Locomotive Scheduling Under Uncertain Demand

Transcription

1 TRANSPORTATION RESEARCH RECORD Locomotive Schedling Under Uncertain Demand ScoTT SMITH AND YosEF SHEFFI Each day, railroads face the problem of allocating power to trains. Often, power reqirements for each train are not known with certainty, and the fleet of locomotives may not be homogeneos. To deal with both of these complications, we formlate a mlticommodity flow problem with convex objective fnction on a timespace network. The convex objective allows s to minimize expected cost nder ncertainty by penalizing trip arcs likely to have too little power. Or soltion heristic sends locomotives down the shortest paths (based on marginal arc costs) in the time-space network and then attempts to improve interchanges of locomotives arond cycles. Two lower bonds are also developed by relaxing the mlticommodity aspect of the problem. In test problems, ranging in size from 5 to 44 arcs, the heristic performed well, with short rnning times and costs averaging within 3 percent of the best of the two lower bonds developed. A problem freqently faced by transportation carriers is the allocation of a fixed spply of vehicles to a given schedle. Examples inclde the allocation of locomotives to freight trains, of bses to transit rotes, and of airplanes to flights. These examples have the following featres in common:. There is a pblished or "committed to" schedle of services that have to be carried ot; 2. The spply of vehicles to <rips can be represented as an integer, mlticommodity minimm cost flow problem over a network of trip, layover, and storage arcs. The problem has mlticommodity aspects becase the vehicle fleet is not homogeneos; for example, locomotives may have different power ratings and airplanes may be of different sizes. (Natrally, however, there are some important differences between the modes. For example, in the rail mode, two or more locomotives typically are sed to meet demand for a given trip, whereas only one airplane is sed for a single airline trip); 3. Even thogh the schedle is fixed, the demand for service may vary. In the rail context, the tonnage of a given train is variable. In the bs or airline context, the nmber of passengers on a given trip will vary. Frther, it may sometimes be desirable not to meet all the demand, for example, by having standees on bses, or refsing airline reservations, or leaving cars behind for the next freight train to pick p. This paper formlates this allocation problem and sggests soltion techniqes in the context of rail. First, backgrond information on both the formlation of the problem and past research in this area is presented. Second, the problem is formlated as a mathematical program. Third, a fast heristic soltion techniqe is presented. Finally, the reslts of the Department of Civil Engineering, Massachsetts Institte of Technology, 77 Massachsetts Avene, Cambridge, Mass. 23. heristic are compared with lower bonds obtained throgh varios relaxations. The techniqes presented here explicitly consider ncertainty in locomotive demand and are able to deal with locomotives of different power ratings. BACKGROUND This section looks at the network representation of the locomotive schedling problem. This formlation nderlies virtally all other attempts in the literatre to develop a soltion for this problem. Some of that research is reviewed in the second part of this section. Time-Space Representation The rail schedling problem is typically formlated as a minimm cost flow problem on a time-space network, which is a graph of locations verss time on which activities are plotted (Figre ). Each node in this network represents a terminal (yard) at a point in time, and arcs are of the following types:. Trip arcs represent trains between the pstream terminal node and the downstream terminal node that the arc connects. dmmy spersorce FIGURE Sample time-space network. Layover arc Trip arc End arc

