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[3&1 Law '~

     
 

L I B R A R Y
Michigan 5mm
University  

 
     

This is to certify that the

thesis entitled

TRUCKLOAD MOTOR CARRIER.RATES IN A
NORMATIVE SPATIAL ENVIRONMENT

presented by

James Patrick Hynes

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in .Markgtingjnd

Transportation

 

__ 7 "
(£2 .3, 2/, mm, ..i--,M4 V
J

Major professor

Date 944%, I] I ? 7/

0-7639

 

 

WM!“ *3

 

ABSTRACT

TRUCKLOAD MOTOR CARRIER RATES IN A
NORMATIVE SPATIAL ENVIRONMENT

BY

James Patrick Hynes

Very little analytical attention has been focused
on directional and spatial effects in truckload freight
rate structures. Directional effects in freight rates
have been widely discussed in the literature, but a general
mathematical resolution of the problem has not evolved.

Directional effects are looked at here in terms of
a profit maximizing motor carrier. Under normative
assumptions, the characteristics of a rate structure
which maximizes profit for the carrier are examined.

Rates on truckload freight traffic in an actual
common motor carrier are examined for directional patterns
which correspond with empty vehicle movement patterns
prescribed by the normative model. The data indicate a
positive statistically significant relationship.

It is concluded that directional effects are
significant and measurable factors in a motor carrier's
rate structure, and therefore should not be ignored in

contemporary rate studies.

TRUCKLOAD MOTOR CARRIER RATES IN A

NORMATIVE SPATIAL ENVIRONMENT

BY

James Patrick Hynes

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Marketing and Transportation

1971

© Copyright by
JAMES PATRICK HYNES

1971

ACKNOWLEDGMENTS

This dissertation was brought to fruition through
the efforts of many people. The author wishes to extend
his sincere appreciation for these contributions.

My gratitude is especially directed toward Dr.
Frank H. Mossman, committee chairman, for providing me with
the basic theoretical framework for implementing this re-
search. His inspirational and material help made it
possible.

I wish to thank Dr. Richard F. Gonzalez, committee
member, for his invaluable critiques of the theoretical
aspects of this thesis. A special thanks to Dr. Gardner M.
Jones for providing insightful interpretations of the
relations encountered in the research. Also, the aid of
the motor carrier firm which provided the data is greatly
appreciated.

My greatest thanks goes to my wife, Donna, whose
willing sacrifices and encouragement contributed to the

successful completion of this project.

ii

 

TABLE OF CONTENTS

Chapter

I. A FRAMEWORK FROM WHICH TO APPROACH TRUCK-
LOAD FREIGHT TRANSPORTATION RATES IN A

MOTOR CARRIER SYSTEM . . . . . . . .
Introduction . . . . . . . . . .
General Considerations . . . . . .
Viewpoints . . . . . . . . . .

The Problem . . . . . . . . . .

Rate Adjustment and Profit Maximization
Literature Review . . . . . . . .
Transportation Equipment Scheduling. .
Objectives and Organization . . . . .

II. CHARACTERISTICS OF PROFIT MAXIMIZATION IN A
NORMATIVE MODEL OF A MULTIPLE TERMINAL MOTOR

CARRIER O O O O I O O I O O O I
Introduction . . . . . . . . . .
General Model Structure . . . . . .

Segmentation of Carrier Activities . .

Freight and Vehicle Assumptions . . .
Mathematical Notation. . . . . .
Number of Terminals. . . . . .
Freight Flows. . . . . . . .
Empty Vehicle Flows. . . . .
External Vehicle Capacity. . . .
Flow Constraints . . . . . . .

iii

Page

13

14

17

17
18

18
19

21
21
24
24
24

25

Chapter
Cost and Revenue Assumptions . . . .

Direct Transportation Costs. . . .
Empty Vehicle Movement Costs . .
Truckload Freight Revenues . . . .
Profits on Other Carrier Activities .

Model Synthesis . . . . . . . .
Necessary Profit Maximization Conditions.

Positive Freight and Empty Vehicle
Flows. . . . . . . . . . . .
Positive Freight Flow and Zero Empty
Vehicle Flow . . . . . . . . .
Positive Empty Vehicle Flow and Zero
Freight Flow . . . . . . . . .
Zero Freight Flow and Zero Empty
Vehicle Flow . . . . . . . . .
Constrained Vehicle Flow at Terminals .

The Interpretation of the Conditions for

Maximized Profits . . . . . . . .
Terminal Multipliers. . . . . . .
Marginal Vehicle Profit. . . . .

Optimal Relationships Among the Values
of the Terminal Multipliers . . . .
Marginal Vehicle Profits and Circuitous
Routes . . . . . . . . . . .
Changes in Terminal Vehicle Capacity
Caused by Other Carrier Activities . .
The Relationship Between Marginal
Revenue and Marginal Direct Cost. . .

The Attainment of a Normative Optimum. .
Convenient Functional Relationships. .

Normative Optimality and the Behavior of

an Actual Motor Carrier Firm. . . . .
Adjustment Processes. . . . . . .
Conclusions . . . . . . . . . .

Marginal Revenues and Marginal Costs .
Terminal Multipliers. . . . . . .
Theoretical Behavior and Actual

Behavior of a Motor Carrier Firm. . .

iv

Page
27
27
29
29
31
32

33

34
34
36
36
36
37

37
38

39
42
44
45
46

47

48
51
52

52
53

54

 

Chapter Page

III. AN EMPIRICAL INVESTIVATION . . . . . . 55
Introduction . . . . . . . . . . 55
Organization . . . . . . . . . 56
Data Source . . . . . . . . . . S6
Intraline Group Ten Freight . . . . 58
Mathematical Notation of Data. . . . 59
Unit Revenue. . . . . . . . . 60
Direct Unit Cost . . . . . . . 60
Empty Vehicle Movement Cost. . . . 61
Direct Unit Profit. . . . . . . 62
Number of Shipments . . . . . . 62
Total Freight Flow. . . . . . . 62
Intraline Group Ten Freight Flow . . 62
Distance Between Terminals . . . . 63

A Note on the Distinction Between
Empirical Variables and Normative

variables 0 O O O O O O O O O 63
A Note on Weighted Observations . . . 63
A General Description of the Hypothesized
Profit Patterns . . . . . . . . . 65
The Carrier and Rate Adjustments. . . 67
Empty Vehicle Flows . . . . . . . 68
General Assumptions . . . . . . . 70
Statistical Models, Tests and Results. . 71
Imputed Empty Vehicle Flows and Differ—
ences Among (t) Values . . . . . . 72
Weighted Observations . . . . . . 75
Unit Profit Regression Models. . . . 75
Concomitant Variables. . . . . . 78
The Analysis of Residuals . . . . 80
Alternative Regression Models. . . . 81
Unit Revenues and Distances. . . . 81
Deletion of Minor Terminals. . . . 82
Alternative Computation of Imputed
Backhaul Cost Variables . . . . . 83
Conclusions . . . . . . . . . . 83

 

Chapter Page

IV. EVALUATION AND CONCLUSIONS . . . . . . 85
An Overview . . . . . . . . . . 85
Significance of the Normative Models . . 87

Shippers, Carriers, and Regulatory
Agencies. . .' . . . . . . . . 87.
Regulation and Economic Theory . . . 88

The Significance of the Empirical

Findings . . . . . . . . . . . 89
Potential Research Extensions . . . . 89
BIBLIOGRAPHY . . . . . . . . . . . . . 91
APPENDICES
Appendix
A. Analysis of Variance Tables. . . . . . 107
B. The Kuhn—Tucker Theorem . . . . . . . 112

vi

 

10.

11.

LIST OF TABLES

A Summary of Mathematical Variables Used in
Chapter II 0 I O I O O O O O O O

A Summary of Necessary Conditions for
Maximized Profit . . . . . . . . .

A List of Normative and Empirical Variables.
Linear Programming Model Used for the
Determination of Imputed Empty Vehicle

Flows O O O I O O O O O I O O O

Assumptions of the Unit Profit Regression
Models . . . . . . . . . . . .

A Summary of Statistical Tests . . . . .

Analysis of Variance Table for Regression
Model WClLP2A . . . . . . . . . .

Analysis of Variance Table for Regression
MOdel WC ZLP 2A 0 O O O O O I O O 0

Analysis of Variance Table for Regression
MOdel WC 3LP 2A 0 O O O O O O O O 0

Analysis of Variance Table for Regression
MOdel WC4LP 2A 0 O O O O O O O O 0

Analysis of Variance Table for Regression
Model WC5LP2A . . . . . . . . . .

vii

Page

22

35

64

73

77

79

107

108

109

110

111

 

LIST OF FIGURES
Figure Page

1. Freight and Vehicle Flows in a Hypothetical
system 0 O O O O O O O O O O O O 7

viii

 

CHAPTER I

A FRAMEWORK FROM WHICH TO APPROACH TRUCKLOAD
FREIGHT TRANSPORTATION RATES IN A

MOTOR CARRIER SYSTEM

Introduction

 

This dissertation focuses on one facet of an issue
in the field of transportation. The issue is truckload
motor carrier freight rates. The intent here is to examine
analytically and to explain spatial interrelationships
among truckload freight rates in a multiple terminal motor
carrier system.

It is recognized in transportation that freight
rates are affected by the direction of freight movement.1
For example, lower domestic air rates generally prevail
from west to east and from south to north.2 Nevertheless,
no explicit analytical explanation to this phenomena has

evolved in the literature. In some cases this phenomena

 

lSee D. P. Locklin, Economics of Transportation
(5th ed.; Homewood, 111.: R. D. Irwin, Inc., 1960),
Chapters 20 and 23.

21bid., p. 779.

 

 

is completely ignored. This dissertation presents an

analytical resolution of the problem.

General Considerations

A volatile question for at least a half century
has been the precise identification of the economic con-
siderations faced by transportation firms and their cus-
tomers, and the interaction processes affecting freight
rates. The Pigou-Taussig controversyl of the early nine-
teen hundreds is evidence of the early differences in the
interpretation of the transportation rate problem with
regard to cost and demand factors. Since then the problem
has been discussed by other economists, accountants, trans-
portation experts, academicians, engineers, politicians,
statisticians, business men, and operations researchers.2
Progress has been realized through these interchanges, and
with each advance it becomes more evident that current
problems in transportation cannot be viewed solely from
one reference point. Rather, these problems are multi-
disciplinary, and acceptable solutions must address and

integrate a variety of contentions.

 

lPrincipal documentary evidence of this contro—
versy can be found in F. W. Taussig, "Railway Rates and
Joint Costs Once More," Quarterly Journal of Economics,
XXVII (February, 1913), 378-84. Also in A. C. Pigou,
"Railway Rates and Joint Costs," Quarterly Journal of
Economics, XXVII (May, 1913), 536-58.

 

 

2Specific citations to arguments put forth by
these various groups are given in the literature review
section.

Some day there may be a "transportation science"1

which untangles and places arguments in integrated and
structured perspectives. Unfortunately, this day is not
at hand; work in the area must still proceed on a com-
partmentalized basis, hypothesizing, identifying, and
measuring isolated interrelationships in transportation

systems.

Viewpoints

What is the appropriate charge for moving a ship-
ment of goods between two points? The answer to this
depends on the vested interests of the individual faced
with the question. The carrier entity providing the
transportation service would be motivated to charge the
most profitable price, or rate. The shipper entity uti-
lizing the transportation service would be motivated to
seek the lowest rate which would adequately fulfill his
shipping needs. A community would be interested in en—
forcing rates which maintain and prepare transportation
resources for national emergencies. An economist might be
interested in setting rates so as to maximize the economic
growth of the country. Another economist may be more con-
cerned with rate stability as it relates to economic

growth. All of these considerations must be taken into

 

1See M. L. Manheim, "Principles of Transport
Systems Analysis," Papers-~Seventh Annual Meeting of the
Transportation Research Forum (December, 1966), 9-21.

 

account when designing systems which adjust and control
freight rates in a changing environment.

This dissertation proceeds on the assumption that
although these and many other factors affect the logical
development of freight rates, it is necessary from a
practical standpoint to divide the problem into component
parts, attack each part separately, and then synthesize
results for further analysis. Of these many steps, one
is undertaken here; the economic interactions among
shippers and a motor carrier are examined with particular
emphasis on the directional orientations of rates. The
interactions are examined in terms of a normative economic
model. Specific mathematical conditions are formulated
which relate the supply of and the demand for transpor-

tation among multiple terminals.

The Problem

 

In a normative sense, freight rate levels are
results of economic interactions between shippers and
carriers. These interactions are often explained in terms
of the cost of service and value of service concepts.1 A
rate is economically viable as long as it is greater than

or equal to the carrier's cost of service and less than or

 

lElaborations of these concepts can be found in
most any basic transportation text. See D. F. Pegrum,
Transportation: Economics and Public Policy (revised
ed.; Homewood, 111.: R. D. Irwin, 1968), pp. 229-32.

 

equal to the shipper's value of the service.1 Neither the
cost of service nor value of service approaches clearly
resolve rate problems. First, it is difficult to identify
quantitatively the dollar value of the transportation ser-
vice. Second, it is difficult to allocate costs in a motor
carrier system. It is generally recognized that common
and joint costs cannot be consistently and logically allo-
cated to specific freight movements without taking into
account some intangible factors.

The distribution of the burden of these joint costs

is a matter requiring an interpretation of the

value—of—service or demand factor by kinds of

freight, size of shipment, and directions of move—

ment.2
Another author goes so far as to say:

. . . that there is no cost curve in the economic

sense for the total output of any transportation

enterprise. There are a variety of costs co-

existing at a moment of time, depending on specific

conditions.3

Cost allocation problems arise in several differ-

. . 4 .
ent operat1ona1 areas of a motor carrier. The major

areas are:

 

lHugh S. Norton, Modern Transportation Economics
(2nd ed.; Columbus, Ohio: Charles E. Merrill Pub. Co.,

 

2Charles A. Taff, Commercial Motor Transportation
(Homewood, 111.: R. D. Irwin, 1950), p. 122.

 

3George W. Wilson, "On the Output Unit in Trans-
portation," Land Economics, XXXV (August, 1959), 276.

 

4Taff' OB. Cite, pp. 119-220

 

l. The allocation of empty vehicle movement costs.

