The Traveling Salesman Problem with Stochastic and Correlated Customers

It is well-known that the cost of parcel delivery can be reduced by designingroutes that take into account the uncertainty surrounding customers’ presences. Thus far, routing problems with stochastic customer presences have relied on the assumption that all customer presences are independent from each other. However, the notion that demographic factors retain predictive power for parcel-delivery efficiency suggests that shared characteristics can be exploited to map dependencies between customer presences. This paper introduces the correlated probabilistic traveling salesman problem (CPTSP). The CPTSP generalizes the traveling salesman problem with stochastic customer presences, also known as the probabilistic traveling salesman problem (PTSP), to account for potentialcorrelations between customer presences. I propose a generic and flexible model formulation for the CPTSP using copulas that maintains computational and mathematical tractability in high-dimensional settings. I also present several adaptations of existing exact and heuristic frameworks to solve the CPTSP effectively. Computational experiments on real-world parcel-delivery data reveal that correlations between stochastic customer presences do not always affect route decisions, but could have a considerable impact on route costestimates

확률최대화 조합최적화 문제에 대한 근사해법

학위논문(석사)--서울대학교 대학원 :공과대학 산업공학과,2019. 8. 이경식.In this thesis, we consider a variant of the deterministic combinatorial optimization problem (DCO) where there is uncertainty in the data, the probability maximizing combinatorial optimization problem (PCO). PCO is the problem of maximizing the probability of satisfying the capacity constraint, while guaranteeing the total profit of the selected subset is at least a given value. PCO is closely related to the chance-constrained combinatorial optimization problem (CCO), which is of the form that the objective function and the constraint function of PCO is switched. It search for a subset that maximizes the total profit while guaranteeing the probability of satisfying the capacity constraint is at least a given threshold. Thus, we discuss the relation between the two problems and analyse the complexities of the problems in special cases. In addition, we generate pseudo polynomial time exact algorithms of PCO and CCO that use an exact algorithm of a deterministic constrained combinatorial optimization problem. Further, we propose an approximation scheme of PCO that is fully polynomial time approximation scheme (FPTAS) in some special cases that are NP-hard. An approximation scheme of CCO is also presented which was derived in the process of generating the approximation scheme of PCO.본 논문에서는 일반적인 조합 최적화 문제(deterministic combinatorial optimization problem : DCO)에서 데이터의 불확실성이 존재할 때를 다루는 문제로, 총 수익을 주어진 상수 이상으로 보장하면서 용량 제약을 만족시킬 확률을 최대화하는 확률 최대화 조합 최적화 문제(probability maximizing combinatorial optimization problem : PCO)을 다룬다. PCO와 매우 밀접한 관계가 있는 문제로, 총 수익을 최대화하면서 용량 제약을 만족시킬 확률이 일정 값 이상이 되도록 보장하는 확률 제약 조합 최적화 문제(chance-constrained combinatorial optimization problem : CCO)가 있다. 우리는 두 문제의 관계에 대하여 논의하고 특정 조건 하에서 두 문제의 복잡도를 분석하였다. 또한, 제약식이 하나 추가된 DCO를 반복적으로 풀어 PCO와 CCO의 최적해를 구하는 유사 다항시간 알고리즘을 제안하였다. 더 나아가, PCO가 NP-hard인 특별한 인스턴스들에 대해서 완전 다항시간 근사해법(FPTAS)가 되는 근사해법을 제안하였다. 이 근사해법을 유도하는 과정에서 CCO의 근사해법 또한 고안하였다.Chapter 1 Introduction 1 1.1 Problem Description 1 1.2 Literature Review 7 1.3 Research Motivation and Contribution 12 1.4 Organization of the Thesis 13 Chapter 2 Computational Complexity of Probability Maximizing Combinatorial Optimization Problem 15 2.1 Complexity of General Case of PCO and CCO 18 2.2 Complexity of CCO in Special Cases 19 2.3 Complexity of PCO in Special Cases 27 Chapter 3 Exact Algorithms 33 3.1 Exact Algorithm of PCO 34 3.2 Exact Algorithm of CCO 38 Chapter 4 Approximation Scheme for Probability Maximizing Combinatorial Optimization Problem 43 4.1 Bisection Procedure of rho 46 4.2 Approximation Scheme of CCO 51 4.3 Variation of the Bisection Procedure of rho 64 4.4 Comparison to the Approximation Scheme of Nikolova 73 Chapter 5 Conclusion 77 5.1 Concluding Remarks 77 5.2 Future Works 79 Bibliography 81 국문초록 87Maste

Modelos logísticos estocásticos: una revisión de la literatura

The present article is done with the aim of to establish the state of the art of location, inventory and routing models with stochastic parameters. The literature reviewed was classified with an operations research approach. A systematic review of the literature in the databases ScienceDirect, ScholarGoogle, SpringerLink, Scopus, SemanticScholar, Scielo, and ResearchGate was done. Through guiding questions, 99 articles were selected, of which 74% is recent literature between 2015 and 2019. Stochastic logistic individual models were classified, a taxonomy with an operations research approach was proposed, from their characteristics, parameters, restrictions, the objective functions and solution methods used. Also, the trends and the future lines of research were identified. As a conclusion new strategies and operating policies that permit improve supply chain performance are identified, also the absence of efficient solution methods has been evident to large instances, according to real-life.El presente artículo se realiza con el objetivo de establecer el estado del arte de los modelos de localización, inventario y ruteo con parámetros estocásticos. Se realizó una revisión sistemática de la literatura en las bases de datos ScienceDirect, ScholarGoogle, SpringerLink, Scopus, SemanticScholar, Scielo y ResearchGate. A través de preguntas orientadoras, se seleccionaron 99 artículos, de los cuales el 74% es literatura reciente entre 2015 y 2019.  Se clasificaron los modelos individuales logísticos estocásticos, se propuso una taxonomía con un enfoque de investigación de operaciones, a partir de sus características, parámetros, restricciones, funciones objetivo y métodos de solución utilizados. Asimismo, se identificaron las tendencias y las futuras líneas de investigación. Como conclusión se identifican nuevas estrategias y políticas operativas que permiten mejorar el desempeño de la cadena de suministro, igualmente la ausencia de métodos de solución eficientes ha sido evidente en grandes instancias, según la vida real

New Routing Problems with possibly correlated travel times

In the literature of operational research, Vehicle Routing Problems (VRP) were and still are subject of countless studies. Under the scope of combinatorial optimization, this thesis analyses some variants of VRP both with deterministic and uncertain travel times. The deterministic problem under study is a drayage problem with characteristics con- cerning service types and requirement seldom investigated all together. The formulations proposed to model this problem are: the node-arc formulation and the Set Partitioning formu- lation. Concerning the solution methods, two heuristics and a branch-and-price approach are presented. The section dealing with uncertain and correlated travel times faces two classes of VRP with time windows using either single or joint chance constraints depending on whether missing a customers time window makes the entire route infeasible or not. From a comparison between deterministic and stochastic methods, these last represent a profitable investment to guarantee the feasibility of the solution in realistic instances