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A regional railroad network optimization model for coal transportation
Tronspn
Res:B
Prmted
an Great
Vol
15B, No. 4. pp. 227-238.
0191.2615/81/040227-12802.00/0 0 1981 Pergamon Press Ltd.
1981
Bntam
A REGIONAL
RAILROAD
MODEL
NETWORK
FOR COAL TRANSPORTATION CHIA-JUCH
Department
OPTIMIZATION
CHANG
of Civil Engineering, Marquette WI 53233, U.S.A.
University,
Milwaukee,
and ROBERT D. MILES School
(Received
and KUMARES
C.
SINHA
of Civil Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. 11 January
1980; in revisedform
25
April
1980)
Abstract-A regional railroad network model is presented to evaluate the system’s response to increased coal traffic. An optimal, multimodal, coal-shipping pattern is developed for the study region to minimize total costs and to efficiently use the existing network. A two-stage, general mode1 allocates resources among demands and then assigns flows to the network according to efficiency criteria. The model is sufficiently general to permit modification for specific needs, assumptions and data. Government agencies and industries can apply the model in resource allocation decisions and transportation policy analysis. INTRODUCTION
Due to the dwindling supplies of petroleum and natural gas, there is much discussion and a growing emphasis on the utilization of coal, of which the United States has vast domestic reserves. The conversion to a coal-based energy system is to become more common in the future. The Carter Administration is committed to the implementation of a National Energy Plan which has outlined regulatory and tax measures proposed to encourage industries to convert to coal due to petroleum and natural gas shortages. It is believed that the coal industry is able to increase production in order to meet these future needs. There are constraints, however, that control the speed at which conversion to coal takes place. The transportation of coal is a primary constraint. There are three types of problems that are most likely to develop: (i) certain coal suppliers (those who produce coal advantageous to most consumers) may not be able to meet the great amount of simultaneous demand for their products; (ii) the radical changes in planned coal production and consumption patterns and quantities may lead to a mismatch with the transportation network patterns presently available to carry coal from producing mines to consumption points; and (iii) the regional traffic may be seriously affected by the greatly increased coal movements. An example is an out-of-balance transportation network due to a non-uniform increase in coal traffic causing an overutilization of a part of the system while other parts are underutilized. Studies indicate that based on the projected demand for transporting coal and other commodities the existing transportation system may not be able to handle the demand over the most direct, low-cost route and by the most cost-effective mode (see, for example, Rifas and White, 1976). In order to achieve the maximum overall benefit for the supply-demand system and the maximum efficiency in the transportation system it is imperative that an effort be made to utilize the existing facilities in an optimal way. The primary purpose of this research was to develop a system optimization model which would serve the following two purposes: (1) to predict the future demand for coal movements on the existing rail system of a region in order to evaluate the rail system’s ability to move the required tonnage; and (2) to develop an optimal, multimodal coal shipping pattern for the region which would minimize the cost and also efficiently utilize the existing facilities. The developed model contains two submodels-a coal allocation submodel and a 227
228
CHIA-JUCH CHANG, ROBERTD. MILES and KUMARESC. SINHA
railroad network submodel. The coal allocation submodel is a linear programming model and is used to determine the alternative distribution of coal from producers to consumers by available transportation modes and routes. The railroad network submodel is a traffic assignment model which assigns the rail-coal traffic as well as other non-coal freight traffic onto the railroad network; it is programmed to produce informagion such as link loadings, routings, and capacity analysis, etc. These two submodels follow certain iterative procedures and jointly operate as a system optimization model. DESCRIPTION
OF
THE
SYSTEM
OPTIMIZATION
MODEL
Coal allocation submodel Objective Function: The main objective for individual utilities and industries in purchasing coal from local and/or external sources is to minimize their individual costs. The objective of a system optimization analysis is, however, to plan an optimal system which minimizes the total system costs subject to a number of system constraints. The total system cost includes the mine-mouth cost, the transportation cost, and for high-sulfur coal, the coal-cleaning cost. The mine-mouth cost and the coal-cleaning cost depend upon the type of coal chosen. The transportation cost is determined in terms of the mode and route used for shipping coal. The transportation cost refers to the total cost incurred in the entire course of shipment, which includes the feeder transportation cost, the main-line transportation cost, the loading and unloading cost, and the delivery cost. The objective is to minimize the sum of the required cost and is formulated as such. However, since there are two options available for coal-users in choosing coal, i.e. to use low-sulfur compliance coal or to use scrubbers to clean high-sulfur coal before burning, the objective for the entire system minimizes the sum of the respective costs from the two different decisions. The system objective function is formulated as follows:
Minimize
Z = T Ei c c 2 XY j meM,, BEG,,
+ 4 mL,, Fi $,z,jxij -( I
mg)
where i is the coal supply zone, j is the coal demand zone, m is the coal transportation mode, e.g. rail, water, mixed-mode, truck, etc., g is the gateway to the study region, e.g. transportation centers, river ports, etc., (gateways are not used for intraregional coal supply), X7 is the quantity of coal (in tons) shipped from supply zone i to demand zone j by mode m via gateway g, Mij is the set of available transportation modes for shipping coal from zone i to zone j, Gij is the set of possible gateways from zone i entering zone j, Ei is the mine-mouth cost at mine i, Fy is the feeder transportation cost from mine i to major transportation mode m, TF is the main line transportation cost from mine i to gateway g by mode m, I!,? is the loading and unloading costs for i, j, m, g combination, 04 is the delivery cost from gateway g to destination zone j, Aij is the scrubbing cost of coal mined at zone i for use at zone j, and J is the set of zones which choose scrubbers as the sulfur dioxide emission control strategy. Pollution Constraint: The pollution constraint is based on the Clean Air Act of 1970 and subsequent amendments which specify the regional environmental regulations on restricting the maximum sulfur dioxide emissions in the burning of coal. For example, the state of Indiana requires that the maximum total sulfur dioxide (SO,) emissions be limited to between 1.2 and 6.0 pounds per million Btu depending upon the combustion ratings of power plants (Air Pollution Board of Indiana, 1976). With this constraint, users who attempt to comply with the air pollution regulation are forced to purchase coal only from certain sources where the sulfur content of coal meets the requirement.
