Location Theory & Decision Analysis
(c) Springer
Probability Distribution & Queuing Module
Name: __________________________
PROBABILITY DISTRIBUTION AND QUEUING MODULE
INTRODUCTION
One of the fundamental insights of the physical and social sciences in the 20th century
is the applicability of probability theory. For example, the probabilism of quantum
mechanics has replaced the determinism of Newtonian mechanics. Present day
analysts speak of deterministic models versus probabilistic models, This module,
entitled “Probability Distribution and Queuing,” is a continuation of the module on
probability. After working through this module, the reader should
a) understand the concepts of probability distributions
b) see the applicabilities of probability distributions in a system of queues.
c) gain some insights on using queuing theory as a decision-making tool.
PROBABILITY DISTRIBUTIONS
Definition. A variable whose value is a number determined by the outcome of an
experiment or process is called a random variable.
Everyone is familiar from high school algebra with x as the independent variable
and y as the dependent variable. Random variable is the name used for the variable,
usually represented by an upper case X, in probability theory.
Illustration 1)
Consider the experiment of tossing two pennies. We could ask this question: how many
of the coins show "head"? The possible outcomes of this experiment is: (HH) , (WT) ,
(TH) , (TT) . Readers not sure of this result please consult page one of Probability
Module. The answer to the question is: "head" appears either zero, or once, or twice. If
we let the random variable, X, represent the number of "head" showing in this
experiment, then X has the set of possible values: 0,1, 2.
We could also ask another question: what is the probability of the random variable,
X, to take on the value of 0 or 1 or 2? X has the value of “0" only when (TT) shows;
therefore the probability is ¼. If needed, the reader is advised to consult pages 5 and 6
of Probability Module. X has the value of "1" only when either (HT) or (TH) shows;
therefore the probability is ½. X has the value of "2" only when (HH) shows; therefore
the probability is ¼. The following table summarizes the experiment:
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possible outcomes
Probability Distribution & Queuing Module
number of heads
probability of outcome
(HH)
2
¼
(HT)
1
¼
(TH)
1
¼
(TT)
0
¼
random variable X = number of heads
associated probability of random variable X
0
¼
1
¼ (one head shows up twice in the possible
outcomes; therefore ¼ + ¼ = ½)
2
¼
Note that the sum of the probabilities is 1.
In shorter notation we write:
P(X=0) = ¼ This reads: the probability of the random variable equals zero is ¼
P(X=1) = ½
P(X=2) = ¼
From this illustration we see that a random variable, in addition to having values (in this
case 0, 1. 2) has an associated probability (this case: ¼, ½, ¼ respectively).
We could also graph the results of this experiment:
The heavy dots emphasize the functional relations, for example, at “0,” P(0) is ¼,
not 1/8 or 1/16, etc. The random variable X takes on the traditional independent
variable axis and P(X), the probability, takes on the traditional dependent-variable axis.
Note that X takes on only the values 0, 1, 2; therefore this is called a discrete probability
distribution. P(X) is sometimes called the probability function of the discrete random
variable X. In technical literature, P(X) is usual represented by f (X).
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Illustration 2)
Consider the experiment of tossing one die. Let the random variable X take on the
value of the number of dots showing on the top face of the die. The possible values of
X are: 1, 2, 3, 4, 5, 6. Complete the chart:
random variable X = number of dots showing
associated prob. of random variable X
1
1/6
2
1/6
3
(
)
4
(
)
5
(
)
6
(
)
Complete the discrete probability distribution graph:
Illustration 3)
Consider the experiment of tossing three coins. Let the random variable X take on the
values of the number of "heads" showing. The set of possible values for X is: 0, 1, 2, 3.
If needed, please consult page 3 of Probability Module. Complete the chart and graph:
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Consider the experiment of measuring the heights of every student on campus. The
random variable in this experiment is the measured heights. What is the probability of
finding a student exactly 6 feet tall? "Exactly 6 feet tall" means 6.0000000... feet tall.