2. The allocation of pickup and delivery costs when
multiple shipments and stops are involved.

3. The allocation of costs when excess equipment and
facilities are utilized only during peak seasons.

4. The allocation of costs which relate to the entire
operation of the firm; for example, executive
salaries would fall in this category.

This dissertation is concerned with the total
costs and revenues in the whole motor carrier system as
opposed to being concerned with assigning costs to a
particular freight shipment. In this context, the problem
of identifying which specific freight shipment gave rise
to some cost is, for the moment, a moot issue. Freight
rates are approached from the point of view of a profit
maximizing carrier. Of principal concern is the inter-
action among rates, freight flows, costs, and empty vehicle
movements. An example illustrating basic interrelation-
ships is presented below to indicate the nature of the

problem.

Rate Adjustment and Profit Maximization
Consider a motor carrier firm which transports
freight among five terminals. These terminals are labeled
A, B, C, D, and E; their relative positions are shown in

Figure l.

 

____Bi-directiona1
Freight Flows

Directional Empty
( Vehicle Flows

Figure l.--Freight and vehicle flows in a
hypothetical system.

Suppose that freight movements into and out of
terminal D and E are relatively balanced; that is, through
time total inbound freight movements are roughly equal to
outbround freight movements and that there is neither a
shortage nor surplus of vehicles at these terminals.
Suppose, on the other hand, that freight traffic is un-
balanced at terminals A, B, and C; terminals B and C have
greater inbound freight movements than outbound freight
movements, and terminal A has greater outbound freight
movements than inbound movements. This imbalance is
rectified in the short run by moving empty vehicles from
B to A, and from C to A; this is commonly referred to as
a backhaul situation. Is this the most profitable oper-
ating level, or should the carrier seek changes in certain
freight rates?

Consider the following ranges of alternatives.

1. The carrier could increase the rates on shipments
moving out of terminal A. This would most likely
reduce outbound traffic and therefore would reduce
the number of empty vehicles required by A. Pre-
sumably, an increase in outbound rates would in-
duce shippers to employ other means of transport.

2. The carrier could reduce the rates on shipments
moving to terminal A. This would most likely
increase inbound traffic and would provide addi-
tional vehicles at terminal A. This would reduce

the number and cost of empty vehicles moving to A.

3. The carrier could increase the rates on shipments
moving to terminals B and C from A, D, and E.
This would most likely reduce the freight traffic
moving to B and C, and automatically provides a
closer source of empty vehicles for A. For
example, if shipments from D to B were curtailed,
then the resulting availability of vehicles at D
could move directly to A as opposed to the longer
B to A route.

4. The carrier could leave freight rates as they are;
this avenue would presume that the most profitable,

or optimum, operating level was in effect.

These are only a few of the many possible rate
adjustments that might take place. It is readily apparent
that freight rates in one portion of the system are inter—
related to freight rates and flows in other portions of
the system. This complex problem is approached in this
dissertation by analyzing the conditions which denote

maximized profit in a normative system.

Literature Review

 

The topic of transportation rates has been widely
discussed in the literature. Besides elementary trans-

portation texts,l there are various publications which

 

1For example: John E. Lansing, Transportation and
Economic Policy (New York: The Free Press, 1966); Locklin,

op. cit.; Norton, op. cit.; Pegrum, op. cit.; Charles A.

 

 

10

examine competition in transportation,1 the feasibility
of cost based rates,2 value of service rates,3 and supply
and demand for transportation,4 and problems in the regu-

lation of transportation.5 These documents recognize and

 

Taff, Management of Traffic and Physical Distribution (3rd
ed.; Homewood, 111.: R. D. Irwin, 1964).

 

1John R. Meyer, Merton J. Peck, John Stenason, and
Charles Zwick, The Economics of Competition in Transpor-
tation Industries (Cambridge, Mass.: Harvard University
Press, 1959.

 

 

2Systems Analysis and Research Corporation, Cost—
Based Freight Rates, Desirability and Feasibility (Wash-
ington, D.C.: U.S. Department of Commerce, Under Secretary
for Transportation, 1966); John R. Meyer and Gerald Kraft,
"Evaluation of Statistical Costing Techniques as Applied in
the Transportation Industry," The American Economic Review,
LI (May, 1961), 313-34; A. A. Walters, “The Allocation of
Joint Costs with Demands as Probability Distributions,"
The American Economic Review, L, 3 (1960), 419-32.

 

 

3Interstate Commerce Commission, Bureau of Trans-
port Economics and Statistics, Value of Service in Rate-
Making, Statement No. 5912 (Washington, D.C.: ICC, Bureau
of Transport Economics and Statistics, 1959).

 

4For example: H. Benishay and G. E. Whitaker, Jr.,
"Demand and Supply in Freight Transportation," Journal of
Industrial Economics, XIV (July, 1966), 243—62; Eugene D.
Perle, The Demand for Transportation: Regional and Com—
modity Studies in the United States (Chicago: University
of Chicago Press, 1966); Robert B. Adams, "An Input-Output
Program for the Department of Transportation," Transpor-
tation Journal, VIII (Winter, 1968), 45-52; Z. S. Zannetos,
The Theory of Oil Tankship Rates (Cambridge, Mass.: The
MIT Press, 1966).

 

 

 

 

 

 

5Ann F. Friedlander, The Dilemma of Freight Trans-
port Regulation (Washington, D.C.: The Brookings Insti-
tution, 1969); James Sloss, "Regulation of Motor Freight
Transportation: A Quantitative Evaluation of Policy,"
The Bell Journal of Economics and Management Science, I
(Autumn, 1970), 327-66; C. F. Phillips, The Economics of
Regulation (rev. ed.; Homewood, 111.: R. D. Irwin, Inc.,
1969)}

 

 

11

give lip service to the existence of unbalanced freight
movements and loosely identify responses of freight rates
to this backhaul condition. In some instances the
directional rate and backhaul problem is sometimes
assumed away.

Obviously, the backhaul problem creates great in-

stability in rate structures. . . . No doubt in the

aggregate this effect balances out, but it may be a

hardship on certain carriers.1
In other instances it is assumed to be an extremely com-
plex problem which has no general solution.2

The approach taken in this dissertation is balanced
between the above two extremes. The problem is formulated
here so as to incorporate the basic factors which give
rise to the problem and at the same time provide a precise
explanation of the resulting directional interrelation-
ships in freight rate structures.
A streamlined and tight examination of directional

rates must logically proceed from an unambiguous foundation
in the form of a mathematical model. The first notable

analytical examination of the directional rate and back—

haul problem was presented by T. C. Koopmans and S. Reiter.3

 

lNorton, op. cit., p. 115.

2See Wilson, op. cit. Also M. L. Burstein, et al.,
The Cost of Trucking: Econometric Analysis (Dubuque, Iowa:
Wm. C. Brown Company Publishers, 1965), pp. 112-13.

 

3T. C. Koopmans and S. Reiter, "A Model of Trans-
portation," in Activity Analysis of Production and A110-
cation, ed. by T. C. Koopmans (New York: John Wiley &
Sons, Inc., 1951), pp. 222-59.

 

12

The cited article identified the normative relationships
between transportation rates and movements of empty ships,
or ballast traffic.
All the major concepts in this dissertation were
developed independently of Koopmans' article. The model
presented in Chapter II is a generalization of this early
transportation model. Specifically, Koopmans and Reiter
assumed that cargo flows between terminals are fixed
whereas here it is assumed that cargo flows between termi-
nals are related to the transportation rate charged for
the service. It is interesting to note that Koopmans was
one of the first to identify the analogy between trans-
portation systems and linear graph theory.
The cultural lag of economic thought in the appli-
cation of mathematical methods is strikingly illus-
trated by the fact that linear graphs are making
their entrance into transportation theory just about
a century after they were first studied in relation
to electrical networks, although organized trans-
portation systems are much older than the study of
electricity.1

In following chapters, the linear graph analogy2 itself

is not emphasized because it tends to detract and focus

attention away from issues relevant to transportation.

 

lKoopmans and Reiter, op. cit., p. 258.

2Other linear graph analogies of economic and
social systems are described in contemporary literature.
For example, see F. Harary, R. Norman, and D. Cartwright,
Structural Models: An Introduction to the Theory of
Directed Grgphs (New York: John Wiley & Sons, Inc., 1966);
H. C. Koenig and T. J. Manetsch, System Analysis of the
Social Sciences (East Lansing, Mich.: College of Engi-
neering, Michigan State University, 1966). (Mimeographed.)

 

 

13

The concepts in Koopmans' and Reiter's basic model
have been used mostly in transportation equipment schedul-
ing. Indirect extensions have been in the areas of geogra-
phy and spatial economics.l However, there have been no
notable attempts to generalize the economic nature of
transportation rates in the manner in which they are

treated in this dissertation.

Transportation Equipment Scheduling
Transportation equipment scheduling problems2 are
generally posed in the following manner. Given the esti-
mated requirements for the movement of freight between
terminals in the system, and given the locations, capaci-

ties and costs of transportation equipment in the system,

 

1There are numerous professional journals and
articles concerned with these areas. Some reference
articles are given here. W. L. Garrison, "Spatial Struc-
ture of the Economy: III," Annals of the Association of
AmericapGeographers, L (September, 1960), 357-73; F. R.
Pitts, "A Graph Theoretic Interpretation of Historical
Geography," The Professional Geographer, XVIII (May, 1956),
15—20; L. J. King, 3‘The Analysis of Spatial Form and Its
Relation to Geographic Theory," Annals of the Association
of American Geographers, LIX (September, 1969), 573—95;
Gunnar Olsson, Central Place Systems, Spatial Interaction
and Stochastic Processes," Pa ers and Proceedings, Regional
Science Association, XVIII (I567), I3-45; R. E. Quandt and
W. J. Baumol, "The Demand for Abstract Transport Modes:
Some Hopes," Journal of Regional Science, IX, 1 (1969),
159-62; A. Stegman, 1rAccessibility Models and Residential
Location," Journal of the American Institute of Planners,
XXXV, l (1969), 22-29.

 

 

 

 

 

 

2For example see J. F. Pierce, "Direct Search
Algorithms for Truck-Dispatching Problems," Transportation
Research, III (1969), 1-42; Burstein, et al., op. cit.,
pp. 47-105; J. L. Saha, "An Algorithm for Bus Scheduling
Problems," Operational Research Quarterly, XXI (December,
1970), 463-74.

 

14

the movements of equipment between terminals are scheduled
in a manner which minimizes total costs while satisfying
service requirements. Note again, as with the Koopmans'
model, it is assumed that freight movement requirements
are fixed; the normative models in this dissertation begin
with the assumption that freight movement requirements are
variable with the price, or rate, being charged for the

service.

Objectives and Organization

 

The primary objective of this dissertation is to
develop a normative model which will aid in the expla-
nation and resolution of directional factors in truckload
transportation rates. The secondary objective is to give
empirical evidence showing that actual truckload motor
carrier freight rates resemble the analytical patterns
suggested in the model.

This dissertation is organized into four chapters.
The first chapter introduces the general nature of the
subject and problem area. It focuses on the literature
and relevant issues which underscore the contemporary
significance of the problem, and places in perspective
the approach taken to solve the problem.

The second chapter develops a model which is used
to examine the fundamental economics of a multiple termi-
nal common motor carrier system. The model identifies

the conditions which will occur when a motor carrier

 

15

facing known demand curves has adjusted truckload freight
rates and empty vehicle flows so as to be at an operating
level which maximizes profits. The conditions show that
the rate structure which maximizes profit is not overly
complex; rather, predictable directional patterns will
result. According to the model, directional rates are
ultimately due to the profit maximization motive of the
motor carrier, and to the differences in demands for trans-
portation among points in the transportation system.
Differences in demand give rise to imbalanced freight
movements which cause directional orientations in the
incremental costs of freight movements. This induces

the carrier to change freight rates, which, in turn,
affect traffic flows and vehicle imbalances. This cycle
is theoretically repeated until the carrier is at a point
in which rates, costs, and traffic flows are such that
profit can no longer be increased through additional
changes.

From the relationships prescribed in the normative
model it is hypothesized in Chapter III that patterns in
the truckload freight rate structure are mathematically
related to the patterns of empty vehicle flows and freight
traffic imbalances in an actual system. This hypothesis
is tested by examining the rates, costs, and freight
imbalances in an actual motor carrier system. A motor

common carrier provided the data for this empirical

16

investigation. The empirical analysis measures the sta-
tistical significance of the system wide relationship be-
tween patterns in freight traffic imbalances and the
freight rate structure in a manner suggested by the
normative model. The analysis provides a basic foundation
for examining the effects of unequal demands and vehicle
balancing on truckload freight rates.

The fourth chapter synthesizes the conclusions
drawn from the normative and empirical investigations,
and shows how these conclusions can be used to resolve
contemporary issues in transportation management, regu-

lation, and planning.

 

CHAPTER II

CHARACTERISTICS OF PROFIT MAXIMIZATION IN
A NORMATIVE MODEL OF A MULTIPLE

TERMINAL MOTOR CARRIER

Introduction

 

A normative model of a motor carrier is developed
in this chapter. The model specifies mathematical con-
ditions for profit maximization. The development and
analysis of this model is analogous to the economic analy-
sis of a monopolistic firm,l modified, of course, to a
motor carrier. The same general assumptions made by the
economist for static equilibrium models are made here.
The model focuses on equilibrium states of the system,
where a carrier attains maximum profits. Time dependent
adjustment processes are not explicitly examined here.

In the latter part of the chapter, the model is inter-
preted with regard to its compatibility with contemporary

economic views of transportation rates.

 

lSee C. E. Ferguson, Microeconomic Theory (Home-
wood, Ill.: R. D. Irwin, Inc., 1966), Chapter X and XI.
Also, J. M. Henderson and R. E. Quandt, Microecgpomic
Theory (New York: McGraw-Hill Book Company, 1958),
Chapter VI.

17

18

General Model Structure

 

This model is designed for examining the optimal
economic characteristics of revenues and costs of general
commodity truckload freight in a motor carrier system. The
model represents a motor carrier facing known revenue, cost,
and freight flow relationships; it is then assumed that the
carrier maximizes profit with respect to these known con-
ditions.

