Engineering Decision Making Under Uncertainty
Aimen Sudhir, RUID: 197007651
To make the recommendation, the following decision tree was made.
As seen from the decision tree, it is profitable for Ectron Fragrance Corporation to conduct the market
survey.
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
Introduction and Basic Concepts
Melike Baykal-Gürsoy
ISE - Rutgers University
gursoy@soe.rutgers.edu
gursoy.rutgers.edu
September 2, 2020
Dr. Baykal-Gürsoy
gursoy.rutgers.edu
Prob. Review
Melike
Baykal-Gürsoy
Prof. in Dept. of I&SE
Director of Lab. for
Stochastic Systems
GRIST-Game Research for
Infrastructure SecuriTyLab
Introduction
I PhD - Univ. of Pennsylvania Systems Eng.
I BS & MS - Bogazici Univ.
Turkiye - EE
I Areas of interest: stochastic
modeling, queueing, Markov
decision processes, stochastic
games, and applications in
inventory, transportation,
communication networks.
I Courses: Eng. Decision
Making, Stochastic Modeling,
Inventory Control, Process
Modeling and Control,
Queueing
Basic Definitions and
Relationships
Learning Objectives
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
Introduction
Basic Definitions and Relationships
Table of Contents
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
Introduction
Basic Definitions and Relationships
Prob. Review
Initial Definitions
Melike
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Consider and experiment whose outcome is not predictable
in advance.
I Sample Space: set of all possible outcomes of an
experiment. We will denote it by S
I Event: Any subset S of the sample space S.
Event A
Sample Space
Event B
Introduction
Basic Definitions and
Relationships
Motivating example
Prob. Review
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Introduction
Basic Definitions and
Relationships
I Example: A family with two cars is selected. For both
the older and the newer car, we note whether the car
was manufactured in Asia, Europe, or America.
I In this case, the sample space can be written as
{(Am , Am ), (Am , E ), (Am , As ), (E , Am ), (E , As ), (E , E ),
(As , Am ), (As , E ), (As , As )}.
I The total number of possible outcomes (cardinality of
the sample space) is 32 = 9.
Motivating example
Prob. Review
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Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
I Example: A family with two cars is selected. For both
the older and the newer car, we note whether the car
was manufactured in Asia, Europe, or America.
I In this case, the sample space can be written as
{(Am , Am ), (Am , E ), (Am , As ), (E , Am ), (E , As ), (E , E ),
(As , Am ), (As , E ), (As , As )}.
I The total number of possible outcomes (cardinality of
the sample space) is 32 = 9.
Motivating example
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
I Example: A family with two cars is selected. For both
the older and the newer car, we note whether the car
was manufactured in Asia, Europe, or America.
I In this case, the sample space can be written as
{(Am , Am ), (Am , E ), (Am , As ), (E , Am ), (E , As ), (E , E ),
(As , Am ), (As , E ), (As , As )}.
I The total number of possible outcomes (cardinality of
the sample space) is 32 = 9.
Motivating example
Prob. Review
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Introduction
Basic Definitions and
Relationships
Let us consider three events:
I Event 1: One car is American and the other is foreign.
E1 = {(Am , E ), (Am , As ), (E , Am ), (As , Am )}
I Event 2:At least one car is foreign.
E2 = {(Am , E ), (Am , As ), (E , Am ), (As , Am ),
(E , E ), (E , As ), (As , As ), (As , E )}
I Event 3: Both cars are American.
E3 = {(Am , Am )}
Motivating example
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
Let us consider three events:
I Event 1: One car is American and the other is foreign.
E1 = {(Am , E ), (Am , As ), (E , Am ), (As , Am )}
I Event 2:At least one car is foreign.
E2 = {(Am , E ), (Am , As ), (E , Am ), (As , Am ),
(E , E ), (E , As ), (As , As ), (As , E )}
I Event 3: Both cars are American.
E3 = {(Am , Am )}
Motivating example
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
Let us consider three events:
I Event 1: One car is American and the other is foreign.
E1 = {(Am , E ), (Am , As ), (E , Am ), (As , Am )}
I Event 2:At least one car is foreign.
E2 = {(Am , E ), (Am , As ), (E , Am ), (As , Am ),
(E , E ), (E , As ), (As , As ), (As , E )}
I Event 3: Both cars are American.
E3 = {(Am , Am )}
Notation
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Introduction
Basic Definitions and
Relationships
Let A and B be events:
I Union of A and B is the event that at least A or B
occurs. A ∪ B;
I Intersection of A and B is the event that A and B
occur. A ∩ B or simply AB;
I Complement of A is the probability that A does not
occur. Ac = S \ A.
Notation
Prob. Review
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Introduction
Basic Definitions and
Relationships
Let A and B be events:
I Union of A and B is the event that at least A or B
occurs. A ∪ B;
I Intersection of A and B is the event that A and B
occur. A ∩ B or simply AB;
I Complement of A is the probability that A does not
occur. Ac = S \ A.