A regional
railroad
network
optimization
model for coal transportation
229
However, it is not the delivery of coal which is constrained, but the actual burning of coal. This allows the blending of high- and low-sulfur coal to obtain a fuel which when burned is, on the average, below the allowable level (LeBlanc, 1976). The constraint is formulated as follows:
i
me&f,,
BEG,,
where Si is the sulfur content (per cent by weight) of coal mined at i, Rj is the maximum allowable sulfur content of coal, which is converted from the maximum SO2 emission limit for each j C+J, and i, j, m, g, Mij, Gij, J and X7 are defined earlier. The above constraint assumes that the blending facility is available for zone j, and that it is economical to have coal shipped from various sources. However, there is a possibility that for some areas it is not economical to do so. In this case, consumers secure their needs from a single source where the sulfur content of coal does not exceed the maximum allowable level. The constraint then becomes that in order to ship coal from zone i to zone j the sulfur content of coal mined at zone i should be less than or equal to the limit of consumer j, otherwise no shipment will be made from i to j. This can be represented as: if Si > Rj, then dij = 0 if Si < Rj, then dij = 1 where dij is the decision variable of the shipment from zone i to zone j. This is a conditional choice problem and is formulated with a set of mixed integer constraints : Mdij + Si - Rj 2 0
if
Si > Rj,
dij = 0
if
Si I Rj,
dij = 1
M(l - dij) + Rj - Si 2 0 0 I dij I
1, dij = integer
and X”g - Md..V 5 0 I 1’ ,vg
1’
2
0
i if *
if
d.. = 0
X!Y = 0
dii
_$g
=
1:
2
0
where M is a very large positive number. The original pollution constraint is substituted for the above mixed integer constraint, whenever applicable. However, due to limits of time and computer capacity, the mixed integer constraint is not applicable in large regional problems. Demand Constraint: The demand constraint assures that the energy requirement for each coal-burning unit is fulfilled, i.e. for each demand zone the total amount of energy generated by coal from all selected sources is equal to its particular need. This is represented in terms of heat requirement (Btu) because of the wide variation in heat content of coal mined from different areas.
where Hi is the heat content of coal (in Btu/ton) mined at i, and Nj is the energy requirement at zone j (in Btu). Supply Constraint: Due to the limited supplies in the mining areas, the total quantity of coal shipped from each supply zone by all modes of transportation should not exceed its available capacity. The available capacity is equal to the output capacity less local consumption. Thus : 1 1 C X;g I Pi - Bi for each i me&f{,
geGi,
j
CHIA-JUCH CHANG, ROBERT D.
230
where Pi is the annual production zone i.
MILES
and
KUMARES
C.
SINHA
at zone i, and Bi is the local consumption
at the same
Gateway Capacity Constraint: The coal allocation submodel allocates the zonal coal demand from external sources to the gateways of the study region, while the network submodel routes coal traffic from the gateways to the final destinations. For intraregional coal supplies the traffic is routed directly to the destinations without using gateways. The shipment of coal from external sources is constrained by the gateway capacities. The gateways are defined as the main entrances to the study region; they are usually the railroad main-line interchanges, the intermodal transfer stations, the water ports, and/or the transportation center cities which are located around the boundary of the region. For railroads, the gateway traffic is constrained by the line capacities, while for waterways the loading and unloading capacities of coal docks are used rather than the navigating capacities. The gateway capacity constraints are delineated as:
1
Xtg I Cr for each g and m
(LA= K,,
where K,, is the set of i, j combinations using gateway g and mode m, and Cy is the transportation capacity at gateway g for mode m. Non-Negativity Constraint: The linear programming coal allocation submodel requires that all decision variables be non-negative numbers. Hence, X;g 2 0 for all i, j, m and g. DUAL
INTERPRETATION
OF
THE
SUBMODEL
The dual formulation of the Coal Allocation Model provides several meaningful economic interpretations of the model. The dual model is formulated as follows: Maximize
subject to :
for all i, j E J, m, g and HiUj - v - Siyj -
C
C
Zr I (El + Fy + Tyg + L;’ + 0;)
gEGi,+ msMii
for all i, j E J, m, g where F is the shadow price of supply i (the change of total system cost per unit change of supply at i), Uj is the shadow price of demand j, E; is the shadow price of sulfur limit at j, Z: is the shadow price of gateway capacity at g for mode m, a is average sulfur content of coal, and N, P, R, C, E, F, 7: L+ D, S, H, A are as defined in the primal formulation. The duality theorem of linear programming states that the minimum cost for the delivery system is equal to the maximum revenue for the price system (Koopmans, 1951). The objective function of minimizing the total cost in the primal formulation is, therefore, equivalent to the maximization of the net revenue, which is equal to the gross revenue less the total constraint cost as expressed in the equation. The dual constraints also have a very interesting meaning in economic interpretation. The constraints can be rearranged to the following form: HiUj-~-EiI(Fi”+TY’g+LY+D~+Aij)+
Six+
for
j E J,
C msM,, 2 Z: BEG,,
A regional
railroad
network
optimization
model for coal transportation
231
and 1 27 Six + 1 geG,,msM,j
HiUj - r/I - Ej I (Fy + 7’7’ + L!g + 0:) + for
j f+J.
The left hand sides of the above inequalities can be interpreted as the gross unit profit for mine i to supply the demand at j, and the right hand sides are the sum of transportation cost and system constraint cost incurred in shipping coal from i to j. It is economically feasible to ship coal from i to j only if the gross profit for i could cover the total cost, otherwise no shipments should be made. This condition can be expressed as follows: (i) if
HiUj-~-Ei=(F~+T~‘+L~‘+DQ+Aij)+
for
Six+
C C Z: gEGllmeMa,
jg J
or - Ei = (Fr _t Tyg + L!’ + 04) +
Six + !
1 1 Zr gsG,, msM,,
then Xyg 2 0. (ii) if
HiUj
-
v
- Ei < (F~ + Tug + Lo
for
+ Oj”+ Aij) +
Sip +
C C Z,m gEG,,msM,,
jE J
Or HiUj
-
r/;: -
Ei<(F~+T~g+L~g+D~)+
for
Six+
1 C Zy BEG,,mEMa,
j+J
then XFg = 0. lJ This is consistent with the well-known “Complementary programming. RAILROAD
NETWORK
Slackness Theorem” of linear
SUBMODEL
An initial set of transportation alternatives is generated by the coal allocation submodel through an analysis of supply, demand and costs as discussed. These alternatives are produced without considerations of the physical constraint and transportation efficiency of the regional railroad network. The purpose of the railroad network submodel is to select, among the alternatives produced from allocation submodel, the optimal modes and routes suitable for the particular region. The relationships between the two submodels and their inputs are explained by the following simple diagram: Exogenous Variable Endogenous ~ Variable
T-k
Transptrt
Costs YM,,,
Selection of Mode, Routes
T\,
Routes
ailroad Network4/Optimal Submodel
Alternative ./ Allocation
232
CHIA-JUCH CHANG, ROBERT D. MILES and KUMARES C. SINHA
The railroad network submodel used in conjunction with the coal allocation submodel is an algorithm designed to develop and analyze traffic loadings on links of a railroad network. The submodel uses a railroad network data base and a computer program to process and assign the coal traffic interchanges generated from the coal allocation submodel. The basic computer program is a subset of the battery of programs first developed at the University of California for urban transportation planning (Hornburger, 1970), and then modified at Purdue University (Mekemson, 1979). The program is used with appropriate modifications to reflect its application to railroad networks. The input to the railroad network submodel includes the network geometry, the network parameters, and the traffic interchange data. Network Geometry: The network geometry is a description of the interconnections and segments of the network representing the railroad system under consideration. Like a highway network, a railroad network can be described by a series of nodes connected by links. However, because of the special railroad characteristics, the rail traffic assignment is different from the highway assignment process. For example, the rail traffic tends to stay on the same railroad as long as possible. Hence, in coding the network a penalty is considered at junctions to indicate that train movements from one railroad to another at certain junctions either are subjected to delay or else are completely prohibited. Since rail cars often transfer between two carriers at specific interchange points, an appropriate cost is established for those interchanges to allow interline services. Figure 1 depicts two simple examples of the schematic representation of junctions, interchanges, and trackage rights in a railroad network in comparison with those in a highway network. It is observed on Fig. l(a) that the intersection point of railroads A and B, although representing only one junction or interchange point, is coded with two different nodes connected by a dummy link. The distance of the dummy link for