A COMPARATIVE EVALUATION OF THE BOSE-EINSTEIN ENTROPY,
SPATIAL EQUILIBRIUM AND THE LINEAR PROGRAMMING TRANSPORTATION MODELS
Cindy Hsiao-Ping Peng
Lecture of Department of Applied Economics, Yu Da College of Business.
No.168, Hsueh-fu Rd, Tanwen Village, Chaochiao Township, Miaoli County,
361 Taiwan, R. O. C.
Tel: +886-37-651188 Ext: 5151
Fax: +886-37-651225
Email: cindy@ydu.edu.tw
Shih-shen Chen
Assistant professor of Department of International Business and Trade,
Shu-Te University
59, Hun Shan Rd., Hun Shan Village, Yen Chau, Kaohsiung County, Taiwan
82445 R.O.C.
TEL: 886-7-6158000 Ext:3309
FAX: 886-7-6158000 Ext:3399
Email:chenss@mail.stu.edu.tw
Pao- Yuan Chen
Associate professor of Department of managerial economics,
Nanhua University
No.32,Chung Keng Li,Dalin, Chia-Yi ,62248,Taiwan R.O.C.
Tel:886-5-2721001 Ext:2530
Fax:886-5-2427194
Email: paoyuan@mail.nhu.edu.tw
A COMPARATIVE EVALUATION OF THE BOSE- EINSTEIN ENTROPY,
SPATIAL EQUILIBRIUM AND THE LINEAR PROGRAMMING TRANSPORTATION MODELS
LITERATURE REVIEW
Laws on the motion of masses date back to the works of Galileo and Newton
in the beginning of the seventeenth century. The latter have been
subsequently refined by several first rank physicists (e.g., Maxwell and
Einstein). On the other hand, shipments of commodities over economic space or
spatial allocation models in general have not advanced much beyond either the
classical linear programming formulations proposed by Hitchcock (1941),
Kantorovich (1942) and Koopmans (1949), or its market-oriented versions (the
spatial equilibrium models) emanating from the works by Enke (1951), Samuelson
(1952), and Takayama and Judge (1964, 1971). The objective of the HitchcockKantorovich - Koopmans formulation is to minimize total energy (transportation
and other production costs) in trans locating masses (interregional commodity
shipments). Similarly, the objective of the spatial equilibrium models (an
analogue of Kirchhoff's law of electric circuits) is to maximize the net
social payoff (Samuelson, 1952) while subject to the same constraints as that
in its linear programming counterpart.
There has been a large body of literature that improves or extends the
original Takayama-Judge model, including: reformulation and a new algorithm by
Liew and Shim (1978), Nagurney (1986); inclusion of income by Thore
(1982); transhipment and location selection problem by Tobin and Friesz
(1983, 1984); sensitivity analyses by Yang and Labys (1981, 1982), Chao and Friesz
(1984), Daffermos and Nagurney (1984), and Tobin (1984, 1987); computational
comparison by Meister, Chen and Heady (1978), and Nagurney
1
(1987a); iterative methods by Pang and Chan (1982), Irwin and Yang (1982);a linear
complementarity formulation by Takayama and Uri (1983); sensitivity analysis
of complementarity problems by Tobin (1984) and Yang and Labys
(1985); applications of the linear complementarity model by Kennedy (1974), Sohl
(1984), Khatri-Chhetri, Hite and Nyankori (1988), Takayama and Hashimoto (1989),
and Uri (1989); a solution condition by Smith (1984); the spatial equilibrium
problem with flow dependent demand and supply by Smith and Friesz (1985);
nonlinear complementarity models by Friesz, etc. (1983), and Irwin and Yang
(1983); variational inequalities by Pang and Chan (1982), Daffermos (1983),
Harker (1984), Tobin (1986) and Nagurney (1987b); a path dependent spatial
equilibrium model by Harker (1986); and dispersed spatial equilibrium by Harker
(1988). For the detailed description of the advances in the spatial equilibrium
models, readers are referred to Labys and Yang (1991).
