Optimal Transportation Patterns of Coal in the Great Lakes Region

L. King; E. Casetti; J. Odland; K. Semple

Economic Geography, Vol. 47, No. 3. (Jul., 1971), pp. 401-413.

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OPTIh4AL TRANSPORTATION PATTERNS OF

COAL IN THE GREAT LAKES REGION"

L. KISG, E. CASETTI. J. ODL.AND, K. SEMPLE

The Ohio State C'nicer.sity

This paper discusses the probleni of deternlining optinial transportation pat- terns of bit~uninous coal throughout the Great Lakes region. The lnodel whicll is developed talces into acco~~n t the follon7- ing empirical features of the coal trans- portation sj7stelu. First, there are given the levels of coal denlanded at numerous centers throughout the region. This de- niand is associated with the coking plants and utility plants in the region. Sccond, the productive capacities and rates on the nzajor transportation routes are given. These particular empirical features of the coal distribution system do not appear to have been described elsewhere in the geographic literature.

It should be stressed that the major emphasis in the paper is on the actual ~noclelling of a real-world problem situa- tion, namely the determination of opti- rnal transportation patterns of coal in a region. The nlodel n7hicll is formulated is partially tested with some real-world <la& in the latter half of the paper. There are many subtleties in the actual rate structure and rail-on7nership patterns that could not be taken into account given the colzstraints ~ulder which this project was carried out. The results pre- sented here, however, illustrate the power of this approach.

The stucly area includes the bitumi- nous coal producing areas and the coal receiving centers of the nlidn~est and Great Lakes region of the Cnited States

" This paper is bascd on work clone uncler contract to 1,ogistics Systems Inc., Boston. Grateful ackno\vledgenient is made of the as-sistnnce Dr. J. L. Heskctt hfr,

and Canada. The area, shown in Figure 1: extends from southern Kentucky to the northernmost exterision of the Great Lakes, and roughly from Harrisburg, Pennsj-lvania in the east to ;\linneapolis- St. Paul in the west. The nlajor coal receiving centers 011 the Atlantic sea-board are excluded.

The major geai:ralized r a1 '1 -routes in- cluded iu the study are sho\~7ri in Figure 2. These include the route\vays from western Pennsylvania to the southeril shores of Lake Erie ( the Baltimore and Ohio, and Bessemer and Lake Erie lines ) ; northern S'l'cst Virginia to the southern shores of Lake Erie ( the Balti- more and Ohio line); central West Vir- ginia to the southern shores of Lake Erie (the Chesapeake and Ohio, and New York Central lines ) ; southern M7est \'irginia and Kentucky to the southern shores of Lake Erie ( the Norfolk and IVestern: and Louisville and Nasllville lines); soutllerl~ Illiriois to the Chicago area (the Illinois Central line); and finally, Kentuck>- and southern Indiana to the Chicago area (the Chicago and Eastern Illiriois lint, ) .

The major Great Laltes rnovelnents of coal originate in the eastern sections of the region. Mayer [ I 1 1 shows, in fact, that this eastern domillance allo\vs the midn7estern mines to dominate coal mar- kets in the lakes area onlv as far north as Milwaukee.

In stud!, solne 104

routes for w]lich rates are readill avail-able, Tjris set of routes

barge- movelllents for lvllich no rates are available, and it collsidcrs lake lnovenlents olll!7 ill coml,illation \vith rail shipments. That is to say, the 104 routes

A. ~ l : ~ i ~ ~~ ~ ~ i ~ xnc,, i D ~ ,~ or transportation arcs arc eitherof systems t and ~ rail H. Gnuthier of Ohio State. links or rail-lake routes.

In the nzodel, each coal produci~lg area or coal receiving center is treated as a node. In the case of the coal pro- ducing areas, the regional groupings given ill the Kegstone Manual [ 8 ] are used to define the nodes. These coal dis- tricts correspond in most cases with the rate districts used bv the railroads serv- ing the areas, and fbr any one such dis- trict the railroads charge a single rate for all coal shipnlents originating at points within the district.'