The crux of the profit maximization problem is
the optimum adjustment of freight rates with respect to
balanced vehicle movements in the system.

A fundamental operating constraint on a motor
carrier is the equalization of inbound and outbound vehicle
flow at each terminal. A surplus of vehicles at certain
terminals and shortages of vehicles at other terminals
are caused by unequal inbound and outbound freight flows
at the terminals. This problem is rectified by adjusting
freight rates and empty vehicle movements in manners which
promote profitable balanced freight and vehicle flows in

a system.

Segmentation of Carrier Activities
The model focuses on the economics of interterminal
truckload freight, as opposed to freight activities such
as less-than-truckload movements, intraterminal freight
movements, and pickup and delivery. These other freight

activities are not ignored in the model; rather, the

 

19

effects of these other freight activities are incorporated
in the model as known variables. This assumption simpli-
fies the understanding of the problem, and provides a
foundation for analyzing the economics of these other
transport activities. For descriptive purposes, carrier
activities are segmented into three mutually exclusive

and exhaustive nominal categories. The first category is
truckload freight movements, the second is empty vehicle
movements, and the third is called other carrier activi-
ties.

Truckload freight movements refer to the movement
of truckload freight between separate terminals in the
system. Empty vehicle movements refer to the movements
of empty vehicles between separate terminals in the system.
Other carrier activities refer to activities which do not
fall into the above two categories. These nominally seg-
mented activities are synthesized in the model by assum—
ing that the revenues, costs, and vehicle imbalances caused
by gphgg carrier activities have known effects in the
system; the model then maximizes remaining profits on

truckload freight with respect to vehicle balancing costs.

Freight and Vehicle Assumptions
The system under consideration is a motor carrier
which transports general commodity freight among spatially

separated stationary terminal facilities. The terminals

 

 

20

in the system are numbered sequentially from (1) to (6),
where (6) represents the number of terminals in the
system.

It is assumed that truckload freight enters the
system or originates, at some terminal, and is transported
by a vehicle directly to its destination terminal without
passing through other terminals. The freight then leaves
the system. Freight and vehicle movements between termi—
nals are designated by the origin and destination of the
movement; they are identified by the terminal from which
the movement originated, and the terminal for which the
movement is destined. Given that there are (0) terminals
in the system, there are (02) pairs of origins and desti-
nations; this includes intraterminal movements. Intra-
terminal movements are movements in which both the origin
and destination are in a region surrounding a terminal.

In most cases they include pickup and delivery operations.
As mentioned above, intraterminal movements are categorized
as other carrier activities and are not explicitly examined.
In all there are (0) intraterminal movements; therefore,
there are (02 — 8) interterminal freight movements in the
model.

Freight and vehicle movements in the system are
measured here as flows; that is, quantity per unit of
time. It is not absolutely necessary that specific units

of freight and vehicle flow be assigned in this model

 

21

because the variables and equations can be defined to
handle any logical unit of measure. However, for descrip-
tive purposes, hundredweight per month is used to specify
freight flows, and hundredweight capacity per month is
used to specify the flow of vehicles in the system. It

is assumed the vehicle capacity flow from one terminal to
another during a monthly period must equal or exceed the
freight flow moving between those two terminals; the excess
of vehicle capacity over freight flow is called empty
vehicle flow. Empty vehicle flows arise because of the
necessity to achieve balanced flows of vehicles in a
system; these and other assumptions are detailed in

following sections.

Mathematical Notation

 

The mathematical symbols used to represent
freight and vehicle flows are identified in this section.
Table l is a summary of these and other mathematical

variables used in this chapter.

Number of Terminals
As mentioned previously, the Greek letter (9)
represents the number of terminals in the motor carrier

system.

Number of Terminals in System = 9 (2-1)

The terminals are indexed by the letters i,j,k,m, and n

in this and later chapters.

 

 

 

22

TABLE l.--A summary of mathematical variables used in
Chapter II.

 

(0) is the number of terminals in the system.

q(i,j) is freight flow from terminal (1) to (j), (hundred-
weight per month).

(q) represents the freight flows in the entire system.

u(i,j) is empty vehicle flow from terminal (i) to (j),
(hundredweight capacity per month).

(u) represents the empty vehicle flows in the entire
system.

(q(i,j) + u(i,j)) is the vehicle capacity flow from
terminal (1) to j), (hundredweight capacity per month).

s(i) is exogenous vehicle capacity available at terminal
(i), (hundredweight capacity per month).

r(qo,i,j) is the first derivative of the total revenue
function with respect to q(i,j) evaluated at the solution
(qo). It is the marginal revenue on freight moving from
terminal (1) to (j).

a(qo,i,j) is the first derivative of the total direct cost
function with respect to q(i,j) evaluated at the solution
(go). It is the marginal direct cost on freight moving
from terminal (1) to (j).

b(uo,i,j) is the first derivative of the total cost
function of moving empty vehicles with respect to u(i,j)
evaluated at the solution (no). It is the marginal cost
of moving empty vehicles from terminal (1) to (j).

p(qo,uo,i,j) is the marginal vehicle profit on vehicles
moving from terminal (1) to (j). It is equal to
(r(qo,i.j) - a(qo.i,j)) or (-b(uo.i.j)). which ever
is greater.

(scmvp) is an informal mathematical variable which repre-
sents the sum of marginal vehicle profits around an
arbitrary circuitous route in the system.

tdc(q) is monthly total direct cost as a function of
freight flow (q).

tec(u) is monthly total cost of empty vehicle movements
as a function of empty vehicle flows (u).

23

TABLE l--Continued

 

 

ttr(q) is monthly total truckload revenue as a function
of freight flow (q).

(tpa) is monthly total profits on other carrier
activities, a constant.

t(i) is the imputed marginal profitability of vehicle
capacity at terminal (i). It is derived from the

Kuhn-Tucker theorem, and is used to describe optimality
conditions.

d,y(i,j) and z(i,j) are "dummy“ variables used to apply
the Kuhn-Tucker theorem to the motor carrier model.
They are not given an explicit interpretation.

 

 

24

Freight Flows

The variable q(i,j) represents the hundredweight
per month flow of truckload freight from origin (1) to
destination (j). The flows of interterminal truckload
freight for the entire system are designated as q(i,j),
(i=l,...,6; j=l,...,0; i#j). The letter q, without
adjacent parentheses, is used to designate the freight

flows in the entire system.
q E q(irj)r (1:11-00:93 j=l,...,0; iié j) (2’2)

This equivalency is made for convenience in order to

shorten the designation of system wide freight flows.

Empty Vehicle Flows

The variable u(i,j) represents the flow of empty
vehicles from terminal (i) to terminal (j). This variable
is measured in terms of hundredweight capacity per month.
The letter 11 is used to represent system wide empty

vehicle flows.
u E u(i,j), (i=l,...,6; j=l,...,8; i#j) (2-3)

External Vehicle Capacity

Vehicle capacity which arises, or is required,
at terminals as a result of other carrier activities in
the system is represented by the variable s(i); it is the
monthly surplus, or shortage, of vehicle capacity at

terminal (i). The variable s(i) is measured in terms of

25

hundredweight vehicle capacity per month. A positive
value for this variable signifies that other carrier
activities provide an excess of vehicles at the indexed
terminal; a negative value signifies that other carrier
activities require vehicles at the indexed terminal.
System wide surpluses and shortages are represented by

the letter s.

s E s(i), (i=1,...,9) (2-4)

In the initial development of profit maximization charac-
teristics, the values in s are assumed to be constant;
during the interpretation of the profit maximization

characteristics, this assumption is relaxed.

Flow Constraints

 

It is assumed that truckload freight is trans-
ported directly from the origin terminal to the desti-
nation terminal, and that the amount of vehicle capacity
used to transport the truckload freight equals, or exceeds,
the amount of freight transported.

The fundamental constraint on freight and vehicle
flows is that the availability of vehicle capacity at any
terminal must equal or exceed the amount of vehicle
capacity required at the terminal through time. This
constraint is informally expressed as follows. A termi-
nal's inbound truckload freight, plus the terminal's

inbound empty vehicles, minus the terminal's outbound

truckload freight, minus the terminal's outbound empty
vehicles, plus the vehicle capacity adjustments due to
other carrier activities must be greater than or equal

to zero.

26

A mathematical expression of the constraint is

given below in (2-5) for terminal (m).

where

8 6 8

I q(i,m) + Z u(i,m) - 2 q(m,i)
i=1 i=1 i=1
i#m i¢m i#m

+ s(m) Z 0

(Inbound truckload freight flow)

(Inbound empty vehicles flow)

(Outbound truckload freight flow)

(Outbound empty vehicles flow)

(Vehicle capacity adjustments
due to other carrier activities)

6

Z u(m,i)
i=1
i#m

6

E q(irm)
i=1
i#m

0

Z u(i,m)
i=1
i¢m

6

Z q(mri)
i=1
i¢m

0

Z u(m,i)
i=1
i#m

s(m)

(2-5)

In all there are (0) of these equations, each expressing

the vehicle flow constraints at each of the (0) terminals.

 

27

In adjusting rates and controlling the flow of freight and
empty vehicles in the system, it is assumed that the
carrier must operate within the confines of these con-

straints.

Cost and Revenue Assumptions

Four categories are identified here; the first
category accounts for truckload transportation costs, the
second accounts for empty vehicle movement costs, the
third accounts for truckload freight revenues, and the
fourth accounts for revenues and costs, or profit, on
other carrier activities.

Mathematical relationships are identified so as
to provide a basis to assess the profitability of feasible,
or admissible, freight and vehicle flows.

The derivatives of the mathematical functions
described in this section are used extensively in the
model and, therefore, derivative notations are described
along with the functions. It is assumed that all functions
described below are continuously differentiable within

the relevant ranges of the flow variables.

Direct Transportation Costs

 

Direct truckload freight transportation costs are
defined here to include all costs related to the movement
of transporting truckload freight among the terminals.

Direct transportation costs do not include any costs

28

related to the movement of empty vehicles. They are
assumed to include the costs of pickup and delivery,
loading and unloading, insuring, billing, linehaul costs,
and all other costs directly related to the movement of
freight.

A mathematical function is used to describe the
relationship between truckload freight flows and total
direct cost. The function describing the relationship

is denoted as tdc(q).
Monthly Total Direct Cost = tdc(q) (2-6)

The symbol q represents freight flows in the system as
defined in (2-2). For given values of q, the function
tdc(q) gives monthly direct transport costs incurred by
the system.

The partial derivative of tdc(q) with respect to

q(i,j) evaluated at a point qO is denoted as a (qo,i,j).

. . Btdc
a(qollrj) = sq lfj)

q=qo (2-7)

In economic terms,1 a(qo,i,j) is called the marginal
direct cost of moving freight from terminal (1) to (j) at

the point qo.

 

lMarginal cost is a standard economic term, see
Co E. Ferguson, 02. Cite, pp. 168-750

29

Empty Vehicle Movement Costs
The cost related to the movement of empty vehicles

in the system is expressed as a function of u.
Monthly Total Empty Vehicle Movement Cost = tec(u) (2-8)

The symbol u represents empty vehicle flows in the system
as defined in (2-3). For given values of u, the function
tec(u) gives monthly total empty vehicle movement cost.
The partial derivative of tec(u) with respect to u(i,j)

evaluated at a point uO is denoted as b(uo,i,j).

3tec(u)

b(uorilj) = 5371737

u=uO (2-9)

In economic terms, b(uo,i,j) is called the marginal cost
of moving empty vehicles from terminal (i) to (j) at the

p01nt uo.

Truckload Freight Revenues

 

In this model it is assumed that the carrier is
facing known demand curves for freight transportation ser-
vice between terminals, and that the carrier can operate
at any desirable point on the demand curve. It is assumed
that the carrier can control the amount of truckload
freight flowing between each pair of terminals by adjust-
ing prices of the transportation service. Presumably, a

high rate, or price will discourage the movement of freight

30

between two terminals; likewise a low freight rate will
increase the flow of freight between two terminals.

It is assumed that there are mathematical functions
describing the relationships between freight rates and
truckload freight flow for each pair of terminals; further-
more, it is assumed that these functions can be inverted
to give the freight rate as a function of the freight flow.
This makes it possible to express revenues as a mathe-
matical function of freight flows in the system; the
function ttr(q) gives the total truckload freight revenue

as a function of q.
Monthly Total Truckload Freight Revenue = ttr(q) (2-10)

The symbol q represents freight flows in the system as
defined in (2—2). For given values of q, the function
ttr(q) gives monthly total revenue.

The assumptions regarding the above function are
made in a mathematical sense in order to examine the
profit maximizing characteristics of the model. Actual
rate and marketing adjustments involved in controlling
freight flows are not explicitly identified; rather it is
assumed that these adjustment processes take place, and
that the equation in (2-10) gives the relationship between
truckload revenue and freight flows in the system.

The partial derivative of ttr(q) with respect to

q(i,j) evaluated at a point qO is denoted as r(q0,i,j).

31
. . = 3ttr(q)
r(qolllj) 3q(i1j)
q=qO (2-11)

In economic terms,1 r(qo,i,j) is called the marginal
revenue on freight moving from terminal (1) to (j) at the
p01nt qo.

Profits on Other Carrier
Activities

 

 

The variable (tpa) is used to represent monthly

profits of other carrier activities.

Monthly Profits on Other Carrier Activities = tpa (2—12)

As previously defined, other carrier activities are those
activities which are neither classified in the inter-
terminal truckload freight category nor the empty vehicle
movement category. For example, less-than-truckload
freight movements and intraterminal freight movements
fall in this category. Overhead items not included in
truckload freight and balancing costs would also be in—
cluded here. The variable (tpa) is assumed to be mathe-
matically independent of q and u, and, therefore, is

treated as a constant in this model.

 

lMarginal revenue is a standard economic term,
see Ferguson, op. cit., pp. 87-95.

