摘要
利用合作博弈理论研究了批处理机生产与成批运输协调调度问题。具有初始调度顺序的工件经过单台相同批处理机加工完成后,由一辆具有容量限制的运输车分批次运输到下一工序或客户端。考虑到工件可以通过结盟,并在联盟内重新调度以达到总成本节省,以工件联盟最大成本节省为特征函数建立合作博弈模型。分析博弈性质,并通过合理稳定的成本节省分配方法降低各工件成本。当运输车满批运输时,证明了合作博弈既是σ0-组可加博弈也是凸博弈,β规则和Shapley值均能得到核心分配,且Shapley值可以表述成一种简单可计算形式。当运输车无需满批运输时,通过反例分析了合作博弈的性质。
In the production process of the process industry,there are many complicated processes.The continuous production processes require the coordination of production and transportation scheduling.The coordinated scheduling problem of batch production and batch delivery(CSP-BPBD)has a wide range of application scenarios in process industries such as iron and steel industry.The CSP-BPBD is described as follows:there are n jobs belonging to different customers,each of which needs to be processed on an identical batching machine with production capacity limits and then transported in batches by a transporter with limited capacity to the downstream process(or a customer).Furthermore,there is an initial processing order on the jobs of customers.It takes the same time for the batching machine to produce a batch of jobs and the same time for the transporter to transport a batch of jobs,which are independent of the number of jobs in the batch.Considering that jobs are willing to form a coalition by cooperation and rearrange their processing and transportation orders to gain cost savings,we can allocate the cost savings reasonably,so that all customers in the coalition can benefit.Therefore,it is of certain practical significance to study a coordinated scheduling problem of production and transportation by taking a cooperative game theoretical approach.In this paper,we take the cooperative game theory to study CSP-BPBD.The customer’s cost is defined as a linear function of the job’s completion time.The cooperative game models are proposed for CSP-BPBD with jobs as players and the maximal cost savings of the coalition as the characteristic function.For two different cases of whether the transporter delivers jobs in full batch or not,the properties of corresponding games are analyzed,and some fair and stable allocations of cost savings are presented.Firstly,the conditions of feasible schedule scheme and optimal schedule scheme are given.For each scheduling order,there can be different feasible schedule schemes corresponding to it,and each feasible schedule scheme corresponds to only one scheduling order.The optimal schedule schemeπ^(*)must satisfy the following conditions:(1)π^(*)is a feasible schedule scheme arranged in non-increasing order according to the cost coefficient of the job;(2)The jobs are processed in full batch on the batching machine with no idle time.Secondly,considering whether the transporter delivers jobs in full batch or not,the sets of admissible schedule schemes are defined for a CSP-BPBD sequencing situation,and the corresponding cooperative games(N,v k)(k=1,2)are defined,where N is the set of players(or jobs),v k(k=1,2)is the maximal cost savings of coalition for two different ways of transportation.Some properties of the games are obtained,and the cost of each job can be reduced by reasonable and stable cost saving allocations.It is proved that,when the transporter delivers jobs in full batch,the cooperative game(N,v^(1))has no externalities(the cost savings of the coalition are not affected by the jobs outside)and isσ0-component additive,and then the characteristic function v^(1) can be written as the unique linear combination of unanimity games.Since unanimity games are convex,we show these coefficients of the linear combination are nonnegative,and the game(N,v^(1))is convex.That is to say,the bigger the coalition,the greater the marginal cost savings for(N,v^(1)).Both theβrule and the Shapley value can give a core element of a cooperative game(N,v^(1)).Moreover,the Shapley value can be expressed in a simple and computable form in this case.Numerical examples are given to verify the correctness of the conclusions.However,when there is no need for full batch transportation,it is obtained that the cooperative game(N,v^(2))has externalities.A counterexample is provided to show the game(N,v^(2))need not beσ0-component additive or convex.In this case,there is no guarantee that the core is nonempty,and these games need to be further studied.For future research,we will pay attention to the general CSP-BPBD and other coordinated scheduling problems in complex production and transportation environment,analyze the properties of cooperative games in different ways of cooperation,and design reasonable and stable allocations of cost savings.
作者
孙文娟
刘鹏
宫华
SUN Wenjuan;LIU Peng;GONG Hua(School of Management,Shenyang University of Technology,Shenyang 110870,China;School of Science,Shenyang Ligong University,Shenyang 110159,China)
出处
《运筹与管理》
CSSCI
CSCD
北大核心
2024年第12期115-121,I0048,I0049,共9页
Operations Research and Management Science
基金
辽宁省“兴辽英才计划”项目(XLYC2006017)
辽宁省教育厅科学研究项目(LG202025,LJKQZ2021057)。
作者简介
孙文娟(1982-),女,安徽怀宁人,副教授,博士研究生,研究方向:生产调度与物流优化;刘鹏(1978-),男,湖北黄冈人,教授,博士生导师,研究方向:生产调度与物流优化;通讯作者:宫华(1976-),女,辽宁桓仁人,教授,博士生导师,研究方向:生产调度与物流优化。
