Module 3
Decision Theory
and the Normal
Distribution
Prepared by Lee Revere and John Large
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-1
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Learning Objectives
Students will be able to:
1. Understand how the normal
curve can be used in performing
break-even analysis.
2. Compute the expected value of
perfect information (EVPI)
using the normal curve.
3. Perform marginal analysis
where products have a constant
marginal profit and loss.
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-2
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Module Outline
M3.1 Introduction
M3.2 Break-Even Analysis and the
Normal Distribution
M3.3 EVPI and the Normal
Distribution
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-3
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Introduction
The normal distribution can be
used when there are a large
number of states of nature and/or
alternatives.
Note: it would be impossible to develop a
decision table or tree if there were 50, 100,
or even more states and/or alternatives!
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-4
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Break-Even Analysis
and the Normal
Distribution
Break-even analysis, also called costvolume analysis, answers managerial
questions relating the effect of a decision
to overall revenues or costs.
Break-even point (units) =
Fixed Costs
Price/unit – Variable Costs/unit
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-5
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Barclay Brothers New
Product Decision
The Barclay Brothers Company is a large
manufacturer of adult parlor games. They are
considering introducing Strategy, a new game.
Their fixed cost for Strategy = $36,000 and
their variable costs are $4 per game produced.
They intend to sell the game for $10. What is
the break-even point?
Break-even point (units) =
$36,000
$10 - $4
= 6,000 games
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-6
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Barclay Brothers New
Product Decision
The Barclay Brothers expect demand to
be 8,000. They believe there is a 15%
chance it will be less than 5,000 and a
15% chance it will be greater than 11,000
~ what is the probability they will lose
money by not selling 6,000 games?
Hint: the normal distribution can be used
if the Barclay Brothers believe demand is
normally distributed.
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-7
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Normal Distribution
for Barclay’s Demand
Mean of the Distribution, µ
15 Percent Chance
Demand Exceeds
11,000 Games
15 Percent
Chance
Demand is
Less Than
5,000 Games
X
5,000
11,000 Demand (Games)
µ=8,000
Z=
Demand - µ

 Use the normal table to find Z
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-8
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Normal Distribution
for Barclay’s Demand
Look up 0.85 in the body of the normal table
and find the associated Z:
Z = 1.04; so
1.04 = 11,000 - 8,000

 = 2,885 units
What is the probability of
selling less than
6,000 – 8,000
6,000 units????
2,885
= -.69
Z = .7549 so, 1 - .7549 = .2451
Or a 24.51% chance of losing $$$ !
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-9
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Barclay’s Break-Even
Analysis
P(loss) = P(demand < breakeven) = 0.2451
= 24.51%
P(profit) = P(demand > breakeven) = 0.7549
= 75.49%
Should Barclay bring its Strategy game
to the market?
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-10
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Expected Value of Perfect
Information and the
Normal Distribution
The opportunity loss function describes
the loss that would be suffered by making
the wrong decision. The opportunity loss
function can be computed by:
Opportunity K (Break-even point - X)
loss =
for X < Breakeven
$0 for X > Breakeven
where
K = the loss per unit when sales are below the
break-even point
X = sales in units.
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-11
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Barclay’s
Opportunity Loss
Function
Opportunity $6 (6,000 - X)
loss =
for X < 6,000 games
$0 for X > 6,000 games
Loss Profit
Loss ($)
µ = 8,000
 = 2,885
Slope = 6
Break-even
point (XB)
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
Normal
Distribution
µ
6,000
M3-12
X
Demand
(Games)
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Expected Opportunity
Loss
The expected opportunity loss is the
sum of the opportunity losses
multiplied by the probability of that
demand occurring.
In the Barclay Brothers example, there
are a large number of possible sales
volumes (0,1,2,…5,999), so the unit
normal loss integral is used.
N(D) is the table value for the normal
loss integral for a given value of D.
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-13
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Expected Value of
Perfect Information
The expected value of perfect
information is equivalent to the
minimum EOL.
EVPI = EOL = K N(D)
Where
EOL = expected opportunity loss
K = loss per unit when sales are below
the break-even point
 = standard deviation of the
distribution
  breakeven
D

µ = mean sales
N(D) = the table value for the unit normal
loss integral
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-14
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Expected Value of
Perfect Information
(continued)
 = 2,885
K = $6
8,000  6,000
D
 0.69  0.60  0.09
2,885
N(.69) = .1453
Therefore
EOL = K N(.69)
= ($6)(2885)(.1453) = $2,515.14
EVPI = $2515.14
To accompany Quantitative Analysis
for Management, 9e
by Render/Stair/Hanna
M3-15
© 2006 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458