junctions without interchange facilities is assigned an assumed large value M to prohibit turning movements between railroads A and B. For approved interchanges an appropriate value is developed from the cost data (Morlok and Waner, 1976) and converted to the equivalent distance. Figure l(b) shows that the trackage rights of railroad A over railroad C on link 2-3 is represented by adding a separate link 7-8 for A with a specified share of capacity. The distance of link 7-8 is adjusted to reflect delay caused by switchings. Figure 2 shows a small network example coded for a hypothetical rail system. This type of network coding is used in the assignment process in the submodel. Network Parameters: The traffic assignment process in the submodel requires network impedance values to allow for the selection of routes through the network under study. These impedance values are either travel time or distance, or some derivative function of these for use in the minimum path calculations. The link capacity data is another parameter. The capacity of a railroad system is extremely difficult to measure; it is a function of a number of elements such as the type of track, the signaling system, the operating speed, as well as the siding characteristics etc (Prokopy and Rubin, 1975). If the detailed information is not available, the theoretical capacity of a railroad link can be estimated on the basis of the number of tracks and the traffic control system (see Hay, 1967, and USDOT, 1974). The capacity can be expressed either by number of trains per day or by tons per a time period. This is used as a check on the assignment process to indicate proper selection of the network and interconnection with load points. Estimated capacities are used to provide a feedback during the assignment process in representing system congestion via a capacity restraint technique. TraJic Interchange Data: The network model requires an origin-destination trip matrix which contains the number of trips from each origin to each destination in the network. The matrix does not contain specific routings, only interchanges of trips throughout the network. The traffic interchange data needed for input to this model includes the coal traffic generated from the coal allocation model as well as the non-coal traffic from survey data. The data is expressed as one-way trips on a zone to zone basis. It is recorded as either number of cars or tons whichever is available. This type of trip
A regional
railroad
network
optimization
model for coal transportation
233
matrix is used for calibration of the network and for loading trips on various networks for plan development purposes. Assignment Process: Traffic assignment techniques are based upon the premise that users of a transportation network wish to optimize some measures of their travel. Generally, the characteristics of travel considered are travel time, distance, and cost. The model utilizes a minimum path algorithm developed by the Road Research Laboratory (RRL) (Whiting and Miller, 1961) to compute a shortest route for each origin-destination pair. After the minimum path trees are found, the model assigns the traffic interchange data to the shortest routes by a capacity restraint procedure.
c
./’ RAILROAD ia)
5ec
TRACKAGE
RIGHTS 6 I
? 4
RAILROAD
HIGHWAY
Fig. 1. Railroad
network
vs Highway
network.
CHIA-JUCH CHANG, ROBERT D. MILES and KUMARES C. SIKHA
234
The traffic assignment procedure used in this model was modified from the capacity restraint method developed by the Traffic Research Corporation (Irwin et al., 1961a), which was part of a nest of models involving trip distribution, modal split and traffic assignment. This procedure was modified by Irwin et al. (1961 b) in developing a capacity restraint procedure which considers the change in link travel impedance in response to loaded volumes. The procedure adopted adjusts the travel distance according to a relationship between assigned volume and link capacity. Thus, the travel distance was increased much in the same way as increasing congestion causes speeds to be lowered in real situations. The all-or-nothing technique, in the railroad case, is not realistic in the
.
LEGEND
EXISTING
SY:TEM
:
. . .
. .
g2 ’
// 12.----
‘\ -1,.
3 .
/
b “1
CODED
’
/’
:/
:.*. I’.
: l
ZONE
CENTROID
YARD/
INTERCHANGE
NODE
WITHOUT
YARD/INTERCHANGE
RAILROAD LARGE
: .
a WITH
TRACKAGE
RIGHTS
IMPEDANCE
. LINK . DUMMY LINK . TRAFFIC . ENTERING . . . . . . . . . . . . . . /
of railroad
.
.
RAILROAD
NETWORK --__Fig. 2. An example
:
*
GATEWAY OWNING
@
network
coding
c:,
i
M
. . .
11111 . . . ... . ... . -b
. . . .
. . . .
:
.
.
A regional
railroad
network
optimization
model for coal transportation
235
AILIL'STTR,\\'EI, 'TTill: ,\I'Pl.Y C,\PAC1TY
Fig. 3. The capacity
restraint
assignment
procedure.
sense that for parallel railroad lines only one will be loaded as long as the capacity is not reached. Figure 3 illustrates the procedure on a flow chart. The use of the capacity restraint procedure to modify assigned loadings provides a more realistic distribution of traffic in the system. SYSTEM
OPTIMIZATION
PROCESS
The objective of the process is to optimally allocate coal from producers to consumers so as to meet the supply and demand requirements without violating the system constraints. The system constraints considered in this model are different from those considered in previous studies (Rifas and White, 1976; LeBlanc, 1976) in that the major emphasis in this model is placed upon the regional railroad systems capacities rather than the interstate trunk lines. Accordingly, the existing facilities in the study region are expected to be more efficiently utilized through the use of the system optimization model. Basic Concept: The optimization model attempts to compare the equilibrium network traffic distributions associated with different coal shipping patterns. The overall network performances and system costs are evaluated for each alternative in choosing a desired
CHIA-JUCH CHANG, ROBERT D. MILES and KUMARES C. SINHA
236
optimal
situation
as follows:
Let Pi = coal shipping pattern for the base condition Pi = new shipping pattern of coal. The system costs associated with Pi and Pi are Ci and Cl respectively, thus, Ci = J(Pi) c; = J(Pi) and ACi = Cl - Ci represents the difference in system cost. The equilibrium volume associated with Pi and PI are, respectively, V,i
=
V’i and V~i,
ftpi)
Vhi =f(Pi) and the respective
service levels are Si and St, si
=
dV,i)
s; = g(V&) The service level is often expressed
in terms of a load factor, i.. By definition,
where Vci is the capacity of the system i. The load factor indicates the utilization facility with respect to its capacity. Then, the change in service level is
of the
ASi = S; - Si The optimization concept is to maximize the favorable changes in service level ASi with a minimum increased system costs ACi. If the design goal for transportation service is set up for the system, then the objective is to minimize the cost in order to achieve the goal. Based on this concept, the optimization model is devised to search the coal shipping pattern which is associated with the optimal combination of the set (ASi, ACi) or (max AS:, min AC:). Optimization Process: The rail system optimization model first uses the coal allocation submodel to determine the possible distributions of coal into and within the study region. The zone to zone coal traffic is then assigned, together with other non-coal traffic, onto the railroad network under study by using the railroad network submodel. The procedure is as follows: (1) The coal supply and demand and their transportation alternatives are input to the coal allocation submodel to generate a set of coal shipping/distribution options. Each coal shipping option is associated with a cost value which indicates the total required system cost. The shipping options are arranged by the ascending order of the cost with the minimum cost one as the base condition for the analysis. (The minimum cost shipping is termed the optimal solution from the “coal allocation submodel”.) (2) The cost associated with each shipping option is analyzed to see how the coal price, transportation costs, and other elements affect the coal distribution. The relative cost changes between each of two options are examined. (3) The zone to zone coal traffic from the base condition coal distribution is added to the non-coal traffic movements to form a combined freight movement matrix. The noncoal traffic is obtained from survey data and is considered as an input to the model. (4) The combined freight traffic is loaded by the use of the railroad network submodel, onto the existing and/or planned railroad networks to investigate the network performances. The critical links, as well as the average link/line capacity uses are examined. The energy efficiencies which are reflected in gross ton-mile movements are also analyzed. (5) The same procedure is repeated for other coal-shipping options. The results are
A regional
railroad
COAL SUPPLY
network
optimization
237
model for coal transportation
COAL DEMAND COAL TRANSPORTATION COAL ALLOCATION SUBMODEL
ALTERNATIVES
COAL DlSTRlBUTlON OPTIONS, DN N=l....M,M=NO.OFlTERATlONS D, = MINIMUM SHIPPING COST DISTRIBUTION
_
l---pINTERZONAL
COAL DISTRIBUTION,
DN
J ANALYSIS OF SYSTEM COST
SHIPPING COSTS SHADOW PRICES OPPORTUNITY COSTS,
)
INTERZONAL NON-COAL (EXOGENOUS)
ETC
FREIGHT
MOVEMENTS
SUMMATION OF MULTICOMMODITY MOVEMENTS
.
