ

)(

)(−−

=x

eXf

! The normal distribution is specified completely when we know the mean, µ, and the standard deviation, σ .


"  The normal distribution is symmetrical, with the midpoint representing the mean.

"  Shifting the mean does not change the shape of the distribution.

"  Values on the X axis are measured in the number of standard deviations away from the mean.

"  As the standard deviation becomes larger, the curve flattens.

"  As the standard deviation becomes smaller, the curve becomes steeper.

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| | |

40 µ = 50 60

| | |

µ = 40 50 60

Smaller µ, same σ

| | |

40 50 µ = 60

Larger µ, same σ

Figure 2.8


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µ Figure 2.9

Same µ, smaller σ

Same µ, larger σ

Using the Standard Normal Table

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Step 1 Convert the normal distribution into a standard normal distribution.

!  A standard normal distribution has a mean of 0 and a standard deviation of 1

!  The new standard random variable is Z

σµ−

=XZ

where X = value of the random variable we want to measure µ = mean of the distribution σ = standard deviation of the distribution Z = number of standard deviations from X to the mean, µ


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For example, µ = 100, σ = 15, and we want to find the probability that X is less than 130.

15100130 −

=−

=σµXZ

dev std 21530

==

| | | | | | | 55 70 85 100 115 130 145

| | | | | | | –3 –2 –1 0 1 2 3

X = IQ

σµ−

=XZ

µ = 100 σ = 15 P(X < 130)

Figure 2.10

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AREA UNDER THE NORMAL CURVE Z 0.00 0.01 0.02 0.03

1.8 0.96407 0.96485 0.96562 0.96638 1.9 0.97128 0.97193 0.97257 0.97320 2.0 0.97725 0.97784 0.97831 0.97882 2.1 0.98214 0.98257 0.98300 0.98341 2.2 0.98610 0.98645 0.98679 0.98713

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Step 2 Look up the probability from a table of normal curve areas.

!  Use Appendix A or Table 2.9 (portion below). !  The column on the left has Z values. !  The row at the top has second decimal

places for the Z values.

Table 2.9 (partial)

P(X < 130) = P(Z < 2.00) = 0.97725

The Exponential Distribution

! The)exponen'al!distribu'on)(also)called)the)nega've!exponen'al!distribu'on))is)a)con.nuous)distribu.on)oIen)used)in)queuing)models)to)describe)the).me)required)to)service)a)customer.))Its)probability)func.on)is)given)by:)

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xeXf µµ −=)(where

X = random variable (service times) µ = average number of units the service facility can

handle in a specific period of time e = 2.718 (the base of natural logarithms)

The Exponential Distribution

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time service Average1

value Expected ==µ

21Varianceµ

= f(X)

X

Figure 2.17 3-34

The Six Steps in Decision Making

1.  Clearly define the problem at hand. 2.  List the possible alternatives. 3.  Identify the possible outcomes or states of

nature. 4.  List the payoff (typically profit) of each

combination of alternatives and outcomes. 5.  Select one of the mathematical decision

theory models. 6.  Apply the model and make your decision.

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Types of Decision-Making Environments

Type)1: )Decision)making)under)certainty)!  The)decision)maker)knows!with!certainty)the)

consequences)of)every)alterna.ve)or)decision)choice.)

Type)2: )Decision)making)under)uncertainty)!  The)decision)maker)does!not!know)the)

probabili.es)of)the)various)outcomes.)Type)3: )Decision)making)under)risk)

!  The)decision)maker)knows!the!probabili'es)of)the)various)outcomes.)

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Decision Making Under Uncertainty

1.  Maximax (optimistic)

2.  Maximin (pessimistic)

3.  Criterion of realism (Hurwicz)

4.  Equally likely (Laplace)

5.  Minimax regret

There are several criteria for making decisions under uncertainty:

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Maximax Used to find the alternative that maximizes the maximum payoff.

!  Locate the maximum payoff for each alternative. ! Select the alternative with the maximum number.

STATE OF NATURE

ALTERNATIVE FAVORABLE MARKET ($)

UNFAVORABLE MARKET ($)

MAXIMUM IN A ROW ($)

Construct a large plant 200,000 –180,000 200,000

Construct a small plant 100,000 –20,000 100,000

Do nothing 0 0 0 Table 3.2

Maximax

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Criterion of Realism (Hurwicz) This is a weighted average compromise between optimism and pessimism.

! Select a coefficient of realism α, with 0≤α≤1. ! A value of 1 is perfectly optimistic, while a

value of 0 is perfectly pessimistic. ! Compute the weighted averages for each

alternative. ! Select the alternative with the highest value.

