European Journal of Operational Research 197 (2009) 307–311

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Innovative Applications of O.R.

A multi-unit tender award process: The case of Transantiago

Juan Carlos Muñoz a,*, Diego Molina b,1

a Departamento de Ingeniería de Transporte, Pontificia Universidad Católica de Chile, Casilla 306, Cod. 105, Santiago 22, Chileb SECTRA, Teatinos 950 Piso 16, Santiago, Chile

a r t i c l e i n f o

Article history:Received 22 August 2006Accepted 16 June 2008Available online 1 July 2008

Keywords:AuctionMulti-unit auctionTransitTransantiago

0377-2217/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.ejor.2008.06.025

* Corresponding author. Tel.: +56 2 354 4270; fax:E-mail addresses: jcm@ing.puc.cl (J.C. Muñoz), dm

1 Fax: +56 2 696 6477.

a b s t r a c t

The Transantiago public transit services were grouped into five trunk service units and 10 feeder units,and were awarded to private operators through a simultaneous tender. To avoid problems of ownershipconcentration, limits were placed on the maximum number of units per operator. This paper describesthe issues involved in such simultaneous multi-unit tenders where various actors bid for different busi-ness units to be awarded in a single tender process. The general problem is formulated as a minimum costnetwork flow problem (MCNFP). Tie-breaking criteria are discussed and modeled. The problem is thensolved for the Transantiago case. There follows a discussion of the utility of handling this tender usingthe tool developed here.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

Consider a simultaneous competitive tender of multiple ser-vices in which tenderers may submit bids for as many of the ser-vices as they please, but various constraints are placed on the setof services any one of them can be awarded (a maximum numberof services, for example). The constraints may be motivated bysuch factor’s as the tendering authority’s concern to ensure diver-sification of providers, or the limits on tenderers’ capacity to pro-vide multiple services. In this type of tender, one would naturallyseek to have the total set of services awarded in a manner thatmaximizes some objective such as the amount of revenue collectedby the process.

Determining the optimal tender award is not a trivial problem.A reasonable solution can be rapidly found using heuristics butthere is no guarantee it will be optimal, and in some cases subop-timality may be costly. To better visualize the difficulties involved,imagine a tender of various services in which multiple biddersmust offer sums of money in the hope of winning the right to pro-vide them. Imagine further that no single bidder may be awardedmore than one of the services. Now suppose that a given biddersubmits the best offer on all of them. The question then arises asto which one of the services this bidder should be awarded. Shouldit be the one for which this tenderer bid more than for any other? Ifthe second best offer received happened to be very similar, littlewould in fact be lost if this other bidder were awarded it. As soonbecomes evident, finding an immediate solution is non trivial.

ll rights reserved.

+56 2 553 0281.olina@sectra.cl (D. Molina).

At a first sight, tender processes like the one just describedinvolving multiple services, may appear to belong to the familyof combinatorial auctions as defined in De Vries and Vohra(2003). However, they are not. Indeed, this paper shows that theircharacteristics make it much simpler to solve. Auctions were firstexamined by Friedman (1956) where interrelated tender processeswere considered. Later, these processes were discussed in Starkand Mayer (1971) and Rothkopf (1977). Rassenti et al. (1982) stud-ied these processes for allocating airport time slots. In the last dec-ade combinatorial auctions have been reported in a wide range oftendering situations, including railway line sections (Brewer andPlott, 1996), school bus services (Letchford, 1996) and the supplyof school meals (Epstein et al., 2002, 2004). Examples of significantachievements from applying combinatorial auctions to industrialprocurement might be reviewed at Hohner et al. (2003), Mettyet al. (2005) and Sandholm et al. (2006).

In the most general case, combinatorial auctions lead to highlycomplex optimization models that are NP-complete and requiregreat computational effort to arrive at a solution. Pekec andRothkopf (2003) provide an overview of developments in this areaand explore the effects of the various characteristics of each type ofproblem on the complexity of their solutions.