 

 

32

Model Synthesis
The model is synthesized here in terms of the
relevant costs, revenues, flow constraints, and profit
maximization objectives. Profits are maximized by find-
ing values for q and u which maximize equation (2-13) with
respect to the constraints in equations (2—14), (2-15),

Monthly Profit = ttr(q) ~ tdc(q) — tec(u) + tpa (2—13)
6
(( Z (q(m,i) + u(m,i) - q(i,m) - U(i.m)))
i=1
i¢m
' s(nn) :3 0, (m=l,...,0). (2—14)
q(i,j) : O, (i=1,...,0; j=1,...,0; i¢j) (2—15)
u(i,j) Z 0, (i=1,...,0; j=l,...,8; i#j) (2—16)

The equation (2-13) was obtained by appropriately summing
equations (2-6), (2-8), (2—10), and (2—12). The equations
(2-14) were obtained by multiplying equation (2-5) by a
minus one and consolidating the summation signs; this re—
arrangement facilitates the application of maximization
conditions. The equations in (2—15) and (2-16) are non-
negative restrictions on freight and vehicle flow. The
necessary conditions for maximization of (2-13) are out-

lined in the next section.

 

33

Necessary Profit Maximization Conditions

This section specifies the mathematical relation-
ships which must occur when maximum profit is attained in
the model presented in the preceding section. The neces-
sary conditions stated here become necessary and sufficient
conditions when (2-13) is concave.1

The application of the Kuhn-Tucker theorem on
non-linear programming, as developed in Appendix B, to
the problem stated in (2—13), (2-14), (2-15), and (2-16)
gives the following necessary conditions. If qo and uO
are solutions which maximize profits with respect to the
constraints, then there must exist values for the vari-
ables t(i). (i=1,...,6), and y(i,j), z(i,j), (i=l,...,0;
j=1,...,6; i#j)r which satisfy (2-17), (2—18). (2—19),

(2'20)! (2’21): (2'22): (2'23), and (2‘24).

t(i) 3 o, (i=1,...,0) (2-17)
6
t(i) - (( 2 qo(i.j) + uo(i.j) - qo(j.i) — uo(j.i))
i=1
175i
- s(i)) = 0, (i=1,...,8) (2-18)
Y(ilj) _>_ or (i=1-1000167 j=ll'°‘le; 175:1) (2‘19)

y(i.j) - qo(i.j) = 0, (i=l,...,e; j=1,...,e; i#j) (2-20)

 

1This aspect is expanded upon in a later section.

34
z(i,j) 3 o, (i=l,...,0; j=l,...,6; i#j) (2-21)
z(i,j) - uo(i,j) = o, (i=l,...,8; j=1,...,0; i¢j) (2-22)

r(qovi.j) - a(q0.i.j) - t(i) + t(j) + y(i,j) = 0.

(i=l,...,9; j=l,...,e; i#j)o (2‘23)

-b(u0.i.j) - t(i) + t(j) + 2(i.j) = o.

(i=l,...,9; j=l,...,9; i#j)o (2-24)

The above conditions are simplified in Table 2;
the table was developed from the following observations.
Positive Freight and Empty
Vehicle Flows
For (i,j) such that qo(i,j) > 0 and uo(i,j) > 0,
then y(i,j) and z(i,j) must equal zero because of (2-20)
and (2-22) respectively. Therefore, y(i,j) can be dropped
from (2—23), and z(i,j) can be dropped from (2-24). The
combined result is condition one in Table 2.
Positive Freight Flow and Zero
Empty Vehicle Flow
For (i,j) such that qo(i,j) > 0 and uo(i,j) = 0,
then y(i,j) = 0 because of (2-20) and can be dropped from
(2-23). Because of (2—21), z(i,j) can be removed from
(2-24) by replacing the equality sign with an inequality.

The combined result is condition two in Table 2.

 

35

TABLE 2.——A summary of necessary conditions for maximized
profit.

 

If q and uO satisfy vehicle flow constraints, and maximize
profit, then there must exist real values for t(i),
(i=l,...,6), which satisfy the following conditions.

Conditions

 

1. For (i,j) such that
qo(ilj) > or and
uo(i,j) > 0, then

r(qorirj) ' a(qolirj) = t(i) ’ t(j) 'b(uorirj)o

2. For (i,j) such that
qo(irj) > DI and
uo(i,j) = 0, then

r(qo.i.j) - a(qo.i.j) = t(i) - t(j)

|v

_b(uopipj) o
3. For (i,j) such that

qo(ilj) = 0' and

uo(i,j) > 0, then

r(qolirj) ‘ a(qorirj)

IA

t(i) - t(j) = -b(uo.i.j).

4. For (i,j) such that
qo(i.j) o. and
uo(i,j) 0, then

[v

r(qolirj) ' a(qolirj) : t(i) - t(j) -b(uorirj)o

5. For (i) such that

0
((jilqo(i.j) + uo(i.j) - qo(j’i) - uo(j.i))
1175i
- s(i) = 0, then t(i) Z 0.

6. For (i) such that

6
((jilqo(i.j) + uo(i.j) - qo(j.i) - uo(j.i))
j#i
- s(i) < 0, then t(i) = 0.

 

 

 

36

Positive Empty Vehicle Flow and
Zero Freight Flow

For (i,j) such that qo(i,j) = 0 and u0(i,j) > 0,
then y(i,j) Z 0 because of (2-19), and z(i,j) = 0 because
of (2-22). Therefore, y(i,j) can be removed from (2-23)
by replacing the equality sign with an inequality, and
z(i,j) can be dropped from (2-24). The result is con-
dition three in Table 2.

Zero Freight Flow and Zero Empty
Vehicle Flow

For (i,j) such that qo(i,j) = O and uo(i,j) = 0,
then y(i,j) Z 0 and z(i,j) Z 0 because of (2-19) and (2—21).
Therefore, y(i,j) and z(i,j) can be removed from (2-23) and
(2-24) by replacing the equality signs with inequalities.
The result is condition four in Table 2.

Constrained Vehicle Flow
at Terminals

From (2-17) the values for t(i) must in all cases
be greater than or equal to zero for an optimal solution.

Furthermore, for (i) such that

qo(ilj) + 110(irj) " qo(jri) " uo(jli))
- 5(i) < 01 (2‘25)

then t(i) must equal zero; this is concluded from (2-18).
In words, this condition is saying that if there is

excess vehicle capacity at a terminal, say (i), at an

 

37

optimal solution then t(i) must equal zero. These are

the fifth and sixth conditions in Table 2.

Interpretation of the Conditions
for Maximized Profit

 

 

Motor vehicles, and hence vehicle capacities are
an economic resource in a motor carrier system. This
intuitively implies that there is an economic value
attached to vehicle capacity and vehicle flows. It so
happens that following this line of thought provides an

interpretation of the equations in Table 2.

Terminal Multipliers

As pointed out in Appendix B, the variables t(i),
y(i,j), and z(i,j) used in equations (2-17) through (2-24)
are called Kuhn-Tucker multipliers and are analogous to
Lagrange multipliers; each multiplier reflects the imputed
effect on the objective function which would result from
an incremental change on the associated constraint equation.
From (2—18) it is apparent that t(i) is associated with the
vehicle flow constraint on the (i)th terminal. Therefore,
the value of t(i), for an optimal qO and uo, equals the
imputed increase in profit that would result from an in-
cremental unit increase in vehicle capacity at terminal
(1).

Mathematically, (t(i) - d) equals the incremental
increase in profit that would result if (d) replaced the

zero in the right hand side of the (i)th equation in

 

38

(2-14), and the problem was resolved for optimum qO and uO.
It is assumed that the absolute value of (d) is arbitrarily
small.

Similarly, (-t(j) +t(i)) equals the imputed in—
crease in profit that would result from removing a unit
of vehicle capacity from terminal (j) and adding it to
terminal (i). Mathematically ((—t(j) +t(i)) - d) equals
the incremental increase in profit that would result if
(-d) replaced the zero in the right hand side of the (j)th
equation in (2-14), and (+d) replaced the zero in the
right hand side of the (i)th equation in (2—14), and the
problem was resolved for optimum qO and uo. The reverse
would also hold; (—t(j) +t(i)) equals the imputed decrease

in profit that would result from adding a unit of vehicle

capacity to terminal (j) and removing it from terminal (i).

Marginal Vehicle Profit

The relationship of the t(i) to marginal revenues
and costs of freight and vehicle movements can be better
interpreted by defining a variable called marginal vehicle
profit. Marginal vehicle profit is defined here to be the
maximum and immediate incremental increase in profit that
would accompany a unit increase in vehicle flow from one
terminal to another. Marginal vehicle profit will be
denoted as p(q,u,i,j), and is mathematically defined as

follows.

39

Terminal (1) to (j) Marginal Vehicle Profit = p(q,u,i,j)

r(q.i.j) ' a(qrilj)

Whichever

or .
15 greater

(2-26)

-b(u'i’j)

The value of p(q,u,i,j) is found by finding the most
profitable (least expensive) method of moving an incre-
mental unit of vehicle flow from terminal (i) to (j).
There are two alternative ways of moving an additional
vehicle from, say, terminal (1) to (j). One would be to
move an empty vehicle, and the other would be to induce
additional freight flow; for the former, marginal profit
(negative cost) would be -b(u,i,j), for the latter it
would be (r(q,i,j) - a(q,i,j)). Marginal vehicle profit
is defined to be the most profitable of the two.
Optimal Relationships Among the Values
of the Terminal Multipliers

By applying the definition of marginal vehicle
profits to the conditions one through three in Table 2,
the following relationship is derived. For (i,j) such

that qo(i,j) > 0, or uo(i,j) > 0, or both, then

t(i) ‘ t(j) = P(qoru lilj)° (2'27)

0

Stated in words, (2-27) is saying that at an
optimal solution the imputed value of vehicle capacity
at terminal (1), minus the imputed vehicle value at

terminal (j), equals the marginal vehicle profit on

 

 

40

vehicles which are flowing from (i) to (j). It is
intuitively logical that this should be true because if
(2-27) did not hold, then there is a direct implication
that qO and 110 are not optimal.

For example, suppose
t(i) - t(j) > p(qo.u0.i.j). (2-28)

Recall from the previous section that (t(i) - t(j)) equals
the imputed decrease in profit that would result from
adding a unit of vehicle capacity to terminal (j) and
removing it from (1). Hence, (2-28) is saying that the
imputed incremental lpgg in profit of shifting vehicle
capacity from terminal (i) to (j) is greater than the
marginal vehicle profit gained from the actual movement.
Clearly, this is marginally unprofitable and vehicle flows
from (i) to (j) should be reduced; the initial solution
could not be optimal.

Consider the opposite situation; suppose

t(i) - t(j) < p(qoruolirj)o (2-29)

Here, the marginal vehicle profit from the actual movement
is greater than the imputed decrease in profit resulting
from the shift in vehicle capacity from terminal (i) to
(j). This is marginally profitable, and therefore mggg
vehicles should be sent from (i) to (j) to further

increase profits; the initial solution could not be

 

41

Optimal. From the counter examples in (2-28) and (2-29),
the relationship in (2-27) is intuitively consistent with
logical analysis.

The relationships in (2-27), (2-28), and (2-29)
were formulated under the assumption that vehicles were
flowing between the relevant terminals. Condition four
in Table 2 is now applied to specify relationships when
there is no vehicle flow between two terminals. For (i,j)

such that qO(i,j) = u0(k,j) = 0, then

t(i) - t(j) _>_ p(qo,uo,i,j). (2-30)

This is also intuitively logical; no vehicles are flowing
from (i) to (j) because the imputed decrease in profit
resulting from a shift in capacity from terminal (i) to
(j) exceeds, or equals, the marginal vehicle profit which
is to be gained.

In an abstract sense, the conditions stated in this
section are implying that if economic returns (profits) are
being maximized on a set of mobile resources (vehicles)
then it is a necessary condition that the incremental
values of the resources at various points in space be
directly related to the returns on moving the resources
between those points. In a loose sense, the model
developed in this chapter provides a structure for identi—

fying the spatial value of transportation resources.l

 

1This also applies to the commodities being trans-
ported. See Paul Samuelson, "Spatial Price Equilibrium and

 

42
Marginal Vehicle Profits and
Circuitous Routes

Special relationships occur on directed circuitous
routes when profit is at a maximum. A directed circuitous
route is defined here as a continuous path which begins at
some terminal, and sequentially proceeds to other termi-
nals, and eventually returns to the terminal from which it
began. A directed circuitous route may contact any finite
number of terminals; the only restriction is that it pro-
ceeds from one terminal to another, and eventually closes
the loop by returning to its originating terminal.

At a profit maximizing solution, the sum of mar-
ginal vehicle profits on any directed circuitous route will
always be less than or equal to zero; furthermore, if there
are positive vehicle flows on each segment, then the sum
of marginal vehicle profits must equal zero.

For example, consider the directed circuitous
route which travels from, say, terminal (1) to terminal
(j) to terminal (k) to terminal (m) to terminal (n), and
back to terminal (i). The sum of marginal vehicle profits

along this directed route is (scmvp). where
scmvp = p(qoruo.i,j) + p(qo.uo.j.k) + p(qo.uo.k.m)

+ p(qo.uo.m.n) + p(qo.uo.n.i) (2-31)

 

Linear Programming," American Economic Review, XLII (June,
1952), 283-303.

 

 

 

43

If vehicle flows are positive along each segment of the
route, then the relationship in (2-27) can be applied to

give

scmvp = t(i) - t(j) + t(j) - t(k) + t(k) - t(m)

+ t(m) - t(n) + t(n) - t(i) = 0 (2-32)

This equals zero because for each positive terminal
multiplier there is also a corresponding negative terminal
multiplier.

For cases where there are some segments with zero
vehicle flows in the circuit, the relationships in (2-30)
are applied to find the value of (scmvp). In this case
(scmvp) will be less than or equal to zero because each
p(qo,uo,i,j) will be less than or equal to t(i) - t(j).

It is intuitively reasonable that such relation—
ships occur in an optimal system because of the nature of
the vehicle flow constraints. Notice that an increase,
or decrease, in vehicle flow around a circuitous route
will 22; cause vehicle imbalances because for each termi-
nal along the route, changes in inbound flow will be
offset by equal changes in outbound flow. Notice also,
that the sum of marginal vehicle profits around a
circuitous route equals the change in profit that would
result from an incremental change in vehicle flow around
the route. Therefore, a solution could not be optimal if

there was some circuitous route in which (scmvp) was

44

greater than zero because this would indicate that profits
could be increased by increasing vehicle circulationl along
the route. Similarly, a solution could not be optimal if
there was some circuitous route in which vehicle flow is
positive along all segments of the route and the value of
(scmVp) was simultaneously not zero because an appropriate
change in flow could increase profit.