2 46 There is a power reqirement for each trip, which may be represented either by a lower bond on the arc flow of locomotives or by a penalty fnction that increases greatly as this flow falls below the minimm desired vale; 2. Layover arcs represent short-term storage at a terminal. They have a lower bond of and some fixed cost per nit; 3. Bypass arcs represent long-term storage of nneeded nits and have very low cost per nit; and 4. End arcs represent locomotive reqirements at each terminal at the end of the planning horizon. Any practical timespace representation has a finite planning horizon. Therefore, if this horizon is short, end effects mst be considered. In the model here, we want to know how many locomotives are needed at each terminal at the end of day, week, or whatever period we are modeling. Ths, the end arcs will have either cost fnctions or lower bonds similar to those for trip arcs. This is a mlticommodity network flow problem with either a "bndle constraint" in the lower bond for each arc or a penalty term in the cost fnction that "bndles" the commodities. We flow locomotive nits of varios types throgh the network, bt a minimm level of motive power mst be met for each arc. Past Work Comprehensive reviews of rail schedling are contained in two papers by Assad (,2) and one by Peterson (3). Some of the earliest analytical work in locomotive assignment is that of Bartlett in 57 ( 4), who presented a pairing algorithm to assign vehicles to a fixed schedle. Later, McGaghey et al. (5) described the distribtion of locomotives and cabooses with a time-space network model. They sed an ot-of-kilter algorithm to find the optimal flow of nits throgh a singlecommodity network with a fixed lower bond on the power spplied to each arc. In 76 Florian et al. (6) considered the mlticommodity aspect of locomotive schedling, with fixed lower bonds. They sed Bender's decomposition to solve this mlticommodity flow problem and reported good reslts with medim-size (abot 2 train movements) problems bt had less sccess with larger problems. In 8, Booler (7) formlated the same mlticommodity flow problem bt obtained an integer soltion sing a heristic method based on linear programming. All of this work assmes deterministic, known lower bonds on the power flows. Frthermore, there has been only limited sccess with mlticommodity flows, particlarly with large problems, as already mentioned. As the first step to the explanation of or approach to the problem, the next section formlates the locomotive allocation problem as a mathematical program. Later we assme that the lower bond is not known with certainty, and we reformlate the problem sing a penalty fnction. FORMULATION This section starts with the "traditional" mathematical programming formlation of the problem. It then incorporates the ncertainty in locomotive reqirements directly into the TRANSPORTATION RESEARCH RECORD 25 power needs. The formlation and notation that follow relate to the time-space representation of the locomotive assignment problem. Define the following: i = arc nmber in the time-space network, j = locomotive type, x = flow of locomotive type j on arc i, H = horsepower rating of locomotive type j s = horsepower flow on arc i (s; = "i. Hr;), x = vector of locomotive flows of type j on all arcs, C (s;) general operating cost fnction on arc i, c, operating cost per nit HP flow on arc i,! demand for power on trip arc i (this may be either a deterministic vale or a random variable), F, cmlative distribtion fnction for l;, f, = probability density fnction for l, µ = average demand for power (expected vale of l;), rr, standard deviation of demand for power, Pls) = general penalty cost fnction for power shortfall on link i, penalty per nit of power shortfall on arc i, p, Z,(s,) cost fnction on arc i (inclding operating cost and penalty), N = node arc incidence matrix for the time-space network, and b = vector of sorces and sinks for locomotive type j. We first formlate the problem with deterministic lower bonds on the power reqirements, and then show how these lower bonds can be modeled as random variables. That formlation leads to the se of penalty fnctions in the objective of the mathematical program. In all cases, we assme that the lower bond is expressed in terms of horsepower, so that combinations of locomotive types with the same total power rating are interchangeable. The mathematical formlation is sbject to "i. Hr, ~ for all i Nx = b for all j x; integer and ~ () (2) (3) (4,5) This is a mlticommodity minimm cost network flow problem. The objective is to