Mathematical Modeling of Earthwork Optimization Problems

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1 Mathematcal Modelng of Earthwork Optmzaton Problems Yang J 1, Floran Sepp 2, André Borrmann 1, Stefan Ruzka 2, Ernst Rank 1 1 Char for Computaton n Engneerng, Technsche Unverstät München ² Optmzaton Group, Department of Mathematcs, Technsche Unverstät Kaserslautern Abstract In the past ths research efforts n optmzng earthwork processes focused manly on mnmzng transportaton costs and mass haul dstances, respectvely. Ths knd of optmzaton problem, well known as earthwork allocaton problem can be solved by applyng lnear programmng technques. As a result, the most cost-effcent cut-to-fll assgnments wll be found. In ths artcle, startng from an optmal cut-to-fll assgnment, we formulate a new correspondng combnatoral optmzaton problem. Ths earthwork secton dvson problem arses when a large road proect s dvded nto several lnear constructon sectons and tendered to dfferent normally non-cooperatng constructon companes. The optmzaton obectve s to partton the optmzed cut-to-fll-assgnments n dfferent earthwork sectons wth mnmal earth movements between them. Ths problem s subected to certan user-defned constrants, lke number of sectons, mnmal and maxmal secton-length, etc. The proposed soluton model wll be ntegrated nto an earthwork modelng and assessment system whch allows performng a quantty take-off from a roadway model to provde the necessary nput data for the optmzaton algorthms. Keywords: earthwork optmzaton, lnear programmng, road constructon, mathematcal modelng. 1 Introducton Earthwork s the maor workng task n road constructon proects and characterzed by large quanttes of earth materal whch have to be excavated, transported, and flled, possbly over a long dstance. Therefore, lnear programmng (LP) technques have been appled n order to mnmze the transportaton costs and the mass haul dstances n the earthwork processes, respectvely. The frst LPmodel of ths earthwork allocaton problem has been formulated and developed by Stark and Mayer (1983), further studes and extensons of ths model have been done by Easa (1987 and 1988), Jayawardane (1990) and Son (2005). As a result, the LP soluton provdes the optmal cut-fllassgnments and determnes the correspondng amount of earth to be hauled. Nowadays, large road constructon proects, such as hghway proects, wll usually be tendered to dfferent constructors or to sub-constructors by a general constructor. The dvded sub-proects can be processed n parallel n order to reduce the overall proect duraton. Normally, t s dffcult to establsh some knd of cooperaton between these constructors. Consequently, f the dvson of earthwork sectons s not optmal, t may happen that one or more constructon sectons suffer from consderable overflow of earth materals whle the other sectons demand addtonal materals from external borrow pts. Although the materal flows can be balanced durng the constructon phase wth

2 a lot of coordnaton efforts, ths wll usually result n addtonal costs for the remtter. Accordngly, t s advantageous to solve the earthwork secton dvson problem at a very early stage n order to support the tender or general constructor to make an optmal decson. The optmzaton results of these two optmzaton problems can be ntegrated n exstng computer-aded earthwork systems whch have been developed n prevous research efforts. Ths ncludes earthwork control systems (Askew et al., 2002, Km et al., 2003), earthwork modelng and smulaton systems (Chahrour 2007, J et al., 2009) and 4D vrtual road constructon frameworks (Söderström and Olofsson, 2007). 2 Earthwork allocaton problem In road constructon proects, cut and fll areas are tradtonally defned by ntersectng the road level wth the terran level vertcally (Fgure 1a). The quanttes of cut and fll areas can be calculated usng numercal methods, dependng on the natonal regulaton n cvl engneerng, such as the Gauß- Ellng-method appled n Germany (REB 1979). The mass haul dstance can be defned as Eucldean dstance between the centre ponts of cut and fll areas. Fgure 1. (a) Example of cut and fll areas n road constructon proect; (b) correspondng bpartte drected graph. To formulate the optmzaton problem, we defne G = (P, E) to denote a bpartte graph whch contans of a vertex set P and the edge set E. The set of vertces P s parttoned nto two dsont subsets U and V of P. The set U conssts of those vertces correspondng to cut areas and, analogously, the set V represents vertces correspondng to fll areas. For each vertex P, the parameter X denotes the amount of materal to be sent (f U ) or to be flled (f V ). We may assume that the total amount to be sent equals the total amount to be flled by ntroducng dump stes and borrow pts: A dump ste s used to dump earth materal due to materal overflow. A borrow pt provdes fllng materals whch have been bought n addton. A drected edge e s ntroduced for each par of vertces (,) where s a vertex correspondng to a cut area and s a vertex correspondng to a fll area. Each of these edges mrrors the possblty of sendng materal from a cut area to a fll area. Addtonally, each edge e has an assocated cost c whch represents the cost of transportng one mass unt of materal from to. A decson varable x s assgned to each of the drected edges n the set E. It denotes the quanttes of earth to be hauled from cut to fll followng the edge drecton (Fgure 1b). We can model the earthwork allocaton problem as a lnear programmng problem (cf. Fgure 2). We assume that the (known) transportaton cost along each edge (,) s non-negatve,.e., c 0. The obectve functon (1) s to mnmze the total transportaton cost. Due to the fact that n the real world only postve materal flows make sense, the decson varables x are restrcted to be non-negatve (see Constrant (4)). Constrant (2) mples that the total quantty of materal to be hauled from some cut area to all fll areas equals the total quantty of materal X provded by cut. Constrant (3) s smlar to (2) for the requrements n.