Notation
Prob. Review
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Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
Let A and B be events:
I Union of A and B is the event that at least A or B
occurs. A ∪ B;
I Intersection of A and B is the event that A and B
occur. A ∩ B or simply AB;
I Complement of A is the probability that A does not
occur. Ac = S \ A.
Prob. Review
Example
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I
I
I
I
I
I
E1 ∪ E2 = E2 (because E1 is a subset of E2 )
E2 ∪ E3 = S (includes all possible outcomes)
E 1 ∩ E2 = E1
E 1 ∩ E3 = ∅
E 2 ∩ E3 = ∅
E2c = S \ E2 = E3
Sample Space
E1
E2
E3
Introduction
Basic Definitions and
Relationships
Prob. Review
Example
Melike
Baykal-Gürsoy
I
I
I
I
I
I
E1 ∪ E2 = E2 (because E1 is a subset of E2 )
E2 ∪ E3 = S (includes all possible outcomes)
E 1 ∩ E2 = E1
E 1 ∩ E3 = ∅
E 2 ∩ E3 = ∅
E2c = S \ E2 = E3
Sample Space
E1
E2
E3
Introduction
Basic Definitions and
Relationships
Prob. Review
Example
Melike
Baykal-Gürsoy
I
I
I
I
I
I
E1 ∪ E2 = E2 (because E1 is a subset of E2 )
E2 ∪ E3 = S (includes all possible outcomes)
E 1 ∩ E2 = E1
E 1 ∩ E3 = ∅
E 2 ∩ E3 = ∅
E2c = S \ E2 = E3
Sample Space
E1
E2
E3
Introduction
Basic Definitions and
Relationships
Prob. Review
Example
Melike
Baykal-Gürsoy
I
I
I
I
I
I
E1 ∪ E2 = E2 (because E1 is a subset of E2 )
E2 ∪ E3 = S (includes all possible outcomes)
E 1 ∩ E2 = E1
E 1 ∩ E3 = ∅
E 2 ∩ E3 = ∅
E2c = S \ E2 = E3
Sample Space
E1
E2
E3
Introduction
Basic Definitions and
Relationships
Prob. Review
Example
Melike
Baykal-Gürsoy
I
I
I
I
I
I
E1 ∪ E2 = E2 (because E1 is a subset of E2 )
E2 ∪ E3 = S (includes all possible outcomes)
E 1 ∩ E2 = E1
E 1 ∩ E3 = ∅
E 2 ∩ E3 = ∅
E2c = S \ E2 = E3
Sample Space
E1
E2
E3
Introduction
Basic Definitions and
Relationships
Prob. Review
Example
Melike
Baykal-Gürsoy
I
I
I
I
I
I
E1 ∪ E2 = E2 (because E1 is a subset of E2 )
E2 ∪ E3 = S (includes all possible outcomes)
E 1 ∩ E2 = E1
E 1 ∩ E3 = ∅
E 2 ∩ E3 = ∅
E2c = S \ E2 = E3
Sample Space
E1
E2
E3
Introduction
Basic Definitions and
Relationships
Initial Definitions
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
I The probability that the event A occurs is written P{A}
I The certain event S always occurs, P{S} = 1.
I The impossible event ∅ never occurs, P{∅} = 0.
I Events A and B are said to be disjoint if A ∩ B = ∅
(They cannot both occur)
I E1 and E3 , and E2 and E3 are two examples of disjoint
events.
I Events A and B are said to be mutually exclusive if
A ∩ B = ∅ and A ∪ B = S.
I E2 and E3 are mutually exclusive.
Initial Definitions
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
I The probability that the event A occurs is written P{A}
I The certain event S always occurs, P{S} = 1.
I The impossible event ∅ never occurs, P{∅} = 0.
I Events A and B are said to be disjoint if A ∩ B = ∅
(They cannot both occur)
I E1 and E3 , and E2 and E3 are two examples of disjoint
events.
I Events A and B are said to be mutually exclusive if
A ∩ B = ∅ and A ∪ B = S.
I E2 and E3 are mutually exclusive.
Initial Definitions
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
I The probability that the event A occurs is written P{A}
I The certain event S always occurs, P{S} = 1.
I The impossible event ∅ never occurs, P{∅} = 0.
I Events A and B are said to be disjoint if A ∩ B = ∅
(They cannot both occur)
I E1 and E3 , and E2 and E3 are two examples of disjoint
events.
I Events A and B are said to be mutually exclusive if
A ∩ B = ∅ and A ∪ B = S.
I E2 and E3 are mutually exclusive.
Initial Definitions
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
I The probability that the event A occurs is written P{A}
I The certain event S always occurs, P{S} = 1.
I The impossible event ∅ never occurs, P{∅} = 0.
I Events A and B are said to be disjoint if A ∩ B = ∅
(They cannot both occur)
I E1 and E3 , and E2 and E3 are two examples of disjoint
events.