The case we will be concerned with might have been formu-lated and solved as an integer programming problem withNP-complete complexity. Fortunately, a proper formulation as aminimum cost network flow problem (MCNFP) considerably sim-plifies the solution process, obviating the need for major computa-tional efforts or heuristics to determine the optimal assignment.The MCNFP family of problems is commonly encountered in basicworks on operational research (see Ahuja et al., 1993) and is wellknown to have a polynomial complexity.

308 J.C. Muñoz, D. Molina / European Journal of Operational Research 197 (2009) 307–311

The application presented here was developed in the context ofTransantiago, one of twelve programs making up the Chilean cityof Santiago’s public transit modernization plan. Its objectives areto promote the use of these public transportation modes and im-prove service levels while reducing the pollution and congestionthey cause. It is hoped that current demand for these services willbe maintained, and if possible increased, thereby reversing thedeclining trend of recent decades. To accomplish all of this, a reg-ulatory framework has been proposed that should enable this mar-ket to achieve sustainable growth from the operators’ and users’points of view as well as from urban development, social and envi-ronmental perspectives.

In this context, the minimization of costs associated with publictransit is fundamental if fares are to be lowered and the systemmade more attractive to the user. As a first step in reducing thesystem’s costs, the transit services were totally redesigned in orderto better match supply to demand. The result was a scheme of ser-vices for structuring and articulating the transit system known astrunk services, plus a local network of feeder services that give ac-cess to the structural trunk network with an integrated faresystem.

In order to configure economic units that are attractive for pri-vate operators and facilitate monitoring, the trunk services weregrouped into five units and the feeder services into ten. Once thedesign stage of the system was complete, the granting of conces-sions to private operators was managed by the transit authoritythrough a public tender process. The tender was conducted withina framework of objective and transparent criteria for choosing thebest alternatives that provide a minimum required level of serviceto riders, and in a manner that guaranteed bidders would competeon price. The goal was to ensure future costs to public transit userswould be minimized (Klemperer, 2002). To avert undue concentra-tion of ownership, the transit authority also required that there bea minimum number of operators and bidders can only be awardeda maximum number of units. Since total fares paid by transit usersin Santiago are on the order of US$700 million per year, any poten-tial savings that can be identified would justify the additional ef-fort required to solve the problem.

In Section 2 below, the general problem is formulated. In Sec-tion 3 a mathematical programming model is introduced, and nec-

1

2

s

q

.

.

.

.

.

.

.

.

.

.

{0,0,min(n,r1)

1

2....

m

{0,0,min(n1,r11)}

{0,0, min(n2,r1

Bidder Classe

{0,0, min(nm,r1m)}

1.....

m

O

Fig. 1. Flow d

essary modifications for discriminating between multiple optimalsolutions by resort to tie-breaking criteria are analyzed. In Section4 this methodology is applied to the real-world case of Santiago’spublic transit system tender held in January of 2005. For analyzinghow representative of the bidding process this result was, in Sec-tion 5 a simulation of the process is presented. The paper con-cludes with a discussion of the results and main conclusions.

2. Mathematical formulation of problem

Consider a multi-unit tender process for a given system. Theprocess thus divides the system into q business units, which in turnare grouped into m classes such that each class k offers qk businessunits ð

Pqk ¼ qÞ. Each interested tenderer may bid on as many

business units as desired but cannot win more than n businessunits. In addition, for each class k a given bidder cannot acceptmore than nk business units.

The tendering authority attempts to assign all of the businessunits among the various bidders, and thus will avoid the non-award of business units while minimizing the total costs of the sys-tem. To this end, each tenderer includes in its bid the total unit costit would require to accept a given business unit. Let s be the totalnumber of bidders and pij the cost required by bidder i for businessunit j (tenderers will not necessarily bid on all business units). Also,each bidder i declares the maximum number ri of business units itis willing to accept plus the maximum number rik in each class k.

The problem as just described may be expressed as a minimumcost network flow problem (MCNFP), and can be visualized as illus-trated in the diagram in Fig. 1.