1 The coal proch~cing regions in Illinois and Iiidiana were modifiecl solliewhat from their definition in the I<e!ystone Mc~nriul tor closer correspondence with the rate grollpc.

The node identified with each coal producing region is the ruajor transpor- tation center within the region. These nodes are shown in Figure 1. The pro- ductive capacity of each node is defined as the sun1 of the bituminous coal ship- ments for 1963 from the counties in-cluded in the region, as published in the Cetrsus of Mineral Inrlustries [3, p. 431. No attempt is ]?lade to differentiate the coal produced according to its grade. For the later testing of the model, only 15 conlposite nodes were used to rep- resent the coal-mining districts. These nodes are listed in Table 1and s1101vn irr Figure 2.

Fig. 1. The coal distrib~ition network.

There are two points which shouId be noted with respect to this aggrt,haation.

First, the three nodes of Clearfield, Broadtop, and Blossburg in central Perin-

sylvania are exchtdecl from the analysis since these centers are too far away fro111 any other production center to be grouped \\,it11 it, and transportation rates are not available for tl~em.Second, Pitts-burgh's own demanc). for coal is sub-tracted from the coal capacity for Pitts-burgh and only thc balancc enters the transportation nctwork.

Two major sources of coal demand are considered, thermal-electric generating plants, and the coking ovens of the steel industry. Together thcse two sets of customers use approxirnatcl!- three-quar-ters of thc coal consumed in the region 1131. Only utility plants with an installed capacity greatei. than 100,000 kw. arc considered; for thcse plants data arc available fro111 the National Coal Asso-ciation [12, pp. 3-41] on their locations

Key to

KAMES AS]> XI

Coal Nodes 29. IV. Kty. 30. Boonville

1. Bntler 31. llan\~ille 2. I'ittsburgh 32. Sullivan 3. Connellsville 3:3. Centralia 4. AIoundsville 34. S. Ill. 5. Fai rn~ont 35. Springfield 6. Kanawha 36. La Salle 7. Pocahonta~ 37. Blosshurg

8. Elkhorn 88. Clearfield 9. Ohio 8 99. Broadtop

10. XIiddle (Ohio) 40. Ul>pev Potonlac 11. S. Ohio 12. Princeton 13. T ~ r r rHaute 14. Belleville

1 9 . Proria 16. Freeport 1. Toronto

17. Solnnset 2 . Bnffalo IS. Prr5:<111 3. Detroit

19. Bandolph .I. Milwaukee 20. Nicholas 5. Crem Bay 21. Logan 6. I~ll l l l t l l

22. Williamson 7. Chicago

23. Raleigh 8. Toledo

24. Hazard 9. Cleveland 23. IIarlan 10. Cincinnati

26. .\liddleshoro 11. Youngstoivn

27. Leetonia 12. Struhrnvillr 28. Cambridge 1 3 . Beverly

and the anlounts of coal used in 1966. Data on the Canadian utilities are pro-vided bv Beier and Heskett [2]. A total of 139 Gtilitj~plants are covered b? the studv.

he coking ovcns, as listed b j thc Amcrican Iron and Stccl Institute [ I ] , are found at 21 different locations in thc study area and all of these arc in-cludcd. Thc alllounts of coal used at thesc locations are cstimatcd here. Thc n~rll~ber ovens per location is kno~vn;of the capacity of ovens at the diffcrerlt locations are assumed uniform and hcnce an average production levcl per oven per year can be obtained by dividing total annual regional coke production 11:- thc total nunlber of ovens-this yields a level of 4,500 tons of coke; it is assurned that the weight of coke produced at each oven is three-fourths t l~cweight of the coal charge and thus, annual coal de-inand per oven should be around 6,000 tons: finally, annual coal demand per

