Research On Optimal Ranking Optimization Of Finite Resources Game Under Constant Speed

Scheduling problem is a kind of combinatorial optimization problems, due to the processor task or assignment is limited, so most of the scheduling problems is to find an optimal solution from limited feasible solutions, as to achieve the minimum of the objective function.In this paper, we investigate resource allocation games for job scheduling when the resource are limited. The resource we considered are identical and the social costs of the games are utilitarian. In terms of machine scheduling, assignment of jobs to machines in which selfish agents, representing individual jobs, select machines for processing the jobs, and each job will be minimize its cost. The structure of this article is as follows:The first chapter is an introduction, it mainly introduces the combinatorial optimization problems, the background of game scheduling and some preliminary knowledge.In the second chapter, we consider the load balancing game in uniform machines. A Nash equilibrium(NE) is a strategy profile, in any NE assignment, no job can reduce its cost by unilaterally changing its machine. But in terms of a given social objective, such an Nash equilibrium is not necessarily, indeed can often be far from optimal. We use the notions of the price of anarchy( POA) and the price of stability( POS) to analyze the quality of NE solutions.when the The objective function is Total completion time,we prove the POA and the POS. When the objective function is the makespan, we prove the POA.In the third chapter, we consider a system with two uniform machines and a fixed number m of uniform machines. The social cost is the sum of the system makespan and activation cost of machines. We assess the quality of Nash equilibrium in terms of the POA.we assume that the speed of the machine is 1 and a respectively, and the activation of each machine cost is equal to its speed when there are two machines. we assume that the machine activation cost is 1 and don’t change with the speed of each machine when there are m machines, we prove the POA of two cases.