However, both linear programming and spatial equilibrium models being
built on the foundation of relatively low total energy levels, are known to
exhibit certain regularities. For instance, there exist very limited numbers
of positive commodity flows (Silberberg, 1971; Gass, 1985). As a result, an
important phenomenon of cross-hauling is precluded from the models. In addition, all optimum commodity flows must obey some symmetrical or reciprocity
condition (Silberberg, 1971; Yang, 1989). Despite these limitations, applications of the linear programming transportation and spatial equilibrium
models have proliferated especially in agricultural and energy markets in
which transportation costs constitute a significant portion of the market
demand price. For instance, one of the earliest applications of the linear
programming model was on the U.s. coal market (Henderson, 1958) and most
applications of the spatial equilibrium models may be found in agricultural
2
and energy market (see Labys, 1989; and Thompson, 1989 for literature review
of such applications).
Another branch of important spatial interaction models is based on
Newton's gravity law. The problem of applying various gravity models to
interregional commodity shipments lies in the choice of an appropriate
functional form and consistency of the estimates (Hwa and Porell, 1979).
While Newton's model may be appropriate for essentially a large-scale
structure, it is not suitable for modeling a small-scale problem from a
physical point of view. Perhaps, this is a reason that one of the successful
applications of the gravity models may be found in the multi-commodity inputoutput formulation by Leontief and Strout (1963)1. Like that in physics,
Newton's model fails in the world of (i) extremely small particles
(insignificant commodity flows) in which the quantum mechanics prevails and
(ii) fast and highly massive objects (extremely strong markets) dealt with by
relativity. It is interesting to note that the large-scale Leontief-Strout
was published one year before Takayama and Judge reformulated the EnkeSamuelson problem into a standard quadratic programming or spatial equilibrium
model. The entropy modeling had not received enough attention until 1970 when
Wilson derived the gravity model from the entropy-maximizing paradigm. By the
middle of the 1970's Evans (1973), and Wilson and Senior (1974) proved the
relationship between the linear programming and the entropy-maximizing
models2. As a matter of fact, Hitchcock-Kantorovich-Koopmans linear
programming transportation problem was shown to be a special case of the
entropy model. The detailed descriptions on these models may be found in
1 Other contributions include Isard and Bramhall (1966), Theil (1967) and Boyce
and Hewings (1980).
2 An alternative formulation was established by Erlander (1977, 1980)
3
Batten (1983), and Batten and Boyce (1986). However, the implementation of
such entropy models to the interregional commodity shipment problem has been
limited despite the recent result by Yang (1990).
The purpose of this paper is to implement the Bose-Einstein entropy model
to the Appalachian steam coal market in which coal flows are viewed identical.
It is interesting to explore that if the coal shipments of the Appalachian
market are more closely related to the law of electric circuits or particles
of quantum mechanics. Such a comparison, to the best of our knowledge, has
never been implemented in the context of interregional spatial commodity
modeling. In the next section, we formulate the model. We then compute the
coal flows from the Bose-Einstein entropy model, and compare them with that of
the Hitchcock-Kantorovich-Koopmans transportation problem, and the EnkeSamuelson-Takayama-Judge spatial equilibrium model.
THE BOSE-EINSTEIN ENTROPY FORMULATION OF THE INTERREGIONAL COMMODITY
MODELING
According to the second law of thermodynamics, in a universe ruled by
entropy, there is a tendency at all times to move inexorably toward greater
and greater disorder (see Georgescu-Roegen, 1971 and Nikjamp and Paelinck,
1974).3 Batten (1983) showed that movements of distinguishable particles
(interregional commodity flows) can be used to formulate the Maxwell-Boltzmann
entropy model while movements of independent and identical particles may be
We do not consider the phenomenon of self-organization in the dissipative
structure where a hexagonal structure arises from heat diffusion and
viscous forces if the temperature difference is maintained in the so-called
Raleigh-Benard hydrodynamic instability. The theory of dissipative
structure does not invalidate the second law of thermodynamics; but rather,
it illustrates a recently developed forfrom-equilibrium process of a
synchronized spatial pattern en route to equilibrium. See Glansdorff and
prigogine (1971) and Coveney and Highfield (1990) for detail.
