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Entropy of freight transportation and price variation
Entropy of freight transportation and price variation Sufi M. NAZEM
initially, towards logical analysis and development of these concepts and later utilizes empirical analyses to test and justify the validity of these postulations.
University of Nebraska, Omaha, NE, U.S.A. Received September 1978 Revised January 1979
2. r~.finition of entropy This article examines possible relationships between network entropy of commodity flow and variation in unit price of freight transportation service. Entropy of seventy-five maj 3r commodity flows through the railroad network in the Unked States have been calculated using data available in the 1974 waybill sample; unit price of transportation service for various networks has been measured as cents per tonmile. The results of empirical analysis confirm that an inverse relationship exists between variation in unit price and network entropy of commodity flow. Commodities with high network entropy appear to exert forces towards a constant price and those commodities with lower entropy, providing greater marginal utility, appear to have a wide variation in unit price. These findings can be used in regulating pricing decisions in the transportation industry.
The concept of network entropy can be related to the flow of goods through a distribution network, where a certain number of distinct points of demand and points or areas of supply can be identified. These points can be represented by an origin-destination of a network system for flow of goods. The logical network and flow of goods can be represented symbolically. Let ~i
be the flGw of commodity from origin i to destination 1; i -- 1, 2, ..., m, and] = 1, 2, "''9 JV~,
ai
be the total amount of commodity traffic originated at area i, bi be the total amount of commodity traffic terminated at area 1. Then the joint probability that one unit of commodity traffic originated at area i will terminate at area ] is ~ven by
I. Introduction The concept of entropy originated from the basic principle of the second law of thermodynamics. Over the years the idea has been borrowed by other disciplines and has been applied in several problem areas within the physical and social sciences, in particular in the field of information theory and network analysis. A limited application of the concept may be found in several transportation studies, notably those of Wilson [5], and Tomlin and Tomlin [3]. A good review of the state of the art may be found in [ 1]. However, much of the application in transportation so far has been limited to urban travel demand. The current research explores how network entropy can be used to measure utility of transportation service. Since price of transportation is related to the concept of utility, ~r value of service, network entropy is likely to have some effect on the price of such serdce. The current research has been devoted,
m
n
where 0 ~ pq < 1 and ~,i~qPii = 1. The network entropy of commodity flow in the entire ~ystem, then, can be defined as
H= - ~
i
~ Pi/ log pq i
if the system has m origins and n destinations.
3. Entropy and equilibrium flow The network entropy H = - Ei~,/Pii log pq is essentially a measure of uncertainty in the commodity flow pattern created because of disequilibrium in distribution patterns. The entropy of a network flow system
© North-Holland Publishing Company European Journal of Operational Research 3 (1979)413-416. 413
414
S.M. Nazem/Entropy of freight transportation and price variation
is maximum under an equilibrium distribution pattern. The maximum value of entropy of a network system can be established by maximizing the entropy function -~
. ~ P , logPii i
i
where 0 < pq < I for all i and L From the above relationsldp and using the general probability condition ~ i ~ p # - 1, it can be proved that under equilibrium condition the value of PO is ltmn and furthermore, the maximum value of entropy is attained only at equilibrium. Thus 1
('Oil)equilibrium
- mn
(/])maximum =
log m n
for all i and 1, and
The result demonstrates the equilibrium distribution pattern that is the flow of goods is distributed every through the network.