Weighted average = α(maximum in row) + (1 – α)(minimum in row)

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Minimax Regret Based)on)opportunity!loss)or)regret,)this)is)the)difference)between)the)op.mal)profit)and)actual)payoff)for)a)decision.)

! Create)an)opportunity)loss)table)by)determining)the)opportunity)loss)from)not)choosing)the)best)alterna.ve.)

! Opportunity)loss)is)calculated)by)subtrac.ng)each)payoff)in)the)column)from)the)best)payoff)in)the)column.)

! Find)the)maximum)opportunity)loss)for)each)alterna.ve)and)pick)the)alterna.ve)with)the)minimum)number.)

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Decision Making Under Risk

!  This)is)decision)making)when)there)are)several)possible)states)of)nature,)and)the)probabili.es)associated)with)each)possible)state)are)known.)

!  The)most)popular)method)is)to)choose)the)alterna.ve)with)the)highest)expected!monetary!value!(EMV).!!  This)is)very)similar)to)the)expected!value!calculated)in)the)last)

chapter.)

EMV (alternative i) = (payoff of first state of nature) x (probability of first state of nature) + (payoff of second state of nature) x (probability of second state of nature) + … + (payoff of last state of nature) x (probability of last state of nature)

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Expected Value of Perfect Information (EVPI)

•  EVPI places an upper bound on what you should pay for additional information.

EVPI = EVwPI – Maximum EMV

•  EVwPI is the long run average return if we have perfect information before a decision is made.

EVwPI = (best payoff for first state of nature) x (probability of first state of nature) + (best payoff for second state of nature) x (probability of second state of nature) + … + (best payoff for last state of nature) x (probability of last state of nature)

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Expected Opportunity Loss

! Expected!opportunity!loss)(EOL))is)the)cost)of)not)picking)the)best)solu.on.)

! First)construct)an)opportunity)loss)table.)! For)each)alterna.ve,)mul.ply)the)opportunity)loss)by)the)probability)of)that)loss)for)each)possible)outcome)and)add)these)together.)

! Minimum)EOL)will)always)result)in)the)same)decision)as)maximum)EMV.

! Minimum)EOL)will)always)equal)EVPI.

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Sensitivity Analysis !  Sensitivity analysis examines how the decision

might change with different input data. !  For the Thompson Lumber example:

P = probability of a favorable market

(1 – P) = probability of an unfavorable market

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Five Steps of Decision Tree Analysis

1.  Define the problem. 2.  Structure or draw the decision tree. 3.  Assign probabilities to the states of nature. 4.  Estimate payoffs for each possible

combination of alternatives and states of nature.

5.  Solve the problem by computing expected monetary values (EMVs) for each state of nature node.

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Structure of Decision Trees

•  Trees start from left to right. •  Trees represent decisions and outcomes in

sequential order. –  Squares represent decision nodes. –  Circles represent states of nature nodes. –  Lines or branches connect the decisions nodes

and the states of nature.

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Expected Value of Sample Information

EVSI = – Expected value

with sample information, assuming

no cost to gather it

Expected value of best decision without sample

information

= (EV with sample information + cost) – (EV without sample information)

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Calculating Revised Probabilities

!  Recall Bayes’ theorem:

)()|()()|()()|()|(

APABPAPABPAPABPBAP

!×!+××

=

where events two any=BA,

AA of complement=!

For this example, A will represent a favorable market and B will represent a positive survey.

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Utility Theory

! Monetary)value)is)not)always)a)true)indicator)of)the)overall)value)of)the)result)of)a)decision.)

! The)overall)value)of)a)decision)is)called)u'lity.!! Economists)assume)that)ra.onal)people)make)decisions)to)maximize)their)u.lity.)

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Regression Analysis

Multiple Regression

Moving Average

Exponential Smoothing

Trend Projections

Decomposition

Delphi Methods

Jury of Executive Opinion

Sales Force Composite

Consumer Market Survey

Time-Series Methods

Qualitative Models

Causal Methods

Forecasting Models Forecasting Techniques

Figure 5.1

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Qualitative Models

! Qualita've!models)incorporate)judgmental)or)subjec.ve)factors.)

! These)are)useful)when)subjec.ve)factors)are)thought)to)be)important)or)when)accurate)quan.ta.ve)data)is)difficult)to)obtain.)

! Common)qualita.ve)techniques)are:)! Delphi)method.)! Jury)of)execu.ve)opinion.)! Sales)force)composite.)! Consumer)market)surveys.)

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Qualitative Models

•  Delphi Method – This is an iterative group process where (possibly geographically dispersed) respondents provide input to decision makers.