The probability is zero because we have no chance at all of finding a student
6.0000000 ... feet tall. We may be able to find some students between the intervals
6.0000000 ... ft and 5.0000000 ... ft; or between the intervals 5.5000000... ft and
5.0000000... ft. This experiment illustrates the concept of continuous probability
distribution. It is called continuous because its random variable takes on continuous
numerical values and not discrete numerical values. Whereas in a discrete probability
distribution, of a random variable is represented by the height of the graph, for a
continuous probability distribution, we represent the probability as the area between
two intervals. The most famous continuous probability distribution is the bell curve,
technically called the normal distribution.
This graph could represent the heights of all males on campus. The average height
is 5'10". The probability of a male 5'10" is zero. The probability of a male between 5' 8"
and 6' is the area of the shaded region. One of the characteristics of a probability
distribution is the sum of the probabilities is one; therefore the total area under the
curve of the normal distribution is one. The reader should verify this characteristic for
the discrete distributions by going back to illustrations 1, 2, 3 and for each distribution,
sum up the probabilities for values of the discrete random variable. They should sum to
one.
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Consider this procedure. We measure the heights of 100 male students. We plot
the results into a histogram where the horizontal axis is the height groups and the
vertical axis is the number of male students. The step-wise discrete histogram can be
approximated by a continuous bell curve, as shown. The continuous curve can be
standardized into the curve shown on page 3 with an area of one.
Height group
5'4" - 5'6"
5'6" - 5'8"
5'8" - 5'10"
5'10" - 6'
6' - 6'2"
6'2" - 6'4"
Number of male students
5
15
30
30
15
5
We could ask this question: what is the probability of picking out from this 100 male
student group a student between 5'8" and 5'10"? The area of this height group on the
histogram divided by the total area of the histogram gives the probability of picking one
student with the height. The answer is 30 / 100 = 0.3
THREE FREQUENTLY USED DISTRIBUTIONS
In this section, we study three frequently used distributions: the binomial, the Poisson,
the exponential. The binomial distribution is one of the most frequently used discrete
probability distributions. This distribution is frequently used to model experiments or
processes in which events occur either "success" or "failure.” There are four
characteristics of a binomial distribution model:
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1) Each trial of the experiment falls into one of two categories: either "success" or
"failure.”
2) There are a fixed number of trials in the experiment. Call the number “n.”
3) The outcomes of the trials are independent of one another. For example, getting a
"head" on tossing a penny will not "influence" the next toss of the penny to get either
a "head" or “tail.”
4) For each trial, the probability of a success is the same constant, p. The binomial
n
 x
distribution, in functional form, is: P(x) =   p x (1 & p) n &x
where X is discrete and is the numerical value of the random variable, and
n
 x  = n! / [x! (n&x)!]
 
n
 x
Notice   is not the same as n / x.
Illustration 4)
Let us use the binomial distribution to model the tossing of a penny. "n" is the number
of times we toss the penny. Let X be the random variable taking on the value 0 when no
head shows, 1 when a head shows, 2 when 2 heads show, 3 when 3 heads show, etc.
We call "a head shows on a toss" a success; therefore p = ½.
a) What is the probability of getting a head in one toss?
n = 1, X = 1, p = ½
 1
 1
P(X =1) =   (1/2)1 (1/2)0
We expect this result from experience: the probability of getting a head on one toss is
½.
b) What is the probability of getting one head in two tosses? Just one head, not two
heads.
n = 2, X = 1, p = ½
2
1
P(X=1) =   (1/2)1 (1/2)1 = 2 x ½ x ½ = ½.
c) What is the probability of getting exactly two heads in five tosses?
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n = 5, X = (
Probability Distribution & Queuing Module
), p = ½
 5
P(X=
) =   (1/2)(
)
 
(1/2)(
)
d) What is the probability of getting exactly five heads in five tosses?
Illustration 5)
The Peekskill Precision Glass Co. is having trouble because only 90% of its glass is
non-defective,
a) If we randomly 10 glass products, what is the probability of finding exactly eight nondefective products?
n=(
), X = (
P(X=
)=
), p = 0.9
( ) (0.9)(
)
(
)(
)
=(
)
b) If we select 10 glass products, what is the probability of finding exactly nine nondefective products?
n=(
), X = (
P(X=
)=
), p = 0.9
( ) (0.9)(
)
(
)(
)
=(
)
In these two examples, the random variable is the number of non-defective products, 8
and 9 respectively.