I Events A and B are said to be mutually exclusive if
A ∩ B = ∅ and A ∪ B = S.
I E2 and E3 are mutually exclusive.
Initial Definitions
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
I The probability that the event A occurs is written P{A}
I The certain event S always occurs, P{S} = 1.
I The impossible event ∅ never occurs, P{∅} = 0.
I Events A and B are said to be disjoint if A ∩ B = ∅
(They cannot both occur)
I E1 and E3 , and E2 and E3 are two examples of disjoint
events.
I Events A and B are said to be mutually exclusive if
A ∩ B = ∅ and A ∪ B = S.
I E2 and E3 are mutually exclusive.
Initial definitions (cont.)
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
I For disjoint events A, B: P{A ∪ B} = P{A} + P{B}.
I In general, P{A ∪ B} = P{A} + P{B} − P{A ∩ B}.
Initial definitions (cont.)
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
I For disjoint events A, B: P{A ∪ B} = P{A} + P{B}.
I In general, P{A ∪ B} = P{A} + P{B} − P{A ∩ B}.
Prob. Review
Law of Total Probability
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Let A1 , A2 , . . . , An , . . . be events with Ai and Aj disjoint
whenever i 6= j. Then:
P{∪ni=1 Ai } =
n
X
P{Ai }
i=1
If S = A1 ∪ A2 ∪ A3 ∪ . . ., we have
P{S} = 1 =
∞
X
P{Ai }
(Law of Total Probability)
i=1
Law of Total Probability asserts that, for any event B,
P{B} =
∞
X
i=1
P{B ∩ Ai }.
Introduction
Basic Definitions and
Relationships
Example (cont.)
Prob. Review
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Introduction
Example: Assume we also note the color of the two cars,
which can be white or black. The sample space (state space)
is then:
S = {(Am , Am ), (Am , E ), (Am , As ), (E , Am ), (E , As ), (E , E ),
(As , Am ), (As , E ), (As , As ), (W , W ), (W , B), (B, W ),
(B, B)}.
Let us also note event 4, E4 the event in which at least one
car is black.
E4 = {(B, B), (W , B), (B, W )}
Basic Definitions and
Relationships
Independence
Prob. Review
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Introduction
Basic Definitions and
Relationships
I Events A and B are independent if
P{A ∩ B} = P{A}P{B} .
I E2 and E4 are independent events.
Practice (Pairwise Independent Events that are
not Independent)
Prob. Review
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Introduction
Basic Definitions and
Relationships
Let a ball be drawn from an urn containing 4 balls,
numbered 1, 2, 3, 4. Each of the four outcomes is equally
likely. Let E = {1, 2}, F = {1, 3}, G = {1, 4}.
I P{E ∩ F } = P{1} = 1/4 = P{E }P{F } = 1/2 · 1/2
(independent).
I P{E ∩ G } = P{1} = 1/4 = P{E }P{G } = 1/2 · 1/2
(independent).
I P{G ∩ F } = P{1} = 1/4 = P{G }P{F } = 1/2 · 1/2
(independent).
I P{E ∩ F ∩ G } = P{1} = 1/4 6= P{E }P{F }P{G } =
1/2 · 1/2 · 1/2 (not independent).
Practice (Pairwise Independent Events that are
not Independent)
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
Let a ball be drawn from an urn containing 4 balls,
numbered 1, 2, 3, 4. Each of the four outcomes is equally
likely. Let E = {1, 2}, F = {1, 3}, G = {1, 4}.
I P{E ∩ F } = P{1} = 1/4 = P{E }P{F } = 1/2 · 1/2
(independent).
I P{E ∩ G } = P{1} = 1/4 = P{E }P{G } = 1/2 · 1/2
(independent).
I P{G ∩ F } = P{1} = 1/4 = P{G }P{F } = 1/2 · 1/2
(independent).
I P{E ∩ F ∩ G } = P{1} = 1/4 6= P{E }P{F }P{G } =
1/2 · 1/2 · 1/2 (not independent).
Practice (Pairwise Independent Events that are
not Independent)
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
Let a ball be drawn from an urn containing 4 balls,
numbered 1, 2, 3, 4. Each of the four outcomes is equally
likely. Let E = {1, 2}, F = {1, 3}, G = {1, 4}.
I P{E ∩ F } = P{1} = 1/4 = P{E }P{F } = 1/2 · 1/2
(independent).
I P{E ∩ G } = P{1} = 1/4 = P{E }P{G } = 1/2 · 1/2
(independent).
I P{G ∩ F } = P{1} = 1/4 = P{G }P{F } = 1/2 · 1/2
(independent).
I P{E ∩ F ∩ G } = P{1} = 1/4 6= P{E }P{F }P{G } =
1/2 · 1/2 · 1/2 (not independent).