I TOTAL INTERZONAL
t
FREIGHT
MOVEMENTS
Ii 2 TRAFFIC
RAILROAD
DISTRIBUTION
NETWORK
INFO. OPERATING
POLICY “B
ON THE RAIL NETWORK t CRITICAL LINKS, EXCESS LINKS/LINES UTILIZATION ENERGY EFFICIENCY
-
OR REDUNDANT
LINKS/LINES
AFTER SECOND ITERATION CHECK IMPROVEMENTS OF THE NETWORK TRAFFIC 4 SELECT
THE DESIRED
OPTIMUM
Fig. 4. The system optimization
process.
compared with each other with respect to the changes in total costs and in network performances. After several interactions an optimal system consistent with the goal is achieved. The entire optimization process is illustrated in Fig. 4.
MODEL
APPLICATIONS
There are many uses to which the rail system Several major uses are listed below.
optimization
model
can be applied.
(1) To predict the future coal shipping patterns and traffic distributions so as to identify the potential bottlenecks or problems which might be encountered. (2) To determine the future (and/or existing) optimal situations which will alleviate the potential problems and efficiently utilize the existing facilities. (3) To investigate the relative advantages or disadvantages of various actions concerning the modifications of railroad networks. The effects of potential abandonment and/or new construction is studied by testing the alternative networks. Similarly, the evaluation
238
CHIA-JUCH CHANG, ROBERT D. MILES and KUMARES C. SINHA
of changing a link characteristic, such as upgraded track or change in traffic control, is possible. (4) The routings determined from the model are used to study minimum paths in relation to existing routing patterns. Recommendations for new routing policies can be developed from such studies. A routing study is of value in determining the relative importance of links in the system and assists in setting priorities for network improvements. CONCLUSIONS
The system optimization model presented in this paper is a general model. The model can be modified to fit the user’s specific needs, assumptions, and data. The model selects the coal source(s), the transportation mode(s) and route(s), as well as the transfer station(s) for each coal demand unit in the study region in order to minimize the overall cost and delay. The existing model is sufficiently efficient to be used for testing many alternatives and doing sensitivity analysis. The preliminary run of the coal allocation submodel used 56 set for 118 iterations to handle 754 variables (X;g) with 77 constraints and 2873 non-zero elements on the CDC 6500 computer system at Purdue University. It took 41 set for the railroad network submodel to assign each traffic pattern onto a coded network with 228 nodes connected by 964 one-way links. This model, when properly modified, can be used by governmental agencies and industries to examine and evaluate the present and future transportation policies and related resource allocation decisions. For example, constraints which recognize the characteristics of certain special consumers (e.g. metallurgic plants) can be added. The model can easily be expanded to include new transportation modes such as pipelines in determining modal split. (To consider pipeline as a new mode in the allocation submodel.) In addition, the two submodels can be combined into one model as long as the problem at hand is not so large that it exceeds the storage capacity of the computer. Another important extension of this model would be to consider the nationwide distribution of coal and freight traffic. Acknowledgements-The research reported Purdue University Research Foundation. necessarily those of the sponsoring agency.
in this paper was sponsored by a David Ross Grant from the The contents reflect the views of the authors only. and are not
REFERENCES Air Pollution Board of the State of Indiana (1976) Limitations for sulfur dioxide emissions from stationary sources. Regulation APC 13, Indianapolis, Indiana. Chang, C. J. (1979) A rail system optimization model for the transportation of coal. Ph.D. thesis. School of Civil Engineering, Purdue University, W. Lafayette. Indiana. Hay W. W. (1967) Arr Introduction to Trunsportatmn Enginrering. Wiley, New York. Homburger W. S. (1970) Traffic estimation computer programs for educational purposes. The Institute of Transportation and Traffic Engineering, University of California, Berkeley. Irwin N. A., Dodd N. and Von Cube H. G. (1961a) Capacity restraint in assignment programs. Transpn Res. Bull. 297, 109-127. Irwin N. A. and Von Cube H. G. (1961b) Capacity restraint in multi-travel mode assignment programs. Transpn Res. Bull. 347, 258-289. Koopmans T. C. (1951) Activity Analysis of Production and Allocation, pp. 317-329. Wiley, New York. LeBlanc, M. R. (1976) A transportation model for the United States coal industry. M.S. thesis, Agriculture Experiment Station, Cornell University, Ithaca, New York. Mekemson J. R. (1979) Purdue network analysis model-user manual. School of Civil Engineering, Purdue University, W. Lafayette, Indiana. Morlok E. K. and Waner J. A. (1976) Approximation equations to ICC costs of rail, TOFC and truck intercity freight systems-development and applications. Department of Civil and Urban Engineering, University of Pennsylvania, Philadelphia, Pennsylvania. Prokopy J. C. and Rubin R. B. (1975) Parametric analysis of railway line capacity. USDOTRep. NO. FRAOPPD-75-1, Washington, D.C. Rifas B. E. and White S. L. (1976) Coal transportation capability of the existing rail and barge network-1985 and Beyond. Electric Power Research Institute, Palo Alto, California. USDOT (1974) Rail service in the midwest and northeast region. A Report by the Secretary of Transportation, Vl, Washington, D.C. Whiting P. D. and Miller J. A. (1961) A method for finding the shortest route through a road network. Oper. Res. Quart. 2, (l-2).
