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Logist. Res. (2010) 2:65–78 DOI 10.1007/s12159-010-0031-8 OR IGINAL PAPER Solving a bi-objective winner determination problem in a transportation procurement auction Tobias Buer Giselher Pankratz Received: 5 February 2009 / Accepted: 14 April 2010 / Published online: 13 June 2010 Springer-Verlag 2010 Abstract This paper introduces a bi-objective winner Keywords Winner determination determination problem which arises in the procurement of Combinatorial auction Multiobjective optimisation transportation contracts via combinatorial auctions where Branch-and-bound Genetic algorithm bundle bidding is possible. The problem is modelled as a bi-objective extension to the set covering problem. We consider both the minimisation of the total procurement 1 Procurement of transportation contracts costs and the maximisation of the service-quality level at which the transportation contracts are executed. Taking Shippers, like retailers as well as industrial enterprises, into account the size of real-world transport auctions, a often procure the transportation services they require via solution method has to cope with problems of up to some reverse auctions, where the objects under auction are hundred contracts and a few thousand bundle bids. To transportation contracts. Usually, such contracts are solve the problem, we propose a bi-objective branch-and- designed as framework agreements lasting for a period of bound algorithm and eight variants of a multiobjective 1–3 years, and deﬁning a pick-up location, a delivery genetic algorithm. Artiﬁcial benchmark instances that location, and the type and volume of goods that are to be comply with important economic features of the transport transported between both locations. Additionally, further domain are introduced to evaluate the methods. The details such as a contract-execution frequency, e.g., deliv- branch-and-bound approach is able to ﬁnd the optimal ery twice a week, and the required quality of service, e.g., a trade-off solutions in reasonable time for very small predeﬁned on-time delivery rate, are speciﬁed in a trans- instances only. The eight variants of the genetic algorithm portation contract. A carrier can bid for one or more con- are compared among each other by means of large tracts. In each bid, the carrier states how much he wants to instances. The best variant is also evaluated using the small be paid for accepting the speciﬁed contracts. instances with known optimal solutions. The results indi- Transportation procurement auctions are of high eco- cate that the performance largely depends on the initiali- nomic relevance. Caplice and Shefﬁ [4] report on the size sation heuristic and suggest also that a well-balanced of real-world transportation auctions in which they were combination of genetic operators is crucial to obtain good involved over a period of 5 years. According to their report, solutions. in a single transportation auction up to 470 (median 100) carriers participated, up to 5,000 (median 800) lanes were tendered, and the annual cost of transportation amounted up to US-$ 700 million (median US-$ 75 million). Elmaghraby T. Buer (&) G. Pankratz and Keskinocak [10] present a case study of a procurement Department of Information Systems, Faculty of Business auction event in which a do-it-yourself chain operating Administration and Economics, University of Hagen, mainly in North America procured transportation services 58084 Hagen, Germany for about a fourth of the in-bound moves to their chain e-mail: tobias.buer@fernuni-hagen.de stores, which corresponds to a number of over 600 lanes. In G. Pankratz the study at hand, the terms lane and transportation contract e-mail: giselher.pankratz@fernuni-hagen.de 123 66 Logist. Res. (2010) 2:65–78 are used interchangeably. Similarly, shippers in Europe As to the multiple-criteria property of the allocation strive to consolidate their transportation procurement problem, there are two ways by which most shippers solve activities by running European-wide tenderings. As a the conﬂict between cost and quality goals: consequence, transportation procurement auctions in Eur- One way is to restrict participation in the auction to ope have signiﬁcantly increased in size and scope over the those carriers that comply with the minimum quality last few years which makes it difﬁcult to manage them standard required to meet the quality demands of any of the without the help of advanced information technology. In contracts. Thus, the service quality performance of all recent years, specialised Internet portals have emerged, remaining carriers is considered equal, and the only which offer contractors a neutral environment for issuing objective is to minimise total procurement costs. Unfortu- their logistics contracts. Sizes of tenderings processed via nately, unless the contract requirements are fairly homo- such platforms reportedly scale up to several hundreds of genous, this approach leads to the quality requirements of contracts [5]. many contracts being exceeded. The second way is to take In the scenario presented here, there are a number of into account service-quality performance differences interesting problems on the carrier’s as well as on the between carriers by applying penalties or bonuses to the shipper’s side. This paper focuses on the allocation prob- bundle bid prices, depending e.g., on a carrier’s service- lem that has to be solved by the shipper after all bids are quality in previous periods. submitted. In particular, two characteristics of the given This paper focuses on a third alternative, which inte- scenario are of interest. grates quality and cost criteria by explicitly modelling the First, from a carrier’s point of view, there are comple- WDP as a bi-objective optimisation problem. This model mentarities between some of the contracts. That is, the extends a previous model presented in [2], which can be costs for executing some contracts simultaneously are seen as a special case of the model presented in this paper. lower than the sum of the costs of executing each of these Previous work does not generally focus on modelling contracts in isolation. The cost effect of such comple- and solving winner determination problems under explicit mentarities is also referred to as economies of scope. consideration of multiple objectives. Different kinds of Second, allocation of contracts to carriers has to be winner determination problems in combinatorial auctions done taking into account multiple, often conﬂicting for transportation contracts are treated in [4, 10, 15, 20, decision criteria. While some of the criteria (e.g., limiting 21]. All these studies focus on bundle bidding to exploit the total number of carriers employed) may be naturally complementarities between contracts and consider mini- expressed as side constraints, other criteria should be misation of total procurement costs to be the only considered explicitly as objectives. In particular, there is objective. The structure of the remaining paper is as follows: usually a trade-off between the classical cost-minimisa- tion goal on the one hand and the desire for high service Sect. 2 deﬁnes the bi-objective winner determination quality on the other. Both objectives are of almost equal problem that is being studied. To