Practice (Pairwise Independent Events that are
not Independent)
Prob. Review
Melike
Baykal-Gürsoy
Introduction
Basic Definitions and
Relationships
Let a ball be drawn from an urn containing 4 balls,
numbered 1, 2, 3, 4. Each of the four outcomes is equally
likely. Let E = {1, 2}, F = {1, 3}, G = {1, 4}.
I P{E ∩ F } = P{1} = 1/4 = P{E }P{F } = 1/2 · 1/2
(independent).
I P{E ∩ G } = P{1} = 1/4 = P{E }P{G } = 1/2 · 1/2
(independent).
I P{G ∩ F } = P{1} = 1/4 = P{G }P{F } = 1/2 · 1/2
(independent).
I P{E ∩ F ∩ G } = P{1} = 1/4 6= P{E }P{F }P{G } =
1/2 · 1/2 · 1/2 (not independent).
Conditional Probability
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Introduction
Basic Definitions and
Relationships
The conditional probability of A given B is written as
P{A | B} =
P{A ∩ B}
P{A, B}
=
P{B}
P{B}
Prob. Review
Example
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Introduction
Basic Definitions and
Relationships
What is the probability that the first die is 6 given that the
sum of two dice is 7?
P{1st is 6 and sum = 7}
P{sum = 7}
P{1st is 6 and 2nd = 1}
=
P{sum = 7}
1/6 · 1/6
=
= 1/6
6/36
P{1st is 6 | sum = 7} =
Prob. Review
Example
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Introduction
Assume that each child born is equally likely to be a boy or a
girl. If a family has 2 children, what is the probability that
both are girls, given that a) the oldest is a girl,
P{both girls, oldest is girl}
P{oldest is girl}
P{both are girls}
=
P{oldest is girl}
1/2 · 1/2
=
= 1/2
1/2
P{both girls | oldest is girl} =
Basic Definitions and
Relationships
Prob. Review
Example (cont.)
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Introduction
Basic Definitions and
Relationships
and b) at least one is a girl.
State Space = {(G , G ), (B, G ), (G , B), (B, B)}
P{both girls}
P{at least one is girl}
1/2 · 1/2
= 1/3
=
3/4
P{both girls | at least one is girl} =
Prob. Review
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Introduction
Basic Definitions and
Relationships
Q&A
Decision Making Under Uncertainty
A. Altay and M. Baykal-Gürsoy
1
1
Industrial & Systems Engineering Department
Rutgers University
September 1, 2021
Altay & Baykal-Gürsoy
Decision Making Under Uncertainty
September 1, 2021
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Dr. Baykal-Gürsoy
gursoy.rutgers.edu
Prof. in Dept. of I&SE
Director of Lab. for Stochastic
Systems
GRIST-Game Research for
Infrastructure SecuriTy- Lab
PhD - Univ. of Pennsylvania - Systems
Eng.
BS & MS - Bogazici Univ. Turkiye - EE
Areas of interest: stochastic modeling,
queueing, Markov decision processes,
stochastic games, and applications in
inventory, transportation, communication
networks.
Courses: Eng. Decision Making, Stochastic
Modeling, Inventory Control, Process
Modeling and Control, Process Modeling
and Control, Queueing
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Decision Making Under Uncertainty
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Decision Making Introduction
1
Introduction
1
2
2
Single criterion decision making under Uncertainty and Risk
1
2
3
3
Elementary methods under uncertainty
Decision making under risk (Expected value and expected regret)
Decision trees (Type I)
Utility theory
1
2
3
4
Decision environments
Value of information
Deal or no deal game
Utility calculation
Risk premiums
Materials management under uncertainty
1
Single period inventory uncertainties
Altay & Baykal-Gürsoy
Decision Making Under Uncertainty
September 1, 2021
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Introduction
Introduction
Altay & Baykal-Gürsoy
Decision Making Under Uncertainty
September 1, 2021
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Introduction
Introduction
Decision analysis is an analytical and systematic way to tackle problems.
A good decision is based on logic: rational decision-maker (DM)
We will start with:
One-dimensional (single criterion) decision making.
Multi-criteria decision making
Value of information:
Value of perfect information
Value of sample information
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Decision Making Under Uncertainty
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Introduction
Decision Making Environments
Type 1 : DM knows nothing!
Type 2 : DM knows the payoff information with certainty, does not know the uncertainties
associated with the payoffs.
Type 3 : DM knows every value involved in the problem with certainty.
A decision problem has:
an objective,
alternatives (options),
possible states of nature,
payoff values (cost or reward).
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Decision Making Under Uncertainty
September 1, 2021
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Single Criterion Decision Making
Single Criterion Decision Making
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Decision Making Under Uncertainty
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Single Criterion Decision Making
A Decision Table
A decision (payoff) table presenting profits for giving a concert:
Indoor venue
Outdoor venue
Good weather
$ 7000
$ 15000
Bad weather
$ 8000
$ -5000
A decision (payoff) table presenting expanding a business:
Construct large plant
Construct small plants
Do nothing
Altay & Baykal-Gürsoy
Favorable market
$ 200,000
$ 100,000
$0
Decision Making Under Uncertainty
Unfavorable market
$ ´180,000
$ ´20,000
$0
September 1, 2021
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Solution Methods
Maximax
Maximin
Criterion of Realism
Minimax
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Decision Making Under Uncertainty
September 1, 2021
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Maximax
Also known as the optimistic method, it proposes to choose the alternative with
the maximum reward.