Each arc of the network in the figure has associated with it aunit cost, a minimum flow and a maximum flow given by a setof three attributes as shown. The network itself is made up of threefamilies of nodes. The first family contains s nodes, each one asso-ciated with a bidder; the second family comprises s groups of mnodes each, with every group associated to a node in the first fam-ily; and the third family has q nodes, each one associated to a busi-ness unit. For greater clarity we assume that a successive count ofthe qk business units in class k begins with bk and ends with ek (thatis, ek � bk = qk � 1 and ek + 1 = bk + 1).

1

2

q

.

.

.

.

.

.

2)}

1

1

1

1

1

1

1

{p12,0,1}

e1

b2

e2

bm

{p11,0,1}

{p1e1 ,0,1}

{p1e2 ,0,1}

{p1b2 ,0,1}

{p1q ,0,1}

{p1bm ,0,1}

iagram.

J.C. Muñoz, D. Molina / European Journal of Operational Research 197 (2009) 307–311 309

As depicted in the diagram, a flow of q units enters the networkthrough a common node O and exits disaggregated into one flowunit for each business unit node, reflecting the requirement thatall business units must be awarded. The path taken by a flow unitfrom O to a given business unit node indicates the bidder who isawarded that business unit. The q flow units begin by branchingoff to each of the s bidders without exceeding either the maximumn for each bidder or the maximum number of business units eachone has declared. Thus, for the arc (O,bidderi) indicating the numberof business units awarded to bidder i, the flow cannot exceed eithern or ri. The various awards won by a bidder are disaggregated alongthe path toward the q business units, but is subject to compliancewith the maximum number of such units in each class that a ten-derer is permitted. Thus, the flow of arc (bidderi, classki) cannot ex-ceed either ni or rik. Finally, arc (classki, unitj) indicates that bidder imade a bid for business unit j. Its flow will be 1 if it is awarded theunit, and 0 if not. To simplify the diagram, only the arcs and nodesassociated with the first bidder are shown; the other tenderers’ arcsand nodes would be drawn in analogous fashion.

Formulating the problem in this way as an MCNFP means wecan be sure the optimal solution to the relaxed problem in whichthe integrality constraints on the variables, that is, on the networkarc flows, are ignored, will in fact satisfy those constraints. This isthe case because all the vertices of the relaxed problem domain arefeasible for the original problem (for a review on MCNFP, see Ahujaet al., 1993).

3. Mathematical programming model for the MCNFP

The objective of the problem is to determine which bidder shallbe awarded each of the business units. We begin by defining A asthe set of pairs (i,j) each of which represents a bid made by bidderi for business unit j. We then define the following decisionvariables:

xij ¼1 if bidder i is awarded business unit j

0 otherwise

�8ði; jÞ 2 A:

We further define the following auxiliary parameter:

djk ¼1 if business unitis a member of class k

0 otherwise

8><>: 8j 2 f1; . . . ; qg; k 2 f1 . . . mg:

The model we must solve is then

Min C ¼Xði;jÞ2A

pijxij

subject toXq

j¼1

xij 6 n 8i 2 f1; . . . ; sg; ð1Þ

Xq

j¼1

djkxij 6 nk 8i 2 f1; . . . ; sg; 8k 2 f1; . . . ;mg; ð2Þ

Xs

i¼1

xij ¼ 1 8j 2 f1; . . . ; qg; ð3Þ

xij 2 f0;1g 8i 2 f1; . . . ; sg; 8j 2 f1; . . . ; qg: ð4Þ

This objective function minimizes total costs associated with theaward. Let C* be the total cost of the optimal assignment. Con-straints (1) and (2) require that no bidder be awarded more thann business units nor more than nk units in class k, while constraints(3) require that all units are awarded. Constraints (4) express thebinary nature of the variables, but given that this model belongsto the MCNFP family of problems, they may be replaced by theinterval 0 6 xij 6 1 without modifying the optimal solution.