Res:B
Prmted
an Great
Vol
15B, No. 4. pp. 227-238.
0191.2615/81/040227-12802.00/0 0 1981 Pergamon Press Ltd.
1981
Bntam
A REGIONAL
RAILROAD
MODEL
NETWORK
FOR COAL TRANSPORTATION CHIA-JUCH
Department
OPTIMIZATION
CHANG
of Civil Engineering, Marquette WI 53233, U.S.A.
University,
Milwaukee,
and ROBERT D. MILES School
(Received
and KUMARES
C.
SINHA
of Civil Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. 11 January
1980; in revisedform
25
April
1980)
Abstract-A regional railroad network model is presented to evaluate the system’s response to increased coal traffic. An optimal, multimodal, coal-shipping pattern is developed for the study region to minimize total costs and to efficiently use the existing network. A two-stage, general mode1 allocates resources among demands and then assigns flows to the network according to efficiency criteria. The model is sufficiently general to permit modification for specific needs, assumptions and data. Government agencies and industries can apply the model in resource allocation decisions and transportation policy analysis. INTRODUCTION
Due to the dwindling supplies of petroleum and natural gas, there is much discussion and a growing emphasis on the utilization of coal, of which the United States has vast domestic reserves. The conversion to a coal-based energy system is to become more common in the future. The Carter Administration is committed to the implementation of a National Energy Plan which has outlined regulatory and tax measures proposed to encourage industries to convert to coal due to petroleum and natural gas shortages. It is believed that the coal industry is able to increase production in order to meet these future needs. There are constraints, however, that control the speed at which conversion to coal takes place. The transportation of coal is a primary constraint. There are three types of problems that are most likely to develop: (i) certain coal suppliers (those who produce coal advantageous to most consumers) may not be able to meet the great amount of simultaneous demand for their products; (ii) the radical changes in planned coal production and consumption patterns and quantities may lead to a mismatch with the transportation network patterns presently available to carry coal from producing mines to consumption points; and (iii) the regional traffic may be seriously affected by the greatly increased coal movements. An example is an out-of-balance transportation network due to a non-uniform increase in coal traffic causing an overutilization of a part of the system while other parts are underutilized. Studies indicate that based on the projected demand for transporting coal and other commodities the existing transportation system may not be able to handle the demand over the most direct, low-cost route and by the most cost-effective mode (see, for example, Rifas and White, 1976). In order to achieve the maximum overall benefit for the supply-demand system and the maximum efficiency in the transportation system it is imperative that an effort be made to utilize the existing facilities in an optimal way. The primary purpose of this research was to develop a system optimization model which would serve the following two purposes: (1) to predict the future demand for coal movements on the existing rail system of a region in order to evaluate the rail system’s ability to move the required tonnage; and (2) to develop an optimal, multimodal coal shipping pattern for the region which would minimize the cost and also efficiently utilize the existing facilities. The developed model contains two submodels-a coal allocation submodel and a 227
228
CHIA-JUCH CHANG, ROBERTD. MILES and KUMARESC. SINHA
railroad network submodel. The coal allocation submodel is a linear programming model and is used to determine the alternative distribution of coal from producers to consumers by available transportation modes and routes. The railroad network submodel is a traffic assignment model which assigns the rail-coal traffic as well as other non-coal freight traffic onto the railroad network; it is programmed to produce informagion such as link loadings, routings, and capacity analysis, etc. These two submodels follow certain iterative procedures and jointly operate as a system optimization model. DESCRIPTION
OF
THE
SYSTEM
OPTIMIZATION
MODEL
Coal allocation submodel Objective Function: The main objective for individual utilities and industries in purchasing coal from local and/or external sources is to minimize their individual costs. The objective of a system optimization analysis is, however, to plan an optimal system which minimizes the total system costs subject to a number of system constraints. The total system cost includes the mine-mouth cost, the transportation cost, and for high-sulfur coal, the coal-cleaning cost. The mine-mouth cost and the coal-cleaning cost depend upon the type of coal chosen. The transportation cost is determined in terms of the mode and route used for shipping coal. The transportation cost refers to the total cost incurred in the entire course of shipment, which includes the feeder transportation cost, the main-line transportation cost, the loading and unloading cost, and the delivery cost. The objective is to minimize the sum of the required cost and is formulated as such. However, since there are two options available for coal-users in choosing coal, i.e. to use low-sulfur compliance coal or to use scrubbers to clean high-sulfur coal before burning, the objective for the entire system minimizes the sum of the respective costs from the two different decisions. The system objective function is formulated as follows:
Minimize
Z = T Ei c c 2 XY j meM,, BEG,,
+ 4 mL,, Fi $,z,jxij -( I
mg)
where i is the coal supply zone, j is the coal demand zone, m is the coal transportation mode, e.g. rail, water, mixed-mode, truck, etc., g is the gateway to the study region, e.g. transportation centers, river ports, etc., (gateways are not used for intraregional coal supply), X7 is the quantity of coal (in tons) shipped from supply zone i to demand zone j by mode m via gateway g, Mij is the set of available transportation modes for shipping coal from zone i to zone j, Gij is the set of possible gateways from zone i entering zone j, Ei is the mine-mouth cost at mine i, Fy is the feeder transportation cost from mine i to major transportation mode m, TF is the main line transportation cost from mine i to gateway g by mode m, I!,? is the loading and unloading costs for i, j, m, g combination, 04 is the delivery cost from gateway g to destination zone j, Aij is the scrubbing cost of coal mined at zone i for use at zone j, and J is the set of zones which choose scrubbers as the sulfur dioxide emission control strategy. Pollution Constraint: The pollution constraint is based on the Clean Air Act of 1970 and subsequent amendments which specify the regional environmental regulations on restricting the maximum sulfur dioxide emissions in the burning of coal. For example, the state of Indiana requires that the maximum total sulfur dioxide (SO,) emissions be limited to between 1.2 and 6.0 pounds per million Btu depending upon the combustion ratings of power plants (Air Pollution Board of Indiana, 1976). With this constraint, users who attempt to comply with the air pollution regulation are forced to purchase coal only from certain sources where the sulfur content of coal meets the requirement.