Less formally, this necessary condition is saying
that any feasible deviation from a profit maximizing
solution cannot result in greater profit.

Changes in Terminal Vehicle Capacity
Caused by Other Carrier Activities

At an optimal solution, the imputed effects on
profits of changes in an s(i) are given by the value of
t(i). An increase of (d) in s(i) has the same effect as
increasing the right hand side of the (i)th equation by
(d); from Appendix B, the imputed increase in profit would
be (t(i) - d). It is assumed that the absolute value of
(d) is relatively small.

This provides a valuable insight to the economics
of other carrier activities, particularly less-than-
truckload movements. The t(i) values prescribe the eco-

nomic effects of vehicle inputs and outputs at each

 

1The term circulation is used by Ford and Fulker—
son to describe methods for maximizing flows and profits
in constrained networks. See L. R. Ford, Jr., and D. R.
Fulkerson, Flows in Networks (Santa Monica, Calif.: The
RAND Corporation, 1962), pp. 50-53.

 

45

terminal on truckload freight profitability. With known
values of t(i), a carrier could evaluate the effects of
less-than-truckload freight on truckload freight profits

and vice versa. For example, effects of partially_loaded

 

vehicles moving between terminals could be evaluated with
respect to economic impacts on truckload freight.

The Relationship Between Marginal Revenue

and Marginal Direct Cost
For qo(i,j) > 0, it is not necessary that r(qo,i,j)

equal a(qo,i,j) at an optimal solution go. In fact,
according to the first condition in Table 2, it would
only occur when t(i) = t(j). This is a peculiar, but
logical, result of the model. This relationship is
pointed out here because in the economic analysis of
production type firms it is repeatedly stated that "maxi-
mum profit is attained at that rate of output and sales
for which marginal cost equals marginal revenue."l
Clearly, this condition does not necessarily hold for
the motor carrier model when the concepts of marginal
costs and revenues are restricted to freight flows be-
tween two terminals. However, recall the relationship
specified in (2-23); it was necessary that (scmvp) equal
zero, or in other words, that the sum of marginal revenue

equals the sum of marginal direct cost around a circuitous

 

lFerguson, op. cit., p. 232.

 

 

46

route. Therefore, the traditional economic notion that
marginal revenue must equal marginal cost in order to
maximize profit also applies here to circuitous routes.

If r(qo,i,j) equals a(qo,i,j) for all relevant (i)
and (j) at an optimal solution, then it is implied, among
other things, that the objective function in (2-13) is
maximized without being actively constrained by (2-14).
That is to say, it was possible to independently maximize
profits on truckload freight moving between each pair of
terminals without encountering vehicle balancing problems.
This would be a very unique situation indeed. Therefore,

it is presumed that for an optimal solution, r(qo,i,j)

 

will generally not equal a(qo,i,j) even if all uO(i,j) = 0
because of differences in the demands for transportation

between terminals.

The Attainment of a Normative Optimum

 

So far, attention has been focused on the neces-
sary conditions for maximized profit. This section briefly
reviews the problems associated with actually maximizing
(2-13) with respect to (2-14), (2-15), and (2—16).

The constraints (2-14), (2-15), and (2-16) are
linear and, therefore, the admissible or feasible values
of q and u are convex sets.1 This simplifies the problem

greatly, however, no general solution algorithm exists

 

lSee G. Hadley, Linear Programming (Reading, Mass.:
Addison-Wesley Publishing Co., Inc., 1962), p. 60.

 

47

for the above problem unless (2-13) is concave, in which
case a local maximum is necessarily a global maximum,1
and the Kuhn-Tucker conditions become necessary and
sufficient for a global maximum.2

The problem is simplified even more by assuming
that (2-13) is a concave quadratic function of q and u;
there are several algorithms available for solving this
type of problem.3 Simple functions which satisfy this

restriction are given below.

Convenient Functional Relationships
Let truckload freight flow from, say, terminal

(1) to (j) be a linear function of the rate;
q(i,j) = c(i,j) - d(i,j) - Rate(i,j) (2-34)

where c(i,j) and d(i,j) are non-negative constants.
Revenue on the (i) to (j) freight flow as a function

of q(i,j) would, therefore, be

 

lSee G. Hadley, Nonlinear Programming (Reading,
Mass.: Addison-Wesley Publishing Co., Inc., 1964), pp.

 

2See J. Abadie, "On the Kuhn-Tucker Theorem,"
Nonlinear Programming, ed. by J. Abadie (Amsterdam:
North-Holland Publishing Co., 1967), pp. 21-36.

 

3This is the standard quadratic programming
problem. See L. Cooper and D. Steinberg, Introduction to
Methods of thimizatio ( Philadelphia, Pa.: W. B.
Saunders Co., 1970): PP. 295-301; F. S. Hillier and G. J.
Lieberman, Introduction to Operations Research (San
Francisco: HoIden-Day, Inc., 1967), pp. 578-80.

 

48

Revenue(i,j) = (c(i,j)/d(i,j)) ' q(irj)

- (1/d<i.j>> - q(i.j>2 <2-35)
and marginal revenue would, therefore, be
r(q.i.j) = (C(irj)/d(ilj)) - (2/d(i.j)) - q(i,j) (2-36)

Let the direct cost be a linear function of q,
and let empty movement costs be a linear function u; this
means that a(q,i,j) and b(u,i,j) would be constants for
all relevant values of (q) and (u). Given the values for
these parameters, profit can be maximized in a finite
number of steps using the cited algorithms.

Normative Optimality and the Behavior
of an Actual Motor Carrier Firm

 

It would be convenient to say at this point that
if differences appear between the model developed in this
chapter and an actual motor carrier system, then the in-
tent and purpose of the model has been misunderstood.
With this statement all real world incompatibilities
could be ignored for the sake of theoretical consistency.
Needless to say, there are some very apparent and glaring
problems which arise when one tries to compare the model
with an actual system. These problems are highlighted
below.

Management would have difficulty ascertaining

cost functions, let alone revenue functions. These

49

functions are often unknown or only estimates. This
problem was avoided in the model by assuming that these
functions were known.

Even if cost and revenue functions were known at
a point in time, they would change rapidly through time.1
This issue was avoided by implicitly assuming that a
carrier makes instantaneous and optimal adjustments.

There are difficulties with regard to the conti-
nuity of cost and revenue functions. Truckload freight
movements can only move in integer units; however, for the
sake of making it possible to identify derivatives and
profit maximization conditions, it was implicitly assumed
that vehicles are continuously divisible.

Transportation is regulated by "a chaotic patch-
work of inconsistent and often obsolete legislation and
regulation."2 The model assumes no regulation and that
the carrier adjusts rates so as to maximize profits.

It is implicitly assumed that freight inputs are
continuously flowing through the system; this avoids the
issue of random fluctuations in freight and vehicle move-

ments 0

 

lSee Wilson, op. cit., pp. 266-76.

2A quote from President Kennedy's special trans-
portation message to Congress in 1962. See the "Antitrust
Head Urges an End to Regulation for Most Transport," Wall
Street Journal, January 29, 1971, p. 6.

 

50

Even the assumption about profit maximization has
dubious overtones; the inability to measure and estimate
costs and revenues precisely, casts doubt on the carrier's
ability to maximize profit.

These problems project the complex and transient
nature of transportation rates. They are sobering, yet
they reinforce a contention that if a model handled all
of these aspects, then its complexity would be of such
magnitude that few individuals, if any, would have the
mental capacity to understand and interpret it.

These issues must be put in perspective with the
original intent of the model developed here; its purpose
is to provide an analytical framework from which to view
and analyze the general nature of truckload freight rates.
To do this it is necessary to step away from reality and
reflect on the general nature of the problem (as opposed
to focusing on specific intricacies), in order to make
progress toward generalizations in behavior.

In this light, only general comparisons between
the model and a real system can be sought; it would be
naive to do otherwise. A motor carrier's behavior should,
at most, loosely parallel the relationships found in the
:model; other factors intervene and disguise the effects

put forth in the model.

51

Adjustment Process

The dominance of truck movement costs cannot be
ignored by management.

Inasmuch as transportation costs make up 50 per cent
of total costs . . . it is savings from this source
that appears to dominate the relation between output
and total costs. It is interesting to note that this
is where one would expect savings from scheduling and
routing advantages to accumulate.1

Therefore, a motor carrier is strongly motivated
to take into account marginal trucking costs when setting
freight rates.

The concept concerning circuitous routes, and the
summation of marginal vehicle profits around those routes
is not only theoretically significant in the normative
model, but is also operationally significant. If it can
be assumed that marginal revenues and costs can be reason-
ably estimated by the carrier, and that they can be com-
puted on circuitous routes, then the carrier has a workable
method of identifying non-optimal conditions and can adjust
operating levels accordingly.

For example, reconsider the situation described in
(2-31), and also assume that vehicle flows are positive
along all constituent segments in the circuit.

If the carrier recognizes that (scmvp) differs from

zero, then it can go about adjusting rates and flows in the

circuit to increase profit. Through time, (scmvp) will

 

1Burstein, et al., op. cit., p. 33.

 

 

52

tend toward zero. This argument would apply to all cir-
cuits in the system; presumably, the circuits which showed
the greatest profit potential would be adjusted first and
the others would follow.

Obviously, it is not likely that all, or even one
(scmvp) will reach zero because of errors in measurement,
errors in judgment, regulatory restrictions, and changes
in economic conditions; nevertheless, the adjustment
toward zero would parallel the characteristics put forth

in the normative model.

Conclusions

 

Three basic conclusions are drawn in this chapter.
The first conclusion concerns the relationships between
marginal revenues and costs with regard to circuitous
routes. The second conclusion concerns the terminal
multipliers and their interpretation. The third con-
clusion concerns the notion of normative optimality and

the actual behavior of a motor carrier.

Marginal Revenues and Marginal Costs
It has been demonstrated in the model that mar-
ginal revenues on interterminal freight movements will
generally 32; equal the marginal direct cost of trans-
portation when a motor carrier is at a profit maximizing
state. On the surface this appears to contradict micro-

economic theory, but further analysis shows that vehicle

53

flow constraints differentiate motor carrier operations
from the production type firm. In general, it is required
in the motor carrier model that the sum of marginal
revenues equals the sum of marginal costs around a
circuitous route for profit to be maximized, as opposed
to the equality of marginal revenue and marginal cost on
a single trip.

In effect, transportation rates and costs in one
portion of a system affect transportation rates in other

portions of the system.

Terminal Multipliers

When a motor carrier maximizes profit, the value
of vehicle capacity at each terminal can be imputed from
the solution. These imputed values are associated with
the terminals; they are the Kuhn-Tucker multipliers which
embody the necessary, and sometimes sufficient, conditions
for profit maximization in the model. The conceptual
explanation of these multipliers revolves around the fact
that when freight is transported from one terminal to
another, vehicle capacity is required at the origin and
is made available at the destination.

The value of a terminal multiplier at an optimal
solution reflects the incremental increase in profit that
would result if a unit of vehicle capacity was added to a

terminal. In a logical manner they interrelate the

 

54

marginal vehicle profits on vehicles which are flowing
between terminals.

In general, the differences between the marginal
values of vehicles at two points in an optimal solution
equals the marginal profit of moving a vehicle from the
one point to the other.

Theoretical Behavior and Actual Behavior
of a Motor Carrier Firm

The model developed in this chapter synthesizes
some fundamental conditions encountered in the transpor-
tation of truckload freight; it is designed to formalize
the relationships which occur when a motor carrier maxi-
mizes profits. In order to accomplish this goal, it is
necessary to make simplifying assumptions; each assumption
pushes the model further into an abstract realm. Never-
theless, the model incorporates basic features of a motor
carrier system, and these same basic features are real
forces and constraints in the management of a motor carrier.
This gives reason to believe that the observed relation—
ships among rates and costs of an actual motor carrier
should parallel the relationships expressed in the
normative model. At present this is only a hypothesis;
in Chapter III this hypothesis is examined, and empirical
evidence is presented to show that relationships which

exist in the model also exist in an actual system.

 

CHAPTER III

AN EMPIRICAL INVESTIGATION

Introduction

 

In Chapter II a model identifies the normative
characteristics of maximized profit on truckload freight
in a multiple terminal motor carrier system. The purpose
of this chapter is to provide empirical evidence showing
that the rate and profit structure of an operating motor
common carrier exhibits characteristics which are similar
to those predicted in the normative model.

Motor carrier freight rates are a result of inter-
actions among a variety of factors. Some of these factors
are economical, some are legal, and others are political
in nature. The rates examined here are a result of these
factors. Some specific factors are analyzed and the others
are assumed to affect the rate structure in a random manner.

Regression models are used to test the statistical
significance of the relationships suggested in the norma-
tive model. The specific assumptions of these models are
detailed in the analytical sections of the chapter.

A general hypothesis investigated in this chapter

is addressed to the empirical relationships between

55

56

truckload freight profitability patterns and freight flow
imbalances in the system. The form of the hypothesis is
related to the normative equations and relationships
developed in Chapter II, and in this sense the normative
model provides an explanation of the statistical results
presented in this chapter. However, the statistical results
presented here also stand independent of the normative

model in that the results are significant in their own
right, and depend on the normative model only for expla-

nation, not justification.

Organization

This chapter is organized into five major sections.
The first section introduces the general nature of the
empirical analyses undertaken. The second section de-
scribes the source of the data and the mathematical
notation used to denote the various observed variables.
The third section is devoted to a verbal and informal
mathematical description of the hypothesis being examined.
The fourth section gives a formal mathematical statement
of the statistical models and hypothesis being tested;
statistical results are also presented. The last section
presents conclusions and summarizes the empirical find-
ings.

Data Source

 

A motor common carrier provided the data for this

investigation. The reasons for using the data from this

 

57

carrier are twofold. First, the carrier has significant
freight traffic imbalances and has empty vehicle flows
which should theoretically be reflected in the freight
rate and profit structure; hence, the data provides an
excellent setting for comparing actual profit inter-
relationships with normative relationships. Second, the
carrier has detailed accounting records of freight traffic
flow, revenues, and costs listed according to freight
origin and destination; this, of course, greatly facili-
tated data collection.