assign all locomotive types j to the network, whose node-arc incidence matrix is N, at minimm cost. The varios locomotive types cannot be assigned separately becase they all contribte to the power on each train link. This bndling of locomotive types appears in the lower bond constraint 2. Recall that the original problem calls for ncertainty in demand. Therefore, the fixed lower bond formlation of Eqations throgh 5 may not be realistic. This is becase a fixed lower bond can be thoght of as an infinite cost penalty on flows below it. Sch a cost fnction for an arc with a lower bond of 5 and cost per nit flow of c is shown in Figre 2. According to this cost fnction, 4 HP on this train has an infinite cost whereas 5 HP has the lowest cost,

3 Smith and Sheffi 47 N (/) 5 Flow s(hp) FIGURE 2 Arc cost fnction with deterministic lower bond. 2. The distribtion of the demand for power, l;. The probability density fnction for l; is denoted by f; and its cmlative distribtion by F This distribtion has mean µ, and variance a}; and 3. The magnitde and shape of the penalty, given a shortage. This will have the form P (l - s;) (recall thats = IjH;c J. In the eqations that follow, the sbscript i is dropped to make the notation easier to read. The cost as a fnction of power spply, s, is Z(s). Given the probability density fnction of l, this cost can be expressed as follows: r Z(s) = C(s) + P(l - s) f(i) di (6) As a tractable approximation to the normal distribtion, we assme that l follows a logistic distribtion where F({) = [ + exp ((al) (µ, - {))] - (7) "' N f({) = (al) exp((a!) (µ, - /)) x [ + exp((a/rr) (µ, - ))]-2 (8) "' where a = -rr/\/j =.8 The cost fnction, Z(s), now becomes FIGURE 3 5 Flow s(hp) Arc cost fnction with ncertain power demand. Z(s) = cs + p r lf(l) di - ps r f({) di = cs + p r If(/) di + ps(f(s) - ) () reqirement is actally a random variable, which can attain vales lower than 5 HP. A more realistic cost fnction (Figre 3) might be obtained by reasoning as follows: After integrating (by parts) the cost becomes Z(s) = cs + (pla)log(l!f(s)) () Or forecast demand is 5 HP bt we know the forecast might be off by-.as mch as HP. Therefore, or safest assignment wold be to have 6 HP on this train; 5 HP wold probably work; 4 HP wold be nacceptable. However, we do not want a fixed lower bond of 6 HP becase 5 HP may be all the power we have available. To yield a cost fnction that looks like the one shown in Figre 3, we move constraint 2 into the objective with a penalty term. By doing so, we acknowledge that (a) power reqirements may vary in a random manner and (b) the lower bond on power is not a hard-and-fast rle; rather, there is a trade-off between service qality and the amont of power spplied. This formlation provides a more realistic representation and, argably, makes the problem easier to solve. Ths, rather than having the demand for power,, as a fixed lower bond, it is modeled as a random variable. The shape of the cost fnction derived this way depends on three elements:. The operating cost of additional power; we assme this is some fnction, C;(s;) of the power spplied, s This cost is assmed independent of the demand,, and locomotive type; In the logistic distribtion mentioned above, negative demand is theoretically possible. However the parameters are sch that this is not a problem in practice. Cost fnctions were also derived for niform and gamma distribted demands, with some examples plotted in Figre 4. Note the following points:. All fnctions approach the CT = case asymptotically as s becomes either very large or very small with respect to f.l; 2. There is not mch difference between the ca es wi.th logistic, niform, and gamma distribtion-. This is reassring, I ecase it indicates that the exact shape of the distribtion for demand may not matter mch, and we can e the logi tic distribtion to form a tractable cost fnction. Althogh the gamma di tribtion i probably the most realistic r presentation of l (it never ha vales bel w zero) it does not yield a clo ed form for Z(s) and, conseqently, i difficlt to work with; 3. All fnctions have the desired shape (as in Figre 3). In the remainder of the paper, we assme that the demand for power I is logistically distribted with mean µ, and standard deviation.