Another discrete distribution that's frequently used to model events is the Poisson
distribution, named after its discoverer, the 19th century French mathematician
Poisson, The Poisson distribution has the form
P(X) = e&m m x / x!
where X is the discrete random variable taking on the values: 1, 2, 3, 4, ... . m is a
positive number representing the average of the occurrence of the modeled event. e is
the base of the natural log, 2.718 ....
This distribution is frequently used to model the occurrences of events in a fixed
period of time or space. For example, the Poisson can be used to model:
• The number of wrong numbers dialed in a telephone exchange in a one-minute
period.
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•
•
•
Probability Distribution & Queuing Module
The number of defects in a newly manufactured airplane.
The number of cars passing a point in a road in an one minute period.
The number of customers arriving at a checkout counter in an one minute period.
Illustration 6)
Suppose a checkout counter at a supermarket has an average of two customer arrivals
per five-minute interval.
a) What is the probability of one customer arrival per five minute?
b) What is the probability of four customer arrivals per five minute?
a) The random variable, X, is the number of customers arriving in the 5-minute interval.
m = 2, the average in a 5-minute interval.
X=1
P (X=1) = (e&2) (21) / (1!) = (e&2) (2) = 0.27
b) m = 2, the average in a 5-minute interval; X = 4.
P (X = 4) = (e&2) (2 4) / (4!) = (e&2) (2) = 0.09
Illustration 7)
Suppose at a toll booth the average arrival of cars is one per one minute interval.
a) What is the probability of two cars arriving in a one-minute interval?
m = l; X = ( )
P(X= ) = (e &1) (1(
)
)/(
!) = (
)
b) What is the probability of three cars arriving in a one-minute interval?
c) What is the probability of four cars arriving in a one-minute interval?
This exercise illustrates our common sense expectation: if on the average one car
arrives per one-minute interval, then the probability would decrease if we expect two
cars to arrive, or three cars to arrive, or four cars to arrive, and so on.
The third probability distribution we examine is the exponential distribution:
P(X) = a e &a x
where x is greater than or equal to zero. x can take on any values equal to or greater
than zero; thus this is a continuous probability distribution. "e" is the base of the natural
log, “1/a” is the average value of the random variable X.
A more useful form of the exponential distribution is:
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P(X # t) = 1 & e &a x
This equation is read: the probability of the random variable taking on values less
than or equal to time t is 1 & e &a x.
Illustration 8)
At a service counter, the average interarrival time of customers is five minutes.
Interarrival time is the time between consecutive arrivals.
a) What is the probability of one customer arriving within a one-minute interval of the
previous customer?
b) What is the probability of one customer arriving within a two-minute interval of the
previous customer?
c) What is the probability of one customer arriving within a three-minute interval of the
previous customer?
The random variable is the time interval in minutes.
a) 1/a = 5 minutes; a = 0.2; P(X #1) = 1 & e &(0.2) (1) = 0.181
b) P(X
) = 1 & e &(
)
= 0.330
c) P(X
) = 1 & e &(
)
= 0.451
These results match our common sense expectation that given a 5 minute average
interarrival time of customers, as time increases from the "0" time, the probability of an
arrival also increases. For this example, there is a probability of 0.181 of an arrival
within a time period of one minute, probability of 0.330 of an arrival within a time period
of two minutes, probability of 0.451 of an arrival within a time period of three minutes,
and so no. In other words, if we wait long enough, eventually we will have an arrival.
Illustration 9)
At a service counter, the average service time per customer is 2 minutes.
a) What is the probability of servicing a customer in less than or equal to 3 minutes?
b) What is the probability of servicing a customer in less than or equal to 5 minutes?
c) What is the probability of servicing a customer in less than or equal to 10 minutes?
The random variable is the service time in minutes.
a) 1/a = 2 minutes; a = 0.5; P(X #3) = 1 & e &(0.5) (3) = 0.777
b) P(X
) = 1 & e &(
)
= 0.918
c) P(X
) = 1 & e &(
)
= 0.993
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These results match our common sense expectation that given an average service time
per customer of 2 minutes, as time increases from the initial time (the time at which the
service begins), the probability of completing the service on the customer increases. In
this example, the probability is 0.777 of servicing the customer in 3 minutes or less;
probability is 0.918 for 5 minutes or less; probability is 0.993 for 10 minutes or less.