14. Cheshire I?. Dayton 16. Canton 17. Eric IS. Dullkirk 19. Rochester 20. Minneapolis 2 1. IVansau 22. 1.a Crosse 23. Dixon

24. Wailkegan 25. Havana 26. Coffeen

27. St. Louis 28. Grand Tower 29. l'nd~lcah 30. Fort TVilliam 31. hfarrlxiette 32. S a ~ t l tSte. blarir 39. Saginaxv

34. Port Huron 15,hl~lskegon 36. Lansing

97. Kalan~azoo 38. Indianapolis 39. Centerton 40. Ln~rrmcehurg 41. Madison 42. Louisville 45. Owenshoro

44. Pranktort 45. 1,exingTon 46. 13urnside

47. Jl'indsor 18. Lorain

49. S t r i~ t ton 50. Conesviile 51. C o l u m h ~ ~ s 52. Philo 53. Portrlnollth 54. Lonise 3,5. C;raha~n 56. St. Mar). 57. Hamilton

,58. O~wego 59. .4sl1tabula 60. Heesclale 61. Tt~mplrton 62. Aliqnippa 6'1. Brownsville 64. )It. Plcasant 6,5. Johnstown

66. Middlc.to\vn

L. Sandnsky 2. Connrar~t 3. Sodus Point

coking oven center is obtained by multi-pl>ing this level of 6,000 tons by the number of ovens at the center.

The utility plants dnd coking-oven cente~s are first aggregated into S2 re-ceiving nodes (Figure 1).Small rcceiv- ers are go~iped with large ones so long as there are no marked differences in the coal trailsportation rates applicable to the centers in question. These 62 nodes then are aggregated further into tlie 19 major centers listed in Table 2 and shown in Figure 2.

As regards transportation rates, the study uses uolernle and truinloucl rates \vhic2l apply to most illaior coal ship- inents. In a fcw cases, carload rates have

to be usec1.Vor each route, the lowest rate for coal shiplneiits is applied. This ensures a consistent basis for the appli- cation of rates althougl~ it is not com-pletely realistic since all shipments do not meet the volui~le or trainload re-quirernents for the lowest rates. The rail rates used are those which were in effect bet\veen September 1, 1962 and August 19, 1967."

? The rate information was generously pro-vicled by officers of the Rrssemer ancl Lake Erie Railroad, Pittsburgh, and the Elgin, Joliet, and Eastern Railroad, Chicago.

T l 1 e rail rates tlsed are at cs parte 223 level. This level was in effect bctwcen Septem- her 1, 1962 ancl August 19, 1967, ancl was used because illost p~il)lished rates were at this ex parte level.

Fig. 2. Coal network with aggregated noctes.

,4 fen7 characteristics of the rate struc- ture for coal shipments are worth stress- ing in the context of this study. First, the rates refiect competition and are not related simply to distance of haul. The volume and unit train rates, especiall>-, have been established consistent with the particular marketing strategies of the companies involved. Second, in the case of volume rates particularly, there are a number of attached conditions re- garding l~linil~lum trainloads, annual volun~es, and in some cases, demurrage charges, which affect the utilization of the service. Finally, rates for rail-hauls to the lake ports are differentiated ac-cording to the ultimate destination of the coal. For example, the rate for coal shipments from Connellsville to the port of Conneaut, assuming the coal ultimate- ly goes to Buffalo by lake vessels, is higher than the corresponding rates for

TABLE 1

Calmcity (1000 tons)

C1 Butlrr (includes Fleeport)

C2 Pittsburgh

C 3 Con~lcllsville (includes Somerset)

C 1 hfoundsville 6,614

C5 Fairmont 24.2.33 (includes Preston, Randolph)

C6 Kanalrha ( includes Nicholas )

C7 Pocahontas 76,569 (includes Logan, Williamson, Raleigh)