solve this problem, an importance to most shippers, cf. Caplice and Shefﬁ [3] exact bi-objective branch-and-bound and a bi-objective and Shefﬁ [20]. genetic algorithm are introduced in Sect. 3. The algorithms In their recent review of the carrier selection literature, are evaluated on newly generated benchmark instances in Meixell and Norbis [17] identiﬁed that the issue of econ- Sect. 4. Finally, Sect. 5 gives an outlook on planned future omies of scope is dealt with in only a few papers and work. should be emphasised in future research. In order to exploit economies of scope (i.e., complementarities) between contracts in the bidding process, the use of so-called 2 A bi-objective winner determination problem combinatorial auctions is increasingly recommended [1, 2, (2WDP-SC) 20]. Combinatorial auctions allow carriers to submit bids on any subset of all tendered contracts (‘‘bundle bids’’). The winner determination problem (WDP) of a combina- Through this, carriers can express their preferences more torial procurement auction with two objectives is a gener- extensively than in classical auction formats. However, alisation of the well-known set covering problem (SC). bundle bidding complicates the selection of winning bids. Hence the problem at hand is called 2WDP-SC. It is for- This problem is known as the winner determination prob- mulated as follows: lem (WDP) of combinatorial auctions. In the procurement Given is a set of transport contracts T. Let t denote a context, the WDP is usually modelled as a variant of a set transport contract with t [ T; a set of bundle bids B where a partitioning or set covering problem, both of which are bundle bid b [ B is deﬁned as triple b: = (c, s, p). This NP-hard combinatorial optimisation problems. For a means a carrier c [ C is willing to execute the subset of survey on winner determination problems, see e.g., [1]. transport contracts s at a price of p. Given is furthermore a 123 Logist. Res. (2010) 2:65–78 67 set Q: = {q |Vc [ C ^ Vt [ T} where q C 0 indicates the contracts. Note that {c(b)|b [ W ^ t [ s(b)} is the set of ct ct quality level by which carrier c fulﬁls the transport contract carriers who have won a bid on transport contract t. Since t. Note that this is a rather expressive way to integrate contracts need to be executed only once but may be part of quality aspects in the model. However, in practice, it may more than one winning bid, it is not appropriate to simply be difﬁcult to capture all the q values. In this case, the add up the respective qualiﬁcation values of all b [ W. ct model also allows to represent quality levels of lower Instead, it appears reasonable to assume that the shipper granularity depending on the granularity of the shipper’s will break ties in favour of the bidder who offers the carrier assessment. At the margin, if the shipper evaluates highest service level for a given contract. Hence, by carrier quality only on a one-value-per-carrier basis, the assumption, for each transport contract t only the maximum quality values for carrier c will be initialised as q ¼ q qualiﬁcation values q with c [ {c(b)|b [ W^ t [ s(b)} are ct ct ct 1 2 for all t , t [ T. added up. Note that this rule might introduce an incentive 1 2 The task is to ﬁnd a set of winning bids W B; such for the carriers towards undesired strategic-bidding that every transport contract t is covered by at least one bid behavior. As this paper does not focus on auction-mecha- b. Furthermore, the total procurement costs, expressed in nism design, we leave this issue to forthcoming research. objective function f , are to be minimised and the total service quality, expressed in objective function f ,is tobe maximised. The 2WDP-SC is modelled as follows: 3 Solution approaches for the bi-objective winner determination problem min f ðWÞ¼ pðbÞð1Þ b2W To solve the 2WDP-SC, this section presents two algorithms. max f ðWÞ The ﬁrst is an exact algorithm based on the idea of branch- and-bound. Taking into account the NP hardness of the ¼ maxfq jc2fcðbÞjb 2 W ^ t 2 sðbÞgg ð2Þ ct bi-objective set covering problem, the non-linear objective t2T function f , and the large size of real world problems, the s.t. sðbÞ¼ T: ð3Þ branch-and-bound approach will probably solve only some b2W of the relevant problems in reasonable time. Therefore, a Each transport contract t has to be chosen at least once second solution approach is presented which is an extension (3). Accordingly, some contracts may be covered by two to a successfully applied multiobjective genetic algorithm. or more winning bids and therefore ‘‘paid more than once’’ Both algorithms aim to ﬁnd all trade-off solutions without by the shipper. Hence, preferring a set covering to a weighting the two objective functions. Thus, the shipper set partitioning formulation might seem at ﬁrst does not have to quantify his preferences, which can be counterintuitive. However, given the same set of bundle challenging [20]. Both algorithms ﬁnd a set of non-domi- bids, the total cost of an optimal solution to the set covering nated solutions (the true Pareto set or a good approximation problem never exceeds the total cost of an optimal set set, respectively). The shipper ﬁnally has to choose a solution partitioning solution and might be even lower. Of course, a from this set according to his subjective preferences. The set partitioning formulation is appropriate if each carrier latter is outside the scope of this study. For notational con- could be forced to submit a bundle bid on each of the venience, the 2WDP-SC is treated in the following as a pure |T| 2 - 1 contract combinations. However, this seems minimisation problem, i.e., the objective function f is unrealistic in practical scenarios due to the high number redeﬁned as f := (-1) f and is to be minimised. 2 2 of possible combinations. For this reason, from the At ﬁrst, the underlying terminology is deﬁned (cf. e.g., shipper’s point of view, the set covering formulation [25]): The set of all feasible solutions of an optimisation problem is denoted by X. A solution x [ X is evaluated by a appears more suitable. Nevertheless, if a contract is covered by more than one winning bid, there is at least vector-valued objective function f(x)= (f (x),…, f (x)) with 1 m m 1 one carrier who must not carry out this contract, although fðxÞ2 R : A solution x [ X dominates another solution 1 2 that carrier’s bid won the auction. In the scenario at hand, x [ X (written x x ), if and only if no component of the this is possible, as it appears reasonable to assume free vector-valued objective function f(x ) is larger and at least disposal [19]. In the transportation, procurement context, one component of f(x ) is smaller than the corresponding free disposal means that a carrier has no disadvantage if he component of f(x ). A solution x* is called Pareto optimal if is asked by the shipper to carry out fewer contracts than he there is no x [ X that dominates x*. The set of all Pareto was paid for. optimal solutions is called Pareto (solution) set X : A set of The ﬁrst objective function (1) minimises the total cost solutions X is called an approximation of X or (Pareto) of the winning bids. The second objective function (2) approximation set, if every solution in X is not dominated by any other solution in X: maximises the total