Given a long enough time, once began, the customer will be serviced.
QUEUING MODEL
The reader should realize by now that the exponential distribution is the most useful
probability distribution in modeling queues. The formulas in the textbook are derived
from this distribution. The two most important statistical properties of a queuing system
are the probability distributions of interarrival times and of service times. In real queuing
systems, the distributions of interarrival times and service times can take on almost any
form. However, to a queuing model, the exponential distribution is used to model the
interarrival times and service tines. From illustrations 8, 9, the reader should convince
himself or herself that the exponential distribution is sufficiently realistic and it also has
the virtue of being mathematically simple.
An elementary queuing model can be diagramed:
One of the assumptions of this model is the input source produces customers
"randomly.” Randomly means each arrival is independent of the other arrivals. An
example of non-random arrivals is the exit from a subway train station. Each person
leaving the train station is "dependent" to each other because they all arrived at the
station on the same train. Another assumption is that the input source has an infinite
supply of customers. This assumption is to allow easier mathematical calculations.
Another assumption is that the generation of new customers is not affected by the
number of customers ready in the queuing system. The point is clear: in every model,
assumptions must be made and clearly explained.
All elementary queuing models are said to have a Poisson input and an exponential
service time. However, to model the input process, the exponential distribution is used
in place of the Poisson distribution because of the former's mathematical simplicity. The
exponential distribution's modeling of interarrival time is equivalent to the Poisson
distribution's modeling of an average arrival rate.
In the service area, the elementary queuing model's service discipline is first come
first served, otherwise, more mathematical complications. The exponential distribution
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again is used to model the service time. The final major assumption for the service-area
component is the distribution is applied to all servers and all customers.
QUEUING THEORY AS A DECISION MAKING TOOL: CASE STUDY1
.
Everyone hates standing in a queue. No one derives any benefits or pleasure from a
queue because the time spent in a queue is wasted time, and to the economist and to
almost everyone, the wasted time represents wasted productivity and lost money.
Consider the case of the Peekskill Country Club. The Club has 500 members each
paying an annual base fee of $300. Its Board of Directors decided to charge no fee for
its limited tennis facilities. The Board also decided not to have an advance sign up
system and decided to have a simple first-come-first-play policy. During the warm
season, everyone wants to play tennis and obviously queues are developed everyday
and result in wasted time.
In the graph, curve A represents the average waiting time as a function of the
number of people who want to play tennis. The shape of the curve is reasonable
because we expect waiting time to increase with an increase in the number of people
who want to play. Curve B represents the number of people who want to play in
response to the average waiting time. It is reasonable to assume that as the average
waiting time increases, fewer people want to play. Point E is the equilibrium, the
intersection of curves A and B. At E, "C" number of people want to play, with a “D”
average waiting time. C is 125 and D is approximately 44 minutes.
We assume everyone values his or her waiting time at $3.00 per hour. In other
words, a person doesn't care whether he or she pays a $3.00 fee to play immediately or
waits an hour to play for free, Accordingly, another vertical scale, M = equivalent dollar
1
Stokey, E.: Zeckhauser, R. (1978). A Primer for Policy Analysis.Norton, New York.
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price, is put on the graph. For example, a person either pays $3.75 to play immediately
or waits 75 minutes; either choice is indifferent to the person. At point E, the average
wait of 44 minutes is worth $2.20 for each person waiting; $2.20 is also the value for the
marginal tennis player. A marginal player would pay either $2.20 to play immediately or
wait 44 minutes; but a more dedicated tennis player may be willing to pay, for example,
$3.75 to play immediately or wait 75 minutes.
The 44-minute wait was too long and everybody started to complain to the Board.
The Board investigated the matter and realized the situation fitted the characteristics of
a queuing model. The 44 minutes waiting time represented a so called "deadweight
loss,” i.e., no one benefits from the waiting time. The Board decided to impose a $2.00
fee per person per hour of play time. With this $2.00 fee, the tennis players naturally will
not be willing to wait as long. Therefore, the curve of number wanting to play in
response to length of wait goes down to B’ (see graph). This results in a new
equilibrium at F. At the point F, 100 people wait on an average of 14 minutes to play.
