Sequencing games with repeated players

Two classes of one machine sequencing situations are considered in which each job corresponds to exactly one player but a player may have more than one job to be processed, so called RP(repeated player) sequencing situations. In max-RP sequencing situations it is assumed that each player’s cost function is linear with respect to the maximum completion time of his jobs, whereas in min-RP sequencing situations the cost functions are linear with respect to the minimum completion times. For both classes, following explicit procedures to go from the initial processing order to an optimal order for the coalition of all players, equal gain splitting rules are defined. It is shown that these rules lead to core elements of the associated RP sequencing games. Moreover, it is seen that min-RP sequencing games are convex. We thank two referees for their valuable suggestions for improvement. Financial support for P. Calleja has been given by the Ministerio de Educación y Ciencia and FEDER under grant SEJ2005-02443/ECON, and by the Generalitat de Catalunya through a BE grant from AGAUR and grant 2005SGR00984.     

Convex games with an infinite number of players and sequencing situations

In this paper we study convex games with an infinite countable set of agents and provide characterizations of this class of games. To do so, and in order to overcome some shortcomings related to the difficulty of dealing with infinite orderings, we need to use a continuity property. Infinite sequencing situations where the number of jobs is infinite countable can be related to convex cooperative TU games. It is shown that some allocations turn out to be extreme points of the core of an infinite sequencing game.     

On games corresponding to sequencing situations with ready times

This paper considers the special class of cooperative sequencing games that arise from one-machine sequencing situations in which all jobs have equal processing times and the ready time of each job is a multiple of the processing time.By establishing relations between optimal orders of subcoalitions, it is shown that each sequencing game within this class is convex.This author is financially supported by the Dutch Organization for Scientific Research (NWO).     

On the Convexity of Precedence Sequencing Games

In this paper we study a class of cooperative sequencing games that arise from one-machine sequencing situations in which chain precedence relations are imposed on the jobs. We show that these sequencing games are convex if the initial order of the jobs is a concatenation of chains. F. Klijn's research is supported by a Ramón y Cajal contract of the Spanish Ministerio de Ciencia y Tecnología. The main part of F. Klijn's work was supported by a Marie Curie Fellowship of the European Community programme “Improving Human Research Potential and the Socio-economic Knowledge Base” under contract number HPMF-CT-2001-01232, carried out at the Departament d'Economia i d'Història Econòmica, Universitat Autònoma de Barcelona. His work is also partially supported by Research Grant BEC2002-02130 from the Spanish Ministerio de Ciencia y Tecnología and by the Barcelona Economics Program of CREA     

Sequencing games without initial order

In this note we study uncertainty sequencing situations, i.e., one-machine sequencing situations in which no initial order is specified. We associate cooperative games with these sequencing situations, study their core, and provide links with the classic sequencing games introduced by Curiel et al. (Eur J Oper Res 40:344–351, 1989). Moreover, we propose and characterize two simple cost allocation rules for uncertainty sequencing situations with equal processing times.     

Sequencing Games with Slack Due Windows and Group Technology Considerations

We study sequencing situations in which the customers are initially sequenced to be served by a single server. We consider both slack due windows and group technology simultaneously. We introduce two division rules to divide among the customers the cost saving from resequencing the customers to follow the optimal sequence and characterize the rules axiomatically. Applying cooperative game theory to analyze the sequencing games corresponding to the sequencing situations, we use the theory’s solution concepts to solve the games.     

On the balancedness of multiple machine sequencing games

This paper takes a game theoretical approach to sequencing situations with m parallel and identical machines. We show that in a cooperative environment cooperative m-sequencing games, which involve n players, give rise to m-machine games, which involve m players. Here, n corresponds to the number of jobs in an m-sequencing situation, and m corresponds to the number of machines in the same m-sequencing situation. We prove that an m-sequencing game is balanced if and only if the corresponding m-machine game is balanced. Furthermore, it is shown that m-sequencing games are balanced if m∈{1,2}. Finally, if m⩾3, balancedness is established for two special classes of m-sequencing games. Furthermore, we consider a special class of m-sequencing situations in a noncooperative setting and show that a transfer payments scheme exists that is both incentive compatible and budget balanced.     

Linear Transformation of Products: Games and Economies

In this paper, we introduce situations involving the linear transformation of products (LTP). LTP situations are production situations where each producer has a single linear transformation technique. First, we approach LTP situations from a (cooperative) game theoretical point of view. We show that the corresponding LTP games are totally balanced. By extending an LTP situation to one where a producer may have more than one linear transformation technique, we derive a new characterization of (nonnegative) totally balanced games: each totally balanced game with nonnegative values is a game corresponding to such an extended LTP situation. The second approach to LTP situations is based on a more economic point of view. We relate (standard) LTP situations to economies in two ways and we prove that the economies are standard exchange economies (with production). Relations between the equilibria of these economies and the cores of cooperative LTP games are investigated.     

Sharing costs in highways: A game theoretic approach

This paper introduces a new class of games, highway games, which arise from situations where there is a common resource that agents will jointly use. That resource is an ordered set of several indivisible sections, where each section has an associated fixed cost and each agent requires some consecutive sections. We present an easy formula to calculate the Shapley value, and we present an efficient procedure to calculate the nucleolus for this class of games.     