4 48 TRANSPORTATION RESEARCH RECORD 25 2 f'=5hp o logistic CT =IOOOHP + niform CT= IOOOHP o gamma CT =I OOOHP o logistic CT = 2HP " niform CT = 2HP o gamma CT = 2HP x CT =O (fixed demand) the incremental assignment method and the Frank-Wolfe algorithm sed to solve the traffic assignment problem. Review of Incremental Assignment and Frank Wolfe N (/) 5 Flow s (HP) FIGURE 4 Arc cost fnction nder varios demand distribtions, The problem we are solving can now be smmarized as follows: Minimize the total cost of flowing power on all arcs, sbject to the following constraints:. Flow conservation constraints are met for each power class and 2. The nmber of locomotives on each arc is integer and nonnegative. In other words: min I;{c,s, + (p,a/a) log(l + exp [(a/µ;)(µ,-s,)])} () sbject to Nxj = bj for all j X;j integer and ;::: (As before, s, is defined as IjHr,i and a =.8.) (2) (3,4) This representation, in addition to having the advantages mentioned earlier, also lends itself well to the treatment of ncertain end effect arcs, which can be modeled like trip arcs with high CJ. The next section looks at soltion approaches for this problem. SOLUTION APPROACHES We have formlated a nonlinear, mlticommodity integer network flow problem. Exact soltion techniqes for sch a problem are likely to be neither easy nor fast. This problem, however, is similar to the (mlticommodity) traffic assignment eqivalent program. The traffic assignment problem deals with the assignment of an origin-destination trip matrix to a transportation network so as to minimize each ser's travel time (or cost). In traffic assignment, the arcs have a fzzy pper bond that arises from highway congestion effects, whereas or problem has a fzzy lower bond arising from power shortfall. Both problems can be formlated as a soltion to a convex program over network flows. The heristrc sed to solve the locomotive schedling borrows from both In incremental assignment, wc start with zero flow on the network and choose a nmber of increments, n. The algorithm, then, in each of n iterations, greedily assigns lln of the total flow along each shortest path from origin to destination. Becase the cost fnction is nonlinear, these shortest paths may change with each iteration (8). Althogh this algorithm can be set to maintain integrality of the soltion, has intitive appeal, and is easy to implement, it has a nmber of shortcomings. First, it does not always work, as shown throgh conterexamples by Ferland et al. (). Second, if it is to prodce reasonable soltions, the nmber of increments may have to be very large, ths ndly increasing the rnning time. The Frank-Wolfe algorithm (8) is a feasible direction method and therefore starts with an initial feasible soltion and moves to improved soltions, maintaining feasibility throghot. It does this by developing linear approximations to the objective fnction and by solving linear sbproblems to find the correct distances to move in improving directions. With cost minimization and a convex objective fnction, the Frank-Wolfe method does converge to the optimal soltion and is easy to implement on networks. Frthermore, a lower bond to the optimal soltion is provided at each iteration. Flows, however, are split between paths; ths integrality is lost. In most traffic assignment problems, convergence is rapid (abot five iterations) bt may be slowed if the soltion is in a highly nonlinear portion of the objective fnction (). The Two-Commodity Heristic Approach The heristic presented here obtains a feasible soltion to the problem throgh incremental assignment, and then obtains improvement. throgh a fea ible direction method. Unlike Frank-Wolfe, it maintains integrality and exploit the int - grality of the problem by moving one locomotive nit at a time, ths obviating the need for line searches in the feasible direction method. The heristic rns in two phases. First, it loads the network by assigning one nit at a time to shortest paths. This is referred to as the GREEDY phase. Second, after the network is loaded, it attempts improvements by sending flows arond agmenting negative cycles in an INTERCHANGE phase. These agmenting cycles are similar to the agmenting paths of maximm flow algorithms in that they inclde both forward and reverse arcs; ths flow can be removed from an arc when going against the flow direction. Both phases are otlined in more detail below: GREEDY Phase Step. Initialization. Start with zero flow, and compte arc marginal costs at zero flow. Step. Send one nit down the shortest path from any sorce to the spersink; pdate arc flows. (Note rhat the order