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However, it is not the delivery of coal which is constrained, but the actual burning of coal. This allows the blending of high- and low-sulfur coal to obtain a fuel which when burned is, on the average, below the allowable level (LeBlanc, 1976). The constraint is formulated as follows:
i
me&f,,
BEG,,
where Si is the sulfur content (per cent by weight) of coal mined at i, Rj is the maximum allowable sulfur content of coal, which is converted from the maximum SO2 emission limit for each j C+J, and i, j, m, g, Mij, Gij, J and X7 are defined earlier. The above constraint assumes that the blending facility is available for zone j, and that it is economical to have coal shipped from various sources. However, there is a possibility that for some areas it is not economical to do so. In this case, consumers secure their needs from a single source where the sulfur content of coal does not exceed the maximum allowable level. The constraint then becomes that in order to ship coal from zone i to zone j the sulfur content of coal mined at zone i should be less than or equal to the limit of consumer j, otherwise no shipment will be made from i to j. This can be represented as: if Si > Rj, then dij = 0 if Si < Rj, then dij = 1 where dij is the decision variable of the shipment from zone i to zone j. This is a conditional choice problem and is formulated with a set of mixed integer constraints : Mdij + Si - Rj 2 0
if
Si > Rj,
dij = 0
if
Si I Rj,
dij = 1
M(l - dij) + Rj - Si 2 0 0 I dij I
1, dij = integer
and X”g - Md..V 5 0 I 1’ ,vg
1’
2
0
i if *
if
d.. = 0
X!Y = 0
dii
_$g
=
1:
2
0
where M is a very large positive number. The original pollution constraint is substituted for the above mixed integer constraint, whenever applicable. However, due to limits of time and computer capacity, the mixed integer constraint is not applicable in large regional problems. Demand Constraint: The demand constraint assures that the energy requirement for each coal-burning unit is fulfilled, i.e. for each demand zone the total amount of energy generated by coal from all selected sources is equal to its particular need. This is represented in terms of heat requirement (Btu) because of the wide variation in heat content of coal mined from different areas.
where Hi is the heat content of coal (in Btu/ton) mined at i, and Nj is the energy requirement at zone j (in Btu). Supply Constraint: Due to the limited supplies in the mining areas, the total quantity of coal shipped from each supply zone by all modes of transportation should not exceed its available capacity. The available capacity is equal to the output capacity less local consumption. Thus : 1 1 C X;g I Pi - Bi for each i me&f{,
geGi,
j
CHIA-JUCH CHANG, ROBERT D.
230
where Pi is the annual production zone i.
MILES
and
KUMARES
C.
SINHA
at zone i, and Bi is the local consumption
at the same
Gateway Capacity Constraint: The coal allocation submodel allocates the zonal coal demand from external sources to the gateways of the study region, while the network submodel routes coal traffic from the gateways to the final destinations. For intraregional coal supplies the traffic is routed directly to the destinations without using gateways. The shipment of coal from external sources is constrained by the gateway capacities. The gateways are defined as the main entrances to the study region; they are usually the railroad main-line interchanges, the intermodal transfer stations, the water ports, and/or the transportation center cities which are located around the boundary of the region. For railroads, the gateway traffic is constrained by the line capacities, while for waterways the loading and unloading capacities of coal docks are used rather than the navigating capacities. The gateway capacity constraints are delineated as:
1
Xtg I Cr for each g and m
(LA= K,,
where K,, is the set of i, j combinations using gateway g and mode m, and Cy is the transportation capacity at gateway g for mode m. Non-Negativity Constraint: The linear programming coal allocation submodel requires that all decision variables be non-negative numbers. Hence, X;g 2 0 for all i, j, m and g. DUAL
INTERPRETATION
OF
THE
SUBMODEL
The dual formulation of the Coal Allocation Model provides several meaningful economic interpretations of the model. The dual model is formulated as follows: Maximize
subject to :
for all i, j E J, m, g and HiUj - v - Siyj -
C
C
Zr I (El + Fy + Tyg + L;’ + 0;)
gEGi,+ msMii
for all i, j E J, m, g where F is the shadow price of supply i (the change of total system cost per unit change of supply at i), Uj is the shadow price of demand j, E; is the shadow price of sulfur limit at j, Z: is the shadow price of gateway capacity at g for mode m, a is average sulfur content of coal, and N, P, R, C, E, F, 7: L+ D, S, H, A are as defined in the primal formulation. The duality theorem of linear programming states that the minimum cost for the delivery system is equal to the maximum revenue for the price system (Koopmans, 1951). The objective function of minimizing the total cost in the primal formulation is, therefore, equivalent to the maximization of the net revenue, which is equal to the gross revenue less the total constraint cost as expressed in the equation. The dual constraints also have a very interesting meaning in economic interpretation. The constraints can be rearranged to the following form: HiUj-~-EiI(Fi”+TY’g+LY+D~+Aij)+
Six+
for
j E J,
C msM,, 2 Z: BEG,,
A regional
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231
and 1 27 Six + 1 geG,,msM,j
HiUj - r/I - Ej I (Fy + 7’7’ + L!g + 0:) + for
j f+J.
The left hand sides of the above inequalities can be interpreted as the gross unit profit for mine i to supply the demand at j, and the right hand sides are the sum of transportation cost and system constraint cost incurred in shipping coal from i to j. It is economically feasible to ship coal from i to j only if the gross profit for i could cover the total cost, otherwise no shipments should be made. This condition can be expressed as follows: (i) if
HiUj-~-Ei=(F~+T~‘+L~‘+DQ+Aij)+
for
Six+
C C Z: gEGllmeMa,
jg J
or - Ei = (Fr _t Tyg + L!’ + 04) +
Six + !
1 1 Zr gsG,, msM,,
then Xyg 2 0. (ii) if
HiUj
-
v
- Ei < (F~ + Tug + Lo
for
+ Oj”+ Aij) +
Sip +
C C Z,m gEG,,msM,,
jE J
Or HiUj
-
r/;: -
Ei<(F~+T~g+L~g+D~)+
for
Six+
1 C Zy BEG,,mEMa,
j+J
then XFg = 0. lJ This is consistent with the well-known “Complementary programming. RAILROAD
NETWORK
Slackness Theorem” of linear
SUBMODEL
An initial set of transportation alternatives is generated by the coal allocation submodel through an analysis of supply, demand and costs as discussed. These alternatives are produced without considerations of the physical constraint and transportation efficiency of the regional railroad network. The purpose of the railroad network submodel is to select, among the alternatives produced from allocation submodel, the optimal modes and routes suitable for the particular region. The relationships between the two submodels and their inputs are explained by the following simple diagram: Exogenous Variable Endogenous ~ Variable
T-k
Transptrt
Costs YM,,,
Selection of Mode, Routes
T\,
Routes
ailroad Network4/Optimal Submodel
Alternative ./ Allocation
232
CHIA-JUCH CHANG, ROBERT D. MILES and KUMARES C. SINHA
The railroad network submodel used in conjunction with the coal allocation submodel is an algorithm designed to develop and analyze traffic loadings on links of a railroad network. The submodel uses a railroad network data base and a computer program to process and assign the coal traffic interchanges generated from the coal allocation submodel. The basic computer program is a subset of the battery of programs first developed at the University of California for urban transportation planning (Hornburger, 1970), and then modified at Purdue University (Mekemson, 1979). The program is used with appropriate modifications to reflect its application to railroad networks. The input to the railroad network submodel includes the network geometry, the network parameters, and the traffic interchange data. Network Geometry: The network geometry is a description of the interconnections and segments of the network representing the railroad system under consideration. Like a highway network, a railroad network can be described by a series of nodes connected by links. However, because of the special railroad characteristics, the rail traffic assignment is different from the highway assignment process. For example, the rail traffic tends to stay on the same railroad as long as possible. Hence, in coding the network a penalty is considered at junctions to indicate that train movements from one railroad to another at certain junctions either are subjected to delay or else are completely prohibited. Since rail cars often transfer between two carriers at specific interchange points, an appropriate cost is established for those interchanges to allow interline services. Figure 1 depicts two simple examples of the schematic representation of junctions, interchanges, and trackage rights in a railroad network in comparison with those in a highway network. It is observed on Fig. l(a) that the intersection point of railroads A and B, although representing only one junction or interchange point, is coded with two different nodes connected by a dummy link. The distance of the dummy link for junctions