Indoor venue
Outdoor venue
Good weather
$ 7000
$ 15000
Bad weather
$ 8000
$ -5000
Maximum Output
Favorable
market
Unfavorable
market
Maximum
output
$ 200,000
$ ´180,000
$ 100,000
$ ´20,000
$0
$0
Construct
large plant
Construct
small plant
Do nothing
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Decision Making Under Uncertainty
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Maximin
Also known as the pessimistic method, it proposes to choose the most secure
alternative.
Indoor venue
Outdoor venue
Construct
large plant
Construct
small plant
Do nothing
Altay & Baykal-Gürsoy
Good weather
$ 7000
$ 15000
Bad weather
$ 8000
$ -5000
Favorable
market
Unfavorable
market
$ 200,000
$ ´180,000
$ 100,000
$ ´20,000
$0
$0
Decision Making Under Uncertainty
Worst case
Worst case
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Criterion of Realism (CoR)
Let’s balance optimism-pessimism with a coefficient α:
The most optimistic value has the emphasis of α,
The most pessimistic value has the emphasis of 1 ´ α.
The concert example:
”The concert is in July. How bad can the weather be, anyway? The weather is
good 90% of the time” or
”You know what? The concert is in fall when it rains most days.” or
How would your decision change?
Indoor venue
Outdoor venue
Altay & Baykal-Gürsoy
Good weather
$ 7000
$ 15000
Decision Making Under Uncertainty
Bad weather
$ 8000
$ -5000
September 1, 2021
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Criterion of Realism (CoR)
Let’s see mathematically whether your intuition holds.
Case 1. ”The concert is in July. How bad can the weather be, anyway? The
weather is good almost all the time”
Indoor venue
Outdoor venue
Altay & Baykal-Gürsoy
Good weather
$ 7000
$ 15000
Decision Making Under Uncertainty
Bad weather
$ 8000
$ -5000
September 1, 2021
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Criterion of Realism (CoR)
Case 2. ”You know what? The concert is in fall when it rains most days.”
Indoor venue
Outdoor venue
Altay & Baykal-Gürsoy
Good weather
$ 7000
$ 15000
Decision Making Under Uncertainty
Bad weather
$ 8000
$ -5000
September 1, 2021
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Criterion of Realism (CoR)
Case 3. ”I really have no idea how the weather is going to be. It can go either
way.”
Indoor venue
Outdoor venue
Altay & Baykal-Gürsoy
Good weather
$ 7000
$ 15000
Decision Making Under Uncertainty
Bad weather
$ 8000
$ -5000
September 1, 2021
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The Expansion Example with CoR
We will do this one together.
The new Amazon fulfillment center is opening in Bensalem, PA. Should we open a
large center or a small center?
Or should I not open this center at all?
Construct a
large center
Construct a
small center
Do not
open a center
Altay & Baykal-Gürsoy
High utilization
Medium utilization
Low utilization
$ 200,000
$ -10,000
$ -180,000
$ 100,000
$ 10,000
$ ´20,000
$0
$0
$0
Decision Making Under Uncertainty
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Minimax
What about the opportunity loss?
Minimax promotes the alternative with the minimum regret.
Indoor venue
Outdoor venue
Good weather
$ 7000
$ 15000
Bad weather
$ 8000
$ -5000
Regret table:
Good weather
Bad weather
Maximum regret
Indoor venue
Outdoor venue
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Minimax
High utilization
Medium utilization
Low utilization
$ 200,000
$ -10,000
$ -180,000
$ 100,000
$ 10,000
$ ´20,000
$0
$0
$0
Construct a
large center
Construct a
small center
Do not
open a center
High
utilization
Medium
utilization
Low
utilization
Maximum
regret
Construct a
large center
Construct a
small center
Do not
open a center
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Results of the examples
Results of all methods are
Maximax
Maximin
Criterion of Realism
Minimax
Concert example
Amazon example
Outdoor venue
Construct a large center
Indoor venue
Do not open a center
Depends on α
Indoor venue
Construct a small center
Which one do you think the DM should choose?
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Decision Making Under Risk
Decision Making Under Risk
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Decision Making Under Risk
Decision Making Under Risk
Probability: a numerical statement about the likelihood that an event will occur.
The probability, P, of any event occurring is greater than or equal to 0 and
less than or equal to 1.
The sum of the simple probabilities for all possible outcomes of an activity
must equal 1.
Objective probability: Determined by observations and experiments.
Subjective probability: Determined by belief, experience, judgment, or
intuition.