4. Pricing in transportation and entropy Transportation is a service, not a commodity, and therefore as such it has no value. Demand for transportation is created by demand for a commodity and thus the value of transportation is reflected as an increase in the price of a commodity between its origin and destination. A shipper is interested in transporting goods only if he can pass on the cost of transportation to the consumer as an increased price for increased consumer utility. Conceptually, if a commodity enjoys widespread availability then utility created by transportation is limited and therefore the value of transportation service is also limited. Commodities with dispersed sources, however, are likely to create much greater utility or value by being tra~lsported to various consumer centers. Therefore, i~ is plausible that one should expect some relatio~ship between the flow of goods and utility or value of service which should reflect dxe price of transportation service. The value of service for transportation is further influenced by the price of a commodity since a lughly valued commodity can easily absorb a substantial increase in transportation price without any noticeable increase in price for consumers, whereas for a commodity of low value a sim~ar increase could have a rather discouraging influence on consume~ osage. Thus a variation in transportation price may be expected
from two counts, firstly because of variation in utility derived from the service and secondly due to the relative price of different commodities. Comm, :lities with widespread availability can approach equilibrium supply and demand flow, forcing the variation on price of transportation service to a lower level, theoretically variation approaches zero with distribution reaching equilibrium state. The reverse situation should generally prevail if commodities are only available from limited sc~,rces. Although regulatory measures have influenced price in the transportation of goods, the pricing strategy is still directly or indirectly related to the notion of value of service or utility. It can therefore be postulated that marginal utility of a commodity with low network entropy is relatively high, whereas commodities with higher network entropy indicate a relatively Iower marginal utility. A commodity with higher marginal utility will generally allow some flexibility in pricing transportation service because shippers can pass on to the customers a substantial increase in price. This in turn will provide a range of prices depending on location, operational efficiencies of transportation companies, bargaining power of shippers and transportation companies. Therefore, the rationale can be extended to relate entropy, utility and price varia~.ion.Commodities with lower entropy are likely to have greater marginal utility which would be reflected in greater variation in price. Commodities which have reached a near equilibrium distribution pattern should thus indicate a low coefficient of variation showing only smaU variations in the range of prices for transportation service. Empirical study in this article is essentially exploratory to f'md evidence with respect to the above postulations about the possible relationship between network entropy of commodity flow and variation in price of freight transportation service. 5. Empirical analysis and results An empirical examination has been carried out using freight transportation data published in the 'Carload Waybill Statistics', published by the Department of Transportation [4]. Data is based on waybill samples of carloads of t~affic hauled by the railroad industry during the year 1974. Sample data has 0een classified and summarized under five regions namely official, southern, western, southwestern and mountain pacific.
S.M. Nazem/Entropy off'eight transportation and price variation
Analysis in this article includes ton-miles of traffic flow and unit price of transportation service, which is measured as price in cents per ton-mile. Entropy of flow of goods has been calculated using inter-regional flow of ton-miles of goods and price variation has been measured as coefficient of variation of inter-regional ulait price. All seventy-five commodities included in this analysis are based on five-digit standard transportation commodity code classifications, which make each commodity group sufficiently homogeneous. Values of coefficient of variation and network entropy for each of the seventy-five commodities have been estimated from the sample data. A summary of these measures has been included in Tables 1 and 2. Two variables have been regressed to test the hypothesis that the entropy of commodity flow influences the variation in price inversily. The regression results were found to be in the form Variation in unit price =
415
Table 2 Summary of variation in price of transportation service Interval in percentage variation
Number of commodities
10 20 30 40
20 30 40 50
18 32 16 7
50 < 60
2
< < < <
75
variation is thus found to be -0.4')8. The test statistic t is -4.907 with 73 degrees of freedom which provides strong evidence in support of the previous results. These results clearly indicate the existence of an inverse relationsl~ip between variation in price of transportation service and network entropy of commodity flOW.