The carrier operates from thirty-four terminals
located in five states. Data from two nonconsecutive
accounting periods were collected. The accounting periods
were approximately a month in length, and were in the
middle of the 1969 calendar year. Company officials
stated that the data were representative of their oper-
ations during the respective periods. During these two
periods, the carrier moved over four hundred million
pounds of freight; approximately half this amount was
moved in each period.

Rate and cost data of select truckload shipments
were used for this investigation. Only shipment sizes
between ten thousand and twenty thousand pounds, and
traffic with both initial origin and final destination
within the carrier's system were used in the rate and

cost analyses. This shipment group is referred to here,

 

58

and in the remainder of the chapter, as intraline group
ten freight. Freight flow data on other freight groups
were also used incidentally in certain parts of the

analyses.

Intraline Group Ten Freight

Intraline group ten freight accounts for approxi-
mately 10 per cent of the total weight moved by the carrier.
This particular group was selected for the following
reasons. First, if a study is to analyze the rates and
profitability of freight traffic between a set of termi-
nals, then it is reasonable to restrict the analyses to
similar shipment sizes because rates, costs, and profita-
bility are related to the size of the shipment.1 This
homogeneous group of shipment sizes was used to avoid the
possibility of attributing variations in profitability to
traffic imbalances and demand factors when such variations
might have been due to variations in shipment size.

Second, freight traffic which either originates
or terminates with another carrier (interline traffic),
is excluded because such traffic has different economic
characteristics than intraline traffic. Revenue allo-
cations between carriers on interline traffic involve
certain regulatory "rules" and are not comparable with

revenue on similar traffic which both originates and

 

1This is commonly recognized by both carriers and
regulatory agencies; see Locklin, op. cit., p. 649.

 

59

terminates within a single carrier's territory. In light
of the need to analyze a homogeneous freight group, rate
and cost data from interline traffic were not included.
Third, the normative model in Chapter II analyzed
the economics of truckload shipments moving directly from
an origin terminal to a destination terminal. Therefore,
in order to compare actual data with the model it is neces-
sary to consider freight traffic which moves directly from
an origin terminal to a destination terminal and intraline
group ten freight does this. Shipment sizes of less than
ten thousand pounds are usually routed via intermediate
terminals for linehaul consolidation with other shipments.
Fourth, intraline group ten freight shipments are
of sufficient size to warrant point to point rate adjust-
ments by the carrier. The rates are, of course, regulated
by government agencies; however, the carrier retains some
influence on rate levels. This influence is exercised in
several ways. The carrier can actively solicit freight
traffic which moves at favorable rates and de-emphasize
unfavorable traffic. The carrier can also seek modifi-

cations of published rates which are disadvantageous.

Mathematical Notation of Data
To avoid ambiguity, data variables are defined
and labeled in this portion of the chapter, and apply to

the remaining portions of the chapter. The following list

 

60

of variables represent the relevant data collected for

analyses.

Unit Revenue

The variable rl(i,j) is the average unit revenue
on intraline group ten freight moving from terminal (1) to
(j) during the first of the two data collection periods.
Similarly, r2(i,j) is the average unit revenue for the
second data collection period. The rk(i,j), (k=l,2),
variables were computed by dividing the appropriate total
revenue by the total weight; these variables are measured

in dollars per hundredweight.

Direct Unit Cost

 

The variable al(i,j) is the average direct unit
cost of transporting intraline group ten freight from
terminal (i) to (j) during the first of the two data col-
lection periods. Similarly, a2(i,j) is the average direct
cost for the second period. The ak(i,j), (k=l,2), vari-
ables were computed by dividing the appropriate total
direct cost by total weight; the measurement units are
in dollars per hundredweight. Direct costs, as defined
here, include the following basic cost centers: (1) pick-
up and delivery costs, (2) dock and platform costs, (3)
linehaul costs, (4) claim costs, and (5) office and bill-

ing costs. The costs in these centers are assigned to

61

freight movements according to standard motor carrier
accounting techniques.l

With the exception of linehaul costs, the costs in
these centers are also computed according to standard
accounting techniques.2 The exception arises because
motor carrier accounting procedures treat the cost of
moving empty vehicles between terminals as an overhead
expense, and distribute it among linehaul costs. In order
to maintain a comparison with the direct cost variable
used in the normative model, it is necessary to treat

empty vehicle movement costs separate from direct costs.

Empty Vehicle Movement Cost

The variable b(i,j) is the average unit cost of
moving empty vehicle capacity from terminal (i) to termi-
nal (j). These variables are computed by taking the
standard accounting cost of moving a tractor and a single
bottom empty trailer from terminal (1) to (j), and divid-
ing it by the standard hundredweight capacity of the
trailer; resulting measurements are in terms of dollars

per hundredweight capacity.

 

1See John C. McWilliams, Motor Carrier_gost
Techniques (Washington, D.C.: National Motor Freight
Traffic Association, Inc., 1956). Also see Interstate
Commerce Commission, op. cit.

 

21bid.

62

Direct Unit Profit

 

The variable pl(i,j) is the average direct unit
profit on intraline group ten freight moving from termi-
nal (i) to (j) during the first data collection period.
This variable is measured in dollars per hundredweight,
and equals (rl(i,j) - al(i,j)). The variable p2(i,j) is

similarly defined for the second data collection period.

Number of Shipments

 

The variables Nl(i,j) and N2(i,j) are the number
of intraline group ten shipments which moved from terminal
(i) to terminal (j) during the first and second data col-

lection periods respectively.

Total Freight Flow

 

The variable w1(i,j) equals the total freight flow
from terminal (1) to (j) during the first data collection
period. The variable w2(i,j) is similarly defined for the
second period. These variables are measured in terms of
total hundredweight per period. Note that these variables
represent all the freight transported by the carrier; this

includes all shipment sizes and interline freight.

Intraline Groupren Freight Flow

 

The variable qk(i,j), (k=l,2), equals the intraline
group ten freight which flowed from terminal (i) to (j)
during the (k)th data collection period. The flows are

measured in terms of total hundredweight per period.

63

Distance Between Terminals

The variable d(i,j) is the over the road distance
in statute miles between terminals (i) and (j). This
variable is used in the analyses to reenforce certain
statistical models and conclusions.

A Note on the Distinction Between
Empirical Variables and
Normative Variables

In this chapter it is necessary to refer to the
data variables defined in the preceding section, and to
the normative variables defined in the preceding chapter.
Notation is similar but not identical. Table 3 is pro-
vided to summarize the notational and conceptual similari-
ties and differences between the empirical and normative
variables. Some of the empirical variables listed in the

table are defined in the later parts of this chapter.

A Note on Weighted Observations

Due to the fact that the rate and cost variables
defined above represent different numbers of freight ship-
ments, it is necessary to give more importance or weight,
to those which represent large numbers of shipments. In
the statistical models used here, there are standard tech-
niques for handling this situation; briefly, observed
variables are weighted according to the number of ship-
ments the variables represent. The procedures for weight-

ing variables in linear statistical models are well

64

 

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.mmanmwum> HoowufimEm pom w>wu68uoc mo umwa 4II.m wands

65

documented.l Assumptions are made about the variance of
the error terms associated with an observation; the assumed
variance of the error term becomes smaller as greater
weight is assigned to the observation. The formal mathe-
matical assumptions used for weighting the variables are
described in the analytical section of this chapter.

A General Description of the Hypothesized
Profit‘Patterns

 

 

The normative model in Chapter II prescribes the
following pattern of marginal revenues and marginal costs
in an optimal system. There will exist values for t(i),
(i=1,...,0), in which

r(qooi.j) - a(qopipj) = t(i) - t(j) for (i,j)

such that qo(i,j) > 0, (3-1)
and

such that uo(i,j) > 0. (3-2)

These equations are a partial list of the conditions in

Table 2; the variables are defined in Chapter II.

 

lSee Henry Scheffé, The_Ana1ysis of Variance
(New York: John Wiley & Sons, Inc., 1959), pp. 19-22.
For weighted regression see N. R. Draper and H. Smith,
Applied Regression Analysis (New York: John Wiley &
Sons, Inc., 1966), pp. 71-81.

66

It is implied from these equations that in an
optimal normative system the marginal direct profitability
patterns, (r(qo,i,j) - a(qo,i,j)), are related to the
patterns of empty vehicle flows1 through the differences
among the t(i) variables. It can be generally assumed
that b(uo,i,j) is always greater than zero, and therefore
if there is some (i,j) such that uo(i,j) > 0, then t(i)
cannot equal t(j) because of (3-2). This difference be-
tween t(i) and t(j) will necessarily be reflected in the
marginal direct profitability of positive freight flows
which connect the two terminals because of (3-1). It is
sufficient to say here that for a normative system which
is interconnected by positive freight flows, the presence
of empty vehicle flows at an optimal operating level im-
plies that marginal revenues will differ from marginal
direct costs on some, if not all, freight flows. With
the above relationship in mind, it would certainly appear
that perceptible relationships should exist between freight
profitability and patterns of empty vehicle flows when
significant empty vehicle flows are present in an actual

system.

 

1Notice that if there are no empty vehicle flows
in an optimal normative system, then all that can be said
is that t(i) - t(j) > - b(u ,i,j) for all (i,j). This
does not imply that t(i) mugt equal t(j). As pointed out
in Chapter II, the t(i), (i=i,...,0), will equal one an-
other only in some very special circumstances.

67

The Carrier and Rate Adjustments

For the motor carrier firm which provided the data,
approximately thirty-four thousand dollars was expensed
each accounting period to empty vehicle movements. Over
a year, this would represent nearly four hundred thousand
dollars. The patterns of empty vehicle flows were similar
in both periods. On a relative basis, twelve terminals
consistently received empty vehicles, nine terminals con-
sistently supplied empty vehicles, and the remaining
thirteen terminals both supplied and received empty
vehicles at various times during the two accounting
periods.

The carrier's management stated that these vehicle
imbalances stimulated rate adjustments in the interest of
financial health. Government regulatory procedures, of
course, monitored the adjustments which were implemented
by the carrier.

In general, the carrier was very well aware of
freight imbalances and empty vehicle flows. Without re-
vealing specific policies, it can be said that special
marketing adjustments were pursued on freight involving
terminals that had major traffic imbalances. In this
light, the data certainly provides a suitable setting
for examining patterns in the direct unit profit

structure.

68

Empty Vehicle Flows

The movements of empty vehicles in an actual motor
carrier system are governed by a variety of factors, not
all of which are compatible with the conceptual nature of
the empty vehicle flow variable used in the normative
model. The various complications which arise are dis-
cussed below.

For the motor carrier under consideration, empty
vehicle routing decisions are made by a central dispatcher
who is in direct contact with the terminals in the system.
His decisions take into account minimum tractor and
trailer requirements at terminals, vehicle availability,
driver availability, and driver constraints. Consistent
patterns of empty vehicle flows emerge in the system, but
it is apparent from the data that empty vehicles are not
routed directly from surplus terminals to deficit termi-
nals. Indirect routings take place in many instances
because of short term driver constraints, the company's
policy on terminal trailer requirements, tractor short-
ages, and random circumstances.

For example, an empty vehicle might be routed out
of surplus terminal A to a central terminal C, and then to
a deficit terminal B. In terms of the data variables,
this would appear as an empty vehicle movement from
terminal A to terminal C and another empty vehicle move-
ment from terminal C to terminal B, when in fact it is

only an empty vehicle movement from terminal A to B.

69

The normative model presumes perfect information,
and therefore, indirect routing of empty vehicles would
never occur because it is clearly uneconomical to move a
vehicle over a greater distance than necessary. Further-
more, if it is assumed that b(uo,i,j) is greater than zero
for all (i,j), then the normative model precludes a
circuitous route of positive empty vehicle flows; this
follows from (3-2) in that no values could exist for the
relevant t(i) in such a situation. In an actual system,
empty vehicles may traverse 311 routes at one time or
another.

These complications require a need for a generally
applicable technique which identifies underlying empty
vehicle flow patterns and which is compatible with the
normative model. In addition, the technique should be
somewhat independent of a carrier's transient empty
vehicle routings and at the same time reflect actual
patterns in the system.

One viable approach attacks the problem in terms
of freight traffic imbalances. The approach imputes empty
vehicle flows from actual freight flow imbalances; the
imputed empty vehicle flows are determined by calculating
an empty vehicle flow pattern which balances the system

1

at the lowest total cost. It is basically a linear

 

1This approach to imputing empty vehicle flow is
identical to the methodology set forth by Koopmans and
Reiter, op. cit., pp. 222-59.

70

programming origin-destination problem, commonly called
the transportation problem.1 The terminals with greater
freight inflow than outflow become the origins of empty
vehicles, and the terminals with greater freight outflow
than inflow become the destinations of empty vehicles.

The resulting minimum total cost flows from origins to
destinations are the imputed empty vehicle flows. Details
of the procedure and its relationship to the normative

model are presented later in the chapter.

GeneraliAssumptions

The following assumptions are made in order to
make it possible to compare empirical data with the norma-
tive model. First, it is assumed that the carrier is
motivated to adjust operating policies and freight rates
so as to provide a favorable return on investment. This
assumption connotes profit maximization with respect to
the physical and economic constraints placed on the carrier.
Second, it is assumed that average unit revenue is viewed
by the carrier as a close approximation to marginal reve-
nue. Third, it is assumed that the average unit costs
of moving freight and empty vehicles between terminals
are viewed by the carrier as close approximations to mar-

ginal costs. Fourth, it is assumed that the empty vehicle

 

1There are a large number of references on the
transportation problem; a broad and complete analysis of
the subject is in Hadley, Linear Programming, Chapter 9.

71

flows imputed from the freight flow imbalances reflect
the overall vehicle balancing problem facing the carrier.
Fifth, it is assumed that factors not explicitly taken
into account by the model affect the data in a normally
random and independent manner. With these assumptions
the empirical data variables can be applied to least
squares regression models which incorporate the equations
in (3-1) and (3-2).

A number of structural variations are developed in
the regression models in order to provide results which
are not dependent on a single set of assumptions. In a
general sense, the regression models test for a statisti-
cally significant system wide relationship between average
unit profit patterns and imputed empty vehicle flow pat-
terns.