3
4
employed to derive the Bose-Einstein entropy model. The commodity units used
in shipments may be considered "homogeneous" especially in the classical
competitive market. Within this context, the Bose-Einstein entropy
formulation would be more appropriate for agricultural or energy commodity
modelings. The objective of the Bose-Einstein model is to maximize the number
of different arrangements (entropy) in assigning Xij (number of units of
commodity shipments from supply region i to demand region j) to hj numbers of
depots in demand region j where i€I, j€J, ij€ IXJ and I, J are finite sets of
positive integers; and IXJ is a cartesian product of I and J. The problem is
equivalent to assigning x number of identical pigeons to y number of holes (or
the well-known pigeonhole theorem). The answer is (x+y-l) Cy-1 (see Batten,
1983; Lewis and Papadimitriou, 1981). Hence, the number of such arrangements
in assigning Xij units of hj numbers of consumption
depots in region j is (Xij + hj -1) Chj –l
Wij
or
[1]
=
Using Stirling's approximation4, equation [1] can be shown as
Wij
=
=
[2]
X  0.5
4 To facilitate computations, we employ X ij ! X ij ij
2 e
 X ij
for large Xij
5
Taking product over i and j, we have
X

W   Wij 
iI
jJ
iI
jJ

iI
jJ
h
 h j  1
X ij  h j  0.5
ij
[3]
 1
h j  0.5
j
X ijX IJ 0.5
2
Since the logarithmic transformation of [3] preserves the properties of
the maximization problem and for mathematical convenience, we take the natural
logarithmic function of W to have
log W    ( X ij  h j  0.5) log X ij  h j  1    h j  0.5log h j  1
iI jJ
iI jJ
   X ij  0.5log X ij    log 2
iI jJ
iI jJ
[4]
The Bose- Einstein entropy model may then be formulated as
Maximize
log W
[5]
Xij
Subject to
X
ij
 Y j for all
X
ij
 X j for
iI
jJ
all
jJ
iI
  t ij X ij  C
Where
[6]
[7]
iI jJ
[8]
X ij  0
[9]
Yj = consumption level in region j
Xi = production level region i
6
tij
= unit transportation cost from supply region i to
demand region j
C
= total transportation cost analogous to total energy
Level imposed in a closed system
Since the number of depots is fixed in each demand region and the constant
drops out in the maximization process, equation [4] may be reduced to
log W    X ij  h j  0.5log X ij  h j  1    X ij  0.5log X ij
iI jJ
iI jJ
[10]
It is evident from equation [lO] that in the limiting case in which there
exists only one depot in each consumption region, or hj = 1, the right hand
side of equation [lO] is zero or W = 1. On the other hand, the larger the
value of hj, the greater the number of different arrangements (potential
interregional commodity flows). Equations [6], [7], [8], [9], and [10]
constitute the Bose-Einstein Entropy-Maximizing Model that generates a
most probable state of the interregional commodity flows given the constraints
on demand, supply and total transportation cost.
COMPARATIVE EVALUATIONS OF THE BOSE-EINSTEIN ENTROPY,
LINEAR PROGRAMMING AND SPATIAL EQUILIBRIUM MODELS
In order to facilitate the comparison among three models, seven demand and
supply regions, as well as consumption depots in the Appalachian steam
coal market, are reported in Table 1. Forty-nine unit transportation costs
and other demand and supply estimates are shown in Tables 2 and 3, respectively. These data are obtained from Labys and Yang (1980). Since the linear
programming model is the building block upon which other allocation models
rest, we begin the descriptions based on the LP model. Given the available
data, the optimum solution to the Hitchcock-Kantorovich-Koopmans model may
7
be derived from solving the standard linear programming transportation problem:
Minimize
Xij
TC    t ij X ij
[10]
iI jJ
subject to equations [6], [7], and [9].
Similarly, the objective of the spatial equilibrium model with a set of
linear demand and supply functions is to maximize the social net payoff (or
NSP).
