time and proposed an iterative decomposition procedure to solve the model. III. MULTI-OBJECTIVE MATHEMATICAL MODEL A. Problem description To find the production sequence and schedule, we develop a new multi-objective mathematical model. This model tries to minimize the idle time of stations and displacing of products, aside from balancing the line. In this way, we consider three criteria, i) balance rate of the line for sequencing, ii) displacement of products for sequencing, and iii) idle time of stations, which is impressive in both balancing and sequencing. For the balance rate of the line, we compare the total working time of each station with the maximum one of them and try to minimize the differences. For the displacement of the product, we define a boundary for each station, which shows the starting and ending points of the station. Since in the assembly lines, most of the tools are light, mobile, and portable, it is assumed that each station can finish the allocated tasks in the next station, if necessary. It means that it is possible that in some stations, the product passes the boundary of the next station; however, this event creates a mess in the next station and increases the faults, which may lead to the danger. Because of that, we minimize this passing, which is called displacement. Finally, idle time is defined as the amount of time, where station ݆ is waiting for receiving product n+1 after finishing product n. Product n+1 must have two criteria together to be able to process by station j. Firstly, product n+1 must be passed station j-1 boundary because it is assumed that station j cannot process the products, which are before station j-1 boundary even they are ready to be implemented station j tasks. Secondly, it must be finished at station j-1. As presented, we consider a paced assembly line, so there is a conveyer at a constant speed (ݒ (for transferring the products between the stations. Since we are trying to find the production sequence and balancing together, two different variables (ݕ for sequencing and ݔ for the balancing) are defined. In customized mass production, there is a mass variety of products type (I) with low demand. On the other hand, the total order of production is N, which means the production sequence has ܰ position, and product with a lower position number produce sooner. The model tries to allocate the type of product that has the order in the production sequence positions. ݕ=1 means that the position number n is allocated to product type i. The demand for each product type (݀) must be satisfied in the sequence. ݔ also defines the production balancing, which ݔ=1 if task k of product type i allocated in sequence position n fulfill by station j and ݔ=0, otherwise. B. Indices The indices used in the model are as follows. Indices: ݅ Product type (݅ 1,2…,ܫ (݊ Sequence position (݊ 1,2…,ܰ, where ܰ is the total count of products order) ݇,݂,ℎ Tasks (݇ 1,2…,ܭ (݆ Station (݆ 1,2…,ܬ (
时间并提出了一种迭代分解程序来求解模型。三、多目标数学模型 A. 问题描述 为了找到生产顺序和进度,我们开发了一个新的多目标数学模型。除了平衡生产线之外,该模型还试图最大限度地减少站点的空闲时间和产品的置换。通过这种方式,我们考虑了三个标准,i) 用于排序的生产线平衡率,ii) 用于排序的产品位移,以及 iii) 站的空闲时间,这在平衡和排序方面都令人印象深刻。对于线路的平衡率,我们将每个站点的总工作时间与其中最大的一个进行比较,并尽量减少差异。对于产品的位移,我们为每个站点定义了一个边界,它显示了站点的起点和终点。由于在装配线上,大多数工具都是轻便、可移动和便携的,因此假设每个工位都可以在必要时在下一个工位完成分配的任务。这意味着在某些站点中,产品可能会通过下一个站点的边界;但是,此事件会在下一站造成混乱并增加故障,从而可能导致危险。正因为如此,我们最小化了这种通过,这称为位移。最后,空闲时间被定义为在完成产品 n 后,站 t 等待接收产品 n+1 的时间量。产品 n+1 必须同时具有两个标准才能被工作站 j 处理。首先,产品 n+1 必须通过站 j-1 边界,因为假设站 j 无法处理产品,即使它们已准备好执行第 j 站任务,它们也在第 j-1 站边界之前。其次,必须在站 j-1 完成。如前所述,我们考虑一条有节奏的装配线,因此有一个匀速的输送机((用于在工作站之间传输产品。由于我们试图找到生产顺序并平衡在一起,两个不同的变量()定义为排序和平衡)。在定制量产中,产品类型(I)种类繁多,需求量低。另一方面,生产的总订单为N,这意味着生产顺序有一个位置,位置编号越低的产品生产越早,模型试图在生产顺序位置分配有订单的产品类型。=1 表示位置编号 n 分配给产品类型 i。必须按顺序满足对每种产品类型()的需求。彔 还定义了生产平衡,如果在序列位置 n 分配的产品类型 i 的任务 k 由站 j 完成,则 =1,否则 =0。B. 指标 模型中使用的指标如下。指数: 产品类型( 1,2..., (序列位置( 1,2..., ), 其中 是产品订单总数) ,, ℎ 任务( 1,2 ……,丙(円站(円 1,2……,丙(
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并提出了一个迭代分解过程来求解模型。三.多目标数学模型a .问题描述为了找到生产顺序和进度,我们开发了一个新的多目标数学模型。除了平衡生产线之外,这种模式还试图最大限度地减少工位的闲置时间和产品的转移。这样,我们考虑了三个标准,I)用于测序的生产线的平衡速率,ii)用于测序的产品的位移,以及iii)站的空闲时间,这在平衡和测序中都令人印象深刻。对于线路的平衡率,我们将每个站的总工作时间与其中最大的一个站进行比较,并试图最小化差异。对于产品的位移,我们为每个工位定义一个边界,它显示了工位的起点和终点。由于在装配线上,大多数工具都是轻便的、可移动的和便携式的,因此假设每个工位在必要时可以在下一工位完成分配的任务。意味着有可能在某些工位,产品通过下一工位的边界;然而,该事件在下一站造成混乱并增加故障,这可能导致危险。正因为如此,我们尽量减少这种传递,这就是所谓的位移。最后,空闲时间被定义为在完成产品n之后,站݆等待接收产品n+1的时间量。产品n+1必须同时具有两个标准,以便能够由站j处理。首先,产品n+1必须通过站j-1边界,因为假设站j不能处理产品,即使它们准备好执行站j任务,它们也在站j-1边界之前。其次,必须在j-1站完成。如前所述,我们考虑了一条有节奏的装配线,因此有一个恒定速度的输送机(ݒ(用于在工位之间传送产品)。因为我们试图一起找到生产顺序和平衡,所以定义了两个不同的变量(ݕ for排序和ݔ for平衡)。在定制大规模生产中,有大量需求低的产品类型(ⅰ)。另一方面,生产的总订单是n,这意味着生产序列有ܰ位置,位置号越低的产品生产越快。该模型试图分配在生产序列位置中具有订单的产品类型。ݕ=1意味着位置号n被分配给产品类型I。每个产品类型(݀)的需求必须在序列中得到满足。ݔ also定义了生产平衡,如果按顺序分配在位置n的产品类型I的任务k由工位j和ݔ=0完成,则ݔ=1定义生产平衡,否则。指数模型中使用的指数如下。指数:݅产品类型(݅ 1,2…,ܫ (݊序列位置(݊ 1,2…,ܰ,其中ܰ是产品订单总数)݇,݂,ℎ任务(݇ 1,2…,ܭ (݆站(⇟1,2…,⇟()
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