C8 East Kentucky 22,016 (includes Elkorn, Hazard, Harlan, Miclcllrsboro~

CO Ohio 8 26,144 (includes Leetonia, Cambriclge)

C10 Middle (Ohio)

C11 South Ohio

C12 Princeton 32,697 (includes \Vest Kentucky, Hoonville)

C13 Terre Haute 1,992 (includrs Danville, Sullivan)

C14 Helleville 10,718 (includes Centralia, Southern Illinois)

C l 5 Peoria 7,891

(includes Springfield, La Salle)

Toronto and Green Ha); as final destina- tions, but lower than those for Detroit and hIil\vaukee as final destinations. This third feature of the rate structure is an overriding coilsideratioil in structuring the coal network flow problem. As will be noted in the next section, the model can be structured in terms of origins, destinations, and intermediate nodes. But in this analysis interlnediate nodes such as the lake ports hardly ~varrant consideration as such, and the net\vork is best handled as a set of only origins and destinations. The costs on the con- necting arcs represent the sum of the special rail rates and the estimated lake carrier rates. One problem \vith this ap- proach is that the lllodel solutioll may result in a port capacity being violated. However, this can be taken care of in the model as \\rill be noted later.

The charges for coal hauling via lake carriers are contractual agreements, and there are no published rates. One piece of available information is that an 834 per ton lake-carrier charge is included in a rail-lake combined rate offered to Duluth. This is probably an unusually

'TABLE 2

Derniz~~d Node (1,000 tons)

K1 Toronto

R2 BnKalo

H 3 Drtroit

R4 hfilwaukce

R5 Green Hay

R6 Duluth R7 Chicago

R8 Toledo

R9 Clevrland

R12 Steubellvillc

R13 Beverly, Ohio

R11 Cheshire, Ohio

R15 Daytoll

R16 Canton

R17 Erie

R18 Dunkirk, Xexv Yolk R19 Rochester

Destinntions Chicago

Duluth -Slilivaukee 60

Green Bay 70

Detroit -Torollto -Bufl'alo

Toledo -

TABLE 3

ESTI~I.ATEI)1 . i 6 E CARRIER CHAKGES ((. P E R TOH)

Ports Connennt Ashtnh~tla Lorarn Sonrlusky Z'olcdo

83

110

100

4 0

6 0

33

33

low vessel charge, but it is used here as the basis for estinlating lake carrier costs as a direct function of distance (Table 3) . Dunlping charges which are levied at some ports are not taken into accou~lt in these estimates.

In order to determine the opti~ual flo\vs of coal in the Great Lakes area under a varietv of situations character- ized by relatively minor variations in de- mands, supply, and costs, a suitable model \i7as required. The immediate al- ternatives were to use an uns~ecialized

I

linear programming model or a special- ized transportation-type linear program- ming model, but unfortunatel!7 at the time this stud\ was undertaken, opera- tional forms of these nlodels for a problem of the size involved in the study \i7ere not readilv available to the authors. Arnong the transportation nlodels con-sldered as an alternative, a form of net- work flow anal!7sis was chosen. This is based on I?ulkerson's "Out of Kilter Al- gorithm" 14, 5, 6, 71, and is capable of handling problenls several times the size of the one at hand.

The "Out of Kilter Algorithm" ( OKA), is particularly appropriate for solving a sequence of flo\v problems differing ollly slightl! fro111 each other. The optimal flows solutioil of each problenl in such a sequence is used to initiate the com- putation for the follo~i ing problem, and

as a result the solution time of the second problem is only a sillall fraction of the time that would be required to solve it jeparately. Also, for each successive run onl! the information concerning the dif- ferences of the particular problem from the preceding one need be provided.

