A multi-unit combinatorial auction based approach for decentralized multi-project scheduling

Abstract

In industry, many problems are considered as the decentralized resource-constrained multi-project scheduling problem (DRCMPSP). Existing approaches encounter difficulties in dealing with large DRCMPSP cases while respecting the information privacy requirements of the project agents. In this paper, we tackle DRCMPSP by formulating it as a multi-unit combinatorial auction (Wellman et al. in Games Econ Behav 35(1):271–303, 2001), which does not require sensitive private project information. To handle the hardness of bidder valuation, we introduce the capacity query which uses different item capacity profiles to efficiently elicit valuation information from bidders. Based on the capacity query, we adopt two existing strategies (Gonen and Lehmann in Proceedings of the 2nd ACM conference on electronic commerce, pp 13–20, 2000) for solving multi-unit winner determination problems to find good allocations of the DRCMPSP auctions. The first strategy employs a greedy allocation process, which can rapidly find good allocations by allocating the bidder with the best answer after each query. The second strategy is based on a branch-and-bound process to improve the results of the first strategy, by searching for a better sequence of granting the bids from the bidders. Empirical results indicate that the two strategies can find good solutions with higher quality than state-of-the-art decentralized approaches, and scale well to large-scale problems with thousands of activities from tens of projects.

Appendix

Here we design a fast polynomial-time algorithm to solve the RCPSP with time-varying resource capacities, as shown in Algorithm 3. This algorithm is based on the parallel generation scheme [20]. The original algorithm in [20] is designed for resources with constant capacity over the whole scheduling horizon. Here we modify it to incorporate the time-varying resource capacities.

Intuitively, Algorithm 3 consists of a series of stages at certain time tic $t_d$. Three sets of activities are maintained during the whole process: Active Set $\overline{{ AS}}$, Complete Set $\overline{\textit{CS}}$, and Decision Set $\overline{{ DS}}$. In each stage, firstly the activities in $\overline{{ AS}}$ that complete before $t_d$ are moved from $\overline{{ AS}}$ to $\overline{\textit{CS}}$. Next, a set of activities that can be scheduled with respect to precedence and resource constraints are identified and put into $\overline{{ DS}}$. Then, an iterative process is imposed on $\overline{{ DS}}$, which includes three steps: 1) choose the activity $a_{{ ij}}$ according to certain priority rule to start at $t_d$, 2) move $a_{{ ij}}$ from $\overline{{ DS}}$ to $\overline{{ AS}}$, and 3) update the activities in $\overline{{ DS}}$. This process terminates when no more activity can be scheduled in $t_d$, i.e. $\overline{{ DS}}=\emptyset$, which leads to the update of $t_d$. If $\overline{{ AS}}$ is not empty, then the new $t_d$ is set to be the minimum complete time of the activities in $\overline{{ AS}}$. In the case $\overline{{ AS}}$ is empty, which indicates no remaining activity can start at $t_d$ (due to insufficient resource), $t_d$ is set to the next time tic. When all activities are scheduled, the algorithm terminates with a feasible schedule.

The operation of updating $\overline{{ DS}}$ in Lines 4 and 8 includes two steps: (1) find the unscheduled activities whose predecessors are all in $\overline{\textit{CS}}$, and (2) exclude those activities $a_{{ ij}}$ that cannot be scheduled to start at $t_d$ due to insufficient resource at some time slot $t_d \le t \le t_d+d_{{ ij}}$. Different priority rules (e.g. latest finish time (LFT), most total successors (MTS), minimum slack (MS)) can be used in Line 6 to select an activity. Among these priority rules, LFT has been empirically shown to be the most effective one in minimizing the project delay [20] and is chosen for our scheduling algorithm. Complexity of Algorithm 3 is $O\left( J_i^2(G+L_i)d^*_i\right)$ (ignore the update of $t_d$ between Lines 10 and 14), where $d^*_i=\text {max}\{d_{i1},\ldots ,d_{iJ_i}\}$ is the maximum duration of the activities of $P_i$.