Approximately Fair Cost Allocation in Metric Traveling Salesman Games

A traveling salesman game is a cooperative game ${\mathcal{G}}=(N,c_{D})$ . Here N, the set of players, is the set of cities (or the vertices of the complete graph) andc D is the characteristic function where D is the underlying cost matrix. For all S⊆N, define c D (S) to be the cost of a minimum cost Hamiltonian tour through the vertices of S∪{0} where $0\not \in N$ is called as the home city. Define Core ({\mathcal{G}})=\{x\in \Re^{|N|}:x(N)=c_{D}(N)\ \mbox{and}\ \forall S\subseteq N,x(S)\le c_{D}(S)\} as the core of a traveling salesman game ${\mathcal{G}}$ . Okamoto (Discrete Appl. Math. 138:349-369, [2004]) conjectured that for the traveling salesman game ${\mathcal{G}}=(N,c_{D})$ with D satisfying triangle inequality, the problem of testing whether Core $({\mathcal{G}})$ is empty or not is $\mathsf{NP}$ -hard. We prove that this conjecture is true. This result directly implies the $\mathsf{NP}$ -hardness for the general case when D is asymmetric. We also study approximately fair cost allocations for these games. For this, we introduce the cycle cover games and show that the core of a cycle cover game is non-empty by finding a fair cost allocation vector in polynomial time. For a traveling salesman game, let \epsilon\mbox{-Core}({\mathcal{G}})=\{x\in \Re^{|N|}:x(N)\ge c_{D}(N) and ∀ S⊆N, x(S)≤ε⋅c D (S)} be an ε-approximate core, for a given ε>1. By viewing an approximate fair cost allocation vector for this game as a sum of exact fair cost allocation vectors of several related cycle cover games, we provide a polynomial time algorithm demonstrating the non-emptiness of the log 2(|N|−1)-approximate core by exhibiting a vector in this approximate core for the asymmetric traveling salesman game. We improve it further by finding a $(\frac{4}{3}\log_{3}(|N|)+c)$ -approximate core in polynomial time for some constantc. We also show that there exists an ε 0>1 such that it is $\mathsf{NP}$ -hard to decide whether ε 0-Core $({\mathcal{G}})$ is empty or no

Stability and fairness in models with a multiple membership

This article studies a model of coalition formation for the joint production (and finance) of public projects, in which agents may belong to multiple coalitions. We show that, if projects are divisible, there always exists a stable (secession-proof) structure, i.e., a structure in which no coalition would reject a proposed arrangement. When projects are in- divisible, stable allocations may fail to exist and, for those cases, we resort to the least core in order to estimate the degree of instability. We also examine the compatibility of stability and fairness on metric environments with indivisible projects. To do so, we explore, among other things, the performance of several well-known solutions (such as the Shapley value, the nucleolus, or the Dutta-Ray value) in these environments.stability, fairness, membership, coalition formation

Stability and Fairness in Models with a Multiple Membership

This article studies a model of coalition formation for the joint production (and finance) of public projects, in which agents may belong to multiple coalitions. We show that, if projects are divisible, there always exists a stable (secession-proof) structure, i.e., a structure in which no coalition would reject a proposed arrangement. When projects are indivisible, stable allocations may fail to exist and, for those cases, we resort to the least core in order to estimate the degree of instability. We also examine the compatibility of stability and fairness in metric environments with indivisible projects, where we also explore the performance of well-known solutions, such as the Shapley value and the nucleolus.Stability, Fairness, Membership, Coalition Formation

Stability and Fairness in Models with a Multiple Membership

This article studies a model of coalition formation for the joint production (and finance) of public projects, in which agents may belong to multiple coalitions. We show that, if projects are divisible, there always exists a stable (secession-proof) structure, i.e., a structure in which no coalition would reject a proposed arrangement. When projects are indivisible, stable allocations may fail to exist and, for those cases, we resort to the least core in order to estimate the degree of instability. We also examine the compatibility of stability and fairness in metric environments with indivisible projects, where we also explore the performance of well-known solutions, such as the Shapley value and the nucleolus.Stability, Fairness, Membership, Coalition Formation

A machine learning approach for allocating route cost to customers for transportation and logistics services.