So now everyone is happy. The average waiting time is diminished by 30 minutes, a
decrease in "deadweight loss.” The country club takes in 100 x $2 = $200 per day,
assuming each person plays one hour. The dedicated tennis players keep paying the
$2.00 while the less dedicated tennis players decide to do other things.
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[prefer]
Class 3 - A CONSUMER'S GUIDE TO MCDM
Review
Multicriteria Decision Making (MCDM):
1. Alternative set x 0 X
2. Criteria set f = ( f1, ... , fq)
3. Outcome of each choice, f(x) = ( f1(x), ... , fq(x)); or Y = {y} = {f(x)*x 0 X }
4. Preference structure, with which the decision maker picks the best outcome
y* 0 Y corresponding to the choice x* 0 f -1(y*).
Example payoff matrix (2 alternatives & 2 overall outcomes)
Criteria set f:
Utility if there is
payoff y
war f1
peace f2
no-missile (x1 = 1, x2 = 0)
30
100
(30)
(100)
build-missile (x1 =0, x2 =1)
90
50
(50)
(90)
Alternative Set X
min
max
8
Y
Question: Build or not, or x = (0, 1) or (1, 0)?
Preference structure (the Z space)
Unless noted, the payoffs are “gains” instead of “losses.”
(1) Max-min:
Maximize the minimum payoff (pessimistic approach)
Best payoff = y* = 50, Decision x* = f &1(50) = (x1 = 0, x2 = 1) i.e., build.
If the payoffs are “losses” (instead of gains), the max-min preference structure is
equivalent to “min-max” (or minimizing the maximum loss).
(2) Max-max:
Maximize the maximum value (optimistic approach)
1
Decision: no; payoff = 100
(3) Regret:
Examining a column in the payoff matrix
Regret = (max possible payoff of any decision) & (payoff of a particular decision)
Figure 1 - Example X-Y-Z space
Regret payoff matrix (Notice it is different from the original payoff matrix):
war
peace
(max)
no-missile ( x1 = 1)
60
0
(60)
build-missile ( x2 = 1)
0
50
(50)
Minimize maximum regret (min-max)
Decision: build, regret = 50.
(4) Expected value
Probability of Occurrence
f1 (War)
f2 (Peace)
0.1
0.9
Expected Outcome
no-missile (x1 = 1)
30
100
(0.1)(30) + (0.9)(100) = 93
build-missile (x2 = 1)
90
50
(0.1)(90) + (0.9)(50) = 54
Decision: no build, expected outcome = 93.
Notice that for different preference structures, the same data lead to different
decisions.
EXERCISE:
If the payoffs in this example is interpreted as “losses” instead of gains,” please recompute the decisions under the four preference structures (1). (2), (3) and (4).
2
Selecting the appropriate MCDM technique
Figure 2 - A taxonomy of MCDM methods
Figure 3 - The decision analysis procedure
3
[DA process]
Example of X, Y, Z Space
Criteria set f:
Utilit if there
Utility
th is
i
payoff y
Alternative Set X
war f1
peace f2
min
max
no-missile ( x1 = 1, x2 = 0)
30
100
(30)
(100)
build-missile ( x1 = 0, x2 =1)
90
50
(50)
(90)
Y
A taxonomy of MCDM methods
Dominance:
Max-min,
max-max
No info
Standard
level
Ordinal
MCDM
Info on
attribute
C di l
Cardinal
Marginal
Substitution
rate
Info on
alternative
Pairwise
p
preference
Order of pairwise proximity
Satisficing method
Compromise prog
Software
Handout on DECISION ANALYSIS SOFTWARE SURVEY
including
“TreePlan for Education” from Decision Toolworks (http://www.treeplan.com/academic.htm)
Multicriteria Decision-making
Utility theory is a key foundation of economics and operations research. The basic premise is that
a number of disparate metrics can be translated into a common unit called utiles. Once this is
done, cross comparison can then be made among alternatives with seemingly incommensurate
attributes or criteria. This is generally accomplished by a multiattribute utility-function,
whichCafter valuationCcombines the incommensurate attributes or criteria through weights and
scaling constants. Once these incommensurate attributes are converted into a "common currency
of exchange," the utiles, the best alternative(s) can then be picked among competing alternatives.