without interchange facilities is assigned an assumed large value M to prohibit turning movements between railroads A and B. For approved interchanges an appropriate value is developed from the cost data (Morlok and Waner, 1976) and converted to the equivalent distance. Figure l(b) shows that the trackage rights of railroad A over railroad C on link 2-3 is represented by adding a separate link 7-8 for A with a specified share of capacity. The distance of link 7-8 is adjusted to reflect delay caused by switchings. Figure 2 shows a small network example coded for a hypothetical rail system. This type of network coding is used in the assignment process in the submodel. Network Parameters: The traffic assignment process in the submodel requires network impedance values to allow for the selection of routes through the network under study. These impedance values are either travel time or distance, or some derivative function of these for use in the minimum path calculations. The link capacity data is another parameter. The capacity of a railroad system is extremely difficult to measure; it is a function of a number of elements such as the type of track, the signaling system, the operating speed, as well as the siding characteristics etc (Prokopy and Rubin, 1975). If the detailed information is not available, the theoretical capacity of a railroad link can be estimated on the basis of the number of tracks and the traffic control system (see Hay, 1967, and USDOT, 1974). The capacity can be expressed either by number of trains per day or by tons per a time period. This is used as a check on the assignment process to indicate proper selection of the network and interconnection with load points. Estimated capacities are used to provide a feedback during the assignment process in representing system congestion via a capacity restraint technique. TraJic Interchange Data: The network model requires an origin-destination trip matrix which contains the number of trips from each origin to each destination in the network. The matrix does not contain specific routings, only interchanges of trips throughout the network. The traffic interchange data needed for input to this model includes the coal traffic generated from the coal allocation model as well as the non-coal traffic from survey data. The data is expressed as one-way trips on a zone to zone basis. It is recorded as either number of cars or tons whichever is available. This type of trip
A regional
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233
matrix is used for calibration of the network and for loading trips on various networks for plan development purposes. Assignment Process: Traffic assignment techniques are based upon the premise that users of a transportation network wish to optimize some measures of their travel. Generally, the characteristics of travel considered are travel time, distance, and cost. The model utilizes a minimum path algorithm developed by the Road Research Laboratory (RRL) (Whiting and Miller, 1961) to compute a shortest route for each origin-destination pair. After the minimum path trees are found, the model assigns the traffic interchange data to the shortest routes by a capacity restraint procedure.
c
./’ RAILROAD ia)
5ec
TRACKAGE
RIGHTS 6 I
? 4
RAILROAD
HIGHWAY
Fig. 1. Railroad
network
vs Highway
network.
CHIA-JUCH CHANG, ROBERT D. MILES and KUMARES C. SIKHA
234
The traffic assignment procedure used in this model was modified from the capacity restraint method developed by the Traffic Research Corporation (Irwin et al., 1961a), which was part of a nest of models involving trip distribution, modal split and traffic assignment. This procedure was modified by Irwin et al. (1961 b) in developing a capacity restraint procedure which considers the change in link travel impedance in response to loaded volumes. The procedure adopted adjusts the travel distance according to a relationship between assigned volume and link capacity. Thus, the travel distance was increased much in the same way as increasing congestion causes speeds to be lowered in real situations. The all-or-nothing technique, in the railroad case, is not realistic in the
.
LEGEND
EXISTING
SY:TEM
:
. . .
. .
g2 ’
// 12.----
‘\ -1,.
3 .
/
b “1
CODED
’
/’
:/
:.*. I’.
: l
ZONE
CENTROID
YARD/
INTERCHANGE
NODE
WITHOUT
YARD/INTERCHANGE
RAILROAD LARGE
: .
a WITH
TRACKAGE
RIGHTS
IMPEDANCE
. LINK . DUMMY LINK . TRAFFIC . ENTERING . . . . . . . . . . . . . . /
of railroad
.
.
RAILROAD
NETWORK --__Fig. 2. An example
:
*
GATEWAY OWNING
@
network
coding
c:,
i
M
. . .
11111 . . . ... . ... . -b
. . . .
. . . .
:
.
.
A regional
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235
AILIL'STTR,\\'EI, 'TTill: ,\I'Pl.Y C,\PAC1TY
Fig. 3. The capacity
restraint
assignment
procedure.
sense that for parallel railroad lines only one will be loaded as long as the capacity is not reached. Figure 3 illustrates the procedure on a flow chart. The use of the capacity restraint procedure to modify assigned loadings provides a more realistic distribution of traffic in the system. SYSTEM
OPTIMIZATION
PROCESS
The objective of the process is to optimally allocate coal from producers to consumers so as to meet the supply and demand requirements without violating the system constraints. The system constraints considered in this model are different from those considered in previous studies (Rifas and White, 1976; LeBlanc, 1976) in that the major emphasis in this model is placed upon the regional railroad systems capacities rather than the interstate trunk lines. Accordingly, the existing facilities in the study region are expected to be more efficiently utilized through the use of the system optimization model. Basic Concept: The optimization model attempts to compare the equilibrium network traffic distributions associated with different coal shipping patterns. The overall network performances and system costs are evaluated for each alternative in choosing a desired
CHIA-JUCH CHANG, ROBERT D. MILES and KUMARES C. SINHA
236
optimal
situation
as follows:
Let Pi = coal shipping pattern for the base condition Pi = new shipping pattern of coal. The system costs associated with Pi and Pi are Ci and Cl respectively, thus, Ci = J(Pi) c; = J(Pi) and ACi = Cl - Ci represents the difference in system cost. The equilibrium volume associated with Pi and PI are, respectively, V,i
=
V’i and V~i,
ftpi)
Vhi =f(Pi) and the respective
service levels are Si and St, si
=
dV,i)
s; = g(V&) The service level is often expressed
in terms of a load factor, i.. By definition,
where Vci is the capacity of the system i. The load factor indicates the utilization facility with respect to its capacity. Then, the change in service level is
of the
ASi = S; - Si The optimization concept is to maximize the favorable changes in service level ASi with a minimum increased system costs ACi. If the design goal for transportation service is set up for the system, then the objective is to minimize the cost in order to achieve the goal. Based on this concept, the optimization model is devised to search the coal shipping pattern which is associated with the optimal combination of the set (ASi, ACi) or (max AS:, min AC:). Optimization Process: The rail system optimization model first uses the coal allocation submodel to determine the possible distributions of coal into and within the study region. The zone to zone coal traffic is then assigned, together with other non-coal traffic, onto the railroad network under study by using the railroad network submodel. The procedure is as follows: (1) The coal supply and demand and their transportation alternatives are input to the coal allocation submodel to generate a set of coal shipping/distribution options. Each coal shipping option is associated with a cost value which indicates the total required system cost. The shipping options are arranged by the ascending order of the cost with the minimum cost one as the base condition for the analysis. (The minimum cost shipping is termed the optimal solution from the “coal allocation submodel”.) (2) The cost associated with each shipping option is analyzed to see how the coal price, transportation costs, and other elements affect the coal distribution. The relative cost changes between each of two options are examined. (3) The zone to zone coal traffic from the base condition coal distribution is added to the non-coal traffic movements to form a combined freight movement matrix. The noncoal traffic is obtained from survey data and is considered as an input to the model. (4) The combined freight traffic is loaded by the use of the railroad network submodel, onto the existing and/or planned railroad networks to investigate the network performances. The critical links, as well as the average link/line capacity uses are examined. The energy efficiencies which are reflected in gross ton-mile movements are also analyzed. (5) The same procedure is repeated for other coal-shipping options. The results are
A regional
railroad
COAL SUPPLY
network
optimization
237
model for coal transportation
COAL DEMAND COAL TRANSPORTATION COAL ALLOCATION SUBMODEL
ALTERNATIVES
COAL DlSTRlBUTlON OPTIONS, DN N=l....M,M=NO.OFlTERATlONS D, = MINIMUM SHIPPING COST DISTRIBUTION
_
l---pINTERZONAL
COAL DISTRIBUTION,
DN
J ANALYSIS OF SYSTEM COST
SHIPPING COSTS SHADOW PRICES OPPORTUNITY COSTS,
)
INTERZONAL NON-COAL (EXOGENOUS)
ETC
FREIGHT
MOVEMENTS
SUMMATION OF MULTICOMMODITY MOVEMENTS
.