service-quality level of all transport 123 68 Logist. Res. (2010) 2:65–78 3.1 A branch-and-bound algorithm based Algorithm 2 LookaheadBB on the epsilon-constraint method 1: input: (b , …, b ), e 1 max 2: bestCost 1 In order to solve the 2WDP-SC exactly, the Epsilon- constraint method [6, 13] is used. The idea of the Epsilon- 3: bestSolution fg constraint method is to optimise a single objective 4: initial problem node PN ffg; 1;1g function, treating the other objective function as additional 5: initialise queue and add PN to queue side constraint whose value is bounded by a particular e.To 6: while queue not empty do obtain the Pareto set, a proper sequence of single objective 7: PN problem node with minimum lower bound from queue optimisation problems has to be solved for different values 8: remove PN from queue of e. Here, the 2WDP-SC is linearised by treating f as side 2 9: constraint. The derived single-objective minimisation 10: contribute false problem is denoted as eWDP-SC and consists of the 11: if f (PN.W [{b }) \ bestCost then 1 PN.i objective function (1) with the covering constraint (3) and 12: if sðb Þn sðbÞ6¼; then PN:i b2PN:W the epsilon-constraint f (W) \ e. 13: contribute true Using a problem-independent branch-and-bound app- 14: else if f (PN.W) C e and f (PN.W [ b ) \ f (PN.W) 2 2 PN.i 2 roach based on linear relaxation, though seeming natural, then proved unsuitable for solving the eWDP-SC. This is due to 15: contribute true the non-linearity of the second objective function f ,in 2 16: end if which for each transport contract, a max{.} term is cal- 17: end if culated and the results are summed up over all contracts. 18: To obtain a linear model, all max{.} terms have to be 19: if contribute = true then replaced by additional side constraints and additional 20: PN 1 fPN:W [fb g; PN:iþ 1; PN:lbg PN:i decision variables (e.g., [22]). Compared to the |B| decision 21: processNode(PN1) variables of the non-linear eWDP-SC, the linearised variant 22: end if of the model contains |B| ? |T| ? |T| |B| decision vari- 23: ables. For example, even for a small problem instance with 24: freeBids fb 2ðb ; .. .; b Þji [ PN:ig i 1 max 40 bundle bids and 20 contracts, there are already 860 25: if PN.W [ freeBids is feasible then decision variables. 26: PN 2 fPN:W; PN:iþ 1; PN:lbg Therefore, a problem-speciﬁc branch-and-bound proce- 27: processNode(PN2) dure is introduced to solve the eWDP-SC. This algorithm, 28: end if referred to as eLookahead-branch-and-bound (eLBB), 29: end while consists of two main components. The ﬁrst component 30: output: bestSolution (repeatLBBForDifferentEpsilons, Alg. 1) iteratively selects a feasible value for e and hands it over to the second component, the actual branch-and-bound procedure Look- aheadBB (Alg. 2). This procedure solves the eWDP-SC to Alg. 1 initially determines the worst and the best ﬁnd the cost minimal solution for the given quality level e. possible values of f , which relate to the maximum and minimum e-values, respectively (keep in mind that f was redeﬁned to a minimisation objective). On the one hand, Algorithm 1 repeatLBBForDifferentEpsilons the maximum (worst) feasible value for e is calculated by solving the eWDP-SC using LookaheadBB with e = 0. 1: input: set of bundle bids B The obtained solution coincides with the minimal cost 2: W LookaheadBBðB; 0Þ solution of the set covering problem. On the other hand, 3: initialise approximation set X fWg the minimum (best) possible value for e, denoted as e*, is 4: f ðWÞ // worst (highest) e simply given by f (B) (generally, B is not in the Pareto 5: f ðBÞ // best (lowest) e set). 6: while e [ e* do After the minimum and maximum bounds for e are 7: W LookaheadBBðB;Þ known, repeatLBBForDifferentEpsilons triggers Look- 8: X X[fWg aheadBB to consecutively calculate the solutions in the 9: f ðWÞ Pareto set. Alg. 1 computes in each iteration of the while- 10: end while loop one solution. The loop starts with the highest (worst) 11: output: X which is the Pareto set e, calls LookaheadBB and then decreases e to the f value of 123 Logist. Res. (2010) 2:65–78 69 the current Pareto solution until e = e*. By this approach, linear relaxed set covering problem. This set covering the number of required while-iterations to ﬁnd the Pareto problem is deﬁned by those contracts not covered by set is minimal, i.e., the number of costly LookaheadBB PN.W which have to be covered by a subset of the calls is as low as possible. bundle bids given by freeBids. The branch-and-bound procedure LookaheadBB (Alg. 2) solves the eWDP-SC for a given e and the set of bundle Algorithm 3 processNode bids B, represented as sequence (b ) with 1 B i B max i b [ B 1: input: problem node PN and max = |B|, by implicitly enumerating the solution 2: if PN.W is feasible then space. The solution space is divided into subspaces which 3: if f (PN.W) \ bestCost then are represented in the branch-and-bound tree as problem 4: bestCost f ðPN:WÞ; nodes. Here, a problem node is a triple PN : = (W, i, lb)in 5: bestSolution PN:W which PN.W represents the current (probably incomplete) 6: delete all problem nodes in queue with lower bound solution, i.e., the set of winning bids, PN.i represents the C bestCost index of the bundle bid investigated in the node, and PN.lb 7: end if is the lower bound of the current solution PN.W for f .All 8: else active problem nodes are saved in a priority queue 9: if PN.i B |B| then according to ascending values of PN.lb. 10: PN.lb f ðPN:WÞþ cost of linear relaxed solution to the The algorithm was developed according to the following residual set covering problem. main ideas: 11: add PN to queue Branching on bundle bids. Each node PN has two 12: end if potential descendants PN1 and PN2. PN1 contains the 13: end if current bundle bid b as winning bid (b [ PN1.W), PN.i PN.i whereas PN2 does not ðb 62 PN 2:WÞ: Two additional PN:i rules are used to decide whether a descendant node should 3.2 A genetic algorithm based on SPEA2 be generated at all: – PN1 is only generated if b contributes to reach a PN.i To heuristically solve the 2WDP-SC, a multiobjective feasible solution. This means that the current bundle genetic algorithm (MOGA) is applied. This approach has bid b has to cover at least one transport contract PN.i been proven suitable for solving hard multiobjective uncovered so far, or, if the epsilon constraint is not yet combinatorial optimisation problems, e.g., [8]. The pro- met, adding b must reduce f . PN.i 2 posed MOGA follows the Pareto approach and searches for – On the other hand, PN2 is only generated if the current a set of non-dominated solutions. winning bids PN.W and the remaining free bids jointly To ﬁnd a Pareto approximation set, a MOGA controls a lead to a feasible solution with respect to both the set of core heuristics. The core heuristics of a MOGA can covering and the epsilon constraints. In checking this be divided into problem-speciﬁc and problem-independent property, the algorithm has to lookahead on future bundle operators. For those problem-independent operators that bids, which led to the labelling Lookahead in eLBB. care for the specialties of population management in the multiobjective case (ﬁtness-assignment strategy, selection Solving a relaxed problem to obtain a lower bound. For of parents and insertion of children in the