= 48.668 - 10.632 (Entropy of commodity flow) (9.628*) (-3.965") (*Student-t statistics). While the above regression coefficient is statistically significant, a secondary analysis using rank correlation method has been applied to re-examine the relationship for supplementary evidence. Both measures, entropy and variation, have been ranked in ascending order starting g ith the lowest value. The lowest and the highest values thus receiving the rank of unity and seventy-five res;~ectively. The rank correlation between the two ranked variables has been calculated and tested for statistical significance. The rank correlation between entropy and price
Table 1 Summary of entropy of seventy-five commodities Intervals
Number of commodities
less than 0.5 0.5 < !.0 1.0 < 1.5 1.5 < 2.9
1 5 10 29
2.0 < 2.5 2.5 < 3.0
27 3 75
6. Conclusion it is postulated in this article that entropy of flow of goods is likely to influence the price of transportation, since the pattern of flow of goods can be used as a measure of utility derived from transporting goods. Price of transportation service is directly related to utility or value of service. The results confirm that the commodities with higher entropy appear to have a lower variation in price. This is primarily because commodities with higher entropy have lower marginal utility and thus provide very little opportunity for increasing price, resulting in little variation in price. The analysis is based on five-digit standard transportation commodity code classifications which include several commodities in each category. This is likely to contribute some variation in commodity mix but as the data u~ed in this ar,tcle are based on a substantiaUy large sample vf seventy-five commodity groups, this eliminates at least some of the errors contributed by heterogeneity of commodity mix. Overall analysis does indicate statistical significance of the results, confirming the hypothesis presented in this article that network entropy of commodity flow influences the price of transportation service. In terms of policy implications, the above find2ngs
416
S.M. lgazem / Entropy of freight transportation and price variation
lead to a new factor which may be considered with respect to pricing philosophy. For example, in the case of commodities with high entropy or a wide distribution pattern with many origins and distribution points, regulatory price control activities may be somewhat relaxed since market force Js likely to stabilize the price by narrowing differelttials. On the contrary, special attention ~.lay be necessary for certain commodities with lower entropy which potentially may form a transportation monopoly. T I ~ is particularly important in transportation since much of the pricing policy in this industry is directly or indirectly influenced by regulatory agencies. While the current analyses are only of an exploratory nature, the results indicate entropy of commodity flow is related to utility and thus to the 'value of service" concept which
influences pricing policy. Therefore, entropy of flow of goods can play an important role in the making of these policy decisions.
References [ 1] R.B. Potts and R.M. Oliver, Flows in Transportation Networks (Academic Press, New York, 1972). |2] R.E. Sonntag and VanWyler, Fundamental of Statistical Thermodynamics (John Wiley, New York, 1966) Chapter ~V. [31 J.A. Tomlin and S.G. Tomlin, Traffic distribution and entropy, Nature 220 (1968) 974-978. [4] U.S. Del-artment of Transportation, 1974 carload waybill statistics (1975). [51 A.G. Wilson,The use of the concept of entropy in system modeling, Operations Res. Quart. 21 (1970) 247-265.
initially, towards logical analysis and development of these concepts and later utilizes empirical analyses to test and justify the validity of these postulations.
University of Nebraska, Omaha, NE, U.S.A. Received September 1978 Revised January 1979
2. r~.finition of entropy This article examines possible relationships between network entropy of commodity flow and variation in unit price of freight transportation service. Entropy of seventy-five maj 3r commodity flows through the railroad network in the Unked States have been calculated using data available in the 1974 waybill sample; unit price of transportation service for various networks has been measured as cents per tonmile. The results of empirical analysis confirm that an inverse relationship exists between variation in unit price and network entropy of commodity flow. Commodities with high network entropy appear to exert forces towards a constant price and those commodities with lower entropy, providing greater marginal utility, appear to have a wide variation in unit price. These findings can be used in regulating pricing decisions in the transportation industry.
The concept of network entropy can be related to the flow of goods through a distribution network, where a certain number of distinct points of demand and points or areas of supply can be identified. These points can be represented by an origin-destination of a network system for flow of goods. The logical network and flow of goods can be represented symbolically. Let ~i
be the flGw of commodity from origin i to destination 1; i -- 1, 2, ..., m, and] = 1, 2, "''9 JV~,
ai
be the total amount of commodity traffic originated at area i, bi be the total amount of commodity traffic terminated at area 1. Then the joint probability that one unit of commodity traffic originated at area i will terminate at area ] is ~ven by
I. Introduction The concept of entropy originated from the basic principle of the second law of thermodynamics. Over the years the idea has been borrowed by other disciplines and has been applied in several problem areas within the physical and social sciences, in particular in the field of information theory and network analysis. A limited application of the concept may be found in several transportation studies, notably those of Wilson [5], and Tomlin and Tomlin [3]. A good review of the state of the art may be found in [ 1]. However, much of the application in transportation so far has been limited to urban travel demand. The current research explores how network entropy can be used to measure utility of transportation service. Since price of transportation is related to the concept of utility, ~r value of service, network entropy is likely to have some effect on the price of such serdce. The current research has been devoted,
m
n
where 0 ~ pq < 1 and ~,i~qPii = 1. The network entropy of commodity flow in the entire ~ystem, then, can be defined as
H= - ~
i
~ Pi/ log pq i
if the system has m origins and n destinations.