3.1. Tie-breaking procedure

Tender processes often insist that bid amounts pij must fallwithin specified valid ranges. This is done to prevent predatorybidding in which pij is below operating costs. This practice is en-gaged in by firms hoping to win a tender and then demandchanges to the contract at some later date when work is alreadyunderway. To handle such scenarios, ties among different solu-tions must be detected and some sort of tie-breaking criteria areessential. Since the problem is solved with a relaxed linear pro-gramming model, ties are easy to identify: any null non-basic re-duced cost indicates the possibility of multiple optimal solutions.The tie-breaking criteria usually employed by public transit sys-tems are minimum fleet age, minimum average fleet emissions,maximum revenue to the government treasury, etc., or some com-bination of these. Note that since all business units are assignedsimultaneously, a tie may occur between different completeassignments, not just between individual bidders. Whatever thetie-breaking mechanism might be, each tenderer must report a va-lue for the tie-breaking variable corresponding to the unit it is bid-ding for. Then, if the tie-breaking variable value presented bybidder i for business unit j is denoted p1

ij, the tie-breaking processinvolves finding which assignment among all those that are opti-mal for the original problem will minimize

Ppij

1xij (without lossof generality for the functions to be maximized). This problem isformulated as follows:

Min C ¼Xði;jÞ2A

p1ijxij

subject toXði;jÞ2A

pijxij ¼ C�;

Xq

j¼1

xij 6 n 8i 2 f1; . . . ; sg;

Xq

j¼1

djkxij 6 nk 8i 2 f1; . . . ; sg; 8k 2 f1; . . . ;mg;

Xs

i¼1

xij ¼ 1 8j 2 f1; . . . ; qg;

xij 2 f0;1g 8i 2 f1; . . . ; sg; 8j 2 f1; . . . ; qg:

In this new model, the original objective function is now one of theconstraints, requiring that the optimal value C* be maintained. Thus,the domain of possibilities is constrained to those assignments thatare optimal on the previous problem. The new objective functionthen consists in minimizing the tie-breaker criteria.

Unlike the original problem, the new one does not have anMCNFP structure; its relaxed domain does, however, conservethe property that all of its vertices are feasible for the non-re-laxed problem. This is due to the fact that the new constraintlimits the problem’s domain to one of the faces of the original do-main, and the set of new vertices is, therefore, a subset of the ori-ginal ones.

If the tendering authority detects a possibility that the optimalsolution to the new problem might not be unique, it could thenimpose a second or third tie-breaking mechanism. In such cases,each bidder would have to report values for p2

ij and p3ij and the pro-

cess would proceed in a manner analogous to the one justdescribed.

4. Application: The tender of public transit services in Santiago

As noted here in the introduction, the Transantiago tender in-volved the award of 5 trunk units and 10 feeder units. The tender

310 J.C. Muñoz, D. Molina / European Journal of Operational Research 197 (2009) 307–311

specifications (Transantiago, 2004) provided that any one biddermay be awarded no more than 2 trunk units and a maximum of4 units in all. Each tenderer was required to indicate what fare itwould charge per rider for services provided on the units it wasbidding on. The amount had to fall within a valid range whoselower limit ensured the offer was serious and whose upper limitwas such that system costs remained under control. In addition,bidders could specify their desire to be awarded no more than acertain maximum of each type of unit due to limitations of fleetsize or financial capacity. Finally, the assignment of operators tothe services up for award sought to minimize system costs. Thus,the problem was of the simultaneous multi-unit tender type. Pack-age bids under which a bidder can bid on a bundle of units wereforbidden in this process for various reasons. First, the authorityneeded not only a good solution (as would be the case if the bid-ding agency was a private organization), but the optimal one. Also,if more than one optimal solution existed, all of them needed to beidentified. Finally, for transparency purposes, all these optimalsolutions needed to be determined very fast. On the other hand,package bids would have been beneficial for the system since therewere some evidences of economies of scope of winning geograph-ically nearby units (i.e. common terminals, time coordination ofbus services, etc.).

The Transantiago tendering authority could have employed atool to assign units in a way that minimized costs using the meth-odology just presented in the previous section. Instead, it opted touse a heuristic. The decision was based on the transparency affor-ded by the process, which involved the sequential assignment ofbusiness units under procedures that could be clearly explainedto bidders step by step, thus eliminating any alternate interpreta-tions. An important factor was the fact that the actors in Santiago’sprivate bus industry are relatively unprofessional and highly atom-ized. Transantiago was an ambitious undertaking and it was feltthat the risks involved should be minimized. Furthermore, fewmultiple offers were expected, significantly reducing the potentialadvantages of an exact method over a heuristic one.