population), the each problem node, a lower bound is calculated by solving a residual set covering problem which is deﬁned through methods proposed by Zitzler et al. in their Strength Pareto Evolutionary Algorithm 2 (SPEA2) are applied [23, 24]. the remaining free bids, the transport contracts still uncovered and by dropping the integrality constraints. The decision to use SPEA2 relies on its competitive per- formance particularly for solving bi-objective combinato- LookaheadBB uses the procedure processNode (Alg. 3) rial optimisation problems [24]. In addition, standard bitﬂip to control how to continue processing a given PN. Pro- mutation and standard uniform crossover [9] have been vided that PN.W is feasible and a new lowest cost chosen as problem-independent mutation and crossover solution is found, the current best solution and the cur- operators, respectively. rent best cost value are updated. Additionally, all prob- As problem-speciﬁc operators, three core heuristics are lem nodes from the queue whose lower bound is less introduced: Simple Insert, Greedy Randomised Construc- than the current best-known cost value are removed. tion and Remove If Feasible. Remove If Feasible is applied Provided that PN.W is infeasible, a new lower bound PN.lb is computed. The lower bound equals f (PN.W) as a problem-speciﬁc mutation operator, whereas Simple Insert and Greedy Randomised Construction are both used plus the cost value of the optimal solution to the residual 123 70 Logist. Res. (2010) 2:65–78 to initialise a population as well as to repair an infeasible Remove If Feasible (RIF) randomly chooses a winning 0 0 0 solution. The latter is necessary because both the uniform bid b [ W, labels b as visited, and removes b from W.If crossover operator and the bitﬂip mutation operator may after this the solution W is still feasible, then another ran- end up with infeasible solutions. domly chosen winning bid (which is also labelled as vis- Since all three problem-speciﬁc core heuristics operate ited) is removed etc. If W becomes infeasible by removing 0 0 on encoded individuals, the chosen encoding is presented b , then b is reinserted in W. RIF terminates if all winning ﬁrst. A binary encoding of a solution seems suitable for set bids are labelled as visited. covering-based problems like the 2WDP-SC. Every gene Via combination of the core heuristics, a set of different represents a bundle bid b.If b [ W the gene value is 1, and algorithms A is obtained (see Fig. 1). Each algorithm A 2 if b 62 W the gene value is 0. A; i ¼ 1...8 is denoted as a triple, e.g., A is represented Simple Insert (SI) in each iteration randomly chooses a by (SI/BF/GRC) which reads as follows: A uses SI to bundle bid b that contains at least one still uncovered construct solutions, bitﬂip mutation (BF) as mutation transportation contract as a winning bid. The transport operator and GRC as repair operator. Since uniform contracts s in bid b are marked as covered. These steps crossover is the only crossover operator, this operator is not are repeated until all contracts T are covered and SI considered as a distinctive feature in the taxonomy of terminates. Fig. 1. In order to refer to a set of algorithms, the wildcard Greedy Randomised Construction (GRC) is inspired by * is used at one or more positions, e.g., (*/BF/GRC) the construction phase of the metaheuristic GRASP [12] identiﬁes A and A . 2 6 and is slightly adapted for the bi-objective case (see Alg. 4). During each iteration, a winning bid is selected randomly from the restricted candidate list (RCL). 4 Evaluation Algorithm 4 GreedyRandomisedConstruction (GRC) The eLBB and the eight MOGA variants are tested on a set of newly generated benchmark instances, which reﬂect 1: input: infeasible solution W some important economic features of the transportation 2: while W infeasible do domain. First, the generation of these instances is descri- 3: best bundle approximation set RCL fg bed. After that, the results of the eLBB and the eight 4: for all b [ B \W MOGA variants are presented. 5: if b not dominated by any b [ RCL then 6: RCL RCL[fbg 4.1 Generating test instances 7: end if 8: end for To the best of our knowledge, no benchmark instances 9: randomly chose a b from RCL exist for a multiobjective WDP like the proposed 2WDP- 10: W W[fbg SC. However, there are several approaches for generating 11: end while problem instances for single-objective winner determina- 12: output: feasible solution W tion problems with various economical backgrounds, e.g., the combinatorial auction test suite ‘‘CATS’’ of Leyton- Note that the RCL is an approximation set of best Brown and Shoham [16] or the bidgraph algorithm intro- bundles, which holds only non-dominated bundles with duced by Hudson and Sandholm [14]. To generate test p q respect to the rating function g: = (g , g ) with instances for the 2WDP-SC, some ideas of the literature are pðbÞ=jsðbÞn sðWÞj for jsðbÞn sðWÞj [ 0 extended to incorporate features speciﬁc to the procure- g ðb; WÞ¼ ment of transportation contracts. 1 for jsðbÞn sðWÞj¼ 0 q 0 As this investigation does not address any game theo- g ðb; WÞ¼ðf ðWÞ f ðW [ bÞÞ= jsðÞ b j: 2 2 0 retical issues like strategic bidding and incentive compat- b 2W[b ibility, it is assumed that carriers reveal their true Both functions assign smaller values to better bundles. preferences. Thus, the terms ‘‘price’’ and ‘‘cost valuation’’ of a contract combination can be used synonymously. While g rates a bundle according to the average additional costs attributed to each new (i.e., still uncovered) contract General requirements of artiﬁcial instances for combina- torial auctions are stated by Leyton-Brown and Shoham. in b, g weights the reduction in f caused by adding b to the solution by the reciprocal total number of procured Both postulations seem self-evident but have not always be accounted for in the past [16]: contracts (in the current solution). 123 Logist. Res. (2010) 2:65–78 71 Fig. 1 Eight possible combinations of core heuristics Initialize Population SI GRC to form an algorithm A Mutation BF RIF BF RIF Repair SI GRCSC I GRC SS I GRC I GR Var iant A A A A A A A A A i 1 2 3 4 5 6 7 8 – Some combinations of contracts are more frequently as essential bids. Since all submitted bids are supposed bid on than other combinations. This is due to usually to be OR-bids, any non-essential bid could always be different synergies between contracts. replaced by an equivalent combination of two or more essential bids. Therefore, bidding on non-essential bids – The charged price of a bundle bid depends on the contracts in this bundle bid. Simple random prices, e.g., is redundant. – The 2WDP-SC was modelled as a set covering drawn from [0,1], are unrealistic and can lead to computationally easy instances. problem, as it appeared reasonable to assume free disposal. Free disposal means that the price charged by Furthermore, it seems reasonable to demand that the carrier c for any subset of a set of contracts s is not following additional requirements speciﬁc to transportation greater than the price carrier c would charge for s. procurement auctions are met: Formally, this is expressed in the following formula, in – All submitted bids are binding and exhibit additive which B denotes the set of bundle bids submitted by valuations (OR-bids, cf. [18]). Hence, a