3. Entropy and equilibrium flow The network entropy H = - Ei~,/Pii log pq is essentially a measure of uncertainty in the commodity flow pattern created because of disequilibrium in distribution patterns. The entropy of a network flow system
© North-Holland Publishing Company European Journal of Operational Research 3 (1979)413-416. 413
414
S.M. Nazem/Entropy of freight transportation and price variation
is maximum under an equilibrium distribution pattern. The maximum value of entropy of a network system can be established by maximizing the entropy function -~
. ~ P , logPii i
i
where 0 < pq < I for all i and L From the above relationsldp and using the general probability condition ~ i ~ p # - 1, it can be proved that under equilibrium condition the value of PO is ltmn and furthermore, the maximum value of entropy is attained only at equilibrium. Thus 1
('Oil)equilibrium
- mn
(/])maximum =
log m n
for all i and 1, and
The result demonstrates the equilibrium distribution pattern that is the flow of goods is distributed every through the network.
4. Pricing in transportation and entropy Transportation is a service, not a commodity, and therefore as such it has no value. Demand for transportation is created by demand for a commodity and thus the value of transportation is reflected as an increase in the price of a commodity between its origin and destination. A shipper is interested in transporting goods only if he can pass on the cost of transportation to the consumer as an increased price for increased consumer utility. Conceptually, if a commodity enjoys widespread availability then utility created by transportation is limited and therefore the value of transportation service is also limited. Commodities with dispersed sources, however, are likely to create much greater utility or value by being tra~lsported to various consumer centers. Therefore, i~ is plausible that one should expect some relatio~ship between the flow of goods and utility or value of service which should reflect dxe price of transportation service. The value of service for transportation is further influenced by the price of a commodity since a lughly valued commodity can easily absorb a substantial increase in transportation price without any noticeable increase in price for consumers, whereas for a commodity of low value a sim~ar increase could have a rather discouraging influence on consume~ osage. Thus a variation in transportation price may be expected
from two counts, firstly because of variation in utility derived from the service and secondly due to the relative price of different commodities. Comm, :lities with widespread availability can approach equilibrium supply and demand flow, forcing the variation on price of transportation service to a lower level, theoretically variation approaches zero with distribution reaching equilibrium state. The reverse situation should generally prevail if commodities are only available from limited sc~,rces. Although regulatory measures have influenced price in the transportation of goods, the pricing strategy is still directly or indirectly related to the notion of value of service or utility. It can therefore be postulated that marginal utility of a commodity with low network entropy is relatively high, whereas commodities with higher network entropy indicate a relatively Iower marginal utility. A commodity with higher marginal utility will generally allow some flexibility in pricing transportation service because shippers can pass on to the customers a substantial increase in price. This in turn will provide a range of prices depending on location, operational efficiencies of transportation companies, bargaining power of shippers and transportation companies. Therefore, the rationale can be extended to relate entropy, utility and price varia~.ion.Commodities with lower entropy are likely to have greater marginal utility which would be reflected in greater variation in price. Commodities which have reached a near equilibrium distribution pattern should thus indicate a low coefficient of variation showing only smaU variations in the range of prices for transportation service. Empirical study in this article is essentially exploratory to f'md evidence with respect to the above postulations about the possible relationship between network entropy of commodity flow and variation in price of freight transportation service. 5. Empirical analysis and results An empirical examination has been carried out using freight transportation data published in the 'Carload Waybill Statistics', published by the Department of Transportation [4]. Data is based on waybill samples of carloads of t~affic hauled by the railroad industry during the year 1974. Sample data has 0een classified and summarized under five regions namely official, southern, western, southwestern and mountain pacific.