In Chapter 5, we pointed out that cross comparison among alternatives can still be possible
without a multiattribute utility function, although in a more limited sense. For example, a shirt
that is cheaper and better quality is always preferred to one that is more expensive and inferior in
quality. Here, no utility function needs to be constructed to combine price and qualityCthe two
different attributesCinto utiles before a decision can be made between them. And there are
occasions where multiattribute utility theory may not apply.
These examples clearly show that, depending on the occasion and the participating discipline,
different technical terms are often used to mean similar ideas. While there are truly different
disciplinary differences in focus and concerns, they often use similar analytical tools to solve
problems. When we found that there were very few places where such analytical parallels are
established, we decided that this bookCand a companion volume, Chan (2005)Cmay just fill part
of that niche in the literature. Aside from the texts, we wish to draw the reader’s attention to the
Glossary of Technical Concepts in Appendix 5, in which we try to cut across the jargons used in
different disciplines one at a time.
Selecting Decision Analysis Software
Due diligence in selecting a decision-analysis package should include thinking about the
following critical questions in addition to cost (Maxwell 2008):
$ Are there a single stakeholder or multiple decision-makers?
$ Will the stakeholders participate in the decision conference, or will they be periodically
presented results only?
$ Will there be a choice of a single alternative or a portfolio of alternatives?
$ Do stakeholders have multiple, conflicting objectives that must be reconciled?
$ Is there significant uncertainty in the decision outcomes?
$ Is it a single, one-time decision or a sequence of decisions over time?
4
Most hard decision situations require decision-maker(s) to make trades among a complicated set
of competing objectives. There is a number of multicriteria decision-making techniques
implemented in the available software. Multi-Attribute Utility Theory (MAUT) and the Analytic
Hierarchy Process (AHP) are the most prevalent. Most of the packages indicate that they
implement MAUT. In addition to these approaches, ordinal ranking techniques are available in
some of the software packages and can be quickly implemented to develop a first-order set of
weights for a decision model. The quick technique just might be good enough to meet the
analysis goals.
More important than the software, it is critical that analysts understand that different
techniques have different underlying axioms and different philosophies about how decision
models should be formulated. The best example is MAUT vs. AHP, two very different
techniques with different theoretical underpinnings. Various approaches have strengths,
weaknesses and limitations that deserve some research before they are applied. Whichever
technique is applied, it is important that analysts ensure that both the relative importance of
attributes and the range each attribute varies are clearly presented to the stakeholder for
consideration as an integral part of the elicitation process. Considering only importance increases
the risk that the model will produce unreliable results. This point has been discussed in Chapter
5.
Uncertainty is also almost always a factor when making hard decisions. How it is addressed
varies among the packages. How it is best addressed depends on the nature of the uncertainty,
how the model is being developed, the data that is available and the resources that are available
for model development. Tools are available for eliciting probability judgments from experts.
Often, these judgments are placed in a decision analytic model called an influence diagram.
These diagrams are designed to combine an intuitive, visual presentation of the relationships
among the variables with a sound underlying mathematical representation of their joint
probability distribution.
Let us go back to the early days of influence diagrams and other decision analytic models
that considered uncertainty explicitly. In that era, solution time and computer memory for models
were very important considerations. As an example, one influence diagram model developed in
the very early 1990s possesses almost two million solution paths. It took approximately two
hours to solve. Today, the same modelCusing newer versions of the influence diagram software
and a current notebook computerCsolves in less than three minutes. This power allows us to
represent and solve increasingly complicated problems. It also allows analysts to exercise the
models we develop more rigorously.
In some cases the analytic team might have large quantities of data that can be used to
formulate the probability model. Some of these packages (as well as some statistical packages)
have learning algorithms that will build the joint probability distribution from the available data.
If this option is available, the analysis team should be certain to supplement the automated effort
and involve subject matter experts in the review of the resulting model. The experts can help find
errors in the data and, just as importantly, they can supplement the model with knowledge they
possess. Combining what is learned from data and what is learned from experts usually yields a
better model and results in a higher likelihood that the effort will be successful.
