무인항공기 운영을 위한 덮개 모델 기반의 대규모 최적화 기법

학위논문 (박사) -- 서울대학교 대학원 : 공과대학 산업공학과, 2021. 2. 문일경.There is increasing interest in the unmanned aerial vehicle (UAV) in various fields of the industry, starting from the surveillance to the logistics. After introducing the smart city, there are attempts to utilize UAVs in the public service sector by connecting individual components of the system with both information and physical goods. In this dissertation, the UAV operation problems in the public service sector is modeled in the set covering approach. There is a vast literature on the facility location and set covering problems. However, when operating UAVs in the system, the plan has to make the most of the flexibility of the UAV, but also has to consider its physical limitation. We noticed a gap between the related, existing approaches and the technologies required in the field. That is, the new characteristics of the UAV hinder the existing solution algorithms, or a brand-new approach is required. In this dissertation, two operation problems to construct an emergency wireless network in a disaster situation by UAV and one location-allocation problem of the UAV emergency medical service (EMS) facility are proposed. The reformulation to the extended formulation and the corresponding branch-and-price algorithm can overcome the limitations and improve the continuous or LP relaxation bounds, which are induced by the UAV operation. A brief explanation of the UAV operation on public service, the related literature, and the brief explanation of the large-scale optimization techniques are introduced in Chapter 1, along with the research motivations and contributions, and the outline of the dissertations. In Chapter 2, the UAV set covering problem is defined. Because the UAV can be located without predefined candidate positions, more efficient operation becomes feasible, but the continuous relaxation bound of the standard formulation is weakened. The large-scale optimization techniques, including the Dantzig-Wolfe decomposition and the branch-and-price algorithm, could improve the continuous relaxation bound and reduce the symmetries of the branching tree and solve the realistic-scaled problems within practical computation time. To avoid numerical instability, two approximation models are proposed, and their approximation ratios are analyzed. In Chapter 3, UAV variable radius set covering problem is proposed with an extra decision on the coverage radius. While implementing the branch-and-price algorithm to the problem, a solvable equivalent formulation of the pricing subproblem is proposed. A heuristic based on the USCP is designed, and the proposed algorithm outperformed the benchmark genetic algorithm proposed in the literature. In Chapter 4, the facility location-allocation problem for UAV EMS is defined. The quadratic variable coverage constraint is reformulated to the linear equivalent formulation, and the nonlinear problem induced by the robust optimization approach is linearized. While implementing the large-scale optimization techniques, the structure of the subproblem is analyzed, and two solution approaches for the pricing subproblem are proposed, along with a heuristic. The results of the research can be utilized when implementing in the real applications sharing the similar characteristics of UAVs, but also can be used in its abstract formulation.현재, 지역 감시에서 물류까지, 무인항공기의 다양한 산업에의 응용이 주목받고 있다. 특히, 스마트 시티의 개념이 대두된 이후, 무인항공기를 공공 서비스 영역에 활용하여 개별 사회 요소를 연결, 정보와 물자를 교환하고자 하는 시도가 이어지고 있다. 본 논문에서는 공공 서비스 영역에서의 무인항공기 운영 문제를 집합덮개문제 관점에서 모형화하였다. 설비위치결정 및 집합덮개문제 영역에 많은 연구가 진행되어 있으나, 무인항공기를 운영하는 시스템의 경우 무인항공기가 갖는 자유도를 충분히 활용하면서도 무인항공기의 물리적 한계를 고려한 운영 계획을 필요로 한다. 우리는 본 문제와 관련된 기존 연구와 현장이 필요로 하는 기술의 괴리를 인식하였다. 이는 다시 말해, 무인항공기가 가지는 새로운 특성을 고려하면 기존의 문제 해결 방법을 통해 풀기 어렵거나, 혹은 새로운 관점에서의 문제 접근이 필요하다는 것이다. 본 논문에서는 재난이 발생한 지역에 무인항공기를 이용하여 긴급무선네트워크를 구성하는 두가지 문제와, 무인항공기를 이용하여 응급의료서비스를 제공하는 시설의 위치설정 및 할당문제를 제안한다. 확장문제로의 재공식화와 분지평가법을 활용하여, 무인항공기의 활용으로 인해 발생하는 문제 해결 방법의 한계를 극복하고 완화한계를 개선하였다. 공공 서비스 영역에서의 무인항공기 운영, 관련된 기존 연구와 본 논문에서 사용하는 대규모 최적화 기법에 대한 개괄적인 설명, 연구 동기 및 기여와 논문의 구성을 1장에서 소개한다. 2장에서는 무인항공기 집합덮개문제를 정의한다. 무인항공기는 미리 정해진 위치 없이 자유롭게 비행할 수 있기 때문에 더 효율적인 운영이 가능하나, 약한 완화한계를 갖게 된다. Dantzig-Wolfe 분해와 분지평가법을 포함한 대규모 최적화 기법을 통해 완화한계를 개선할 수 있으며, 분지나무의 대칭성을 줄여 실제 규모의 문제를 실용적인 시간 안에 해결할 수 있었다. 수치적 불안정성을 피하기 위하여, 두 가지 선형 근사 모형이 제안되었으며, 이들의 근사 비율을 분석하였다. 3장에서는 무인항공기 집합덮개문제를 일반화하여 무인항공기 가변반경 집합덮개문제를 정의한다. 분지평가법을 적용하면서 해결 가능한 평가 부문제를 제안하였으며, 휴리스틱을 설계하였다. 제안한 풀이 방법들이 기존 연구에서 제안한 벤치마크 유전 알고리즘을 능가하는 결과를 나타내었다. 4장에서는 무인항공기 응급의료서비스를 운영하는 시설의 위치설정 및 할당문제를 정의하였다. 2차 가변반경 범위제약이 선형의 동치인 수식으로 재공식화되었으며, 강건최적화 기법으로 인해 발생하는 비선형 문제를 선형화하였다. 대규모 최적화 기법을 적용하면서, 평가 부문제의 구조를 분석하여 두 가지 풀이 기법과 휴리스틱을 제안하였다. 본 연구의 결과는 무인항공기와 비슷한 특징을 가지는 실제 사례에 적용될 수 있으며, 추상적인 문제로써 다양한 분야에 그대로 활용될 수도 있다.Abstract i Contents vii List of Tables ix List of Figures xi Chapter 1 Introduction 1 1.1 Unmanned aerial vehicle operation on public services 1 1.2 Facility location problems 3 1.3 Large-scale optimization techniques 4 1.4 Research motivations and contributions 6 1.5 Outline of the dissertation 12 Chapter 2 Unmanned aerial vehicle set covering problem considering fixed-radius coverage constraint 14 2.1 Introduction 14 2.2 Problem definition 20 2.2.1 Problem description 22 2.2.2 Mathematical formulation 23 2.2.3 Discrete approximation model 26 2.3 Branch-and-price approach for the USCP 28 2.3.1 An extended formulation of the USCP 29 2.3.2 Branching strategies 34 2.3.3 Pairwise-conflict constraint approximation model based on Jung's theorem 35 2.3.4 Comparison of the approximation models 40 2.3.5 Framework of the solution algorithm for the PCBP model 42 2.4 Computational experiments 44 2.4.1 Datasets used in the experiments 44 2.4.2 Algorithmic performances 46 2.5 Solutions and related problems of the USCP 61 2.6 Summary 64 Chapter 3 Unmanned aerial vehicle variable radius set covering problem 66 3.1 Introduction 66 3.2 Problem definition 70 3.2.1 Mathematical model 72 3.3 Branch-and-price approach to the UVCP 76 3.4 Minimum covering circle-based approach 79 3.4.1 Formulation of the pricing subproblem II 79 3.4.2 Equivalence of the subproblem 82 3.5 Fixed-radius heuristic 84 3.6 Computational experiments 86 3.6.1 Datasets used in the experiments 88 3.6.2 Solution algorithms 91 3.6.3 Algorithmic performances 94 3.7 Summary 107 Chapter 4 Facility location-allocation problem for unmanned aerial vehicle emergency medical service 109 4.1 Introduction 109 4.2 Related literature 114 4.3 Location-allocation model for UEMS facility 117 4.3.1 Problem definition 118 4.3.2 Mathematical formulation 120 4.3.3 Linearization of the quadratic variable coverage distance function 124 4.3.4 Linear reformulation of standard formulation 125 4.4 Solution algorithms 126 4.4.1 An extended formulation of the ULAP 126 4.4.2 Branching strategy 129 4.4.3 Robust disjunctively constrained integer knapsack problem 131 4.4.4 MILP reformulation approach 132 4.4.5 Decomposed DP approach 133 4.4.6 Restricted master heuristic 136 4.5 Computational experiments 137 4.5.1 Datasets used in the experiments 137 4.5.2 Algorithmic performances 140 4.5.3 Analysis of the branching strategy and the solution approach of the pricing subproblem 150 4.6 Summary 157 Chapter 5 Conclusions and future research 160 5.1 Summary 160 5.2 Future research 163 Appendices 165 A Comparison of the computation times and objective value of the proposed algorithms 166 Bibliography 171 국문초록 188 감사의 글 190Docto