Maximize
NSPY j , X i , X ij    a j Yi  1  b j Y j2   ci X i  1  d i X i2    t ij X ij
2 jJ
2 iI
jJ
iI
iI jJ
subject to equations
[6 ],
[7] ,
[11]
and [9].
where aj = estimated intercept of the demand function in region j
bj = estimated slope of the demand function in region j
ci
i=
estimated intercept of the supply function in region i
di = estimated slope of the supply function in region i
Note that if the demand requirement equals optimum consumption level, and
the capacity constraint equals production level for each region, the linear
programming and spatial equilibrium model can be made identical. The optimum
solutions of the Hitchcock-Kantorovich-Koopmanns linear programming, EnkeSamuelson-Takayama-Judge spatial equilibrium, and Bose-Einstein entropy maximizing models are reported in Table 4. The spatial equilibrium solution is
reported from Labys and Yang (1980). All three models employed identical
transportation costs, and the linear programming and entropy models have
identical restrictions on consumption and production levels. However, the
optimum consumption and production levels are endogenous to the spatial
8
equilibrium model, and hence, may well be different from actually observed
levels. For this reason, we restrict the analysis to interregional commodity
flow since they are endogenous to all three models using identical
transportation data. The total transportation cost
  t ij X ij  C
iI jJ
in the Bose-Einstein entropy model is set at the total observed transportation
cost in such a way that the most probable set of interregional commodity flows
may be derived for comparison purposes5.
A perusal of Table 4 indicates that the Bose-Einstein entropy model
generates 14 positive steam coal flows in the Appalachian market while the
spatial equilibrium and Hitchcock-Kantorovich-Koopmans models produce 13 and
12 positive steam coal flows. The Bose-Einstein entropy model correctly
describes the existence of 13 out of 22 actually observed coal shipments in
the market whereas the spatial equilibrium and linear programming models
correctly predict 10 and 8 coal shipments, respectively. Therefore, the
entropy model clearly out-performs the other two models in terms of the number
of positive coal shipments. Further, of the ten actually observed major steam
spatial equilibrium models predict 8 major flows while the linear programming
model predicts 7 flows. That is, the Bose-Einstein model predicts all major
shipments but X23 (from Ohio to Indiana and Michigan) and X57 (from Virginia
to South Atlantic); the spatial equilibrium model predicts all major coal
shipments except X35 (from Northern West Virginia to Ohio Valley) and X66 (from
East Kentucky and Tennessee to South Central); and the Hitchcock-Kantorovich-
The LINDO software package (R. Schrange, 1989) was used to obtain the
optimum solution of the linear programming model; a quadratic programming
package by Cutler and Pass (1971) was employed to solve for the optimum
solution to the spatial equilibrium model; and the GINO (J. Liebman, L.
Lasdon, L. Schrage and A. Warren, 1988) program was used to derive the
Bose-Einstein solution.
5
9
Koopmans linear programming model correctly predicts all major shipments
except X23, X35, and X66• As a consequence, the linear programming is
relatively less satisfactory in predicting the existence of major coal
shipments.
In order to compare the magnitudes of interregional commodity flows under
the three models with that actually observed, we employ the following
performance index:

I   
X ij  X ij
iI jJ
X ij

Where X ij is the predicted coal shipment from supply region i to demand region

j and Xij is the observed value of the corresponding X ij . Of the 10 major
coal shipments, the performance index for the Bose-Einstein entropy, EnkeSamuelson-Takayama-Judge spatial equilibrium, and Hitchcock-KantorovichKoopmans linear programming models are 38.97%, 34.34% and 58.34% respectively.
In other words, these three models explain 61.03%, 65.66% and 41.66% of the
volumes of ten major coal shipments. While the spatial equilibrium and
entropy models are comparable in terms of explaining the volumes of major coal
shipments, the performance of the linear programming transportation model is
clearly less satisfactory in this respect. This comes as no surprise since in
a market-oriented environment, the linear programming model in its simplest
form cannot adequately explain the steam coal shipments by a fixed vector of
unit transportation costs, demand requirements, and capacity constraints.
However, with additional information included in the model such as sulfur
content and pollution controls, the linear programming model may better
explain the coal shipments in the market.
Although the entropy and spatial equilibrium models are comparable in
10
terms of performances for explaining commodity shipments, they are genuinely
different models. The performance of the spatial equilibrium model is
contingent to a large extent on the statistical accuracy in estimating
regional demand and supply functions for a given set of transportation costs.