Processing games with restricted capacities

This paper analyzes processing problems and related cooperative games. In a processing problem there is a finite set of jobs, each requiring a specific amount of effort to be completed, whose costs depend linearly on their completion times. The main feature of the model is a capacity restriction, i.e., there is a maximum amount of effort per time unit available for handling jobs. There are no other restrictions whatsoever on the processing schedule.Assigning to each job a player and letting each player have an individual capacity for handling jobs, each coalition of cooperating players in fact faces a processing problem with the coalitional capacity being the sum of the individual capacities of the members. The corresponding processing game summarizes the minimal joint costs for every coalition. It turns out that processing games are totally balanced. The proof of this statement is constructive and provides a core element in polynomial time.     

Balancedness of multiple machine sequencing games revisited

We study m-sequencing games, which were introduced by [Hamers, H., Klijn, F., Suijs, J., (1999). On the balancedness of multiple machine sequencing games. European Journal of Operational Research 119, 678–691]. We answer the open question whether all these games are balanced in the negative. We do so, by an example of a 3-sequencing situation with 5 jobs, whose associated 3-sequencing game has an empty core. The counterexample finds its basis in an inconsistency in [Hamers et al., ibid], which was probably overlooked by the authors. This observation demands for a detailed reconsideration of their proofs.¹     

Single-machine scheduling of proportionally deteriorating jobs by two agents

We consider a problem of scheduling a set of independent jobs by two agents on a single machine. Every agent has its own subset of jobs to be scheduled and uses its own optimality criterion. The processing time of each job proportionally deteriorates with respect to the starting time of the job. The problem is to find a schedule that minimizes the total tardiness of the first agent, provided that no tardy job is allowed for the second agent. We prove basic properties of the problem and give a lower bound on the optimal value of the total tardiness criterion. On the basis of these results, we propose a branch-and-bound algorithm and an evolutionary algorithm for the problem. Computational experiments show that the exact algorithm solves instances up to 50 jobs in a reasonably short time and that solutions obtained by the metaheuristic are close to optimal ones.     

Multi-stage sequencing situations

This paper introduces the idea of dynamics in cooperative games. The concrete case of multi-stage sequencing situations and the difficulties involved in defining stable cost savings allocations for the games arising from these situations is studied. The MEGS-rule is defined and proven to yield stable allocations. A characterization for the MEGS-rule is given.     

同时最小化最大费用和最大完工时间的双代理无界平行分批排序

有两个代理A和B, 每个代理都各自有一个工件集. 同一个代理的工件可以在同一批中加工, 而且每一个代理都有一个需要最小化的函数. 研究在无界平行分批处理机上同时最小化代理A的最大费用和代理B的最大完工时间问题, 并给出一个算法, 它可在多项式时间内找到关于这个问题的所有Pareto最优点.     

A note on two-agent scheduling on an unbounded parallel-batching machine with makespan and maximum lateness objectives

This paper studies the two-agent scheduling on an unbounded parallel-batching machine. In the problem, there are two agents A and B with each having their own job sets. The jobs of a common agent can be processed in a common batch. Moreover, each agent has an objective function to be minimized. The objective function of agent A is the makespan of his jobs and the objective function of agent B is maximum lateness of his jobs. Yazdani Sabouni and Jolai [M.T. Yazdani Sabouni, F. Jolai, Optimal methods for batch processing problem with makespan and maximum lateness objectives, Appl. Math. Model. 34 (2010) 314–324] presented a polynomial-time algorithm for the problem to minimize a positive combination of the two agents’ objective functions. Unfortunately, their algorithm is incorrect. We then dwell on the problem and present a polynomial-time algorithm for finding all Pareto optimal solutions of this two-agent parallel-batching scheduling problem.     

Partitioning sequencing situations and games

The interaction between sequencing situations and cooperative games starting from the paper of Curiel et al. [Curiel, I., Pederzoli, G., Tijs S., 1989. Sequencing games. European Journal of Operational Research 40, 344–351], has become an established line of research within the theory of operation research games.     

Processing games with shared interest

This paper introduces processing problems with shared interest as an extension of processing situations with restricted capacities (Meertens, M., et al., Processing games with restricted capacities, 2004). Next to an individual capacity to handle jobs, each player now may have interest in the completion of more than one job, and the degrees of interest may vary among players. By cooperating the players can bundle their capacities and follow an optimal processing scheme to minimize total joint costs. The resulting cost allocation problem is analyzed by considering an associated cooperative cost game. An explicit core allocation of this game is provided.     

原始工件的完工时间限制下的最小化最大延误时间的重新排序

考虑由两个代理引起的重新排序问题,其中每个代理都在公共的加工资源下完成各自的不可中断加工的工件.每个代理要求在仅依赖工件的完工时间时最小化某一个特定的目标函数.考虑在原始工件的完工时间限制下的两个代理的单机最小化最大延误时间的重新排序问题.证明了该问题能在多项式时间或者拟多项式时间内解决.