McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

Locating a shopping mall in a coastal area
•Villages are located East to West along the coast (Ocean to the North)
•Problem for the developer is to locate the mall at a place which minimizes total travel miles (TTM).
10
Number of Customers per Week (Thousands)
10 10 5 20 10
West
15
A B x
C D E F G
3.0
3.5
2.0
2.5
4.5
2.0
Distance between Towns (Miles)
4.5
15
East
H
3-2

Minimizing TTM by enumeration
•The developer selects one site at a time, computes the
TTM, and selects the site with the lowest TTM.
•The TTM is found by multiplying the distance to the mall by the number of trips for each town (beginning with town A and ending with town H).
•For example, the TTM for site X (a mile west of town
C) is calculated as follow:
(5.5)(15) + (2.5)(10) + (1.0)(10) + (3.0)(10) + (5.5)(5) +
(10.0)(20) + (12.0)(10) + (16.5)(15) = 742.5
3-3

Marginal analysis is more effective
Enumeration takes lots of computation. We can find the optimal location for the mall
easier using marginal analysis —
that is, by assessing whether small changes at the margin will improve the objective (reduce
TTM, in other words).
3-4

Illustrating the power of marginal analysis
1. Let’s arbitrarily select a location—say, point X
. We know that TTM at point X is equal to 742.5—but we don’t need to compute TTM first.
2. Now let’s move in one direction or another (We will move East, but you could move West).
3. Let’s move from
location X to town C
. The key question: what is the change in TTM as the result of the move?
4. Notice that the move reduces travel by one mile for everyone living in town C or further east.
5. Notice also that the move increases travel by one mile for everyone living at or to the west of point X..
3-5

Computing the change in
TTM
To compute the change in total travel miles (

TTM) by moving from point X to C:

TTM = (-1)(70) + (1)(25) = - 45
Reduction in TTM for those residing in and to the East of town C
Increase in TTM for those residing at or to the west of point X.
Conclusion: The move to town C unambiguously decreases TTM— so keep moving East so long as
TTM is decreasing.
3-6

Rule of Thumb
Make a “small” move to a nearby alternative if, and only if, the move will improve one’s objective
(minimization of TTM, in this case). Keep moving, always in the direction of an improved objective, and stop when no further move will help.
•
Check to see if moving from
town C to town D
will improve the objective.
•
Check to see if moving from
town E to town F
will improve the objective.
3-7

• An optimization problem involves the specification of three things:
• Objective function to be maximized or minimized
• Activities or choice variables that determine the value of the objective function
• Any constraints that may restrict the values of the choice variables
3-8

• Maximization problem
• An optimization problem that involves maximizing the objective function
• Minimization problem
• An optimization problem that involves minimizing the objective function
3-9

• Unconstrained optimization
• An optimization problem in which the decision maker can choose the level of activity from an unrestricted set of values
• Constrained optimization
• An optimization problem in which the decision maker chooses values for the choice variables from a restricted set of values
3-10

• Choice variables determine the value of the objective function
• Continuous variables: Can assume an infinite number of values within a given range — usually the result of measurement.
• Discrete variables
3-11

• Continuous variables
• Can choose from uninterrupted span of variables
• Discrete variables
• Must choose from a span of variables that is interrupted by gaps
3-12

• Net Benefit
(NB)
• Difference between total benefit
(TB) and total cost (TC) for the activity
•
NB = TB – TC
• Optimal level of the activity
(A
*
) is the level that maximizes net benefit
3-13

(Figure 3.1)
4,000
3,000
2,310
2,000
C •
B •
•
B’
NB* = $1,225
•
•
D
D’
F •
1,085
1,000
•
C’
0 200 350 = A* 600 700
Level of activity
Panel A – Total benefit and total cost curves
TC
G •
TB
1,000
A
1,225
1,000
600
• c’’
0
Panel B – Net benefit curve
200
M • d’’
•
350 = A* 600
Level of activity
• f’’
NB
1,000
A
3-14

Marginal Benefit & Marginal Cost
• Marginal benefit
(MB)
• Change in total benefit
(TB) caused by an incremental change in the level of the activity
• Marginal cost
(MC)
• Change in total cost
(TC) caused by an incremental change in the level of the activity
3-15

Marginal Benefit & Marginal Cost
MB



TB

A
MC



TC

A
3-16

• Marginal variables measure rates of change in corresponding total variables
• Marginal benefit & marginal cost are also slopes of total benefit & total cost curves, respectively
3-17

(Figure 3.2)
TC
4,000
3,000
100
320
100
D’
•
•
D
F •
820
G •
520
100
•
B
100
2,000
640
•
C B’
•
520
1,000
8
C’ •
100
100
340
0 200 350 = A* 600
Level of activity
Panel A
– Measuring slopes along TB and TC
800 1,000
MC (= slope of TC)
• d’ (600, $8.20)
6
5.20
4
• c (200, $6.40)
• b
• c’ (200, $3.40)
• d (600, $3.20)
2
MB (= slope of TB)
TB
A g •
1,000 0 200
Panel B
– Marginals give slopes of totals
350 = A* 600
Level of activity
800
A
3-18

Using Marginal Analysis to Find
Optimal Activity Levels
• If marginal benefit > marginal cost
• Activity should be increased to reach highest net benefit
• If marginal cost > marginal benefit
• Activity should be decreased to reach highest net benefit
3-19

Using Marginal Analysis to Find
Optimal Activity Levels
• Optimal level of activity
• When no further increases in net benefit are possible
• Occurs when
MB = MC
3-20

Using Marginal Analysis to Find A *
(Figure 3.3)
0
300
MB > MC
100
• c’’
200
MB = MC
MB < MC
• M
350 = A*
100 d’’
•
600
500
800
NB
Level of activity
1,000
A
3-21

Unconstrained Maximization with
Discrete Choice Variables
• Increase activity if
MB > MC
• Decrease activity if
MB < MC
• Optimal level of activity
• Last level for which
MB exceeds MC
3-22

Irrelevance of Sunk, Fixed, and
Average Costs
• Sunk costs
• Previously paid & cannot be recovered
• Fixed costs
• Constant & must be paid no matter the level of activity
• Average (or unit) costs
• Computed by dividing total cost by the number of units of the activity
3-23

Irrelevance of Sunk, Fixed, and
Average Costs
• These costs do not affect marginal cost & are irrelevant for optimal decisions
3-24

• The ratio
MB/P represents the additional benefit per additional dollar spent on the activity
• Ratios of marginal benefits to prices of various activities are used to allocate a fixed number of dollars among activities
3-25

• To maximize or minimize an objective function subject to a constraint
• Ratios of the marginal benefit to price must be equal for all activities
• Constraint must be met
MB
A

MB
B
P
A
P
B
MB
Z
P
Z
3-26