•  Jury of Executive Opinion – This method collects opinions of a small group of high-level managers, possibly using statistical models for analysis.

•  Sales Force Composite – This allows individual salespersons estimate the sales in their region and the data is compiled at a district or national level.

•  Consumer Market Survey – Input is solicited from customers or potential customers regarding their purchasing plans.

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Time-Series Models

•  Time-series models attempt to predict the future based on the past.

•  Common time-series models are: –  Moving average. –  Exponential smoothing. –  Trend projections. –  Decomposition.

•  Regression analysis is used in trend projections and one type of decomposition model.

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Causal Models

! Causal!models)use)variables)or)factors)that)might)influence)the)quan.ty)being)forecasted.)

! The)objec.ve)is)to)build)a)model)with)the)best)sta.s.cal)rela.onship)between)the)variable)being)forecast)and)the)independent)variables.)

! Regression)analysis)is)the)most)common)technique)used)in)causal)modeling.)

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Measures of Forecast Accuracy

•  We compare forecasted values with actual values to see how well one model works or to compare models.

Forecast error = Actual value – Forecast value

!  One measure of accuracy is the mean absolute deviation (MAD):

n∑= errorforecast

MAD

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Measures of Forecast Accuracy

•  There are other popular measures of forecast accuracy.

•  The mean squared error:

n∑=

2error)(MSE

!  The mean absolute percent error:

%MAPE 100actualerror

n

∑=

!  And bias is the average error.

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Time-Series Forecasting Models

•  A time series is a sequence of evenly spaced events.

•  Time-series forecasts predict the future based solely on the past values of the variable, and other variables are ignored.

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Components of a Time-Series

A time series typically has four components: 1. Trend (T) is the gradual upward or downward

movement of the data over time. 2. Seasonality (S) is a pattern of demand fluctuations

above or below the trend line that repeats at regular intervals.

3. Cycles (C) are patterns in annual data that occur every several years.

4. Random variations (R) are “blips” in the data caused by chance or unusual situations, and follow no discernible pattern.

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Moving Averages

!  Moving averages can be used when demand is relatively steady over time.

!  The next forecast is the average of the most recent n data values from the time series.

!  This methods tends to smooth out short-term irregularities in the data series.

nnperiods previous in demands of Sum

forecast average Moving =

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Moving Averages

!  Mathematically:

nYYYF nttt

t11

1+−−

+

+++=

...

Where: = forecast for time period t + 1 = actual value in time period t n = number of periods to average tY1+tF

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Weighted Moving Averages !  Weighted moving averages use weights to put

more emphasis on previous periods. !  This is often used when a trend or other pattern is

emerging.

∑∑=+ )(

))((Weights

period in value Actual period inWeight 1

iFt

!  Mathematically:

n

ntnttt www

YwYwYwF++++++

= +−−+ ...

...21

11211

where wi = weight for the ith observation

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!  Exponen'al!smoothing)is)a)type)of)moving)average)that)is)easy)to)use)and)requires)liWle)record)keeping)of)data.)

New forecast = Last period’s forecast + α(Last period’s actual demand – Last period’s forecast)

Here α is a weight (or smoothing constant) in which 0≤α≤1.

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Mathematically:

)( tttt FYFF −+=+ α1

Where: Ft+1 = new forecast (for time period t + 1)

Ft = pervious forecast (for time period t) α = smoothing constant (0 ≤ α ≤ 1) Yt = pervious period’s actual demand

The idea is simple – the new estimate is the old estimate plus some fraction of the error in the last period.

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Exponential Smoothing with Trend Adjustment

•  Like all averaging techniques, exponential smoothing does not respond to trends.

•  A more complex model can be used that adjusts for trends.

•  The basic approach is to develop an exponential smoothing forecast, and then adjust it for the trend.

Forecast including trend (FITt+1) = Smoothed forecast (Ft+1) + Smoothed Trend (Tt+1)

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Exponential Smoothing with Trend Adjustment

•  The equation for the trend correction uses a new smoothing constant β .

•  Tt must be given or estimated. Tt+1 is computed by:

)()1( 11 tttt FITFTT −+−= ++ ββwhere

Tt = smoothed trend for time period t Ft = smoothed forecast for time period t

FITt = forecast including trend for time period t α =smoothing constant for forecasts

β = smoothing constant for trend

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Selecting a Smoothing Constant

!  As with exponential smoothing, a high value of β makes the forecast more responsive to changes in trend.

!  A low value of β gives less weight to the recent trend and tends to smooth out the trend.

!  Values are generally selected using a trial-and-error approach based on the value of the MAD for different values of β.

