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Edited by: Jianke Zhang, Xi'an University of Posts and Telecommunications, China

Reviewed by: Rudy Raymond, IBM, Japan; P. Senthil Kumar, Mohan Babu University, India

This article was submitted to Optimization, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The aim of this study was to explore the information disclosure (ID) problem, which involves selecting pairs of two sides before matching toward user-oriented optimization. This problem is known to be useful for mobility-on-demand (MoD) platforms because drivers' choice behaviors are appropriately modeled, but solving the problem is still under development, although heuristic solvers have been proposed. We develop new branch-and-bound-based (BnB) solvers and a new heuristic solver based on a quadratic unconstrained binary optimization (QUBO) formulation. Our numerical experiments show that the QUBO-based solver indeed works within the limit of available bits, and the BnB solver performs slightly better than existing heuristic ones.

Matching between two sides (e.g., items and users, items and markets) is an essential task in many real-world applications. Bipartite graph matching has been investigated as a fundamental problem to model such a matching between two sides [

Global optimization task on bipartite graphs for the transportation domain. Out of possible driver–order assignments, pairs among drivers and orders are decided in terms of some global metric (e.g., total profit).

Focusing on individual participants on each side of bipartite matching is now an emerging topic for human-oriented decision-making. In contrast to the traditional scenario illustrated in

Concepts of the information disclosure, where probabilities, drivers, and orders are illustrated. In the first step, choice probabilities are optimized by selecting disclosures in the ID problem to clarify the structure among drivers and orders. In the second step, we try to solve the matching problem.

Along with the formulation of the ID problem, Yang et al. developed two heuristic solvers, which are reviewed in Section 2.2 and [

Following an MoD scenario, let _{1}, …, _{m}} and _{1}, …, _{n}} be sets of drivers and customer orders, respectively. We assume that each driver _{d, o}^{1}^{2}_{d, o} is the sum of a deterministic term _{d, o} and a random term ϵ_{d, o}, that is, Û_{d, o}: = _{d, o}+ϵ_{d, o}. Therefore _{d, o} is evaluated for each transportation service with the fee _{o} of an order _{d, o} of a pair (_{d, o} is defined as _{d, o}: = β_{0}+β_{1}_{o}+β_{2}τ_{d, o} with three preference parameters _{0}, β_{1}, β_{2}}. In particular, these parameters ^{3}

Let _{d}⊂^{4}_{d}. Yang et al. adopted the following _{d, o} with the parameter α [^{5}

Where

We prepare the binary decision variables _{d, o} = 1 means the system displays _{d, o} when _{d, o} and _{d} = {_{1}, _{2}}; that is, we have (_{d,o1}, _{d,o2})∈{(0, 0), (0, 1), (1, 0), (1, 1)}. The probabilities computed with different

Examples of evaluating probabilities _{d, o} with _{d} = {_{1}, _{2}}, α = 1 under different four (_{d,o1}, _{d,o2}) values [i.e., (0, 0), (0, 1), (1, 0), and (1, 1)].

_{d,o1} |
0.0 | 0.0 | 0.982 | 0.867 |

_{d,o2} |
0.0 | 0.881 | 0.0 | 0.117 |

_{d, {s}} |
1.0 | 0.119 | 0.018 | 0.016 |

To reproduce these examples, we can use _{d,o2} = 11.

Four possible disclosure instances. _{d, o} = 0 if _{d, o} = 0, i.e., a driver cannot select an order that is not displayed.

The ID problem studied in Yang et al. [

Where the term

The solvers studied in Yang et al. [

The basic idea of the two methods is to design a greedy method by evaluating the increments of objective values. Let

With Equation (3), IEC and MLEC are executed as follows:

IEC: Begin with

MLEC: When computing Δ in IEC, filter (

Recall that Yang et al. stated that (1) MLEC can be faster than IEC, (2) MLEC can find better solutions within limited computational resources, and (3) the use of ID problem enhances matching results (e.g., more beneficial for MoD platforms) [

To deeply understand and conduct experimental comparisons to analyze the ID problem, we develop new solvers. For complex combinatorial optimization problems such as the ID problem in this article, the standard approach to design a solver is to follow the branch-and-bound idea [

Quantum annealing (QA) is a heuristic optimization method that is to solve combinatorial optimization problems [

Although the current objective function in Equation (2) is not a quadratic function, any higher order terms are essentially transformed into linear combinations of quadratic terms once the problem is expressed in terms of binary variables. We, therefore, try to develop a QUBO-based solver, considering that the upcoming performance of QUBO solvers, including the QA should be promising for optimization [

From

and

We then need to minimize _{d}), we apply a technique [

Where _{+}. This technique is based on the fact that minimizing

As mentioned earlier, any higher order unconstrained binary optimization problem (HUBO) is essentially transformable to a QUBO problem; see Zaman et al. [

