A note on solving DiDi’s driver-order matching problem

Abstract

This paper investigates the combinatorial nonlinear programming model that DiDi proposed for solving their driver-order matching problem. The model is reformulated to an equivalent continuous nonlinear program which is amenable to efficient commercial solvers. A backward induction procedure for computing the lower bound is also proposed. Computational experiments demonstrate that the local solution produced by the reformulation becomes increasingly close to the global solution, thereby suggesting a diminishing marginal benefit of pursing global optimality as the problem size increases.

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Notes

  1. 1.

    The original formulation in [34] has a multiplicative factor (1/N), to represent the arithmetic mean of order-level success rates. We ignore that factor here since it does not alter the mathematical problem.

  2. 2.

    DICOPT was configured to call CONOPT and CPLEX for solving the NLP and MIP subproblems, respectively. By default, DICOPT stops as soon as the NLP subproblems stop improving, which is a heuristic criterion. We have also tested setting the options to “stop=1 convex=1 maxcycles=1000” in an attempt to defer stopping until the global solution is obtained. However, in this setting the algorithm failed to converge in 30 min for all but one instances.

  3. 3.

    This is an empirical statement in light of Theorem 1. In theory, a solver could return a fractional solution when the optimal solution is not unique and both fractional and integral solutions exist. Example: \(\min \{(1-0.5 x_{11}) + (1-0.5 x_{21}) \mid x_{11} + x_{21} \le 1, x_{11} \ge 0, x_{21} \ge 0\}\). However, a fractional solution (away from 0 or 1 by at least \(10^{-7}\)) of FLC has never been encountered at the default solver option for an array of NLP solvers.

  4. 4.

    For KNITRO, the option “bar_maxcrossit=10” was used to invoke the cross over steps.

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Acknowledgements

The author would like to thank two anonymous referees for their constructive comments that helped identify the connection to submodular functions and provided insights into global optimization algorithms, and thank Lingyu Zhang (DiDi Chuxing) and Dr. Ignacio Grossmann (Carnegie Mellon University) for helpful discussions.

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Correspondence to Yanchao Liu.

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Liu, Y. A note on solving DiDi’s driver-order matching problem. Optim Lett 15, 109–125 (2021). https://doi.org/10.1007/s11590-020-01590-3

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Keywords

  • Combinatorial optimization
  • Submodular welfare problem
  • Mixed integer nonlinear programming
  • Transportation applications