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Trend Projections

!  Trend projection fits a trend line to a series of historical data points.

!  The line is projected into the future for medium- to long-range forecasts.

!  Several trend equations can be developed based on exponential or quadratic models.

!  The simplest is a linear model developed using regression analysis.

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Trend Projection

The mathematical form is

XbbY 10 +=ˆ

Where = predicted value

b0 = intercept b1 = slope of the line X = time period (i.e., X = 1, 2, 3, …, n)

Y

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Seasonal Variations with Trend

!  When)both)trend)and)seasonal)components)are)present,)the)forecas.ng)task)is)more)complex.)

!  Seasonal)indices)should)be)computed)using)a)centered!moving!average)(CMA))approach.)

!  There)are)four)steps)in)compu.ng)CMAs:)1.  Compute)the)CMA)for)each)observa.on)(where)

possible).)2.  Compute)the)seasonal)ra.o)=)Observa.on/CMA)for)

that)observa.on.)3.  Average)seasonal)ra.os)to)get)seasonal)indices.)4.  If)seasonal)indices)do)not)add)to)the)number)of)

seasons,)mul.ply)each)index)by)(Number)of)seasons)/(Sum)of)indices).)

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The Decomposition Method of Forecasting with Trend and Seasonal Components

!  Decomposi'on)is)the)process)of)isola.ng)linear)trend)and)seasonal)factors)to)develop)more)accurate)forecasts.)

!  There)are)five)steps)to)decomposi.on:)1.  Compute)seasonal)indices)using)CMAs.)2.  Deseasonalize)the)data)by)dividing)each)number)by)its)

seasonal)index.)3.  Find)the)equa.on)of)a)trend)line)using)the)

deseasonalized)data.)4.  Forecast)for)future)periods)using)the)trend)line.)5.  Mul.ply)the)trend)line)forecast)by)the)appropriate)

seasonal)index.)

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Using Regression with Trend and Seasonal Components

! Mul'ple!regression)can)be)used)to)forecast)both)trend)and)seasonal)components)in)a).me)series.)!  One)independent)variable)is).me.)!  Dummy)independent)variables)are)used)to)represent)the)seasons.)

!  The)model)is)an)addi.ve)decomposi.on)model:)

where X1 = time period X2 = 1 if quarter 2, 0 otherwise X3 = 1 if quarter 3, 0 otherwise X4 = 1 if quarter 4, 0 otherwise

44332211 XbXbXbXbaY ++++=ˆ

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Monitoring and Controlling Forecasts

!  Tracking signals can be used to monitor the performance of a forecast.

!  A tracking signal is computed as the running sum of the forecast errors (RSFE), and is computed using the following equation:

MADRSFE

=signal Tracking

n∑= errorforecast

MAD

where

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Adaptive Smoothing

! Adap've!smoothing)is)the)computer)monitoring)of)tracking)signals)and)self^adjustment)if)a)limit)is)tripped.)

!  In)exponen.al)smoothing,)the)values)of)α)and)β)are)adjusted)when)the)computer)detects)an)excessive)amount)of)varia.on.!

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Importance of Inventory Control

•  Five uses of inventory: –  The decoupling function –  Storing resources –  Irregular supply and demand –  Quantity discounts –  Avoiding stockouts and shortages

•  Decouple manufacturing processes. –  Inventory is used as a buffer between stages in a

manufacturing process. –  This reduces delays and improves efficiency.

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Importance of Inventory Control

•  Storing resources. –  Seasonal products may be stored to satisfy off-

season demand. –  Materials can be stored as raw materials, work-in-

process, or finished goods. –  Labor can be stored as a component of partially

completed subassemblies. •  Compensate for irregular supply and demand.

–  Demand and supply may not be constant over time.

–  Inventory can be used to buffer the variability.

6-75

Economic Order Quantity

! The)economic!order!quan'ty)(EOQ))model)is)one)of)the)oldest)and)most)commonly)known)inventory)control)techniques.)

!  It)is)easy)to)use)but)has)a)number)of)important)assump.ons.)

! Objec.ve)is)to)minimize)total)cost)of)inventory.)

6-76

Economic Order Quantity Assump.ons:)

1.  Demand)is)known)and)constant.)2.  Lead).me)is)known)and)constant.)3.  Receipt)of)inventory)is)instantaneous.)4.  Purchase)cost)per)unit)is)constant)throughout)the)

year.)5.  The)only)variable)costs)are)the)cost)of)placing)an)

order,)ordering!cost,)and)the)cost)of)holding)or)storing)inventory)over).me,)holding)or)carrying!cost,)and)these)are)constant)throughout)the)year.)