Basically, there are two types of situa- tions that can be handled by the OKA iuodel. In the contest of this coal-flow stud\, the first type of problem \i7ould assuille that:

1) coal is required in specified anlounts at a set of destination nodes;

2 ) coal is available at a set of origin nodes;

3 ) for each origin there is a maximurn amount of coal that it can provide;

4 ) the sum of the maximum amounts of coal available at the set of origins is greater than the sum of the amounts re- quired at the set of destinations; and

5 ) thc origins and the destinatiolls are embedded in a network which in- cludes also nodes that are neither origins nor destinations, has pairs of nodes con- nected by one or Inore directed arcs (each admitting a flow7 in one directiol~ only), and has specified for each arc a tra1;sportation cost, an upper capacit!.. and a lower capacity.

The second of differst ~ ~ p e problem fro111 the first in that it involves 0111~ nodes that are either origins or destina- tions and only arcs connecting origins to destitlations. That is to say, intermediate

nodes and the related transportation arcs are not considered.

In both problems, the din1 i~ to deter- mine a set ot flows through the arcs that: involves a mininlum total cost; makes available at the various destinations the aniounts of coal required there; involves shipments from the various origins equal to or snlaller than the arrlounts of coal available; and does not violate the cnpa- city limitations on the transportation arcs. In fact, by identifying optimal flows, the solutions to these problenls in- directly determine which origins ship to capacit~r, xvhich ones ship less coal than they have available, and which ones, if any, are "inactive."

The second type of problem is a spe-cial case of the first and, therefore, no generality is lost l J 7 discussing the appli- cation of the OKA to the first t)pe of problem. A simple example is pro;ided by the network in Figure 3. Each trans- portation arc in the network and its di- rection are identified by directed lines joining pairs of nodes. The open arrows originating at nodes 2 and 3 identity destinations with coal denlands of 8 and 6, respectively. The open arrows ter-minating at nodes 1and 4 and the asso- ciated members identify these nodes as origi~ls capable of supplying up to 10 and 8 units of coal, respectively. In order to determine by the OKA the set of illininla1 cost Aows that satisfy the de-

mands and do not violate the various capacit~ constraints, the problem is re- forlllulated as follorvs. Two new nodes are added and called the dummy desti-ftntio~t ( DD ) and the cltu~zmy origin ( D O ) . These are shown in Figure 4. Each destination is connected to the dummy destination by a directed arc \vhich is given a cost of zcro, an a n -limited upper capacity, and a lower capacity equal to the amount of coal required at the destination. The dummy origin, in turn, is connected to all the origins by clirected arcs which have zero transportation costs, upper capacities equal to the maximum anlounts of coal that the origins can provide, and lower capacities of zeros. The three rlu~nbcrs associated with each of the other arcs identity in order: the cost of one unit of flow along thc, arc; an upper bound on the amount ot flow along the arc; and a lo\ver bound on the amount of flow along the arc. The letter L in Figure 4 as well as in Figures ,7 and 6 indicates n very large nuinber and is the co~nputational equivalent of unlinlited upper capacity. An arc connecting the dummy destina- tion to the dumii~y origin with zero costs, unlimited upper capacity, and zero lower capacity "closes" the network.

10 Fig. 4. Simple n t , t ~ v o ~ kw ~ t h;Ilunmy ilcstina-

Fig. 3. Siml~lenetwork. tion ant1 dum~il>origin.

DO - I 0 10

DO - i C 0 4

4 - 1 0 4

1- 3 0 0

1 - 5 0 14

5 - 3 0 6

5 - 2 0 5

2 - 1 0 0

2 - DD 8 Y

3 - IID (7 G

Dl) - DO 0 i l.

Tot71 \oIutiotl cost

Kode 1'rioi.s arc: DO, $0: 4. SO: 1, $2; 5 . S f ; 2 , $9; 3, 5 5 , DD, $0. * Yegligihie.