The random properties are generated by the estimated regional equations. On
the other hand, the foundation of the Bose-Einstein model, much like that
found in Brownian motion, rests on the randomness of particle movements fora
given level of total energy in a closed system. The properties of randomness,
unlike the spatial equilibrium model, are due to the assumption of the
statistics mechanics6. The random nature of interregional commodity
shipments in the Bose-Einstein entropy model gives rise to the most probable
set of commodity flows as "optimum" solutions given the constraints on demand,
supply and total observed transportation cost. The unavoidable property of
randomness of interregional commodity shipments in the entropy model gives a
more general theoretical foundation than the other two models which are
basically built on the criterion of minimum total energy or maximum
efficiency. Quantum mechanics is known to embody the electric circuit problem
and other physical phenomena. However, the gravity model, quantum mechanics
and relativity are not compatible (see Hawking, 1988). Consequently, the
solution of the Bose-Einstein model does not need to obey reciprocity
conditions and hence, it allows the phenomenon of cross-hauling. However,
policy implications of the entropy model are not as readily available as the
spatial equilibrium model. For instance, a pollution tax on coal produced in
the Appalachian production region could be detected and traced out through the
6 Einstein did not believe that the universe was governed by chance;
however, he won the Nobel prize for his contribution to the quantum
theory (S. Hawkings, 1988).
11
markets in terms of changes in delivered, mouth of mine prices, consumptions,
productions, economic surpluses and all coal shipments in a non-degenerated
model (Silberberg, 1970; Yang, 1983). The same type of tax imposed on the
entropy model may be used to show that the entropy in a regulated system with
a tax is less than that of the unregulated system (Yang, 1990). As a result,
the policy implications of the entropy model is rather limited in the context
of traditional economic analysis.
CONCLUDING REMARKS
Advances in theoretical physics such as Newton's gravity, Kirchhoff's
laws of electric circuits, and particle movements in quantum mechanics have
contributed significantly to modeling interregional commodity shipments. The
foundation of these spatial models is the Hitchcock-Kantorovich-Koopmans
linear programming transportation problem upon which other spatial allocation
models are built. The linear programming model in its simplest form has some
properties due to the efficiency criterion (e.g., minimizing total costs).
Consequently, its solution must exhibit certain characteristics. First, there
are only a limited number of positive commodity shipments allowed in the
solution set in order to minimize transportation and production related costs.
Second, the phenomenon of cross-hauling is precluded from the linear
programming transportation model since more spatial diffusion (commodity
flows) would not be cost efficient. Third, the reciprocity condition,
X
ij
t k 1

X
k1
t ij
is typically observed. Similarly, the Samuelson-
Takayama-Judge spatial equilibrium model also shares these solution
properties. Nonetheless, the spatial equilibrium model may better explain the
market due to the improvement made from statistically estimated demand and
supply functions. On the other hand, the Bose-Einstein entropy model does not
12
possess these solution properties because it seeks the most probable
interregional commodity shipments based on the assumption of maximizing the
entropy. These two types of models are genuinely different, and hence a choice
must be made in order to implement the models for a given market.
With a given set of transportation costs, demand requirements and
capacity constraints of the Appalachian steam coal market, the solution of the
Bose-Einstein entropy model indicates that it better explains the spatial
diffusion (the number of commodity shipments) as compared to the other two
models. In predicting the volumes of major coal shipments, the spatial
equilibrium model is comparable to that of the entropy model, but it out
performs the linear programming model. Moreover, the policy implications may
be easily modeled in the spatial equilibrium model. Conversely, such policy
implications are limited in the entropy model. The observational evidence in
our study suggest that the entropy or spatial equilibrium models may be
preferred to the linear programming transportation model for the Appalachian
steam coal where transportation costs compose a significant portion of the
delivered price. Finally, the performance of the linear programming model may
be greatly enhanced with additional information included since it has the
greatest computational advantage of the three models investigated
13
TABLE 1
Supply Region
Base Point
l.
Pennsylvania, Maryland
Pittsburgh
2.