I TOTAL INTERZONAL
t
FREIGHT
MOVEMENTS
Ii 2 TRAFFIC
RAILROAD
DISTRIBUTION
NETWORK
INFO. OPERATING
POLICY “B
ON THE RAIL NETWORK t CRITICAL LINKS, EXCESS LINKS/LINES UTILIZATION ENERGY EFFICIENCY
-
OR REDUNDANT
LINKS/LINES
AFTER SECOND ITERATION CHECK IMPROVEMENTS OF THE NETWORK TRAFFIC 4 SELECT
THE DESIRED
OPTIMUM
Fig. 4. The system optimization
process.
compared with each other with respect to the changes in total costs and in network performances. After several interactions an optimal system consistent with the goal is achieved. The entire optimization process is illustrated in Fig. 4.
MODEL
APPLICATIONS
There are many uses to which the rail system Several major uses are listed below.
optimization
model
can be applied.
(1) To predict the future coal shipping patterns and traffic distributions so as to identify the potential bottlenecks or problems which might be encountered. (2) To determine the future (and/or existing) optimal situations which will alleviate the potential problems and efficiently utilize the existing facilities. (3) To investigate the relative advantages or disadvantages of various actions concerning the modifications of railroad networks. The effects of potential abandonment and/or new construction is studied by testing the alternative networks. Similarly, the evaluation
238
CHIA-JUCH CHANG, ROBERT D. MILES and KUMARES C. SINHA
of changing a link characteristic, such as upgraded track or change in traffic control, is possible. (4) The routings determined from the model are used to study minimum paths in relation to existing routing patterns. Recommendations for new routing policies can be developed from such studies. A routing study is of value in determining the relative importance of links in the system and assists in setting priorities for network improvements. CONCLUSIONS
The system optimization model presented in this paper is a general model. The model can be modified to fit the user’s specific needs, assumptions, and data. The model selects the coal source(s), the transportation mode(s) and route(s), as well as the transfer station(s) for each coal demand unit in the study region in order to minimize the overall cost and delay. The existing model is sufficiently efficient to be used for testing many alternatives and doing sensitivity analysis. The preliminary run of the coal allocation submodel used 56 set for 118 iterations to handle 754 variables (X;g) with 77 constraints and 2873 non-zero elements on the CDC 6500 computer system at Purdue University. It took 41 set for the railroad network submodel to assign each traffic pattern onto a coded network with 228 nodes connected by 964 one-way links. This model, when properly modified, can be used by governmental agencies and industries to examine and evaluate the present and future transportation policies and related resource allocation decisions. For example, constraints which recognize the characteristics of certain special consumers (e.g. metallurgic plants) can be added. The model can easily be expanded to include new transportation modes such as pipelines in determining modal split. (To consider pipeline as a new mode in the allocation submodel.) In addition, the two submodels can be combined into one model as long as the problem at hand is not so large that it exceeds the storage capacity of the computer. Another important extension of this model would be to consider the nationwide distribution of coal and freight traffic. Acknowledgements-The research reported Purdue University Research Foundation. necessarily those of the sponsoring agency.
in this paper was sponsored by a David Ross Grant from the The contents reflect the views of the authors only. and are not
REFERENCES Air Pollution Board of the State of Indiana (1976) Limitations for sulfur dioxide emissions from stationary sources. Regulation APC 13, Indianapolis, Indiana. Chang, C. J. (1979) A rail system optimization model for the transportation of coal. Ph.D. thesis. School of Civil Engineering, Purdue University, W. Lafayette. Indiana. Hay W. W. (1967) Arr Introduction to Trunsportatmn Enginrering. Wiley, New York. Homburger W. S. (1970) Traffic estimation computer programs for educational purposes. The Institute of Transportation and Traffic Engineering, University of California, Berkeley. Irwin N. A., Dodd N. and Von Cube H. G. (1961a) Capacity restraint in assignment programs. Transpn Res. Bull. 297, 109-127. Irwin N. A. and Von Cube H. G. (1961b) Capacity restraint in multi-travel mode assignment programs. Transpn Res. Bull. 347, 258-289. Koopmans T. C. (1951) Activity Analysis of Production and Allocation, pp. 317-329. Wiley, New York. LeBlanc, M. R. (1976) A transportation model for the United States coal industry. M.S. thesis, Agriculture Experiment Station, Cornell University, Ithaca, New York. Mekemson J. R. (1979) Purdue network analysis model-user manual. School of Civil Engineering, Purdue University, W. Lafayette, Indiana. Morlok E. K. and Waner J. A. (1976) Approximation equations to ICC costs of rail, TOFC and truck intercity freight systems-development and applications. Department of Civil and Urban Engineering, University of Pennsylvania, Philadelphia, Pennsylvania. Prokopy J. C. and Rubin R. B. (1975) Parametric analysis of railway line capacity. USDOTRep. NO. FRAOPPD-75-1, Washington, D.C. Rifas B. E. and White S. L. (1976) Coal transportation capability of the existing rail and barge network-1985 and Beyond. Electric Power Research Institute, Palo Alto, California. USDOT (1974) Rail service in the midwest and northeast region. A Report by the Secretary of Transportation, Vl, Washington, D.C. Whiting P. D. and Miller J. A. (1961) A method for finding the shortest route through a road network. Oper. Res. Quart. 2, (l-2).