carrier is carrier c: supposed to be able to execute any combination of his 0 0 0 c pbðÞ pðbÞj8sðÞ b sðbÞ^ b; b 2 B : submitted bids at expenses which do not exceed the sum To be an instance suited to the 2WDP-SC, the bundle of the corresponding bid prices. Extra costs do not arise. bids of each carrier should also feature the free dis- Due to the medium-term contract period of 1–3 years in posal property. the scenario at hand, capacity adjustments are possible in order to avoid capacity bottlenecks. Furthermore, the – Finally, it is assumed that the carrier-speciﬁc costs of a transport contract depend on both the contract’s carrier has the opportunity to resell some contracts to other carriers who guarantee the same quality of service. resource requirements and the service-quality level at which the carrier is able to perform the contract. – From the previous assumption, it follows that a rational carrier c does only bid on combinations of contracts The bids are generated using Algorithm 5, which takes that exhibit strictly subadditive cost valuations. The four values as input: the number nBids of bids to be gen- cost valuation of a set of contracts s is called strictly erated, the sets C and T that represent carriers and transport subadditive, if for each partition T of the set s, the cost contracts, respectively, and the density q of the synergy valuation of s is strictly lower than the sum of the cost matrix. The synergy matrix consists of binary values, valuations of all parts of the respective set partition. which indicate the pairwise synergies between contracts. Formally, the carrier-speciﬁc set P of all strict Synergies between contracts imply that the respective subadditive bids can be deﬁned as expressed in the contract combination is cost subadditive. A higher density following formula, in which P(s) denotes all set tends to result in more and larger contract combinations a partitions of s and P(s) denotes the power set of s: carrier has to consider. () First of all, BidGeneration (Alg. 5) initialises some c c c c 0 P ¼ s T j8T 2 PðsÞ : p ðsÞ\ pðÞ s ; c variables. For each carrier, a subset of contracts T is s 2T () determined as the set of contracts that the carrier is sup- [ \ 0 0 posed to be willing to bid for. While it is not necessary that with PðsÞ¼ T PðsÞj s ¼ s^ s ¼; all T are disjoint, they must jointly cover all contracts in T. 0 0 s 2T s 2T After that, the following steps are performed for each carrier. First, the carrier-speciﬁc synergy matrix is ran- Strict subadditivity in terms of cost is due to synergies domly ﬁlled according to density q. The service-quality q between contracts. Bids composed of contracts which ct at which carrier c is able to execute contract t is chosen exhibit strict subadditive cost valuations are referred to 123 72 Logist. Res. (2010) 2:65–78 randomly from the integer values one to ﬁve, with higher according to two criteria: average cost per contract p(b)/ values indicating a higher service level. Furthermore, to |s(b)| and average quality per contract q /|s(b)|. t [ s(b) c(b)t each contract, a resource demand r is assigned. This is an Then, the best contract combinations with respect to these ct abstract indicator for the resources required by a carrier c to criteria are selected according to the dominance concept. In carry out contract t. The resource demand of a given doing so, select makes sure that on the one hand, the total contract may vary from carrier to carrier, as carriers might number of bids submitted by all bidders is nBids, and on the have, e.g., different locations of their depots, different other hand, each t [ T is covered by at least one bundle bid types of vehicles or existing transportation commitments to obtain a solvable instance. which inﬂuence the required resources. The values r are The SubadditiveBidGraph algorithm (cf. Alg. 6) is applied ct chosen randomly between 0.1 and 0.5. to determine prices for the essential contract combinations, which comply with the assumptions of free disposal and strict subadditivity. The algorithm is based on the approach of Algorithm 5 BidGeneration Hudson and Sandholm [14], which generates bids with free 1: input: nBids, density of synergy matrix q, T, C disposal. This approach is extended, such that all generated 2: V c [ C: randomly select relevant contracts T , T, such that bids also show strictly subadditive cost valuations. T ¼ T c2C 3: for all carriers c [ C do 4: V i, j [ T : set s 1 with probability q, indicating that ij Algorithm 6 SubadditiveBidGraph between contracts i and j exist synergies c c 5: V t [ T : randomly set contract quality q [ {1, 2, 3, 4, 5} 1: input: set of essential contract combinations P ; carrier c ct sup c 6: V t [ T : randomly set resource demand r [ [0.1, 0.5] 2: A ðfg i; jÞji; j 2 P and i j ct sub c 7: determine essential contract combinations P 3: A ðfg i; jÞji; j 2 P and i j c sup sub 8: subadditiveBidGraphAlgorithmðP Þ to calculate prices p(s) 4: initialise bidgraph BG P ; A ; A for each s 2 P : 5: 8s 2 P with jsj¼ 1 : 9: end for UBðsÞ LBðsÞ pðsÞ RandomBasePriceðs; cÞ c c 10: 8c 2 C : B selectðÞ P 6: initialise lower bounds c c 11: output: all carrier bids B ¼ B 8s 2 P with jsj¼ 1 : UpdateLowerBoundsðBG; sÞ c2C 7: initialise upper bounds 8s 2 P with jsj [ 1 : UBðsÞ¼ pðtÞ t2s To obtain the set of essential contract combinations in 8: k 2 line 7, assume for each carrier c a synergy graph 9: while kjPj do c c c c SG = (T , E ). Let the vertices be the contracts T carrier 10: for all s2fs 2 Pjjsj¼ k^ LBðsÞ 6¼ UBðsÞg do c is interested in. If two contracts i, j [ T feature syner- 11: set price randomly LBðsÞ UBðsÞ pðsÞ2 LBðsÞ;UBðsÞ½ gies, that is s ¼ 1; then both contracts are connected ij 12: UpdateLowerBounds(BG, s) no c c c 13: UpdateUpperBounds(BG, s) via an edge, that is E ¼ði; jÞjs ¼ 1^ i; j 2 T : It is ij 14: end for assumed that any number of contracts can be combined in a 15: k kþ 1 single bid, as long as the sum of the corresponding resource 16: end while demands does not exceed a maximum total resource 17: output: prices p(s) for each s 2 P consistent to the free demand of 1. This capacity limit is motivated by the fact disposal and the subadditivity assumption that unfolding of complementarities generally is subject to resource limitations. For example, contracts often feature synergies if they are carried out conjointly in the same tour, The idea of the original bidgraph algorithm as proposed which, however, is subject to vehicle capacity restrictions. by Hudson and Sandholm is to deﬁne lower bounds LB(s) The resource demand of each contract t [ T is given by r . and upper bounds UB(s) for each considered contract ct Then, the set of feasible essential combinations of contracts combination s such that free disposal holds. Then the equals the set of all possible induced subgraphs of SG with procedure successively draws a price for each contract r B1. combination between its lower and upper bounds; this price t ct In the next step, a price for each combination of contracts is propagated through the bidgraph to sharpen the lower is determined using the SubadditiveBidGraph algorithm, and upper bounds of the remaining contract combinations. which is explained below. After that, the select operator In order to extend this approach to support contract choses among all