3 Fgure 2. Mathematcal formulaton of earthwork allocaton problem. Ths formulaton s a smplfed mnmal cost flow problem and can be solved effcently usng network flow algorthms (see Ahua, 1993). Havng solved the optmzaton model above, the amount of earth x to be moved from a cut area to a fll area, such that the overall transportaton cost s mnmal, s known (Fgure 1b). A real-world example wll be presented n Secton 4. 3 Earthwork secton dvson problem The earthwork secton dvson problem emerges when a large road constructon proect s dvded nto several separate earthwork sectons. The obectve of ths optmzaton problem s to obtan a reasonable dvson of the proect such that n each earthwork secton the quanttes of excavated materal and fllng materal are preferably balanced n order to avod nteractons between the sectons. As mentoned before, we propose a two-step optmzaton algorthm for ths problem. At frst we solve the earthwork allocaton problem n order to fnd a mnmal cost cut-to-fll assgnment. In the second step we dentfy the secton dvson havng the least necessary overall earth movement between dfferent earthwork sectons among all secton dvsons meetng the demands (such as desred number of earth sectons or maxmal length of a secton). Fgure 3. (a) Example of a feasble secton dvson wth three resultng earthwork sectons A 1 v 1, p 2 u 1, p 3 v 2, B 4 u 2, p 5 v 3, p 6 u 3 and C 7 v 4, p 8 u 4, p 9 v 5. (b) Cut-to-fll assgnment matrx ( x ) obtaned from solvng the earthwork allocaton problem optmally. In order to be able to formulate ths problem, we consder the set P U V together wth the ndex set I 1,, n, representng the possble postons for a secton dvson,.e., the cut and fll areas ordered accordng to ther actual appearance along the constructon proect. Hence, an earthwork secton ES from poston p to poston p conssts of all cuts and flls located n between: ES k P : k. The requred materal flow between poston p and poston p s exactly the value x kl obtaned from the earthwork allocaton problem, gven that p uk s a cut and p vl s a fll, and zero otherwse. An example of a secton dvson s llustrated n Fgure 3.

4 The earthwork secton dvson problem can be formulated as to fnd a feasble dvson of the proect nto earthwork sectons, such that the total materal flow between dfferent sectons s mnmal. Ths combnatoral optmzaton problem can be expressed by a bnary lnear program (BP) wth decson varablesb, representng the earthwork sectons, whch are nterpreted as follows: 1, f an earthwork secton begns at poston and ends at poston b 0, else Obvously, not all possble combnatons of those varables correspond to a feasble earthwork secton dvson, e.g. t s not allowed to have gaps between the sectons, and dfferent sectons must not overlap. Therefore, we need a number of constrants makng sure that we obtan a soluton whch fulfls our requrements. Fgure 4: Mathematcal formulaton of earthwork secton dvson problem. In the followng we want to deduce the constrants appearng n our BP formulaton (n Fgure 4). As mentoned before, we do not want earthwork sectons to overlap. In partcular ths means that at each poston p, 1,, n, at most one secton can begn or end, whch s expressed by nequaltes (2) and (3). The specal cases to consder, namely the frst and last poston, where an earthwork secton has to begn and end, respectvely, are captured n constrant (4) and (5). Snce we want to cover the whole road proect wth our secton dvson, t s necessary that a new secton begns rght after a secton ends. Conversely, no secton can start at poston p 1 wthout the precedng secton