Conseci~(uently, the original problcnl in- the i-j"' arc, Tlie solutioll is subject to volving the determination of the optimal the constraints, flows from a set of origins to a set of clestinations is translated into a nroblem 1 1 1 < X I , < L L , , for all i and i. of determining a set of optinla1 circular flows. The lower bo~uld on each destina- whcre I , , ai~riu , , arc the lower and up-tion-dummv destination arc ensures that per capacities of the i-jth arc, and the dc,stination receives an inflow of coal ey~ial to its dcmand. The upper bound on each dmnnly origin-origin arc guar- antees that no origin has an orrtflo~v greater than the amount of coal avail- \v11ich establishes the circulation princi-

ablrk there. The arcs linking destinations ple that what fbws iuto a node must also

to the dumnlv destinatiovt and the dlml- ltxavc it. 111 the solution given in Table

m! origin to 'the origi-tls carr! zero costs 4. the total cost of the flow pattern is

and t.ottsc.quentlv do not influence the 82 ~ ~ n i t s .

optimal flo1~7s ~Githii~ thc original net- Each "CBAR" teriu in Table 4 is cor-

work. Flcuce, the solution to the refoml- recth interpreted as a net arc cost, which

ulated problem i\ also at1 optimal solu- is aJ overall cost to the system of one

tion for the origii-ial prol>lem. lmit of flow on the arc in question. The

The OKA solution to the illustrative iioc'ie prices, in t t ~ m . can be corzsiderect

problem depicted in Figure 4 is givcn as \hadow prices for a unit of the flow

in T'11>le 4. Tlie solution vector of opti- commodit~ at the different nodes. nlal ilo1~7s is the column hcaded "S." These arc the flows that nlininlize total co\t defined as.

The coal flow pro1)lem was first solved

S C l j X c j for all i and ii as a single reginleri problem without the possiiblit>~ of seasolla1 storage at the Great Lakes ports. The problen~ was

-\t.here c,, is the traospo~.tation cost on thcu reforrnulatecl as a spatial-temporal

11

Nodes* C1 C2 C3 C4 C5 C6 C7 CR C9 C10 C11 C12 C13 C14 C l 5

* Identifying codes are given in Tables 1-2.

transportation network problen~ and solved for three different levels of storage costs.

The results for a single regime11 trans- portation system, which is simply the summer regimen operated on a year-long basis, are shown in Table 5. Four coal nodes (Butler, Pittsburgh, Middle, and South Ohio) operated at full capa- city, whereas four others (Connellsville, ~~;undsville,Pocahontas, and Belleville) were completely inactive. Coal nodes in the mid-continent section (Princeton, Terre Haute, Belleville, and Peoria) served only the Chicago and Milwaukee nodes, whereas coal producing nodes in the eastern section were able to capture the remainder of the lake coal ~ilarkets. This finding is in accord with Mayer's results noted earlier.

It n7as possible to perform a variety of sensitivity analyses on the model by changing transportation costs and delet- ing different nodes. These results, liow- ever, are not reported here but instead

attention is given to the problenl of storage and seasonal regimens and the modifications which these features re-quire in the model.

Coal flows in the Great 1,aker area take place under two distinct regimens, the first correspoiiding to the months when the lakes are navigable (summer regimen), and the second to those when the lakes are not navigable (winter reg- imen). During the suniliier the coal moves through a network which include? railroad and lake route?. but in winter only the more costl\f railroad network is available. The existence of two transpor- tation regimens involving higher co?ts for the winter makes it convenient in niany cases to collect coal at storage points, either at the destinations where coal received in the sunimer is reserved for minter use. or at the origins or lake " ports where coal mined in the winter is stored for shipment during the summer. If ontinial coal flom7s are to be deter-niin&l in a network which involves two

regimens and the possibility of storage, then a spatial-temporal transportation or transhiprnent forillat has to be adopted. Kriebel [ lo ] and King and Henry [9] also discuss the general problem of inter- temporal optin~al flon7s.