feasible contract combinations those combinations that exhibit both free disposal and strictly combinations on which each carrier is supposed to place his subadditive cost valuations, the bidgraph is initialised as bids. Therefore, all contract combinations in P are rated follows: The vertices of the bidgraph BG represent all 123 Logist. Res. (2010) 2:65–78 73 To ensure strictly subadditive valuations, the while-loop essential contract combinations P: There are two sets of sup sub sup arcs, A and A . The arcs in A indicate a superset of Alg. 6 sets the bid prices for all k-combinations in the order of non-decreasing k, starting with k = 2. For all relation, i.e., an arc (i, j) from vertex i to j means that the k-combinations with LB(s) =UB(s), a price is drawn contracts in j are a superset of the contracts in i. Similarly, sub randomly between LB(s) and UB(s) and propagated the arcs in A represent all subset relationships. through the bidgraph to adjust the lower and upper bounds In line 5 through 8 of Alg. 6, the lower and upper bounds of the other contract combinations. of all k-combinations of contracts are initialised. For a In doing so, it must be assured that the upper bound never given k 2 N; let the set of all k-combinations of contracts exceeds the costs of any partition of s since this may lead to be deﬁned as fs 2 P : jsj¼ kg: The lower bounds LB for inconsistencies with respect to the subadditivity requirement. all single contracts (k = 1) are initialised by Algorithm 7. The price p({t}) of a single contract t is a random variable Therefore, Alg. 9 solves a set partitioning problem to opti- mality. The instance of the set partitioning problem is given by that is normal distributed with mean l and variance r . The sub values of p({t}) are forced into the interval [minPrice, the sets {j |(s, j) [ A }and theassociatedcosts UB(j). maxPrice] with minPrice = 0.5 and maxPrice = 1.5. As stated above, higher resource requirements and a higher Algorithm 9 UpdateUpperBounds service level should tend to result in a higher price. Thus, l 1: input: BG, s depends on the resource demand r and the service quality ct 2 0 0 sup 2: for all s 2 BG:PjðÞ s; s 2 BG:A do q of contract t. The variance r is set to 1.0. ct 0 0 3: p price of optimal set partitioning solution to {s |(s, s ) After RandomBasePrice (Alg. 7) has initialised the LB sub [ BG.A } and associated UB(s ) of all 1-combinations, Alg. 8 recursively propagates these 4: if p* \ UB(s ) then prices through the bidgraph and updates the lower bounds 5: UBðÞ s p of all superset contract combinations if necessary. By now, 6: UpdateUpperBounds(BG, s ) the upper bounds for the k-combinations, k [ 1, can be 7: end if calculated as the sum of the prices of all respective 8: end for 1-combination contracts. Algorithm 7 RandomBasePrice The BidGraphAlgorithm continues until the prices of all essential bids are set. After that, the select-Operator of Alg. 5 1: input: single-contract set {t}, carrier c is applied as described above. The procedure keeps gener- 2: minPrice 0:5 ating bids for all carriers, until the test instance is complete. 3: maxPrice 1:5 4: resources_multiplier r =0:3 //expected mean of r (Alg. 5) ct ct 4.2 Measuring the quality of an approximation set 5: qualiﬁcation_multiplier q =3 //expected mean of q (Alg. 5) ct ct 6: l 1:0þ resources_multiplier qualiﬁcation_multiplier 2 To compare the performance of single-objective heuristics 7: r 1:0 in terms of achieved solution quality, a major step is to 8: pðftgÞ normal distributed random variable with mean l and compare the objective function values of the best found variance r solutions, respectively. The matter is more complicated in 9: if p({t}) [ maxPrice OR p({t}) \ minPrice then the bi-objective case, as approximation sets have to be 10: RandomBasePrice({t}, c) compared. Often there are no clear dominance relations 11: end if between the solutions of different approximation sets, see 12: output: p({t}) e.g., Fig. 2a. Therefore, various indicators to measure the quality of approximation sets are discussed in the literature, cf. [25] for a detailed discussion of the state of the art. Algorithm 8 UpdateLowerBounds To evaluate the solution quality of an approximation set, the popular hypervolume indicator I is used [23]. I HV HV 1: input: BG, s measures the dominated subspace of an approximation set, 0 0 sup 2: for all s 2 BG:PjðÞ s; s 2 BG:A do bounded by a reference point RP. RP must be chosen such 3: if LB(s ) \ p(s) then that it is dominated by all solutions of the approximation 4: LB(s ) /p(s) set. Furthermore, the reference point has to be identical for 5: UpdateLowerBounds(BG, s ) all compared heuristics on the same problem instance. 6: end if max max Here, for each instance, RP is deﬁned as f ; f ¼ 1 2 7: end for ðf ðBÞ; 0Þ; respectively. 123 74 Logist. Res. (2010) 2:65–78 Preliminary testing gave evidence that computation (a) 2 solution times of eLBB rapidly increase with the number of bundle algorithm A bids. Even moderate problem sizes caused the eLBB to run solution several hours before terminating. Therefore, a set of eight algorithm B rather small test instances was generated according to Sect. 4.1 in order to evaluate eLBB. The instances vary only in the number of bundle bids (up to 80) and in the number of transport contracts (up to 40). The number of participating carriers and the density of the synergy matrix are held min constant with values of 10 and 50%, respectively. The results of these instances are reported in Table 4 in Sect. 4.4.2. The table shows the number of solutions in the Pareto set and the required runtime in seconds. In addition, min the table contains results from the MOGA, which will be discussed in more detail in Sect. 4.4.2. (b) RP The ﬁndings demonstrate that eLBB is suited to solve small instances with up to 60 bundle bids in less than an hour. For solving problem instances with 80 bundle bids, eLBB consumes several hours of runtime. The test of the instance with 80 bundle bids and 40 contracts was aborted after a runtime of 24 h. These results strongly suggest that exact approaches like the eLBB are inappropriate as a solution approach for practical procurement scenarios min hyper volume which easily reach problem sizes of several hundreds of bundle bids. Nevertheless, for small instances, the optimal solutions obtained by the eLBB provide a valuable min benchmark for evaluating the quality of heuristic approa- ches like the MOGA (cf. Sect. 4.4.2). Fig. 2 Illustration of hypervolume indicator I . a Solutions of two HV approximation sets found by two algorithms A and B. b The shaded 4.4 Evaluation of the genetic algorithm areas of each algorithm depict the dominated subspace, respectively. Note that the light-shaded area is overlapping the dark-shaded area in part. The volume of the dark-shaded area is greater than the The eight genetic algorithms A to A , (cf. 2) were tested on 1 8 volume of the light-shaded area, therefore algorithm B is considered the same hardware platform as the eLBB (Pentium 4, better than algorithm A 2.0 GHz, 500 MB Ram available to the Java Virtual Furthermore, the objective values of all solutions Machine). The problem-speciﬁc heuristics were coded in min max min are normalised according to f ¼ f f = f f Java 6; for the problem-independent parts the SPEA2 i i i i max min max with i ¼ 1; 2; f ¼ f ðBÞ; f ¼ f ðBÞ 1; f ¼ 0: distribution coded in C was used [11]. 