S.M. Nazem/Entropy off'eight transportation and price variation
Analysis in this article includes ton-miles of traffic flow and unit price of transportation service, which is measured as price in cents per ton-mile. Entropy of flow of goods has been calculated using inter-regional flow of ton-miles of goods and price variation has been measured as coefficient of variation of inter-regional ulait price. All seventy-five commodities included in this analysis are based on five-digit standard transportation commodity code classifications, which make each commodity group sufficiently homogeneous. Values of coefficient of variation and network entropy for each of the seventy-five commodities have been estimated from the sample data. A summary of these measures has been included in Tables 1 and 2. Two variables have been regressed to test the hypothesis that the entropy of commodity flow influences the variation in price inversily. The regression results were found to be in the form Variation in unit price =
415
Table 2 Summary of variation in price of transportation service Interval in percentage variation
Number of commodities
10 20 30 40
20 30 40 50
18 32 16 7
50 < 60
2
< < < <
75
variation is thus found to be -0.4')8. The test statistic t is -4.907 with 73 degrees of freedom which provides strong evidence in support of the previous results. These results clearly indicate the existence of an inverse relationsl~ip between variation in price of transportation service and network entropy of commodity flOW.
= 48.668 - 10.632 (Entropy of commodity flow) (9.628*) (-3.965") (*Student-t statistics). While the above regression coefficient is statistically significant, a secondary analysis using rank correlation method has been applied to re-examine the relationship for supplementary evidence. Both measures, entropy and variation, have been ranked in ascending order starting g ith the lowest value. The lowest and the highest values thus receiving the rank of unity and seventy-five res;~ectively. The rank correlation between the two ranked variables has been calculated and tested for statistical significance. The rank correlation between entropy and price
Table 1 Summary of entropy of seventy-five commodities Intervals
Number of commodities
less than 0.5 0.5 < !.0 1.0 < 1.5 1.5 < 2.9
1 5 10 29
2.0 < 2.5 2.5 < 3.0
27 3 75
6. Conclusion it is postulated in this article that entropy of flow of goods is likely to influence the price of transportation, since the pattern of flow of goods can be used as a measure of utility derived from transporting goods. Price of transportation service is directly related to utility or value of service. The results confirm that the commodities with higher entropy appear to have a lower variation in price. This is primarily because commodities with higher entropy have lower marginal utility and thus provide very little opportunity for increasing price, resulting in little variation in price. The analysis is based on five-digit standard transportation commodity code classifications which include several commodities in each category. This is likely to contribute some variation in commodity mix but as the data u~ed in this ar,tcle are based on a substantiaUy large sample vf seventy-five commodity groups, this eliminates at least some of the errors contributed by heterogeneity of commodity mix. Overall analysis does indicate statistical significance of the results, confirming the hypothesis presented in this article that network entropy of commodity flow influences the price of transportation service. In terms of policy implications, the above find2ngs
416
S.M. lgazem / Entropy of freight transportation and price variation
lead to a new factor which may be considered with respect to pricing philosophy. For example, in the case of commodities with high entropy or a wide distribution pattern with many origins and distribution points, regulatory price control activities may be somewhat relaxed since market force Js likely to stabilize the price by narrowing differelttials. On the contrary, special attention ~.lay be necessary for certain commodities with lower entropy which potentially may form a transportation monopoly. T I ~ is particularly important in transportation since much of the pricing policy in this industry is directly or indirectly influenced by regulatory agencies. While the current analyses are only of an exploratory nature, the results indicate entropy of commodity flow is related to utility and thus to the 'value of service" concept which
influences pricing policy. Therefore, entropy of flow of goods can play an important role in the making of these policy decisions.
References [ 1] R.B. Potts and R.M. Oliver, Flows in Transportation Networks (Academic Press, New York, 1972). |2] R.E. Sonntag and VanWyler, Fundamental of Statistical Thermodynamics (John Wiley, New York, 1966) Chapter ~V. [31 J.A. Tomlin and S.G. Tomlin, Traffic distribution and entropy, Nature 220 (1968) 974-978. [4] U.S. Del-artment of Transportation, 1974 carload waybill statistics (1975). [51 A.G. Wilson,The use of the concept of entropy in system modeling, Operations Res. Quart. 21 (1970) 247-265.