6.  Orders)are)placed)so)that)stockouts)or)shortages)are)avoided)completely.)

6-77

Inventory Costs in the EOQ Situation

Mathematical equations can be developed using:

Q = number of pieces to order EOQ = Q* = optimal number of pieces to order

D = annual demand in units for the inventory item Co = ordering cost of each order Ch = holding or carrying cost per unit per year

Annual ordering cost = × Number of

orders placed per year

Ordering cost per

order

oCQD

=

6-78

Inventory Costs in the EOQ Situation

Mathematical equations can be developed using:

Q = number of pieces to order EOQ = Q* = optimal number of pieces to order

D = annual demand in units for the inventory item Co = ordering cost of each order Ch = holding or carrying cost per unit per year

Annual holding cost = × Average inventory

Carrying cost per unit

per year

hCQ2

=

11/16/12

14

6-79

Economic Order Quantity (EOQ) Model

hCQ2

cost holding Annual =

oCQD

=cost ordering Annual

h

o

CDC

Q2

== *EOQ

Summary of equations:

6-80

Purchase Cost of Inventory Items

•  Total inventory cost can be written to include the cost of purchased items.

•  Given the EOQ assumptions, the annual purchase cost is constant at D × C no matter the order policy, where –  C is the purchase cost per unit. –  D is the annual demand in units.

•  At times it may be useful to know the average dollar level of inventory:

2level dollar Average

)(CQ=

6-81

Purchase Cost of Inventory Items

•  Inventory carrying cost is often expressed as an annual percentage of the unit cost or price of the inventory.

•  This requires a new variable. Annual inventory holding charge as

a percentage of unit price or cost I =

!  The cost of storing inventory for one year is then

ICCh =

thus, ICDC

Q o2=*

6-82

Sensitivity Analysis with the EOQ Model

!  The)EOQ)model)assumes)all)values)are)know)and)fixed)over).me.)

!  Generally,)however,)some)values)are)es.mated)or)may)change.)

!  Determining)the)effects)of)these)changes)is)called)sensi'vity!analysis.)

!  Because)of)the)square)root)in)the)formula,)changes)in)the)inputs)result)in)rela.vely)small)changes)in)the)order)quan.ty.!

h

o

CDC2

=EOQ

6-83

Reorder Point: Determining When To Order

! Once)the)order)quan.ty)is)determined,)the)next)decision)is)when!to!order.!

! The).me)between)placing)an)order)and)its)receipt)is)called)the)lead!'me)(L))or)delivery!'me.)

! When)to)order)is)generally)expressed)as)a)reorder!point)(ROP).)

Demand per day

Lead time for a new order in days ROP = ×

= d × L 6-84

EOQ Without The Instantaneous Receipt Assumption

•  When inventory accumulates over time, the instantaneous receipt assumption does not apply.

•  Daily demand rate must be taken into account. •  The revised model is often called the production run

model.

Figure 6.5

11/16/12

15

6-85

Annual Carrying Cost for Production Run Model

!  In)produc.on)runs,)setup!cost)replaces)ordering)cost.)!  The)model)uses)the)following)variables:)

Q = number of pieces per order, or production run

Cs = setup cost Ch = holding or carrying cost per unit per

year p = daily production rate d = daily demand rate t = length of production run in days

6-86


Maximum inventory level = (Total produced during the production run) – (Total used during the production run) = (Daily production rate)(Number of days production)

– (Daily demand)(Number of days production) = (pt) – (dt)

since Total produced = Q = pt

we know pQt =

Maximum inventory

level !"

#$%

&−=−=−=pdQ

pQd

pQpdtpt 1

6-87


Since the average inventory is one-half the maximum:

!"

#$%

&−=pdQ

12

inventory Average

and

hCpdQ!"

#$%

&−= 1

2cost holding Annual

6-88

Annual Setup Cost for Production Run Model

sCQD

=cost setup Annual

Setup cost replaces ordering cost when a product is produced over time.

replaces

oCQD

=cost ordering Annual

6-89

Determining the Optimal Production Quantity

By setting setup costs equal to holding costs, we can solve for the optimal order quantity

Annual holding cost = Annual setup cost

sh CQDC

pdQ

=!"

#$%

&−1

2

Solving for Q, we get

!"

#$%

&−

=

pdC

DCQh

s

1

2*

6-90

Production Run Model

Summary of equations

!"

#$%

&−

=

pdC

DCQh

s

1

2 quantity production Optimal *

sCQD

=cost setup Annual

hCpdQ!"

#$%

&−= 1

2cost holding Annual

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