Statistical Models, Tests,
and Results

 

 

The numerical differences among the t(i) values
are calculated from the imputed empty vehicle flows, and
then these differences are "fitted" to average direct unit
profit in a least squares regression model. Statistical
inferences are developed by making assumptions about the
distributions of the error term associated with the

empirical observations.

72
Imputed Empty Vehicle Flows and
Differences Among (t) Values

Imputed empty vehicle flows are determined from
the linear programming model shown in Table 4. The linear
programming model determines the least cost method of
routing empty vehicles in the system so as to provide
each terminal with sufficient vehicle capacity to trans—
port known freight flow requirements.

For notational purposes, the variables u*(i,j),
(i=1,...,34; j=l,...,34; i#j), are used to represent the
minimum total cost solution of the linear programming
model. With known values for u*(i,j) and b(i,j), imputed
differences among the t(i) values can now be determined.
For notational purposes, the variables t*(i), (i=1,...,34),
are used to represent the t(i) values determined from the

imputed empty vehicle flows. From (3-2),

t*(i) - t*(j) = -b(i.j) for (i,j)

such that u*(i,j) > 0. (3-3)

In the minimum total cost solution there are thirty-three
positive u*(i,j), and it so happens that these thirty-
three non-zero values are sufficient to uniquely define
the differences among 311 t*(i), (i=1,...,34). This is
not a coincidence; in fact the differences among the t*(i)
embody the necessary and sufficient conditions for the

optimal solution to the linear programming model depicted

73

TABLE 4.--Linear programming model used for the determi-
nation of imputed empty vehicle flows.

 

Minimize Total Vehicle Balancing Cost:

34 34
Total Vehicle Balancing Cost = Z 2 b(i,j) . u(i,j)
i=1 j=l
ja‘i
Constraints:
2 34 34 34 34
Z [ Z wk(i,m) - Z wk(m,i) ‘3 Z u(m,i) - 2 u(i,m),
k=1 i=1 i=1 i=1 i=1
i¢m i#m i¢m i¢m

(m=1,...,34).
u(i,j) Z 0 for all (i,j)

Definition of Variables:

wk(i:j) is the known total flow, of all freight groups,
from terminal (1) to terminal (j) during period

(k).

b(i,j) is the known unit cost of moving a unit of
empty vehicle capacity from terminal (1) to
(j).

u(i,j) is the unknown empty vehicle flow required to
balance the system.

 

74

1 It stands to reason that the t(i) values in

in Table 4.
the normative model are analogous to the t*(i) values com-
puted from the linear programming model because a motor
carrier system operating at an optimal level would neces-
sarily employ a minimum total cost empty vehicle flow.

The calculated values of the (t*(i) - t*(j)),
(i=1,...,34; j=l,...,34; i#j), have an economic interpre-
tation in the linear programming model which aids in
understanding the nature of the regression models; there-
fore, this interpretation is elaborated upon here. The
calculated value of (t*(i) - t*(j)) is the imputed in-
crease in total empty vehicle movement cost that would
result from one unit of vehicle capacity being removed
from terminal (1) and added to terminal (j), and then
having the problem resolved for a new optimum.2 In effect,
(t*(i) - t*(j)) reflects the change in total vehicle
balancing, or backhaul cost that results when a unit of
vehicle capacity is shifted from terminal (1) to (j);
henceforth, (t*(i) - t*(j)) will be referred to as the
imputed backhaul cost variable. In this light, it is

reasonable to suspect that direct unit profits on freight

(r (i,j) - a (i,j)), or simply p (i,j), will have a
k k k

 

1See Ibid., pp. 309-12. For additional coverage
in a network context, see S. E. Elmaghraby, "The Theory of
Networks and Management Science, Part I," Management
Science, XVII (September, 1970), 24-25.

21bid.

75

positive relationship with the imputed backhaul cost
(t*(i) - t*(j)) which is an indirect cost of freight move-
ment. Stated in another way, the larger the value of
(t*(i) - t*(j)), the larger would be the value of pk(i,j)
because above average indirect backhaul costs must be com-
pensated for by above average direct unit profits on the

freight flow which gives rise to the backhaul cost.

Weighted Observations

The variance of a numerical average of independent
homoscedastic random variables is equal to the variance of
the individual random variable divided by the number of
variables that comprises the average.1 Hence, it is
assumed that the variance of the error term associated
with a dependent regression variable is 02 divided by the
number of shipments which the dependent variable repre-
sents, where o2 is the unknown error variance of a single
shipment. This is a standard technique in weighted re—

. I 2
gre551on analy31s.

Unit Profit Regression Models
These models measure the statistical relationship

between the unit profit variable pk(i,j), and the imputed

 

lSee Emanuel Parzen, Modern Probability Theory
and Its Applications (New York: John Wiley & Sons, Inc.,
1960), p. 371.

 

2See Draper and Smith, op. cit., pp. 71-81.

76

backhaul cost variable (t*(i) - t*(j)). Five least squares
regression models are presented here; the models differ
according to the number of concomitant variables which are
fitted to the unit profit variable before the imputed back-
haul cost variable. These different models are developed
in order to preclude the possibility of attributing vari-
ations in unit profit to the imputed backhaul cost vari-
able when the variations may have been due to known
concomitant variables. It so happens that fitting these
additional concomitant variables does not affect the
relevant statistical inferences made here.

The forms of the five models are described in
Table 5; they are labeled WClLP2A, WC2LP2A, WC3LP2A,
WC4LP2A, and WC5LP2A.

The statistical hypothesis examined here has the
following form.

Null Hypothesis: h=0 (3-4)

Alternative Hypothesis: h # 0 (3-5)
The variable (h) is the regression coefficient associated
with imputed backhaul cost variable; it is expected that
the estimate of this variable will be positive.

The null hypothesis is rejected in all five models
at the 10 per cent significance level using the analysis

1

of variance F-test. The analysis of variance tables for

 

lSee Scheffé, op. cit., pp. 31-32.

77

 

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78

these models are contained in Appendix I. Specific sig-
nificance levels, and the estimates of (h) are listed in
Table 6; the estimates of (h) range approximately between

(.05) and .07).

Concomitant Variables

 

The regression model WClLP2A fits only the grand
mean to the direct unit profit variables before the back-
haul cost variables.

The regression model WC2LP2A fits the grand mean
and the direct unit cost variables before the backhaul
cost variables. It is reasoned here that some variations
in direct unit profit are due to direct unit cost; specifi-
cally, it is reasonable to suspect that large direct unit
profits will be accompanied by large direct unit costs.

In regression model WC3LP2A, the grand mean, direct
unit cost, and the total freight flow variables are fitted
to direct unit profits before the imputed backhaul cost
variables. It is reasoned here that some variations in
direct unit profits are also due to the total volume of
traffic which moves with the intraline group ten freight.
From an economic theory standpoint, a large volume of
traffic from one terminal to another implies that the
average direct cost variable is a more accurate approxi-
mation to marginal direct cost than in the case of small
traffic volumes. In addition, large volumes of traffic

imply a greater potential for competition; hence, unit

79

TABLE 6.--A summary of statistical tests.

 

 

Model F-Statistic Sigiiiigifiie Eggi?fife
WClLP2A 3.000 .08 .048
WC2LP2A 4.618 .04 .058
WC3LP2A 5.191 .025 .059
WC4LP2A 5.908 .02 .063
WC5LP2A 6.418 .015 .066

 

80

revenues may be a more accurate approximation of marginal
revenue. In any case, the total traffic variables are
fitted to the data without adversely affecting the sta-
tistical significance of the backhaul cost variables.

In the regression models WC4LP2A and WC5LP2A,
curvilinear direct unit cost and total traffic variables
are fitted in the models respectively, again without
adverse affects on the backhaul cost variables.

The regression coefficients associated with the
direct unit cost and total traffic flow variables are not
revealed here for obvious proprietary reasons; however,
it is sufficient to say that the coefficients were con-

sistent with expectations.

The Analysis of Residuals

An analysis of the residual errors in the models
did not reveal characteristics which would tend to dis-
count the validity of the statistical inferences. The
residuals were distributed in a bell shaped pattern, and
this conforms with the normality assumption.

There was no significant correlation between the
residuals of the two periods; this is accounted for by
the fact that the data were from two non-consecutive
periods, and that there were some variations in the
traffic mix due to the yearly "model changes" in the
plants which were served by the carrier. Although inter-

temporal correlation among the residuals was not a

81

problem here, it could be a potential problem in other
data.1 Ways of handling this are to either combine the
data as one period, or analyze each period separately.
The latter was tried here, for the sake of curiosity, and

the results were very similar to those already presented.

Alternative Regression Models

Several variations to the principal regression
models were constructed to examine the stability of the
statistical inferences under alternative formulations.
In the interest of maintaining the continuity of the
chapter,only the general aspects of these alternative
regression models are presented. In general, the backhaul
cost variable has an equal or stronger significance in
these alternative models. Three general groups of vari-
ations to the regression models were implemented, and

they are briefly described here.

Unit Revenues and Distances

 

Unit revenues, instead of direct unit profit, were
treated as the independent variable, and the distance
variables, d(i,j), were used, instead of direct unit cost,
as concomitant variables. This variation was developed
from a contention that the direct cost variables might
have had hidden directional influences on unit profita-

bility. By using unit revenue and the distance variables,

 

1See J. Johnston, Econometric Methods (New York:
McGraw-Hill Book Company, Inc., 1963), pp. 177-200.

 

r“

82

direct unit cost variables were not present in the regres-
sion model. The inferences with regard to the signifi-
cance of the backhaul variable were unchanged; the actual
estimates of (h) in this alternative model were not
appreciably different than the estimates in the unit

profit models.

Deletion of Minor Terminals

For the data under consideration, each of the
thirty-four terminals had at least six intraline group ten
shipments moving in and out of the terminal during the
combined accounting periods. However, to insure that
small terminals did not have a disproportionate effect on
the analysis, they were removed. An arbitrary cutoff of
twenty-five group ten intraline shipments was established
as a minimum inbound and outbound limit. This criterion
eliminated nine of the original thirty-four terminals.
All remaining terminals had at least twenty-five inbound
intraline group ten shipments, 229 at least twenty-five
outbound intraline group ten shipments. The data for
these twenty-five terminals were applied to models identi-
cal to those previously described. In general, the sig-
nificance of the backhaul cost variables were much
stronger in these models than in the models where data
from all terminals were included; this is a logical re-
sult in light of the erratic freight flows associated

with the small terminals.

83

Alternative Compupation of
Imputed Backhaul Cost
Variables

 

 

Instead of aggregating the freight imbalances over
the two accounting periods and then computing the backhaul
cost values from the linear programming model, backhaul
cost values were computed for each period from the freight
imbalances which occurred during the period. The result-
ing imputed backhaul cost values were not identical for
each period, but did have a linear correlation coefficient
greater than (.90). These backhaul costs were averaged
from both periods, and then applied to the regression
models in place of the backhaul costs which were imputed
from aggregate freight imbalances. Here again, the re-

sults were not appreciably different than before.

Conclusions

 

The major conclusion of this empirical analysis is
that the data exhibits a measurable statistically signifi-
cant relationship between direct unit profit patterns on
intraline group ten freight and the imputed empty vehicle
flow patterns calculated from freight flow imbalances.
Three implications result from this conclusion:

(1) the relationships prescribed in the normative
model bear a significant resemblance to

empirical data;

84

(2) freight imbalances are a perceptible concomitant
factor in unit profit and rate patterns;
(3) backhaul effects are evident in the rate structure

and do not necessarily "average out" for a carrier.

These implications are often ignored in contemporary trans-
portation literature. For example, the transportation rate
studies cited in the first chapter ignore the problem com-
pletely, or assume that it "averages out." This latter
assumption is somewhat dubious when it can be shown that
there are measurable patterns in the rate structure which
are related to freight imbalances.

The empirical analysis indicates that the motor
carrier under consideration is operating in a manner pre-
dicted by the normative model, and this supports the
general hypothesis that the normative model provides a
practical framework from which to View spatial effects in

truckload freight rate structures.

CHAPTER IV

EVALUATION AND CONCLUSIONS

An Overview

 

This dissertation views motor common
truckload freight rates from the position of
carrier firm. It examines the normative and
interactions among three basic factors. The

(1) the physical necessity of equalizing

flows into and out of each terminal;

carrier

a motor
empirical
factors are:

vehicle

(2) the dependency of freight flows on the freight

rate being charged for the transportation service

and the relative demand for the transportation

service;

(3) the profit maximization motive which stimulates

the motor carrier to adjust freight rates for

economic gain.

A theoretical model consolidates these factors in

order to characterize the behavior of a motor carrier firm.

Principal attention is focused on the rate structure

which maximizes profit for the carrier. In essence, the

model describes how vehicle imbalances which arise from

85

86

unequal demands affect the rate structure in a profit
maximizing motor carrier firm. It is shown that marginal
revenues and costs are interrelated by imputed variables
associated with each terminal in the system; these vari-
ables reflect the marginal value of vehicle capacity at

a terminal. General patterns result in the rate structure
and they are directionally related to origins and desti-
nations through the variables associated with the termi-
nals.

It was also found that circuitous vehicle flows in
the system are very significant from an economic theory
standpoint. In general, it is theoretically incorrect to
assume that marginal revenue will equal marginal cost on
individual freight movements when profit is at a maximum.
Rather, the correct relationship is that the sum of mar-
ginal revenues will equal the sum of marginal costs on
circuitous vehicle flows in the system when profit is at
a maximum.

Rate, cost, and freight flow data were collected
from a motor common carrier, and applied to a statistical
regression model in order to determine the extent to which
the relationships prescribed by the theoretical models
compare to the imperfections of the practical world.
Imputed backhaul cost patterns were calculated from
actual freight flow imbalances in a manner prescribed by

the normative model, and then correlated with direct unit

87

profits on truckload freight. A statistically significant
relationship was found between the two. It is clearly
indicated in the analysis that this cannot be interpreted
to prove or disprove the normative model. However, the
observed empirical relationships certainly enhance the
credibility of the normative model, and strongly indicate
that backhaul patterns have predictable effects in freight

profitability patterns in a multiple terminal system.