Storage may be viewed as a flow through time and be handled in a manner identical to flows through space. The points in time are identified as nodes, and storage costs and capacities are associated with the flows through time. Since flows through space and flon7s through time are espressed in n~atheinaticall~identical terms, they may in fact be handled simultaneously.

The fornlulation of spatial-temporal flon7s in an OKA framework is illustrated in Figures 5 and 6. In Figure 5 a hypo- thetical interteislporal transhiplllent problem is depicted. The nodes in the graph are associated with two n~mlbi:rs separated by a dot. The first nunlber identifies the node, a ~ l d the second has the value of 1or 2 corresponding to the summer and winter regimens, respec-tively. The arcs shown as dashed lines correspond to flows through time, that is, storage. The three numbers associated with these arcs indicate in order: the storage costs, the upper capacity of the

storage facilities, and a value identifying the lower limit allowed for utilization of the storage facilities.

In Figure 6 there is illustrated an in- terten~poral transpol-tation problem. Each node is identified as an origin (0) or destination ( D) , along with the nuill- ber of the node and a code identifying the summer or winter regimens as in Figure 5. Again, the dashed lines incii- cate flows through time (storage).

The intertemporal transhipment prob- lei11 (Figure 5 ) is the inore appealing fornlulation for handling the analysis of Great Lakes coal flo\vs, since it allows storage to take place at intermediate destinations and, in fact, large an~ouots of coal are stored at the Great Lakes ports. Again, however, data considera- tioils dictate the use of the transporta- tion problem. As was noted earlier, vol- unle rail rates for movement of coal to lake ports differ according to the ulti- mate destination of the coal and it is not possible to consider intermediate nodes and transportation routes, given the data that are available.

Each node, therefore, was separated into a winter node and a sumnler node and the supplies at the coal origins and the denlands at the destinations were

Fig. 5. Int?rt?mpornl transhipment problem Fig. 6, Intertenlporal transportation problem (with storage). (with storage).

divided equally between these winter The solution for 50C per ton storage and summer nodes. The possibility of charges is presented in Table 6. Storctge storage way allowed by placing an arc of coal received in the summer takes l~etween the winter and sunliner nodcs place at Toronto (1,766,000 tons), Ruf- at each origin or de~tination. falo ( 1,510,000 tons), Detroit (4,967,-

The rate on each arc that involve^ a 000 tony ) , and Duluth ( 513,000 tons ) , lake shipment was doubled for the win- ancl there is no storage at any of the ter period. Rates on all-rail shipm~ilts origins. remain the same during the winter. Apparently, aome of the origin nodes

LEA~T-COST COAL HECIZILP,AIALOCATION OF FLO\T.S* FOB DUAL

30C STORAGE COSIS (IIILI~IOXSOF . loss)

Nodes

* Figures in parentheses are winter slripment?; other\ are summer shipments.

~vl-tichhave more favorable rates are able during the sumnier and others ship less to ship during, the sulrmrner an entire coal during the winter than they ship year's supply of coal to receiving nodes during the summer. For example, Butler where coal is then stored for the winter. ships its entire sumlner production to During the winter, the producing ilodes Toronto and Buffalo, and during the win-serve a different set of receiving nodes. ter ships to Toledo, Youngstown, and According11-. some iiodes are active 0111~- Erie. During the summer, Toledo is sup-

LGIST-( .O~T A L L O C A T I O S O F C O \ I , FLO\\ ' I 'OH DUAL REC;I\II.Y,

25,,', s-ronacx COSTS ( ~ I I L L I O S ~ o r - r o w s )

plied from Moundsville, while Youngs- tow11 receives coal from Pittsburgh, Con- nellsville, and Ohio 8, and Erie is supplied from Kanawha.

If storage costs are lowered to 25$ per ton the storage at destinations re-~nains the same. One instance of storage at an origin occurs: Butler stores 2,633,- 000 tons of winter coal for summer use. This allo~vs Butler to expand its doini- ilance of the lake coal market, control- ling all supplies to Buffalo and part of the supply to Detroit (Table 7).