5 Smith and Sheffi 4 in which nits are selected will affect the otcome. For the experiments here, the largest nits were arbitrarily selected first). Step 2. Recompte arc marginal costs along that path. Step 3. If all nits have been sent, go to the INTER CHANGE phase, otherwise, go to Step. Note that for large problems, this phase can be speeded p by sending more than one nit at first. Also, saving the shortest path tree rooted at the sink node and reoptimizing it after every assignment (rather than recompting the shortest path at each iteration) offers another opportnity to speed p this phase of the heristic. INTERCHANGE Phase tep. Identify arcs thal are ca ndidates for improvement. In the pre enl implementation the e are arc with large negative marginal co t (i.e., trip arcs with insfficient pow r). Step L. earcb for a flow-agmenting negative cycle involving ome candidate arc. lf no negative cycle can be fond in the network, stop. Otherwise, go to Step 2. Step 2. Interchange flows arond this cycle and pdate arc marginal costs. Go back to Step. The interchanges performed in the second phase are generally more complicated than simply sending one nit of flow arond the cycle. This is becase the interchanges often involve minor HP changes and ths may involve the exchange of two Joe motive type. For example, if or two locomotives types have 2 I P and 3 HP, respectively two interchanges that wo ld prodce a small hor epower change wold be:. Add one high-power and remove one low-power nit on the arc that needs additional power (net change of HP) or 2. Add two I.ow-power and remove one high-power nit (net change of HP). Within the heristic, these interchanges are performed in the following manner:. Create an ordered list of arcs that will benefit from more power. 2. Attempt to find an improving interchange involving one of the arcs on the ordered list. This is done as follows: 2a. Select the first interchange type. 2b. REPEAT. Select the first arc. REPEAT. Try to find an improving interchange (flowagmenting negative cycle) with this arc and interchange type. If one is fond, go to Step 3. Otherwise, select the next arc UNTIL all arcs examined. Consider the next interchange type. UNTIL all interchange types have been considered. 3. If we have fond an interchange Perform the interchange and pdate arc marginal costs. Example Update the ordered list of arcs, go back to Step 2. Otherwise, we terminate, becase no interchange can be fond. Consider a two-node network with two trip arcs and one bypass arc (Figre 5). The arc costs are showrt in Table. The locomotive spply incldes one high-powered (3 HP) and two low-powered (2 HP) locomotives. The greedy phase of the heristic performs as follows:. Send the high-powered nit down arc. 2. Send a low-powered nit down arc Send a low-powered nit down arc. We now have 5 HP on arc and 2 HP on arc 2. Arc 2 is short of power. E... arc,? 3HP [!] bypass arc 2 HP - -:::;r FIGURE 5 Sample network. TABLE COSTS FOR THE EXAMPLE NETWORK Parameters Flow (HP) arc arc 2 3 c p Costs bypass arc arc 2 bypass

6 5 Moving to the interchange phase, we see that the highpowered nit on arc can be exchanged with the low-powered nit on arc 2. After this exchange is performed, we are finished, as no more improving interchanges can be seen. The progress of the heristic in terms of the arc flows and the objective fnction is plotted in Figre 6. The method works by first moving in big jmps (whole nits) toward the optimal soltion, then refining the soltion by making smhller jmps (interchanges). Advantages and Disadvantages of the Heristic Approach This dob!e-ph3se heristic has several advantages. First, it maintains feasibility throghot. Second, by always moving in an improving direction, the method is intitively appealing. Therefore, it may lend itself well to interactive se. Third, it is easy to incorporate other side constraints into the framework of this heristic. Some of these are the following:. Prohibition of certain locomotive types from certain sections of track, 2. Assigning newer, more reliable, locomotives to highpriority trains, and 3. Sending locomotives to home shops for schedled maintenance. Finally, the heristic is also qite fast and prodces close to optimal reslts in several test problems. The disadvantages of this method are, first, its heristic natre: optimality is not garanteed. In addition, the complexity of the interchange phase increases as the nmber of the commodities is increased beyond two. This was not a problem in the case stdy reported later bt may present difficlty in other applications. LOWER BOUNDS Several lower bonds were derived to test the performance of the heristic. A lower bond may be (a) an optimal soltion to a relaxed version of the primal problem, (b) a dal feasible Ii. :::c