Then, we implement a method labeled _{1} and _{2} can be exponentially shifted by a constant, such as exp(

We develop a solver labeled

We have the set ^{mn} if we branch the problem into two sub-problems, namely, with _{d, o} = 0 or _{d, o} = 1. There are different possibilities for splitting the problem, e.g., driver-wise branching for ^{n} with

We need to assess the upper/lower bounds of a sub-problem to prune any redundant sub-problems generated. Since the upper bound of _{d, o}, denoted by _{d, o} = 1 and _{o}, denoted by

Together with the branching phase, we can discuss the computational aspect of branch-and-bound solvers for the ID problem. That is, the computational complexity of BnB solvers is straightforward

Since the set of 2^{nm} decision variables is quite large, the aforementioned branch-and-bound-based solver is still inefficient for some ID problem instances. We then incorporate a new perspective into our branch-and-bound-based solver for the ID problem using implicit constraints. Typical solutions obtained with IEC/MLEC often satisfy _{d, o}, and so the value of ℙ_{o}.

Based on the aforementioned implicit constraints, we propose a heuristic, range-constrained branch-and-bound-based solver. Here, a range [_{d}, _{d}] should be given for each _{d}, _{d}] are assumed to be displayed. If [_{d}, _{d}] = [1, 1], then _{d}<_{d} = 1 for _{d} =

We experimentally evaluate methods for the ID problem.

We prepare two sets of randomly generated instances and name them ^{0}, while

Summary of two datasets,

^{0} |
||||
---|---|---|---|---|

UnifID | Unif(8, 14) | Unif(8, 14) | 2, 3, …, 8 | 50 |

NormID [ |
TrunNorm(20, 10, 5, 40) | 15 | 2, 3, …, 8 | 50 |

Value

Here, we compare the following methods:

^{nm} variables,

^{25} binary variables in

Results for

Results for

Furthermore, the QUBO-based methods did find optimal solutions for

Compared with _{ID} does not suit the branch-and-bound-based solver. Since ℙ_{o} depends on the whole _{ID}, and _{ID}. Linear relaxations of _{d, o} (i.e., relaxation of _{d, o}∈{0, 1} to 0 ≤ _{d, o} ≤ 1) do not accelerate the bounding phase as _{ID} is the sum of high-dimensional products. Therefore, developing an efficient bounding method would be an important next step in elucidating the branch-and-bound-based method for the ID problem. Such a development could be valuable also for evaluating the performance of

To estimate the performance of _{d, o} in the computation _{d, o}, i.e., the possible ranges in _{d, o} values are similar (i.e., modifications in _{d, o} change objective values drastically).

Objective values comparison with

NormID | UnifID | |||||
---|---|---|---|---|---|---|

BnB1 |
BnB1 |
|||||

> | = | < | > | = | < | |

2 | 0 | 47 | 3 | 1 | 32 | 17 |

3 | 9 | 40 | 1 | 6 | 26 | 18 |

4 | 17 | 33 | 0 | 8 | 21 | 21 |

5 | 24 | 26 | 0 | 12 | 14 | 24 |

Columns (>, =, <) indicate the number of instances whose objective values are better than counterparts.

Objective values comparison with

NormID | UnifID | |||
---|---|---|---|---|

BnB |
BnB |
|||

> | = | > | = | |

2 | 0 | 50 | 1 | 49 |

3 | 9 | 41 | 7 | 43 |

4 | 17 | 33 | 9 | 41 |

5 | 24 | 26 | 18 | 32 |

Columns (>, =, <) indicate the number of instances whose objective values are better than counterparts.

In the present article, we have studied the ID problem by revisiting the methods proposed by Yang et al. [

These results for a class of optimization problems based on decision models (i.e., probabilities of selecting items) are valuable for discussions of human-centric optimization tasks. In our future study, we will investigate the theoretical background of problems and other objective functions for real applications involving MoD services.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

KO conceived of the presented idea and developed the method and performed the computations. AO and HY verified the analytical methods and results, encouraged KO to investigate quantum annealing solvers and quadratic unconstrained binary optimization (QUBO) problems to discuss the targeting optimization problem, and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript. All authors contributed to the article and approved the submitted version.

KO, AO, and HY are employed by Toyota Central R&D Labs., Inc.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

The Supplementary Material for this article can be found online at:

^{1}The original notation in Yang et al. [_{o, d}, but we write it as _{d, o} for the convenience.

^{2}The term

^{3}Yang et al. [_{0} = 0, β_{1} = 1[USD], β_{2} = −0.7[USD/km], α = 1.0,

^{4}Intuitively, the value

^{5}The choice becomes at random (α → 0) and seems to be deterministic (α → ∞).