1 2 1 2 2 Thus, values of I range from zero to one, and larger For the evaluation of the genetic algorithms, two data HV values indicate better approximation sets. However, as RP sets were considered. can be chosen freely to a large degree, I is an interval- On the one hand, problem instances of practical size as HV based measure. Therefore, the quality gap between algo- reported in Sect. 1 were generated. These instances are rithms can only be expressed via absolute differences of referred to as large instances. The instances vary in the I , but not via percentage ratios of I . number of bids (500–2,000), the number of contracts (125– HV HV 500) and the number of carriers (25–100). In addition, the 4.3 Evaluation of the eLookahead–branch-and-bound density q of the synergy matrix was varied (25–75%). With respect to the observation that auctions with fewer transport The eLBB was implemented in Java 6. A ﬂoating point contracts usually tend to attract fewer bidders, it appeared precision of ten digits is used. The lower bounds are cal- reasonable to restrict the combinations of instance parameter culated by Dantzig’s Simplex Algorithm in the imple- values to those shown in Table 2. Since for the large mentation of the Apache Commons Math Library (version instances, absolute benchmarks in the form of optimal 2.1). The algorithm was tested on an Intel Pentium 4 solutions are not available, the relative performance of the (2.0 GHz) with 500MB RAM available to the Java Virtual eight MOGA variants on these instances is compared Machine. instead. The results of these tests are discussed in Sect. 4.4.1. 123 Logist. Res. (2010) 2:65–78 75 On the other hand, the small instances described in Sect. – The impression that a weak initial population signiﬁ- 4.3 were also used to evaluate the eight genetic algorithms. cantly compromises ﬁnal solution quality even if more The results for these instances are compared to those of the elaborate mutation and repair operators are used exact eLBB algorithm in Sect. 4.4.2. intensiﬁes by considering test no. 1 in Table 3. The approximation sets derived by the class of algorithms 4.4.1 Results and discussion for large instances which use GRC as initialisation heuristic clearly outperform the class of algorithms which use SI as In this section, the relative performance of the eight initialisation heuristic. This is true even on a signiﬁ- cance level of 0.0001. MOGA variants is evaluated using the large problem instances. The parameter values of the genetic algorithms – From the fact that the overall performance strongly were derived from some preliminary testing. Two to ﬁve depends on the initialisation heuristic, one can assume alternative values for each parameter were tested on three that any effort invested here will be rewarded. randomly selected instances. The values that gave the best – Tests 2 and 3 give no hints that the more intelligent results in manageable time are those presented in Table 1. operators RIF and GRC (applied in the repair phase) The same values were applied to all MOGA variants and promise better results than BF and SI in the general case. However, the performance of RIF signiﬁcantly were kept constant through all experiments. The results for the hypervolume indicator are presented improves if it is applied to an intelligently initialised population (test 5, test 4). in Table 2. The last column indicates the I value of the HV reference approximation set X ¼ X : – Tests 6 and 7 give evidence that the mutation operators A2A The results in Tables 2 and 3 were statistically evaluated BF and RIF do not show different behaviour, even if with the Kruskal–Wallis and the Mann–Whitney rank sum the repair operator is changed. However, if RIF is test. All statistical conclusions are stated at a signiﬁcance applied successfully to an individual, then there is no level of 5%. With respect to the given test instances, the need to apply any repair operator, as the operator leaves compared heuristics and the applied quality indicator, the the individual feasible by deﬁnition. following conclusions may be drawn. – Interestingly, an inﬂuence of the repair heuristic on the performance of all algorithms is not observable (test – The probability distributions of the I values of the HV 8–11). This result gets emphasised as we could not eight algorithms differ signiﬁcantly. The ranks given in prove a signiﬁcant performance advantage of A over Table 2 are derived by a systematic pairwise compar- A (both differ only in the applied repair operator). This ison of the hypervolume values using the Kruskal– followed from the Kruskal–Wallis Test, which takes Wallis rank test. into account all 240 observations (30 instances, 8 algo- – A performs very well, as could be expected, since it rithms). However, statistics paint a different picture if incorporates three problem-speciﬁc heuristics. Accord- only the 60 observations resulting from A and A are 7 8 ing to the Kruskal–Wallis rank test, taking into account compared with a signed rank test. Then, A clearly all 240 outcomes, A dominates all other algorithms but outperforms A . Hence, in well-balanced algorithms, A . A computes the best results for 25 out of 30 test 7 8 the repair operator may be of importance. instances, followed by A which achieves the highest value 5 times, and A which scores 4 times the best value. – The variants A , A , A , A that belong to the class 1 2 3 4 4.4.2 Results and discussion for small instances (SI/*/*) never achieve a best value in any one of the instances (cf. Table 2). For the set of small instances, the solutions found heuristi- cally by the GA are now compared to the Pareto optimal Table 1 Chosen parameter values for the test solutions found by eLBB. This is done to gain more insights Parameter Value into the performance of the MOGA, especially whether the MOGA is capable of ﬁnding optimal solutions and how close Size of population 50 individuals the approximate solutions are to the Pareto front. Uniform crossover-probability 15% The seven small instances that could be solved by eLBB Bit-exchange-probability in uniform-crossover 50% (cf. Sect. 4.3) were computed by all eight variants of the GA. Mutation-probability 100% As before, the computing time was ﬁxed to 5 min. In Bitﬂip-probability 10% accordance with the results for the large instances, the variant Runtime 300 s A performed best, i.e., in all seven instances it reached the No. of parents l for creating k offspring 4 best hypervolume value. For this reason, Table 4 compares No. of offspring k generated by l parents 4 only the results of A to the Pareto optimal solutions. 