Q C\J... ~ c J: E Feosi ble Points Flow on Arc I ( HP) FIGURE 6 Progress to the best heristic soltion. TRANSPORTATION RESEARCH RECORD 25 soltion, or (c) some combination of the foregoing, sch as a dal feasible soltion to a relaxed version of the primal. Two lower bonds were derived for the problem discssed here. Frank-Wolfe Relaxation By relaxing the integrality constraint in the original problem, we obtain a single-commodity (horsepower) network flow problem with convex objective fnction. This can be solved with the Frank-Wolfe (convex combinations) method already reviewed. The Frank-Wolfe method provides both a feasible soltion and lower bond on the relaxed problem at every iteration. This lower bond on the relaxed problem will, natrally, also provide a lower bond on the original minimization problem in the following manner: heristic soltion ~ optimal soltion ~ optimal soltion to relaxed problem ~ lower bond to relaxed problem Unfortnately, a complete relaxation of the integrality constraint in this manner may lead to a large gap between the optimal soltion and the optimal soltion to the relaxed problem. Sch a gap makes it difficlt to evalate the performance of the heristic. Greatest Common Factor (GCF) Relaxation This relaxation is based on the following observation: Any feasible soltion will have a horsepower flow in each arc that is a mltiple of the greatest common factor (GCF) of the horsepower ratings. For example, if there are two locomotive types rated at 2 HP and 3 HP, the flow on each arc will be a mltiple of HP. If there are three locomotive types with ratings of 75 HP, 2 HP, and 3 HP, the flow on each arc mst be a mltiple of 25 HP. Any other horsepower flow is infeasible becase it cannot possibly be prodced as a combination of locomotive flows. We can se this observation to transform the original network problem with convex nonlinear cost fnction to a conventional linear network flow problem with integral pper bonds on the arc flows. This latter problem is easy to solve. The steps in the transformation are. Let b = GCF of the locomotive power ratings. 2. Transform the cost fnction by making it piecewise linear with breakpoints at mltiples of b. See Figre 7. Note that the cost fnction remains convex and we change its vale only at points that cannot be generated by any combination of locomotives. 3. Create n arcs (one for each division in the cost fnction) for each original arc in the network (Figre 8). (We do not need to add additional constraints becase the cost fnction is still convex, ths the arcs will be loaded in the correct order). 4. Becase or sorces, sinks, and bonds are all mltiples of b, we can scale flows down by a factor of b withot losing integrality. The soltion to this problem, when scaled back p, wiil be a mitipie of b.

7 Smith and Sheffi 5 en ~ NIO en ~ \ d \ \ ' " b d3 2 b 3 b Flow s(hp} FIGURE 7 Piecewise linearization of the arc cost fnction. from to FIGURE 8 fnction. Z(s) pper bond = b 4b pper bond = infinity ~ Transformation to a piecewise linear cost "barely adeqate." The basic networks had pie =, e =.5, er/µ =.2, bt these were systematically varied in some of the test problems. Table 2 shows these parameters and reslts for the varios networks. The heristic-optimal vales of the objective fnction behaved reasonably, with the following configrations having increased costs over the baseline (Problems Sl, Ll): Higher penalty term: Increasing the penalty term tenfold approximately dobled the objective fnction (Problems S3, L3). Higher standard deviation: Increasing er/µ from. to also increased the cost sbstantially (Problems S5, LS). Lower power spply: Becase the initial power spply was "barely adeqate," redcing it increased costs somewhat as more trains were nderpowered (Problems S6, L6). We wold expect average rnning times to be a fnction of both the nmber of locomotives spplied and of the size of the network. In the cases here, as the networks became larger, rnning times seemed to be O(nt) where n is the nmber of nodes and t the nmber of locomotives. They were reasonable in all cases, ranging from O.S sec for the smallest network to 63 sec for the largest. This is acceptable becase it is envisioned that in an operating environment, the model will be rn abot once per 8-hr shift rather than continosly. The nmerical reslts were normalized to the best lower bond fond, which was the GCF lower bond. The reslts of the heristic were, on average, within 3 percent of this bond. These normalized reslts are shown in Table 2. Problems with a flat objective fnction (low pie and high er/µ - problems, SS, L2, LS) tended to perform better with reslts, on average, within abot percent of the lower bond. Conversely, problems with a highly nonlinear objective (S3, S4, L3, L4) give reslts that were on average only within 6-7 percent of the lower bond. The Frank-Wolfe algorithm also tended to have poor convergence on these problems. 