endng at poston p, f we dsallow gaps. These propertes are guaranteed by condton (6) n our program, snce by (2) and (3) both of the sums n (6) can only admt the values zero or one.

5 endng at poston p, f we dsallow gaps. These propertes are guaranteed by condton (6) n our program, snce by (2) and (3) both of the sums n (6) can only admt the values zero or one. Observe that these constrants also prevent overlappng sectons. Hence, the aforementoned (n-)equaltes along wth the varable defnton (12) are suffcent to descrbe the feasble secton dvsons. Nevertheless, t may be useful to add several other constrants n order to avod trval solutons, such as the secton dvson only consstng of one secton. The addton of (7) and (8) wth user defned ntegers A max and Amn flters out all secton dvsons n whch the number of resultng earthwork sectons exceeds A max or s below A mn. Let d denote the actual dstance between postons p and p. Then n a smlar manner constrants (9) and (10) ensure that each earthwork secton has a mnmal length of D mn and a maxmal length of D max, where D mn and D max are user defned values. Optonally, the addton of condton (11) makes sure that all earthwork sectons nclude more than ust a sngle cut or fll area. As stated before, the obectve of our optmzaton problem s to fnd a secton dvson wth mnmal ntersectonal earth movement. Therefore t s straghtforward dea to defne the obectve functon value for a feasble dvson by smply summng up all materal flows between dfferent earthwork sectons. However, the materal flow between two non-adacent sectons also nfluences all ntermedate sectons and therefore should be especally punshed. Fgure 5. Cut-to-fll assgnment matrx ( x ) obtaned from the earthwork allocaton problem and matrx ( b ) represent the earthwork secton dvsons. (a) Soluton wth three earthwork sectons ES 1 1, p 2, ES 2 3, p 4, p 5, p 6 and ES 3 7, p 8, p 9, represented byb 12 b36 b79 1, wth obectve value x12 x34 x35 ; (b) soluton b 12 b34 b59 1 wth correspondng sectons ES 1 1, p 2, ES 2 3, p 4 and ES 3 5, p 6, p 7, p 8, p 9, havng the better obectve value x 12. In our obectve functon (1), for each earthwork secton n the dvson ( b 1) we add the 1 n term ( x kl x lk ), whch expresses the sum of materal flows passng the startng poston p. By k 1 l dong so, we also count the materal to be transported beyond the endng poston of a secton, f

6 exstent, snce n a feasble dvson another secton has to begn n the subsequent poston. Consequently, a materal flow x kl s counted each tme t crosses the border of a secton. An example for feasble earthwork secton dvsons wth dfferent obectve value s presented n Fgure 5. 4 Real-world example A large federal hghway constructon proect has been planned to be constructed n Germany n the next year. The lnear constructon ste whch conssts of 41 cut and fll areas s about 20 klometers long. As we can see n the followng fgure, the cut and fll areas are dstrbuted along the entre proect constructon ste. Fgure 6. Part of vertcal algnment of the road constructon proect. The lst of earthwork quanttes correspondng to the cut and fll areas n Fgure 6 are presented n Table 1 of Fgure 7, as well as the optmal cut-to-fll assgnments resulted from solvng earthwork allocaton problem. Fgure 7.Part of earthwork quanttes and result of earthwork allocaton problem. The large road proect wll be tendered to 3 dfferent constructon companes, and each constructon secton must have a length of 4 klometers, at least. Ths earthwork secton dvson problem can be formulated and solved usng the bnary lnear program provded n ths paper. In Fgure 8, the optmal earthwork secton dvson regardng user gven parameters s llustrated. Fgure 8. Optmal earthwork secton dvsons: 3 earthwork secton, each secton at least 4 klometers long. Wth the ntegraton of the powerful open-source lnear and mx-nteger programmng solver (GLKP v.4.3, 2009) nto the earthwork assessment system ForBAU Integrator (J et al., 2009), the solutons of the two optmzaton problems can be found n acceptable runnng tme, e.g. for dvdng 41 earthwork areas, the solver computes the optmal soluton wthn 3 seconds on a common machne.