5
A final consideration whether the model is addressing a single decision in time or a sequence
of decisions over time. Virtually all of the packages will consider a single decision. The influence
diagram packages and some of the Monte Carlo packages will also support considering multiple
decisions that might unfold over time. In influence diagrams, this situation is represented as a
sequence of decisions, likely with uncertainties that will resolve over time spaced in between
decisions. Sometimes a hard decision actually consists of a set of smaller decisions that either
occur over time or can be thought of as a package. A technique for representing this type of
situation is to use a sequence of decisions or an alternative-generation table. This technic is
virtually supported by all of the software packages.
Looking toward the future, here is an observation according to Buede (undated). Future
improvements that will be least likely to make substantial advances involve embedding the
analyst’s wisdom and knowledge in the software for the users who are less skilled analysts.
These improvements should include problem structuring and elicitation features. Buede believed
these capabilities might exist in the distant future, but are not very likely in the near future.
Important as it may be, this software market will always be a small, specialty market until the
less skilled analysts are better prepared.
When shopping for a software, here are some factors for consideration (Maxwell 2008):
G Operating Systems:
$ Is your operating system Windows, Mac OS, Unix, or others?
G Applications:
$ What is the best software if multiple objectives are considered?
$ How does the software represent and analyzes uncertainty for a probabilistic application?
$ How does it represent and analyze probabilistic dependencies?
$ Where applicable, how does it model sequential decision making, portfolio decision making,
and/or multiple stakeholders collaboration?
G Software Features:
$ Can the software import a database or spreadsheet?
$ Does it export presentation graphics?
$ Does it interface with the EXtensible Markup Language (XML) that facilitates transport and
storage of data?
$ Does it accept Application Program Interface (API), such as embedding a decision support
system?
$ Can model segments be copied or moved easily?
$ Can model structure be displayed on screen or printed?
$ Can a user protect his/her data from other users?
$ Does the software support explicitly group elicitation? How?
$ Does it support simultaneous viewing?
$ Is a record of model evolution kept?
G Does the software support
$ Multi-objective Decision Analysis?
$ Multiattribute Utility Theory?
$ Analytic Hierarchy Process?
$ Or other algorithms?
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G Is Pricing Information available for Commercial, Education, Enhanced/High Performance
licenses?
G Are there size limitations in the following:
$ Number of alternatives?
$ Number of levels in value or decision tree?
$ Number of states of a node in a tree?
G Is graphical elicitation techniques available for the following:
$ Model structure / Brainstorming?
$ Value functions / scores?
$ Value weights, Probabilities, Risk preference?
$ Can probabilities or weights be defined as variables that can be operated on?
$ Are graphical sensitivity analyses possible on either weights or probabilities?
$ Can analytical results be portrayed graphically?
$ Can the user document structure or judgments with text?
Decision Analysis Software Survey
The information provided here on the CD are intended to help analysts select a tool set that fits
the specific challenge they face or maybe even a general-purpose package for the long run
(Maxwell 2008). When shopping for decision analysis software, it is advisable to focus on the
potential tool's ability to fit the specific problem or class of problems. Carefully evaluate the
software in relation to the situational factors that are relevant. If the goal is to add a package or
two to the general tool kit, then a package or combination of packages that provide balanced
support across the spectrum of situations and the entire decision analysis process is the best
investment. If the problem calls for involving multiple stakeholders and multiple competing
attributes, then tools that emphasize group support and value elicitation are worth exploring.
Problems involving large uncertainties, diagnosis, complex interdependencies or risk analysis
would benefit most from tools such as influence diagrams, Bayesian networks or one of the
Monte Carlo modeling tools. Whichever tool(s) are selected, they should be intuitive to the user,
explainable to the client and support easy iterations among the various stages of the decision
analytic process.
Here are some decision analysis and multicriteria decision-making products to choose from.
Most of them cover multiobjective decision analysis (MODA) and multiattribute utility theory
(MAUT). From there, additional features vary from one to the other.
G
G
1000Minds - 1000Minds Ltd. (The software includes MODA / MAUT. It also provides
a procedure called PAPRIKA, which is based on the fundamental principle that any
ranking of alternatives is uniquely defined by all possible pairwise rankings of the
alternatives vis-à-vis each other B hence the acronym ‘PAPRIKA. Another prominent
building block is Conjoint Analysis, which involves surveying stakeholders about the
relative importance of a product’s (or service’s) features.)