Ohio
Cadiz
3.
Northern West Virginia
Morgantown
4.
Southern West Virginia
Beckley
5.
Virginia
Bristol
6.
East Kentucky,
7.
Alabama
Tennessee
Pikeville
Birmingham
Demand Region
Consumption Depots
1.Connecticut, Maine, Massachusetts, Boston, Hartford, Providence, Portland
New Hampshire, Rhode Island, Vermont
2.New Jersey, New York, Pennsylvania, New York, Newark, Philadelphia
Washington, D.C., Maryland, Delaware
3.Indiana, Michigan
Detroit, Indianapolis, Fort Wayne
4.Illinois, Wisconsin, Minnesota
Chicago, Milwalkee, Minneapolis
5.West Virginia, Ohio, Kentucky
Cincinnati,
Cleveland,
Columbus,
Lexington, Charleston
Birmingham, Nashville, Jacksonville
6.Tennessee, Alabama, Mississippi
7.Virginia, North Carolina,
Carolina, Georgia, Florida
South Columbia, Raleigh, Charlotte, Miami
14
TABLE 2
UNIT TRANSPORATAION COST
(Cents per 10 6 BTU)
From
1
2
3
4
5
6
7
19.1
14.5
13.2
18.8
15.1
21.1
21.0
22.2
18.3
13.8
19.8
13.3
22.5
24.0
20.2
13.5
11.8
19.6
13.7
21.8
20.9
20.5
16.9
16.6
22.5
12.6
19.2
16.5
21.2
17.6
20.3
21.8
17.2
14.1
14.7
24.3
20.4
16.7
17.1
13.4
16.9
13.6
26.8
23.3
19.8
18.1
15.7
7.0
14.3
To
1
2
3
4
5
6
7
Source: See Mutchler, etc. and Labys and Yang (1980).
TABLE 3
ESTIMAES ON REGIONAL DEMAND AND SUPPLY
Demand
Demand
Supply
Supply
Actual
Actual
Intercepts = (69.8, 86.4,57.5,49.7,61.3,47.4,61.6)
Slopes = (-462.73, - 33.03,-20.5, -40.08, -12.71, -17.79, -22.13)
Intercepts = (27,26.3, 25, 30.4,23,27,28.2)
Slopes = (4.46,4.221,14.259,25.07,24.35,2.6028,59.62)
Production levels* = (1.2,0.93,0.75,0.05,0.24,1.73,0.32)
Consumption level =(0.03,1.29,0.66,0.03,1.72,0.52,0.97)
*Quantities are measured in 1015 BTU, see Labys and Yang (1980).
15
OPTIMUM SOLUTION (in 10
From
To
New England
Mid-Atlantic
PA
MD
(0.03)
«1.025»
1.03
(0.895)
[1. 2]
IN-MI
0.07
(0.304)
North Central
OH
«0.304»
0.240
0.01
Ohio Valley
0.09
«0.708»
0.690
(0.93)
[0.93]
15
TABLE 4
BTU) OF THE APPALACHIAN STEAM COAL MARKET
Northern
Southern
WV
WV
0.03
(0.03)
«0.195»
0.21
(0.155)
[0.09]
«0.336»
0.12
(0.355)
[0.66]
0.01
0.02
VA
[0.03]
«0.042) )
0.03
0.24
(0.2)
[0.2]
«0.138»
0.21
[0.01]
Total Supply
1.200
0.93
« » = Spatial Equilibrium Solution
[]
= Linear Programming Solution
0.75
( ) =
0.05
0.66
0.01
[0.03]
«0.053»
(0.03)
«0.628»
0.56
(0.531)
[0.74]
«0.081) )
South Atlantic
0.24
Bose-Einstein Solution
Those without parenthesis are actual flows.
Total
Demand
0.03
1. 29
0.2
(0.05)
[0.05]
South Central
AL
0.03
(0.239)
«0.035»
0.38
(0.21)
East KY
TN
«0.653»
0.72
(0.97)
[0.96]
1. 73
0.03
1.72
«0.138»
0.28
(0.32)
[0.32]
0.52
0.04
0.97
0.32
5.22
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