Advancements in big data enabled management practices inspire logistics companies to study deeper into their transportation operations with a data driven approach. One such question asks: How can a logistics firm identify high-cost customers in their service network? In the presence of rich data on routes involving many customers, this thesis develops a framework to allocate a route cost among customers that the route serves, where each route is associated with multiple route features related to the transportation cost. Cost is allocated using the proportional allocation approach in combination with the random forest method in machine learning. First, this framework ensembles random forest regression models to determine the importance values of all route features. Next, the importance values of route features are used to allocate cost among customers. Finally, posterior analysis identifies customers in a route or in general that are most costly to serve. Several additional analyses are performed to show potential uses of this cost allocation output. Results of the framework and analyses on three simulated case and two industry cases show the validity of the model and the potential for actionable operational analysis and changes

Spartan Daily, March 20, 1981

Volume 76, Issue 38https://scholarworks.sjsu.edu/spartandaily/6742/thumbnail.jp

Optimization and Mechanism Design for Ridesharing Services

Ridesharing services, whose aim is to gather travelers with similar itineraries and compatible schedules, are able to provide substantial environmental and social benefits through reducing the use of private vehicles. When the operations of a ridesharing system is optimized, it can also save travelers a significant amount of transportation cost. The economic benefits associated with ridesharing in turn attract more travelers to participate in ridesharing services and thereby improve the utilization of transportation infrastructure capacity. This study addresses two of the most challenging issues in designing an efficient and sustainable ridesharing service: ridesharing optimization and ridesharing market design. The first part of the dissertation formally defines the large-scale ridesharing optimization problem, characterizes its complexity and discusses its relation to classic relevant problems like the traveling salesman problem (TSP) and the vehicle routing problem (VRP). A mixed-integer program (MIP) model is developed to solve the ridesharing optimization problem. Since the ridesharing optimization problem is NP-hard, the MIP model is not able to solve larger instances within a reasonable time. An insertion-based heuristic is developed to get approximate solutions to the ridesharing optimization problem. Experiments showed that ridesharing can significantly reduce the system-wide travel cost and vehicle trips. Evaluation of the heuristic solution method showed that the heuristic approach can solve the problem very fast and provide nearly-optimal (98%) solutions, thus, confirming its efficiency and accuracy. From a societal perspective, the ridesharing optimization model proposed in this dissertation provided substantial system-wide travel cost saving (25%+) and vehicle-trip saving (50%) compared to non-ridesharing situation. However, the system-level optimal solution might not completely align with individual participant interest. The second part of this dissertation formulates this issue as a fair cost allocation problem through the lens of the cooperative game theory. A special property of the cooperative ridesharing game is that, its characteristic function values are calculated by solving an optimization problem. We characterize the game to be monotone and subadditive, but non-convex. Several concepts of fairness are investigated and special attention is paid to a solution concept named nucleolus, which aims to minimize the maximum dissatisfaction in the system. However, finding the nucleolus is very challenging because it requires solving the ridesharing optimization problem for every possible coalition, whose number grows exponentially as the number of participants increases in the system. We break the cost allocation (nucleolus finding) problem into a master-subproblem structure and two subproblems are developed to generate constraints for the master problem. We propose a coalition generation procedure to find the nucleolus and approximate nucleolus of the game. When the game has a non-empty core, in the approximate nucleolus scheme the coalitions are computed only when it is necessary, and the approximate nucleolus scheme produces the actual nucleolus. Experimental results showed that, when the game has an empty core, the approximate nucleolus is close to the actual nucleolus. Results also showed that, regardless of the emptiness of the game, by using our algorithm, only a small fraction (1:6%) of the total coalition constraints were generated to compute the approximate nucleolus, and the approximate nucleolus is close to the actual nucleolus

An Extension of the Core solution Concept.

A solution concept for cooperative games, the extended core, is introduced. This concept is always nonempty yet coincides with the core whenever it is nonempty. Moreover, a non-cooperative framework can generate the extended core. Every transferable utility game is associated with a two-player zero-sum non-cooperative game. The min-max values of the associated zerosum games characterize when cooperative games have nonempty cores. If the core is empty, the min-max value determines how an exogenous regulator can impose costs on proper coalition formation so that there are no incentives to deviate from extended core imputations, which are necessarily feasible in the original game. In order to choose among the imputations belonging to the extended core, a proportional version of the nucleolus is proposed as a selection device.

Combinatorial Optimization (hybrid meeting)

Combinatorial Optimization deals with optimization problems defined on combinatorial structures such as graphs and networks. Motivated by diverse practical problem setups, the topic has developed into a rich mathematical discipline with many connections to other fields of Mathematics (such as, e.g., Combinatorics, Convex Optimization and Geometry, and Real Algebraic Geometry). It also has strong ties to Theoretical Computer Science and Operations Research. A series of Oberwolfach Workshops have been crucial for establishing and developing the field. The workshop we report about was a particularly exciting event - due to the depth of results that were presented, the spectrum of developments that became apparent from the talks, the breadth of the connections to other mathematical fields that were explored, and last but not least because for many of the particiants it was the first opportunity to exchange ideas and to collaborate during an on-site workshop since almost two years