Crystal Ball Standard, Professional & Premium Editions - Oracle (The generalized
software package includes, among other analytical features, MODA / MAUT)
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G
G
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G
G
G
G
G
G
G
G
G
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DEA-Solver-Pro 6.0 - SAITECH, Inc. (It includes AHP and Data Envelopment
Analysis)
DPL - Syncopation Software (Features include MODA / MAUT, Decision tree roll
back and Monte Carlo)
ForeTellÆ - DecisionPath, Inc. (ForeTellÆ provides hybrid synthesis of Monte Carlo,
system dynamics, agents/game theory, process and event models.)
GoldSim - GoldSim Technology Group (Features include MODA / MAUT.)
Hiview3 - Catalyze Ltd. (Features include MODA / MAUT)
Logical Decisions, version 6.1 - Logical Decisions (MODA / MAUT, AHP)
Opinions-Online - Systems Analysis Laboratory, Helsinki University of Technology
(MODA / MAUT, Voting, surveys)
Quantitative Methods Software (QMS) - QuantMethods (MODA / MAUT)
RICH Decisions - Systems Analysis Laboratory, Helsinki University of Technology
(Features include MODA / MAUT, Value tree analysis, considering incomplete
information, dominance structure, and decision rules)
RPM-Decisions - Systems Analysis Laboratory, Helsinki University of Technology
(The software is built upon Multiattribute Value Theory, considering incomplete
information, non-dominated portfolios and core indexes. RPM defines a project's core
index as the share of non-dominated portfolios that include the project.)
Smart-Swaps - Systems Analysis Laboratory, Helsinki University of Technology
(Features include MODA / MAUT and the Even Swaps Method, where preferences are
elicited by an elimination process based on value trade-offs.)
TreePlan - Decision Toolworks (TreePlan is a MS Excel add in that performs Decision
Tree Analysis aided by graphic display of the tree.)
Vanguard System - Vanguard Software (The software is a comprehensive package that
includes MODA / MAUT, AHP, Decision Tree Analysis, Monte Carlo Simulation,
Simulation, Optimization, Deterministic Optimization, Forecasting, Sensitivity
Analysis)
Web-HIPRE - Systems Analysis Laboratory, Helsinki University of Technology
(Web-HIPRE is a web-version of the HIPRE 3+ software, featuring MODA / MAUT
and AHP.)
WINPRE - Systems Analysis Laboratory, Helsinki University of Technology (This
decision support program is an implementation of techniques based on the propagation
of imprecise preference statements in hierarchical weighting. It includes MODA /
MAUT, AHP, preference programming, and interval techniques.)
As far a "freeware" is concerned, SMILE (Structural Modeling, Inference, and Learning
Engine) is a fully portable library of C++ classes implementing decision-theoretic methods, such
as Bayesian networks and influence diagrams. Its Windows user interface, GeNIe, is a versatile
and user-friendly development environment for graphical implementation. Both modules,
developed at the Decision Systems Laboratory, University of Pittsburgh, have been made
available to the community since 1998 and have now several thousand users worldwide. Contact
Marek J. Druzdzel for more details.
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ANSWER TO EXERCISE
in
A Consumer’s Guide to MCDM
(1) min-max (As a pessimist, she minimizes maximum loss)
y* = 90
Then build.
(2) min-min (as an optimist who is always blessed, she goes all the way to
minimize minimum loss)
y* = 30
Then no-build.
(3) max-min (As a “regretist,” she maximizes the minimum regret when
regret is savings in cost. This is equivalent to minimizing maximum regret, when
regret is the loss in “brownie points.”)
Regret payoff matrix (where entries are “costs”)
no-missile ( x1 = 1)
war
peace
(min cost)
0
50
(0)
0
(0)
build-missile ( x2 =
60
1)
y* = 0
Either build or no-build will be okay.
(4) min (expected cost) - As an “accountant,” she simply minimizes the
average cost between the two options.
y* = 54
Then build missiles.
Another way is to work with negative payoffs by putting a negative sign in
front of the payoffs and retain the max-min, max-max, min-max, and max
(expected gain) logic.
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