As a further sensitivity test of the inodel another variation on storage cost was considered. At 10$ per ton storage costs even more coal is stored at Butler and some coal is also stored at another origin, Middle ( 150,000 tons). As a re- sult, Middle increases its summer ship- ineats to Green Bay at the expense of Princeton.

This study has investigated least cost flows of bituminous coal from 15 mining districts to 19 receiving centers surround- ing the Great Lakes area. The demands at-the receiving centers represented the aggregated coal consumptions of 139 thermal-electric generating plants and 21 steel lnlants. Translnortation costs were

L L

rail-lakes rates over a 104 link trans~or- tation network. The study has empha- sized the application of the "Out-of-Kilter Aleorithm" in the determination of

0

the optimal least cost flow patterns, and with reference to selected real-world data has illustrated some of the sensi- tivity analyses that can be performed on the model. In the formulation of the problem and the developme~lt of the inodel, significant insights into the spatial organization of this particular transpor- tation system have been realized. 'In other words, some traditional economic geography thenles have been structured

in terms of an analytic framework in- corporating a reasonable llumber of real world features.

1. Alnerican Iron and Steel Institute. Direc-tory of Iron and Steel Works i n the United States and Canada. New York, 1964.

2. Beier, F. J , ancl 3. L. Heskett. A Major hlar- liet for U.S. Bit~iininous Coal i n Canada. Cambridge: Harvard Business School, 1966.

3. Census of hlineral Indt~stries, 1963, j701. I, Summary and Industry Statistics. Wash-ington: Government Printing Ofice.

4. Durbin, E. P, and D. hl. Kroenke. "The Out-of-Kilter Algorithm: A Primer," RAND Corporation: Memorandum RM 5472-PR. December, 1967.

5. Gauthier, I-I. L. "Least Cost Flows in a Capacitated Network: A Brazilian Exam-ple." Geograplzic Studies of Urban Trails- polfation and Network Analysis. Edited by F . Horton. Evanston: Northwestern Uni-1-ersity Press, 1968. pp. 102-27.

6 , Ford, L. K. and D. R. Fulkerson. Flows in Nettuorks. Princeton: Princeton University Press, 1962.

7. Fulkerson, D . R. "An Out-of-Kilter hlethod for Minimal Cost Flow Problems," Journul Soc. Indust. Appl. Mathematics, 9 ( 1961), p p 18-27.

8. Keystone Coal Buyers Manr~al. New Pork: XlcGraw Hill, 1968.

9. King, R. A. ancl 'iV. R. Henry. "Transpor- tation klodels in Studies of Interregional Competition," Journal of Farn Economics, 41 (1939), pp. 997-1011.

10. Kriebel, C. I-I. "Warehousing with Tran-shipment uncler Seasonal Demand," Jozcr-nal of Regional Science, 3 ( 1961), pp. 57-69.

11. hlayer, R. Competition in the Miduestern Coal Industry. Cambridge: Harvard Uni-\-ersity Press, 1964.

12. National Coal Association. Steam-Electric Plant Factors, 1966. i'i~ashington: Division of Econon~ics ancl Statistics, 1967.

13. U.S. Bureau of Slines. hlinerals Yearbook, 1966. Vol. 11, "Fuels." Washington: Gov-ernrnent Printing Office, 1967. p. 620.

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Optimal Transportation Patterns of Coal in the Great Lakes RegionL. King; E. Casetti; J. Odland; K. SempleEconomic Geography, Vol. 47, No. 3. (Jul., 1971), pp. 401-413.Stable URL:

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Literature Cited

9 Transportation Models in Studies of Interregional CompetitionRichard A. King; William R. HenryJournal of Farm Economics, Vol. 41, No. 5, Proceedings of the Annual Meeting of the AmericanFarm Economic Association. (Dec., 1959), pp. 997-1011.Stable URL:

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