5. We now have a conventional mmmm cost network flow problem that will provide a valid lower bond on the original problem becase (a) all feasible soltions to the original problem are feasible in this problem and (b) the cost fnction was changed only at infeasible points in the original problem. Note, however, that a feasible soltion in this problem may not be feasible in the original problem. An example wold be an arc flow that is greater than zero bt less than the power rating of the smallest locomotive. TEST PROBLEMS AND RESULTS The heristic was implemented in FORTRAN on a Micro Vax II rnning Micro VMS 4.5 and tested on problems of for time-space network configrations (Table 2). The smallest of these networks is shown in Figre, and Problems Ll-LS were drawn from an actal 3-day train schedle for the Grand Trnk Western Railroad. The logistic form of the cost fnction was sed in all cases. The power reqirements varied from 3 HP to 5, HP on the trip arcs, and locomotive spplies were fixed to be FURTHER WORK We have developed a model that deals explicitly with the ncertainty in power reqirements. Moreover, the heristic sed to solve this model is promising becase it is both fast and fairly accrate. Frther research shold focs on improvements to the heristic and incorporation of schedle variability. The present implementation of the heristic does not optimize speed. Some improvements, mentioned earlier, inclde sending more than one nit at a time in the early stages of the heristic, and keeping and reoptimizing a shortest path tree rooted at the spersink, rather than recalclating shortest paths for each iteration. Another improvement to the heristic wold be the incorporation of additional side constraints and provision for more than two commodities. Of possibly greater interest is the incorporation of schedle variability. Althogh the model now assmes a fixed schedle, one way to do this wold be to incorporate the heristic into an interactive system that displays where and when shortages of locomotives are likely to occr, and then allowing the ser to adjst the schedle accordingly before rnning the heristic

8 TABLE 2 TEST RESULTS GCF Trial Trips arcs nodes khp p/c ''/,,µ. Cost greedy Normalized Costs int GCF Tl 3 ls T2 3 ls Sl S3 so S SS S S Ml 3S S M2 3S S Ll L L so L LS L L L average Trips - nmber of trips in this network khp = total horsepower spply (thosands HP) p/ c - ratio of penalty to cost term <r/p - coefficient of v ariation for power demand GCF Cost = Total cost (thosands $) for GCF relaxation greedy - total cost after GREEDY phase / GCF Cost int - FW - total cost after INTERCHANGE phase / GCF Cost total cost of Frank-Wolfe soltion / GCF Cost FWLB - GCF - total cost of Frank-Wolfe lower bond / GCF Cost total cost of GCF relaxati on / GCF Cost rn time - total rnni ng time for the heristic, exclding inpt/otpt and comptation time for relaxations.

9 Smith and Sheffi again. However, to adjst schedles within the algorithm will reqire consideration of system wide train schedling and cstomer demand, both of which are very difficlt to qantify. ACKNOWLEDGMENT This research was fnded by the Grand Trnk Western Railroad. REFERENCES. A. A. Assad. Models for Rail Transportation. Transportation Research, Vol. 4A, 8, pp A. A. Assad. Analytical Models in Rail Transportation: An Annotated Bibliography. INFOR, Vol., No., 8, pp E. R. Peterson. Operations Research and Rail Transportation. In Scientific Management of Transport Systems (N. K. Jaiswal, ed.), Elsevier North-Holland, New York, 8, pp T. E. Bartlett. An Algorithm for the Minimm Nmber of Transport Units to Maintain a Fixed Schedle. Naval Research Logistics Qarterly, Vol. 4, No. 2, 57, pp R. S. McGaghey, K. W. Gohring, and R. N. McBrayer. Planning Locomotive and Caboose Distribtion. Rail International, Vol. 4, 73, pp M. Florian, G. Bshnell, J. Ferland, G. Gerin, and L. Nastansky. The Engine Schedling Problem in a Ra.ilway Network. INFOR, Vol. 4, No. 2, 76, pp J. M. P. Booler. The Soltion of a Railway Locomotive Schedling Problem. Jornal of the Operational Research Society, Vol. 3, 8, pp Y. Sheffi. Urban Transportation Networks. Prentice-Hall, Englewood Cliffs, N.J., 85.. J. A. Ferland, M. Florian, and C. Achim. On Incremental Methods for Traffic Assignment. Transportation Research, Vol., 75, pp S. E. Mimas. Eqilibrim Traffic Assignment Models for Urban Networks. Master's thesis. Department of Civil Engineering, Massachsetts Institte of Technology, Cambridge,