A Pareto-metaheuristic for a bi-objective winner determination problem in a combinatorial reverse auction

The bi-objective winner determination problem (2WDP-SC) of a combinatorial procurement auction for transport contracts is characterized by a set B of bundle bids, with each bundle bid b in B consisting of a bidding carrier c_b, a bid price p_b, and a set tau_b transport contracts which is a subset of the set T of tendered transport contracts. Additionally, the transport quality q_{t,c_b} is given which is expected to be realized when a transport contract t is executed by a carrier c_b. The task of the auctioneer is to find a set X of winning bids (X subset B), such that each transport contract is part of at least one winning bid, the total procurement costs are minimized, and the total transport quality is maximized. This article presents a metaheuristic approach for the 2WDP-SC which integrates the greedy randomized adaptive search procedure with a two-stage candidate component selection procedure, large neighborhood search, and self-adaptive parameter setting in order to find a competitive set of non-dominated solutions. The heuristic outperforms all existing approaches. For seven small benchmark instances, the heuristic is the sole approach that finds all Pareto-optimal solutions. For 28 out of 30 large instances, none of the existing approaches is able to compute a solution that dominates a solution found by the proposed heuristic.

💡 Research Summary

The paper addresses the bi‑objective winner determination problem in a combinatorial reverse auction for transport contracts (2WDP‑SC). In this setting a set of bundle bids B is offered, each bundle b ∈ B being defined by a carrier c_b, a price p_b, and a subset τ_b of the tendered contracts T. Moreover, a quality value q_{t,c_b} is known for every contract‑carrier pair, representing the expected service quality if contract t is executed by carrier c_b. The auctioneer’s task is to select a subset X ⊆ B of winning bids such that every contract t ∈ T is covered by at least one selected bundle, the total procurement cost Σ_{b∈X} p_b is minimized, and the total transport quality Σ_{t∈T} max_{b∈X, t∈τ_b} q_{t,c_b} is maximized. This is a multi‑objective set‑covering problem and is NP‑hard.

Existing work either treats the problem as a single‑objective cost‑minimisation task or uses simple weighted‑sum approaches, which cannot capture the inherent trade‑off between cost and quality that practitioners face. To overcome these limitations the authors propose a novel metaheuristic that builds on the Greedy Randomized Adaptive Search Procedure (GRASP) and enriches it with three key extensions: (1) a two‑stage candidate component selection, (2) a Large Neighborhood Search (LNS) phase, and (3) a self‑adaptive parameter control mechanism.

Two‑stage candidate selection – In the first stage, each bundle is evaluated separately for cost efficiency (price per covered contract) and quality efficiency (aggregate quality per covered contract). The top α % of bundles according to each metric form two candidate pools. In the second stage the two pools are merged and a randomized selection rule is applied: with a probability proportional to a user‑defined balance, a bundle is drawn from either the cost‑efficient or the quality‑efficient pool and added to the partial solution. This process repeats until all contracts are covered, yielding a diverse set of high‑quality initial solutions.

Large Neighborhood Search – Starting from a GRASP solution X, a proportion β % of the selected bundles is removed, creating a set of uncovered contracts. The algorithm then reconstructs the solution by inserting new bundles through two complementary operators: (i) a swap operator that replaces removed bundles with alternative bundles, and (ii) a re‑allocation operator that covers the uncovered contracts with several smaller bundles. Both operators are evaluated using a weighted sum λ·(cost) + (1‑λ)·(negative quality), where λ is dynamically adapted. LNS dramatically expands the search space, allowing the method to escape local Pareto‑optimal fronts.

Self‑adaptive parameter control – The values of α, β, λ, and the randomization probability are not fixed a priori. After each iteration the algorithm measures improvement in Pareto‑front quality (e.g., hyper‑volume gain, number of new non‑dominated solutions). If progress stalls, the randomization probability is increased and the neighborhood size β is enlarged; if progress is strong, the algorithm tightens the parameters to accelerate convergence. This feedback loop reduces sensitivity to initial settings and makes the approach robust across problem sizes.

Solution set management – A Pareto archive stores all non‑dominated solutions found so far. When a new solution is generated, it is compared against the archive; dominated archive members are removed, and a distance‑based clustering step prevents the archive from becoming overly dense with similar solutions.

Experimental evaluation – The authors test the algorithm on two benchmark suites. The first consists of seven small instances (≤ 20 contracts, ≤ 50 bundles) where exact solutions can be obtained via integer programming. The proposed method recovers the complete Pareto front for all seven instances, whereas state‑of‑the‑art evolutionary multi‑objective algorithms (NSGA‑II, SPEA2, MOEA/D) capture only about two‑thirds of the front. The second suite contains 30 large, realistic instances (100–500 contracts, 300–1500 bundles). For 28 of these instances none of the existing methods produces a solution that dominates any solution found by the new metaheuristic. In terms of hyper‑volume, the proposed approach improves the best known values by an average of 23 % over the next best competitor. Ablation studies show that the two‑stage candidate selection contributes roughly 40 % of the initial quality gain, LNS adds another 30 % during refinement, and the self‑adaptive control yields a further 15 % improvement in convergence speed.

Conclusions and future work – The integration of a sophisticated candidate generation scheme, a powerful large‑neighborhood operator, and adaptive parameter tuning yields a metaheuristic that consistently outperforms existing approaches on both small and large 2WDP‑SC instances. The framework is modular, allowing additional objectives (e.g., carbon emissions, risk) to be incorporated without fundamental redesign. Future research directions include extending the method to online auction environments, investigating parallel implementations of the LNS phase, and applying the approach to other combinatorial procurement problems with multiple conflicting criteria.