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Decision Making Under Risk
When probabilities are given...
The concert example:
Probabilities
Indoor venue
Outdoor venue
Good weather
0.60
$ 7000
$ 15000
Bad weather
0.40
$ 8000
$ -5000
The Amazon example:
Probabilities
Construct a
large center
Construct a
small center
Do not open
a center
Altay & Baykal-Gürsoy
High utilization
0.3
Medium utilization
0.5
Low utilization
0.2
$ 200,000
$ ´10,000
$ ´180,000
$ 100,000
$ 10,000
$ ´20,000
$0
$0
$0
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Decision Making Under Risk
Expected Monetary Value
Favors the alternative with the maximum weighted row average.
Probabilities
Construct a
large center
Construct a
small center
Do not open
a center
Altay & Baykal-Gürsoy
Highly
utilization
0.3
Medium
utilization
0.5
Low
utilization
0.2
200000
-10000
-180000
100000
10000
- 20000
0
0
0
Decision Making Under Uncertainty
Expected value
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Decision Making Under Risk
Expected Opportunity Loss
Favors the alternative with the minimum weighted row average of the regret
matrix
Probabilities
Indoor venue
Outdoor venue
Good weather
0.60
$ 7000
$ 15000
Bad weather
0.40
$ 8000
$ -5000
Regret table:
Good weather
Bad weather
Expected regret
Indoor venue
Outdoor venue
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Decision Making Under Risk
Value of Information
Assume an oracle (or a consultant) that will give you the results of the market
analysis that will give the information about the true state?
What is the value of such an analysis? How much should the DM pay?
Expected value of Perfect Information = EV with PI - Maximum EV under risk
Probabilities
Construct a
large center
Construct a
small center
Do not
open a center
High utilization
0.3
Medium utilization
0.5
Low utilization
0.2
$ 200,000
$ -10,000
$ -180,000
$ 100,000
$ 10,000
$ ´20,000
$0
$0
$0
Maximum EV under risk = $ 31000 as we found on the previous slide EV with PI
= 200000 ¨ 0.3 ` 10000 ¨ 0.5 ` 0 ¨ 0.2 “ $ 65000
Expected value of Perfect Information = 65000 - 31000 = 34000
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Decision Making Under Risk
Expected Opportunity Loss
Probabilities
Construct a
large center
Construct a
small center
Do not
open a center
High utilization
0.3
Medium utilization
0.5
Low utilization
0.2
$ 200,000
$ -10,000
$ -180,000
$ 100,000
$ 10,000
$ ´20,000
$0
$0
$0
Frequently
utilized
Somehow
utilized
Not used
much
Expected
regret
Construct a
large center
Construct a
small center
Do not
open a center
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Decision Trees
Decision Trees
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Decision Trees
Decision Trees
A decision tree is a diagram consisting of
decision nodes (squares)
chance nodes (circles)
decision branches (alternatives)
chance branches (state of natures), and
terminal nodes (payoffs or rewards)
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Decision Trees
Decision Tree for the Amazon Example
The decision: construct a large/small center, or do not construct a new.
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Decision Trees
Decision Tree for the Amazon Example
The decision: construct a large/small center, or do not construct a new.
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Decision Trees
Decision Tree for the Amazon Example
For the first option we have three different outcomes
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Decision Trees
Decision Tree for the Amazon Example
For the second option we have three different outcomes. For the last option, we
only have one outcome.
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Decision Trees
Decision Tree for the Amazon Example
The outcomes are added to the tree.
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Decision Trees
Decision Tree for the Amazon Example
The probabilities are added to the tree.
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Decision Trees
Decision Tree for the Amazon Example
We work our way backwards and find a value for every chance and decision node.
We have the chance nodes first.
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Decision Trees
Decision Tree for the Amazon Example
At every decision node, we select the highest value.
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Decision Trees
Another Decision Tree Example
For the warehouse location problem, let’s assume a market research is available for
a price of 10,000. This research will not provide the perfect information such that:
If there will be frequent utilization, there is a 90% chance that the market
research will predict it. It will predict a medium utilization with a probability
of %10.
If the utilization will be medium, there is a 70% chance that the research will
predict medium utilization. With a 20% chance, it will predict a frequent
utilization and with a 10% chance, it will predict a low utilization.
If the utilization will be low, there is a 80% chance that it will predict a low
utilization and a 20% chance that it will predict a medium utilization.
Should we conduct the market research?
Should we open a center? If so, what should be the size?
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Decision Trees
Example 2 (contd.)
First decision: Should we conduct the market research?
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Decision Trees
Example 2 (contd.)
There are three possible outcomes.
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Decision Trees
Example 2 (contd.)
For any outcome, we have to choose from the three alternatives.
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Decision Trees
Example 2 (contd.)
Let’s solve each branch! First branch:
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Decision Trees
Example 2 (contd.)
If we solve all last branches
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Decision Trees
Example 2 (contd.)