123 76 Logist. Res. (2010) 2:65–78 Table 2 Comparison of I for eight MOGA variants applied to the set of 30 large test instances (speciﬁed by columns 1–4) HV |B||T||C| q A A A A A A A A X 1 2 3 4 5 6 7 8 500 125 25 25 0.8473 0.8476 0.8482 0.8252 0.8663 0.8663 0.8878 0.8914 0.8914 50 0.8623 0.8627 0.8624 0.8605 0.8827 0.8827 0.9028 0.9038 0.9038 75 0.8614 0.8612 0.8693 0.8667 0.8759 0.8759 0.8937 0.8983 0.8983 1,000 125 25 25 0.9167 0.9170 0.8943 0.8509 0.9371 0.9371 0.9466 0.9479 0.9480 50 0.9223 0.9220 0.8754 0.8652 0.9499 0.9499 0.9523 0.9508 0.9523 75 0.9341 0.9340 0.9233 0.9117 0.9490 0.9490 0.9533 0.9535 0.9536 250 25 25 0.8623 0.8623 0.8725 0.8725 0.8818 0.8818 0.8961 0.9021 0.9021 50 0.8627 0.8625 0.8647 0.8579 0.8720 0.8720 0.8948 0.9001 0.9001 75 0.8555 0.8553 0.8573 0.8648 0.8736 0.8736 0.8953 0.8961 0.8967 50 25 0.8482 0.8488 0.8235 0.8199 0.8864 0.8865 0.8924 0.8927 0.8974 50 0.8498 0.8482 0.8417 0.8357 0.8811 0.8811 0.8935 0.8943 0.8943 75 0.8500 0.8497 0.8431 0.8407 0.8800 0.8800 0.8937 0.8937 0.8937 2,000 125 25 25 0.9553 0.9547 0.8843 0.8812 0.9748 0.9748 0.9720 0.9720 0.9772 50 0.9586 0.9584 0.8944 0.8751 0.9778 0.9778 0.9785 0.9786 0.9786 75 0.9615 0.9614 0.9277 0.9213 0.9757 0.9757 0.9745 0.9746 0.9764 250 25 25 0.9268 0.9267 0.9150 0.9052 0.9516 0.9516 0.9531 0.9531 0.9531 50 0.9282 0.9277 0.9148 0.9130 0.9471 0.9471 0.9522 0.9532 0.9532 75 0.9261 0.9262 0.9317 0.9315 0.9440 0.9440 0.9486 0.9510 0.9510 50 25 0.9150 0.9148 0.8387 0.8229 0.9472 0.9469 0.9331 0.9337 0.9498 50 0.9228 0.9223 0.8775 0.8780 0.9494 0.9494 0.9530 0.9530 0.9546 75 0.9221 0.9222 0.8993 0.8983 0.9471 0.9471 0.9505 0.9508 0.9516 500 25 25 0.8700 0.8700 0.8880 0.8972 0.8911 0.8911 0.8988 0.9022 0.9022 50 0.8601 0.8601 0.8785 0.8863 0.8837 0.8837 0.8933 0.8942 0.8942 75 0.8579 0.8579 0.8807 0.8848 0.8777 0.8777 0.8885 0.8944 0.8944 50 25 0.8503 0.8499 0.8490 0.8510 0.8810 0.8810 0.8901 0.8902 0.8907 50 0.8605 0.8600 0.8587 0.8667 0.8826 0.8826 0.8938 0.8947 0.8947 75 0.8532 0.8529 0.8694 0.8666 0.8776 0.8776 0.8886 0.8887 0.8894 100 25 0.8367 0.8347 0.8263 0.8165 0.8726 0.8715 0.8708 0.8708 0.8809 50 0.8433 0.8416 0.8254 0.8305 0.8798 0.8770 0.8825 0.8827 0.8900 75 0.8468 0.8448 0.8465 0.8370 0.8803 0.8803 0.8939 0.8939 0.8939 Rank 5.5 5.5 7.5 7.5 3.5 3.5 1.5 1.5 Mean 0.8848 0.8848 0.8699 0.8676 0.9081 0.9079 0.9160 0.9166 Standard dev. 0.0405 0.0405 0.0308 0.0310 0.0378 0.0378 0.0311 0.0311 All I values were obtained in a single run for each of the eight MOGA variants A to A . All runs were terminated after 5 min (300 s) HV 1 8 In Table 4, column DI shows the difference of the Note that in general, a higher number optimal solutions HV hypervolume attained I (eLBB) - I (A ) by the two found by the GA does not necessarily imply that the cor- HV HV 8 algorithms. The third column to the right states for each responding approximation set is closer to the true Pareto instance the number of solutions in the Pareto set. In addition, set. In particular, an approximation set that does not con- the second column to the right speciﬁes the number of tain any optimal solution still may be quite close to the solutions found by A which are Pareto optimal, i.e., which Pareto frontier. For example, consider the 60 bundle, bids/ are members of the Pareto solution set derived by eLBB. 40 contracts instance for which A does not ﬁnd any The GA variant A is able to ﬁnd optimal solutions for optimal solution. Nevertheless, DI indicates that A 8 HV 8 six out of seven instances. No optimal solution was found obtains a good approximation of the true Pareto frontier. for the 60 bundle-bids/40 contracts instance. For the The Pareto frontier and the approximation frontier attained smallest instance (for which it is trivial to generate all by A for this instance are simultaneously visualised in possible solutions), all solutions of the Pareto set are found Fig. 3. Though being close to the optimal points, the and DI equals zero. solutions of A appear slightly shifted to the right. HV 8 123 Logist. Res. (2010) 2:65–78 77 Table 3 Statistical comparison of selected sets of algorithms Obviously, A is indeed able to ﬁnd solutions at the same level of f like eLBB, but at the cost of higher values of f . 2 1 No. A vs. A H ^ a (%) i j 0 This effect seems to intensify for decreasing values of f . 1(GRC/*/*) vs.(SI/*/*) reject 0.01 This provides an indication that developing the cost- 2 (*/RIF/*) vs. (*/BF/*) – 73.85 reducing abilities of the problem-speciﬁc core heuristics 3 (*/*/GRC) vs. (*/*/SI) – 91.41 could further improve the GA’s performance. 4(SI/BF/*) vs.(SI/RIF/*) – 16.03 5(GRC/RIF/*) vs.(GRC/BF/*) reject 0.01 6 (*/RIF/SI) vs. (*/BF/SI) – 75.48 5 Conclusions and outlook 7 (*/RIF/GRC) vs. (*/BF/GRC) – 67.65 In this study, a model for a bi-objective winner determi- 8(SI/*/GRC) vs.(SI/*/SI)) – 88.52 nation problem in combinatorial transportation procure- 9(GRC/*/GRC) vs.(GRC/*/SI) – 80.11 ment auctions was presented. The model, which is based on 10 (*/BF/GRC) vs. (*/BF/SI) – 93.10 a set covering formulation, simultaneously minimises total 11 (*/RIF/SI) vs. (*/RIF/GRC) – 99.58 procurement costs and maximises the service-quality level The null hypothesis H says that the hypervolume indicators of the of the execution of all transportation contracts. approximation sets obtained by A and A have the same distribution. i j To solve this model, two algorithms were introduced. The signiﬁcance level a of all rejections is 5%. Based on the given results, a is the minimum level of signiﬁcance level at which H On the one hand, an exact bi-objective branch-and-bound would be rejected algorithm was proposed following the epsilon constraint approach. On the other hand, the well-known multiobjec- tive evolutionary algorithm SPEA2 was extended by a set Table 4 Comparison of heuristic approach A with exact approach eLBB on eight small instances of problem-speciﬁc evolutionary operators to solve the 2WDP-SC. By differently combining these operators, eight |B||T| I eLBB I A DI jj X Found Time (s) eLBB HV HV 8 HV by A variants of this genetic algorithm were constructed. The performance of the algorithms was evaluated on a 20 5 0.8576 0.8576 0.0000 7 7 1 set of newly generated test instances. The test instances 20 0.6095 0.6029 0.0066 11 6 2 were designed to reﬂect important economic properties of 40 20 0.8169 0.8126 0.0043 13 6 44 the transportation domain, e.g., free disposal and strict 40 0.5677 0.5639 0.0038 12 3 112 subadditivity of the submitted bids. 60 20 0.8652 0.8537 0.0115 17 5 2,975 The exact branch-and-bound algorithm ﬁnds optimal 40 0.6988 0.6913 0.0075 10 0 362 solutions only for small instances in reasonable time and 80 20 0.8915 0.8870 0.0045 17 2 19,461 therefore proved unsuitable for transportation procurement 40 – – – – – [86,400 auctions of practical dimensions. The relative performance All runs of A were terminated after 5 min (300 s). The runs of eLBB of the eight MOGA variants was evaluated on the large were terminated after 24 h (86,400 s), if the computation of the problem instances. The results show a strong dependence Pareto set has not been ﬁnished by then of the MOGA performance on the quality of the initial population. Unless the population is initialised using the more elaborated heuristics, even the intelligent operators do not compensate for the losses in solution quality. The best genetic algorithm was also compared to the results of the exact algorithm for the small instances. For these instances, the genetic algorithm was able to generate solutions in or close to the true Pareto solution set. Our ongoing and future work on this topic takes the following directions. In order to improve the performance of the exact approach, calculation of lower bounds is being enhanced using heuristics. 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Logistics Research – Springer Journals
Published: Jun 13, 2010
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