Algorithms for discrete, non-linear and robust optimization problems with applications in scheduling and service operations

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2011.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 101-107).This thesis presents efficient algorithms that give optimal or near-optimal solutions for problems with non-linear objective functions that arise in discrete, continuous and robust optimization. First, we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function. Examples of such problems include those in which the objective function is a product or ratio of two or more linear functions, parallel machine scheduling problems with the makespan objective, robust versions of weighted multi-objective optimization problems, and assortment optimization problems with logit choice models. For many of these problems, we give the first fully polynomial time approximation scheme using our framework. Next, we present approximation schemes for optimizing a rather general class of non-linear functions of low rank over a polytope. In contrast to existing results in the literature, our approximation scheme does not require the assumption of quasi-concavity of the objective function. For the special case of minimizing a quasi-concave function of low-rank, we give an alternative algorithm which always returns a solution which is an extreme point of the polytope. This algorithm can also be used for combinatorial optimization problems where the objective is to minimize a quasi-concave function of low rank. We also give complexity-theoretic results with regards to the inapproximability of minimizing a concave function over a polytope. Finally, we consider the problem of appointment scheduling in a robust optimization framework. The appointment scheduling problem arises in many service operations, for example health care. For each job, we are given its minimum and maximum possible execution times. The objective is to find an appointment schedule for which the cost in the worst case scenario of the realization of the processing times of the jobs is minimized. We present a global balancing heuristic, which gives an easy to compute closed form optimal schedule when the underage costs of the jobs are non-decreasing. In addition, for the case where we have the flexibility of changing the order of execution of the jobs, we give simple heuristics to find a near-optimal sequence of the jobs.by Shashi Mittal.Ph.D