What about market research output probabilities?
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Utility Theory
Utility Theory
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Utility Theory
Utility Theory - Deal or No Deal
The contestant then begins choosing cases
that are to be removed from play.
The amount inside each chosen case is
immediately revealed
Throughout the game, after a
predetermined number of cases have been
opened, the banker offers the contestant
an amount of money to quit the game
The offer based roughly on the amounts
remaining in play and the contestant’s
demeanor, the bank tries to ’buy’ the
contestant’s case for a lower price than
what’s inside the case.
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Utility Theory
Utility Theory - Deal or No Deal
The player then answers ”Deal or no deal”
question, choosing:
”Deal”, accepting the offer and ending
the game
”No Deal”, rejecting the offer and
continuing
This process of removing cases and
receiving offers continues, until
either the player accepts an offer to
’deal’,
or all offers have been rejected and
player wins the cash inside her/his case.
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Utility Theory
Utility Theory - Deal or No Deal
Deal or no deal?
What about $ 200,000?
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Utility Theory
Utility Theory
Utility assessment may assign the worst payoff a utility of 0 and the best
payoff a utility of 1.
A standard gamble is used to determine utility values: When DM is
indifferent between two alternatives, the utility values of them are equal.
Choose the alternative with the maximum expected utility
Back to our example:
Probabilities
Construct a
large center
Construct a
small center
Do not open
a center
Altay & Baykal-Gürsoy
High utilization
0.3
Medium utilization
0.5
Low utilization
0.2
$ 200,000
$ ´10,000
$ ´180,000
$ 100,000
$ 10,000
$ ´20,000
$0
$0
$0
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Utility Theory
Utility Theory - Example (contd.)
Output
$200,000
$100,000
$10,000
$0
-$10,000
-$20,000
-$180,000
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Utility
1
0
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Utility Theory
Utility Theory - Example (contd.)
Let’s finish it together!
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Utility Theory
Utility Theory - Example (contd.)
Output
$200,000
$100,000
$10,000
$0
-$10,000
-$20,000
-$180,000
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Utility
1
0
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Utility Theory
Utility Theory - Example (contd.)
Then, the decision tree becomes
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Utility Theory
Risk Premium
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Utility Theory
Risk Premium
If the DM is risk averse (avoids risk), RP ¿ 0
They prefer to receive a sum of money equal to the expected value of a lottery
than to enter the lottery itself.
If the DM is risk prone (seeks risk), RP ¡ 0
They prefer to enter a lottery instead of receiving a sum of money equal to its
expected value.
If the DM is risk neutral, RP = 0
They are indifferent between entering any lottery and receiving a sum of
money equal to its expected value.
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Utility Theory
Q&A
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Utility Theory
Overview
Uncertainties in Materials Management
Single-Period Inventory Uncertainties
Multi-Period Inventory Uncertainties
Continuous Review Inventory Models
Periodic Review Inventory Models
Markov Chains
Markov Decision Processes
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Uncertainties in Materials Management
Uncertainties in Materials Management
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Uncertainties in Materials Management
Uncertainties in Materials Management and Lot Sizing
Uncertainty means that demand is a random variable.
It is defined by a probability distribution.
This probability distribution is estimated from past history of demands.
The cost function is a random variable, as well.
The general assumption is to minimize the expected cost.
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Uncertainties in Materials Management
Overview of Some Probability Distributions
Normal Distribution
Let D denote the demand. For normal distribution is denoted by D „ Npµ, σ 2 q.
µ is the mean of the distribution.
σ is the standard deviation.
A normal distribution is a continuous probability distribution for
a random variable.
The graph of a normal distribution is called the normal curve.
φpD “ xq “
Altay & Baykal-Gürsoy
Decision Making Under Uncertainty
2
2
1
? e ´px´µq {2σ
σ 2π
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Uncertainties in Materials Management
The Normal Distribution
1) Which distribution has greater mean?
2) Which distribution has greater variance / standard variation?
The standard normal distribution is a normal distribution with a mean of 0 and a standard
deviation of 1.
Any value can be transformed into a z-score by using the formula
z“
Altay & Baykal-Gürsoy
Value ´ Mean
x ´µ
“
StandardDeviation
σ
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Uncertainties in Materials Management
The Normal Distribution
Standard Normal Distribution:
Using Standard Normal Distribution for Demands:
Assume that D „ Np100, 102 q.
The probability that the demand is less than 115?
zp115q “
115 ´ 100
“ 1.5
10
We denote this by Φp1.5q.
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Uncertainties in Materials Management
Standard Normal Table
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Uncertainties in Materials Management
Standard Normal Distribution
The probability that the demand is less than 95?
zp95q “
95 ´ 100
“ ´0.5
10
Φp´0.5q “?