The heuristic itself involved choosing the winning bid for eachunit in a predetermined order. The 5 trunk units were awardedfirst in order of significance (beginning with the ‘‘largest” one asmeasured by expected fare revenues), followed by the feeder units,also by order of significance. More precisely, bids were initially ta-ken from all tenderers interested in the first unit to be awardedand the best bid was chosen, resorting where necessary to tie-breaking mechanisms. Once the first unit was so awarded, the pro-cess continued with the next unit. If a bidder reached its capacitylimit for accepting a given type of unit, its bids for any as yet una-warded units of that type were discarded.

Most of the bids received were aimed exclusively at one type ofunit, either feeder or trunk. The assignment of the trunk unitsawarded them at the minimum possible cost, and in the case of tiesthe heuristic generated the optimal solution. This good perfor-

1 2 3Red Bus Urbano 1,357 905 452Compañía de Servicios para la Loc. Servicio de Transporte de Personas TurMaipo Futuro Transporte Urbano de Santiago Unión del Transporte 2,753 2,595 2,183Buses Gran Santiago 3,694 3,389SuBus 1,712Comercial Nuevo Milenio 4,167TransAraucarias Transportes Metropolitanos de Chile

Buses La Capital Buses Metropolitana

Fig. 2. Summary of tie-breaking bids for Transantiago feed

mance of the heuristic was favored by the fact that almost all thetenderers bid only for the number of units within their declaredcapacity or which was permitted by the tender specifications.

In the case of the feeder units, however, the heuristic’s assign-ment did not lead (initially) to an optimal solution. Due to the nar-rowness of the above-mentioned valid bidding range, all of theunits were assigned at the range’s lower limit. Thus, the tie-break-ing criteria played an extremely important role in the awarding ofalmost all of the units. The criteria actually used for the feederunits was a direct contribution to the system declared for each unitby bidders.

Fig. 2 summarizes the bids received on this tie-breaking criteriafrom the 13 participating companies for each of the 9 feeder units(a 10th unit received no bids and was not awarded). The columnsrepresent the units and are arranged in the order in which theywere awarded. The winning bids are shown in bold type. The firstseven units were awarded to the bidder making the best offer. Inthe case of the eighth unit, however, the bidder making the bestbid (Unión del Transporte) had declared that it was capable ofaccepting only one unit, and had already won Unit 7. If, however,units 7 and 8 had been awarded to tenderers Buses La Capitaland Unión del Transporte, respectively, additional direct revenueto the system of US$ 350,000 would have been collected (1.6% re-spect total revenue). A few days later it transpired that the bid sub-mitted by Buses La Capital was incomplete and was thereforedisqualified. Thus, though the heuristic award mechanism finallygenerated an optimal solution, the process nevertheless illustratesthe advantages of an exact method.

5. Simulation

The exact method could have yielded savings of US$ 350,000 forthe system in the case of the Transantiago bidding process. How-ever, it is reasonable to ask how representative this result is ofthe process and how likely is that the heuristic will yield a differentresult than the exact approach. To answer these questions, a sim-ulation was developed mimicking the offers presented in this case.

Since the nine feeder units received quite different average of-fers, a different normal distribution was fitted (consistent withthe offers received) for each one. Similarly, another distributionwas fitted to represent the number of offers each bidder wouldplace (consistent with those observed in the Transantiago case)and a random process to select which ones. Finally, bidders wouldaccept at most one unit with probability 1/3 and two with proba-bility 2/3.

The simulation was developed assuming that 5, 10, 15 or 20bidders participate in the process. Table 1 presents the results ofrunning 200 times each of these four cases.

The Table presents the average optimal earnings obtained bythe exact method, the average loss suffered of assigning units bya greedy heuristic in comparison with an exact method, its

4 5 6 7 8 1,357 1,810

6161,854 925 1,377

53783 0

204 1,842 231 1,159 1,927

5343,591

883 930 883 88383

959872

9

0

er units (expressed in thousands of American dollars).