Sigpificance of the Normative Models

 

Actual motor carrier systems do not operate
according to a simplified set of theoretical rules;
rather, the real world tends to be a transient complex
process. In order to make a complex problem manageable
it is necessary to revert to simplified models which re-
semble the major elements of the problem. This is the
situation here. The normative model encompasses only a
portion of the total freight rate problem, and is designed
to examine the interaction between the supply of and the
demand for the motor carrier transportation service.

Shippers, Carriers, and Regulatory
Agenc1es

The models are relevant to the problems of
shippers, carriers, and regulatory agencies because they
indicate how rate and cost structures respond to spatial
differences in the demand for the transportation service.

This is of concern to shippers in terms of plant location

88

and physical distribution policies. It concerns the
carrier in terms of its pursuit of profitable freight
rate adjustments, equitable allocation of costs, and the
identification of profitable and unprofitable freight
traffic. It is of obvious concern to the regulatory
agencies; they should be capable of recognizing the
characteristics of an efficient transportation firm in

order that they may better regulate the entire system.

Regulation and Economic Theory

The model identifies a problem with regulatory
policies which treat empty vehicle movement costs as if
they were an overhead item; they are typically allocated
in this nondirectional manner in a motor carrier system.1
A fundamental tenet of price theory is that prices must
be commensurate with economic costs if there is to be an
efficient allocation of resources under the pricing
mechanism. The model clearly shows that empty vehicle
movement costs have a significant directional effect in
the rate structure, and, therefore, nondirectional freight
rates cannot reflect the true economic costs of motor
freight transportation. It is implied that a regulated
freight rate structure which conforms to costs derived

from the nondirectional allocation of empty vehicle

 

lSee McWilliams, op. cit. Also see Interstate
Commerce Commission, op. cit.

89

movements costs will preclude proper economic adjustments

in a system.

The Significgnce of the
Empirical Findings

 

 

The empirical investigation indicates that freight
rates and direct unit profits tend to polarize with respect
to freight imbalances. These findings have definite impli—
cations with regard to motor carrier freight rate studies.
The measured relationship between the rate structure and
freight flow imbalances in a regulated carrier indicates

that these factors should be recognized in rate studies.l

Potential Research Extensions

 

The framework presented in this dissertation can
be extended in two general directions. One extension can
be oriented toward the motor carrier firm, and the other
directed toward regulation of the motor carrier industry.

Using the basic structure of the normative model,
operational accounting methods can be developed for im-
puting interterminal backhaul costs. They could be
imputed from freight and vehicle flows. Implementation
of such a procedure would demand, of course, certain
assumptions regarding overall carrier service policies,
the stability, randomness, seasonal factors, and the

forecasted trends of freight inputs. Once a procedure

 

lSee Walter Y. Oi and Arthur P. Hurter, Economics
of Private Truck Transportation (Dubuque, Iowa: Wm. C.
Brown, 1965), p. 187.

 

 

90

is established for imputing backhaul movement costs on a
system wide basis, the profitability of specific freight
movements can be ascertained.

A cost assignment system which reflects the in-
herent directional cost characteristics of motor carriers
would provide a logical foundation for rate regulation
policies. This aspect deserves special attention in view
of the recent and strong criticisms aimed at the motor
carrier regulatory policies of the Interstate Commerce
Commission.1 Some critics have called for complete
deregulation of the industry. This dissertation provides
a framework for examining the general economic implications

of such proposals.

 

lFriedlaender, op. cit., Chapter 7. Also "Anti-
trust Head Urges An End to Regulation for Most Trans-
port," Wall Street Journal, January 29, 1971, p. 6;
"Council of Economic Advisers Join Ranks of Those Assail-
ing Transport Regulation," Wall Street Journal, February 2,
1971, p. 2.

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The Northwestern University Press, 1964.

TRINC's Blue Book of the Trucking (Industry. Washington,
D. C.: TRINC Associates, Ltd., 1967.

Traffic World. "American Assembly Adopts Proposals to
Relax Transport Rate Regulation." April 26, 1971.

 

. "ICC Replies to Ash Council, Defending Inde-
pendent Role and Regulatory Effectiveness."
April 12, 1971.

"Reevaluation of Transport Regulation Favored
by Joint Economic Committee." April 5, 1971.

"Trucking Rolls Into An Age of Giants." Business Week.

Wall Street Journal. "ICC Contends It Is Doing a Good
JOB Considering Handicaps That It Faces."
March 18, 1971.

 

106

Wall Street Journal. "Nader Group Says ICC Fails Public,
Urges Single Transportation Regulatory Agency."
March 17. 1970.

 

APPENDICES

APPENDIX A

ANALYSIS OF VARIANCE TABLES

107

 

 

 

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APPENDIX B

THE KUHN-TUCKER THEOREM

APPENDIX B

THE KUHN-TUCKER THEOREM

Introduction

 

The Kuhn—Tucker theorem1 is a generalization of
the well-known theorem on Lagrange multipliers. The
Lagrange multiplier theorem specifies certain necessary
conditions for the optimization of a mathematical function
subject to equality constraints; the Kuhn-Tucker theorem
specifies certain necessary conditions for the optimi-
zation of a function subject to inequality constraints.

It should be noted that the Lagrange theorem can be de-
rived from the Kuhn-Tucker theorem.

Under general assumptions, the Kuhn-Tucker theorem
is a necessary condition for an optimal solution; that is
to say, an optimal solution will always satisfy the con-
ditions of the theorem, along with possibly other certain
solutions. In special cases, the theorem becomes both a
necessary and sufficient condition for optimality; that is

to say, the only feasible solution which can satisfy the

 

1H. W. Kuhn and A. W. Tucker, "Nonlinear Program-
ming, " in Proceedings of the Second Berkeley Symposium on
Mathematical Statistics and Probability, ed: by J. Neyman
(Berkeley: University of California Press, 1951).

112

113

theorem is the optimal solution. This special case
occurs when the mathematical functions in the problem
satisfy certain convex and concave requirements.

The development of the Kuhn-Tucker conditions
here begins with the general form of the theorem, and

then focuses on the special case of linear constraints.

General Form of the Kuhn-Tucker Theorem

Consider the following mathematical program.

Maximize f = f(x) (1)
Subject to gi(x) 5.0, (i=1,...,m) (2)
where

(x) is a vector composed of the elements x(j),
(j=l' 00 0’11).

x(j), (j=1,...,n), are the (n) controllable vari-
ables in the mathematical program; the problem
is to find values for these (n) variables which
maximize (f) with respect to the given con-
straints.

f(x) is a known mathematical function of the vector
(x). It is assumed that this function is con-
tinuously differentiable.

g.(x),(i=1,...,m), are (m) known functional inequality
constraints on (x). It is assumed that each
function is continuously differentiable, and that
any set of nonlinear gi(x) functions are not
tangent to one another at the optimal solution.

The Kuhn-Tucker theorem says that if x0 is a feasible

optimal solution to the mathematical program specified

 

lThe "tangent" condition is dealt with by what is
called the constraint qualification property, and is dis—
cussed in later references.

114

above in (l) and (2), then there exists a set of variables

t(i), (i=1,...,m), which satisfy (3), (4), and (5).

t(i) Z 0, (i=1,...,m) (3)
t(i) ' gi(Xo) = or (i=lroo-Im) (4)
3 .(x)

8f(x) _ m . . g1 _ ._

W 1:1(t(l) W ) — 0, (j—l,...,n) (5)
X =X X =X
O 0

where

xO(j), (j=1,...,n), is an optimal solution to the
mathematical program. Of course, this optimal
solution must satisfy the (m) constraint
equations specified in (2).

t(i), (i=1,...,m), are called the Kuhn-Tucker multi-
pliers in the same context as are Lagrange
multipliers. The (i)th multiplier is associated
with the (i)th constraint.

3f(x) is the partial derivative of the objective
3ij5 function (1) with respect to the (j)th
controllable variable evaluated at x0.
x =x
o
agi(x) is the partial derivative of the (i)th
§§T+T— functional constraint with respect to the
3 (j)th controllable variable evaluated at
xo=x Xo'

The equations in (4) show that if the value of
the (i)th functional constraint evaluated at x0 is strictly
less than zero, then the (i)th multiplier must equal zero;
on the other hand, if the value of the (i)th functional

constraint evaluated at x0 is equal to zero, then the

115

(i)th multiplier need not be equal to zero. The equation
in (3) shows that each t(i) must be greater than or equal
to zero. In summary, the (i)th multiplier may be greater
than or equal to zero if the (i)th constraint is at its
upper bound; however, if the (i)th constraint is not at
its upper bound, then the (i)th multiplier must equal
zero.

The equations in (5) show that at the optimal
solution the partial derivative of the objective function
with respect to the (j)th variable must equal the sum-
mation of the products of the multipliers times the
partial derivatives of the constraints with respect to
the (j)th variable.

It is assumed in (l) and (2) that the functions
are continuously differentiable, and that the constraint
functions satisfy the constraint qualification property.
However, there are variations to these assumptions, for
example, it could be assumed that the functions f(x) and
gi(x), need only be differentiable in the neighborhood

of the optimal solution.2

 

lSee C. Almons, Matrix Methods in Economics
(Reading, Mass.: Addison-Wesley Pub. Co., I967), pp.
101-04. Also, G. Hadley, Nonlinear and Dynamic Program—
ming (Reading, Mass.: Addison-Wesley Pub. Co., 1964),
p. 195.

2J. Abadie, "On the Kuhn-Tucker Theorem," in Non-
linear Programming, ed. by J. Abadie (Amsterdam: North-
Holland Pub. Co., 1967), p. 22.

 

 

116

The Values of the Multipliers
It so happens that a value of t(i) can be used to
reflect the effect of incremental changes in the (i)th
constraint on the optimal value of the objective function.
This characteristic of the multipliers is demonstrated
below.l

Suppose the program in (l) and (2) was revised

as follows.

Maximize f* = f(x) (6)
Subject to gi(x) : e(i), (i=1,...,m) (7)
where

e(i), (i=1,...,m) have values which are in a
sufficiently small neighborhood of zero.

f(x) and gi(x), (i=1,...,m), are functions identical
to those defined in (l) and (2).
After solving this revised program, the value of f* will

differ from the value of f by approximately

( Z e(i) - t(i)). (8)

i 1

where

f is the optimal value of the objective function of
the original problem in (1) and (2).

f* is the optimal value of the objective function of
the revised program in (6) and (7).

 

1For greater detail, see J. Abadie, Op. cit.,
p. xiii.

117

t(i), (i=1,...,m), are a suitable set of multipliers in
the optimal conditions of the original problem.

The Kuhn-Tucker Theorem with Linear Constraints
andiondgegativity'RestrictionS'on'the
"Controllable‘variables

In the case where all the constraint functions in
(2) are linear, and the controllable variables (x) are re-
stricted to being non-negative, then the Kuhn-Tucker
theorem is applied as follows.

Consider the following mathematical program.

Maximize f = f(x) (9)
Subject to Gx - s i 0 (10)
x :_0 (11)

where

f(x) and (x) are as defined in (1) and (2).

(G) is a matrix which has (m) rows and (n) columns.
The elements in (G) are known constants.

(s) is a column vector which has (m) rows. The
elements of (s) are known constants.

The constraints in (10) and (11) can be written in

algebraic form as follows.

G(i,j) - x(j)) - s(i) _<_ o, (i=1,...,m) (12)

"v15
H

j
X(j) Z 0, (j=110001n) (13)

where

G(i,j) is the element in the (i)th row and (j)th
column of the matrix (G).

118

s(i) is the element in the (i)th row of the vector (5).
x(j) is the element in the (j)th row of the column
vector (x).
The Kuhn-Tucker theorem says that if x0 is a feasible
optimal solution to the problem stated in (9), (10), and
(11), then there must exist a (m) by (1) column vector
(t), and a (n) by (1) column vector (2) which satisfy

(14), (15), (l6), (l7), and (18).

t i 0 (14)
t' . (G xO - s) = 0 (15)
z 1 0 (16)
z' - x0 = 0 (l7)
3f(x) , _
'_'ax I'G‘t+Z-0 (18)
x =x
o
where

t',z', and G' are the transposes of the vector t,
the vector 2, and the matrix G respectively.

3f(x)

is a column vector representing the deriva-
3x

tives of (f(x)) with respect to x(j),
(j=1,...,n); the (j)th element is
x =x (3f/3x(j)).

 

 

The above matrix expressions are expressed in algebraic

form as follows in equations (19) through (23) respectively.

119

t(i) 3 o, (i=1,...,m) (19)
n
t(i) ° (( Z G(i,j) ‘ XOUH - s(i)) = 0:
j=1
(i=1,...,m) (20)
2(j) :- O, (j=1,...,n) (21)
z(j) ‘ Xo(j) = or (j=lrooorn) (22)
3£%¥} - ( § t(i) - G(i j)) + z(j) = 0
3x 3 . i=1 I I
x =X (j=1,...,n) (23)
O

The equations in (19) and (20) show that if the (i)th
constraint of (G) is at its upper bound, then t(i) can be
greater than or equal to zero; however, if the (i)th con-
straint is not at its upper bound, then t(i) must equal
zero. The equations in (21) and (22) show that if xo(j)
is strictly greater than zero, then z(j) equals zero; on
the other hand, if xo(j) is equal to zero then z(j) may

be either equal to or greater than zero.

Note here that if the objective function in (9)
is linear, then the program becomes a linear programming
problem, and the z(j) and t(i) are the elements in the

"Z-C" row of the simplex method;l each z(j) is asSociated

 

1See Hadley, Linear Programming, pp. 108-48.

120

with an x(j), and each t(i) is associated with the slack
variable in the (i)th constraint.

The constraint qualifications, which were mentioned
previously, are always satisfied when the constraints are
linear.1

The Kuhn-Tucker conditions become necessary and
sufficient conditions for global optimality when the
objective function is concave and the constraint functions
are convex. Linear constraints always define a convex
feasible solution space; therefore, if the objective
function in (9) is concave, then a solution which satis-
fies the Kuhn-Tucker conditions is a global optimum to

the program.2

 

lSee Hadley, Nonlinear and Dynamic Programming,
’p. 201.

2Ibid., p. 212.