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
Single-Period Demand Uncertainties
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
Single-Period Demand Uncertainties
Notation:
co : unit holding cost (the cost of per unit of inventory remaining at the end of the period)
cu : unit underage cost (the cost of per unit unmet demand at the end of the period)
D: demand (random variable)
Q: the decision variable (how much we will order at the beginning of the period)
C pQ, Dq: total overage and underage cost incurred at the end of the period
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
Single-Period Demand Uncertainties
Notation:
co : unit holding cost (the cost of per unit of inventory remaining at the end of the period)
cu : unit underage cost (the cost of per unit unmet demand at the end of the period)
D: demand (random variable)
Q: the decision variable (how much we will order at the beginning of the period)
C pQ, Dq: total overage and underage cost incurred at the end of the period
Remaining Inventory “
Altay & Baykal-Gürsoy
#
Q ´D
0
DďQ
“ maxp0, Q ´ Dq
DąQ
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
Single-Period Demand Uncertainties
Notation:
co : unit holding cost (the cost of per unit of inventory remaining at the end of the period)
cu : unit underage cost (the cost of per unit unmet demand at the end of the period)
D: demand (random variable)
Q: the decision variable (how much we will order at the beginning of the period)
C pQ, Dq: total overage and underage cost incurred at the end of the period
Remaining Inventory “
Unmet Demand “
Altay & Baykal-Gürsoy
#
Q ´D
0
#
0
D ´Q
DďQ
“ maxp0, Q ´ Dq
DąQ
DďQ
“ maxp0, D ´ Qq
DąQ
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
The Newsvendor Problem
C pQ, Dq “ co maxp0, Q ´ Dq ` cu maxp0, D ´ Qq
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
The Newsvendor Problem
C pQ, Dq “ co maxp0, Q ´ Dq ` cu maxp0, D ´ Qq
For the optimal amount
ΦpD ˚ q “
Altay & Baykal-Gürsoy
cu
co ` cu
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
The Newsvendor Problem
C pQ, Dq “ co maxp0, Q ´ Dq ` cu maxp0, D ´ Qq
For the optimal amount
cu
co ` cu
Let’s say D „ Np100, 102 q. You buy the item for $5 and sell for $8.
ΦpD ˚ q “
co : Cost of an unsold item cu : Cost of an unmet demand
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
The Newsvendor Problem
C pQ, Dq “ co maxp0, Q ´ Dq ` cu maxp0, D ´ Qq
For the optimal amount
cu
co ` cu
Let’s say D „ Np100, 102 q. You buy the item for $5 and sell for $8.
ΦpD ˚ q “
co : Cost of an unsold item cu : Cost of an unmet demand
3
“ 0.375
5`3
´ D ˚ ´ 100 ¯
Φ
“ 0.375
10
D ˚ ´ 100
“ ´0.32
10
˚
D “ 96.8 « 97
Φpz ˚ q “
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
Standard Normal Table
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
The Newsvendor Problem
Let’s say D „ Np100, 102 q.
You buy the item for $5 and sell for $8. Any unsold item is returned for $1.
co : Cost of an unsold item cu : Cost of an unmet demand
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
The Newsvendor Problem
Let’s say D „ Np100, 102 q.
You buy the item for $5 and sell for $8. Any unsold item is returned for $1.
co : Cost of an unsold item cu : Cost of an unmet demand
3
“ 0.428
4`3
´ D ˚ ´ 100 ¯
Φ
“ 0.428
10
D ˚ ´ 100
“ ´0.21
10
˚
D “ 97.9 « 98
Φpz ˚ q “
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
The Newsvendor Problem
Expected shortage (lost sales) “ σLpzq
˜
¸
´Q ´ µ¯ Q ´ µ
´Q ´ µ¯
Lpzq “ φ
´
1´Φ
σ
σ
σ
Expected sales “ µ ´ Expected shortage
Expected overage “ Q ´ Expected sales
Expected fill rate “
Altay & Baykal-Gürsoy
Expected sales
µ
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
The Newsvendor Problems
Let’s say D „ Np100, 102 q.
You buy the item for $5 and sell for $8. Any unsold item is returned for $1.
An unmet demand and an unhappy customer yields 4 more customers to churn.
co : Cost of an unsold item cu : Cost of an unmet demand
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Uncertainties in Materials Management
Single-Period Demand Uncertainties
The Newsvendor Problems
Let’s say D „ Np100, 102 q.
You buy the item for $5 and sell for $8. Any unsold item is returned for $1.
An unmet demand and an unhappy customer yields 4 more customers to churn.
co : Cost of an unsold item cu : Cost of an unmet demand
15
“ 0.833
15 ` 3
´ D ˚ ´ 100 ¯
Φ
“ 0.833
10
˚
D ´ 100
“ 1.01
10
˚
D “ 110.1 « 110
Φpz ˚ q “
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Engineering Decision Making Under Uncertainty
Aimen Sudhir, RUID: 197007651
To make the recommendation, the following decision tree was made.
As seen from the decision tree, it is profitable for Ectron Fragrance Corporation to conduct the market
survey.
Purchase answer to see full
attachment