Table 1Simulation results

Number ofbidders

Average totalearnings

Averageloss

Loss std.dev.

Loss probability(%)

5 13,096 651 946 5310 18,921 451 679 5115 21,248 179 409 3820 22,897 142 358 29

J.C. Muñoz, D. Molina / European Journal of Operational Research 197 (2009) 307–311 311

standard deviation, and the fraction of instances in which the heu-ristic did not yield the optimal solution. As can be observed, in thiscase the benefits of using an exact method over the greedy heuris-tic are greater when few bidders participate, since then it is morelikely that a single participant offers the highest bid in more thanone unit. Also, if more bidders participate, the total earnings ofthe bidding process are higher reducing even further the impor-tance of the exact method gains.

6. Conclusions and discussion

In this paper a problem was set out for a simultaneous multi-unit award of services by competitive tender. The formulation de-vised belongs to the minimum cost network flow family of prob-lems, and thus its complexity is significantly reduced. Solutionsto both the problem itself and any ties that may arise between bid-ders can be obtained by solving a continuous linear optimizationproblem.

The methodology was developed specifically for Transantiago,the tender of public transit bus routes in Santiago, but can be ap-plied to any such simultaneous-assignment multi-unit tenderprocess.

In the Transantiago case, the Santiago public transit authoritiesdecided to pass up this tool, opting instead to award the routesusing a simple heuristic. Their decision gave priority to the trans-parency and clarity of such a process over considerations of addi-tional costs resulting from the possibility of a suboptimalassignment. The rationale for this approach was the many politicaland technical risks posed by Transantiago, which involved a totalredesign of the system for the entire city using procedures neverbefore employed in Chile. Furthermore, the process had been heav-ily resisted by the incumbent operators who, although very expe-rienced in running bus services, had little experience with formaltechniques of business management. It was felt that they wouldsimply not trust a method that made assignments via a ‘‘blackbox”. Thus, the decision was made to avoid an additional risk inan already delicate process, a choice that under the circumstanceswas perfectly understandable.

Fortunately, the heuristic that was employed yielded an optimalsolution. Still, had one of the bids not been disqualified for techni-cal reasons, the revenue collected for the treasury would have beenUS$350,000 lower than the amount actually collected. This amplydemonstrates the usefulness of an optimal assignment mechanismsuch as the one presented here. The simulation developed showedthat a similar loss would be expected if the process were to be runagain under similar circumstances. The simulation also showedthat the gains obtained of using an exact method instead of thisgreedy heuristic are greater when few bidders participate. Thisresult is interesting since problems of moderate size are well with-in the capabilities of modern codes for solving mixed integerprograms.

As already noted, the tender award was expressed as a mini-mum cost network flow problem. The process may, however, in-volve additional constraints that prevent the use of such aformulation. During the drawing up of the Transantiago specifica-tions, for example, it was frequently requested that in addition tobids for individual units, tenderers be permitted to submit com-bined offers. Because of the benefits resulting from economies ofscale or horizontal integration, such offers would not merely bethe sum of individual bids. A number of tenderers announced theirintention of bidding for a trunk route together with its main feed-ers. Such an award would have allowed them to consolidate thethree units under a single operator, facilitating transfers betweenroutes and lowering operating, fleet and terminal costs. If tenderspecifications accept such bids, which would reduce system costseven more, the problem (in general) would no longer be express-ible as a minimum cost network flow problem. Formulating amathematical problem as such (which would now belong to thecombinatorial auction family) could be fairly simple. However, itscomplexity would grow significantly. Also, its usability would bereduced, since guaranteeing a unique optimal solution within acouple of minutes (as it was expected) would be more adventurousif the number of bidders was not known in advance.

Acknowledgements

We would like to thank the National Fund for the Developmentof Scientific and Technological Research (FONDECYT Project1040604 and Anillos Tecnológicos ACT-32) for the valuable contri-bution to financing this study.

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