7 5 Concluson and future research Ths paper ntroduces two maor problems arsng n optmzng earthwork processes: fndng the most cost-effcent cut-to-fll-assgnments (earthwork allocaton problem) and dvdng a large earthwork proect nto sectons wth mnmal nter-sectonal materal flows (earthwork secton dvson problem). Ths paper also presents the mathematcal formulaton and soluton model of these two problems usng (bnary) lnear programmng technque. The ntroduced models and ther solutons are appled n a real-world constructon proect, a hghway constructon ste n Germany, to enhance the productvty n constructon proect. In future research, we am at solvng two further optmzaton problems focusng on mnmzng earth transport equpments and the proect duraton: Wth a gven number of transporters, what s the mnmal earthwork duraton? To the gven earthwork duraton, what s the mnmal number of transporters requred to execute all transportaton wthn the prescrbed duraton? 6 Acknowledgements The research work presented n ths paper was carred out wthn the ForBAU proect (Borrmann et al., 2009) whch s funded by the Bavaran Research Foundaton (Bayersche Forschungsstftung). We also gratefully acknowledge partal fnancal support by DFG Graduertenkolleg 753 Mathematk und Praxs. 7 References AHUJA, R.K., MAGNANTI, T.L., ORLIN, J.B., Network Flows. Prentce Hall, Inc. ASKEW, W.H., AL-JIBOURI, S.H., MAWDESLEY, M.J., PATTERSON, D.E., Plannng Lnear Constructon Proects: Automated Method for the Generaton of Earthwork Actvtes. Automaton n Constructon, 11( ). BORRMANN, A., JI, Y., WU, I-C., OBERGRIESSER, M., RANK, E., KLAUBERT, W., GÜNTHNER W., ForBAU - The Vrtual Constructon Ste Proect. In: The 26th CIB-W78 Conference on Managng IT n Constructon, 2009, Istanbul, Turkey. CHAHROUR, R., Integraton von CAD und Smulaton auf Bass von Produktmodellen m Erdbau. Kassel Unversty Press GmbH, Kassel. EASA, M.S., Earthwork Allocatons wth Non-constant Unt Costs. Journal of Constructon Engneerng and Management, 113(1). EASA, M.S., Earthwork Allocatons wth Lnear Unt Costs. Journal of Constructon Engneerng and Management, 114(4). GLPK, GNU Lnear Programmng Kt, v4.3. Avalable onlne: Last accessed Jan JAYAWARDANA, A.K.W., HARRIS, F.C., Further Development of Integer Programmng n Earthwork Optmzaton. Journal of Constructon Engneerng and Management, 116(1). JI, Y., BORRMANN, A., RANK, E., WIMMER J., GÜNTHNER W., An Integrated 3D Smulaton Framework for Earthwork Processes. In: The 26th CIB-W78 Conference on Managng IT n Constructon, 2009, Istanbul, Turkey. KIM, S-K., RUSSEL, J.S., Framework for an Intellgent Earthwork System Part II. Task dentfcaton/schedulng and resource allocaton methodology. Automaton n Constructon, 12(15-27). REB, Massenberechnung aus Querproflen (Ellng). In: REB-Verfahrensbeschrebung, STARK, R., MAYER, R., Quanttatve Constructon Management: Uses of Lnear Optmzaton. John Wley and Sons, Inc. New York, U.S.A. SON, J., MATTILA, K.G., MYER D.S., Determnaton of Haul Dstance and Drecton n Mass Excavaton. Journal of Constructon Engneerng and Management, 131(3). SÖDERSTRÖM, P., OLOFSSON, T., Vrtual Road Constructon - A Conceptual Model. In: The 24th CIB-W78 Conference, 2007,Marbor, Slovenen.

An Optimal Algorithm for Prufer Codes *

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