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摘要: 针对目前综合调度算法不能兼顾产品工艺树中并行工序的并行性和串行工序之间紧密度，影响调度结果的问题，提出考虑后续工序的择时综合调度算法.该算法提出工序序列排序策略，从工艺树的整体结构出发，将其划分成若干内部工序只具有串行关系的工序序列，并按路径长度从长到短的顺序确定其调度次序；提出择时调度策略和考虑后续工序策略，根据工艺树自身特点，从来自不同工序序列的并行工序的不同组合方案中，选择最接近调度目标的方案作为工序调度方案，若该工序调度方案不唯一，则在其中选择该工序加工开始时间最早的调度方案.该算法既保证了工序的并行处理，又提高了串行工序的紧密度，优化了综合调度的结果.最后通过实例说明本文算法对解决综合调试问题具有普遍意义.Abstract: Integrated scheduling algorithms currently neglected the compactness of serial processes when handling a general integrated scheduling problem, and it influenced the scheduling result. Aiming at this problem, an time-selective integrated scheduling algorithm considering posterior processes was presented. The strategy of process sequence sorting was proposed. From the overall structure of the process tree, it was divided into several sequence of processes in which the processes only had a serial relationship. According to the path length to determined the order of its scheduling. The strategy of time-selective and considering posterior processes was proposed. According to the characteristics of the process tree, selected the most close to the scheduling objectives as a process scheduling scheme from the different combination of parallel process from different process sequence. If the process scheduling scheme was not unique. selected the process scheduling scheme in which The processing start time of the process was the earliest. This algorithm promises to proceed together the parallel processing of processes, and effectively raises the compactness of serial processes. The results of integrated scheduling are optimized. Finally illustrated by examples.
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Key words:
- Process sequence sorting /
- posterior processes /
- time-selective /
- integrated scheduling
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图 6 工序序列排序策略序列划分示意图
Fig. 6 The schematic of sequence divided by operation sequence sorting strategy
图 8 处理工序$i$试调度后工序发生冲突示意图
Fig. 8 The diagram of process conflict when process $i$ tried to scheduled
图 10 仅使用择时策略在不同时间点调度工序$B_{1}$方案比较
Fig. 10 Scheme comparison of Scheduling $B_{1}$ in different time when only use timing strategy
图 11 采用考虑后续工序策略在不同时间点调度工序$B_{1}$方案比较
Fig. 11 The comparison of scheme of scheduling $B_{1}$ in different time
表 1 综合调度问题参数表
Table 1 Parameter list of integrated scheduling problem
参数 含义 $M_{i}$ 工序$i$的加工设备 $T_{Si}$ 工序$i$的加工开始时间 $T_{i}$ 工序$i$的加工用时 $T_{Ei}$ 工序$i$的加工结束时间 $T_{Mki}$ 调度工序$i$后第$k$台设备上当前加工完成时间, $1\leq k\leq m$ $T_{Mi}$ 调度工序$i$后当前最晚加工完成的已调度工序加工完成时间 $T_{Tij}$ 工序$i$的第$j$个"准调度时间点", $j\geq 1$ $T_{Ti}$ 工序$i$的"准调度时间点"集合 $F_{ij}$ 工序$i$在工艺树中的第$j$个紧前工序, $j\geq 1$ $N_{if}$ 工序$i$在工艺树中的第$f$个紧后工序, $f\geq 1$ $F_{Mi}$ 工序$i$在其加工设备上的紧前工序 $N_{Mig}$ 工序$i$在其加工设备上的第$g$个后工序 $N_{Qik}$ 工序$i$在其工序序列上的第$k$个后续工序 
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表 2 本文算法调度产品$A$过程
Table 2 Scheduling the product $A$ by the algorithm proposed
工序号 加工设备 准调度时间点 试调度方案加工总时间 后续工序静态调度时间 确定调度时间点 当前方案中每个工序调度时间点 $A_{1}$ $M_3$ - - - 0 $A_{1}$: 0; $A_{2}$ $M_4$ - - - 2 $A_{1}$: 0, $A_{2}$: 2; $A_{10}$ $M_3$ - - - 4 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 4; $A_{11}$ $M_2$ - - - 6 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 4, $A_{11}$: 6; $A_{18}$ $M_1$ - - - 7 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 4, $A_{11}$: 6,
$A_{18}$:7;$A_{15}$ $M_3$ - - - 8 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 4, $A_{11}$: 6,
$A_{18}$:7, $A_{15}$: 8;$A_{21}$ $M_2$ - - - 11 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 4, $A_{11}$: 6,
$A_{18}$:7, $A_{15}$: 8, $A_{21}$: 11;$A_{26}$ $M_3$ - - - 14 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 4, $A_{11}$: 6,
$A_{18}$:7, $A_{15}$: 8, $A_{21}$: 11, $A_{26}$: 14;$A_{25}$ $M_2$ - - - 16 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 4, $A_{11}$: 6,
$A_{18}$:7, $A_{15}$: 8, $A_{21}$: 11, $A_{26}$: 14,
$A_{25}$: 16;$A_{28}$ $M_1$ - - - 17 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 4, $A_{11}$: 6,
$A_{18}$:7, $A_{15}$: 8, $A_{21}$: 11, $A_{26}$: 14,
$A_{25}$: 16, $A_{28}$: 17;$A_{3}$ $M_3$ 2|6|11|16 19|19|18|19 18|22|27|32 2 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$:7,
$A_{18}$: 8, $A_{15}$: 9, $A_{21}$: 12, $A_{26}$: 15,
$A_{25}$: 17, $A_{28}$: 18, $A_{3}$: 2;$A_{9}$ $M_4$ 5|19 19|22 19|32 5 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$:7,
$A_{18}$: 8, $A_{15}$: 9, $A_{21}$: 12, $A_{26}$: 15,
$A_{25}$: 17, $A_{28}$: 18, $A_{3}$: 2, $A_{9}$: 5;$A_{13}$ $M_3$ 8|12|17 20|19|19 18|22|27 8 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$:7,
$A_{18}$: 8, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8;$A_{16}$ $M_2$ 10|16|19 20|21|22 18|24|27 10 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$:7,
$A_{18}$: 8, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10;
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表 3 本文算法调度产品$A$过程(续表 2)
Table 3 Scheduling the product $A$ by the algorithm proposed (continued Table 2)
工序号 加工设备 准调度时间点 试调度方案加工总时间 后续工序静态调度时间 确定调度时间点 当前方案中每个工序调度时间点 $A_{23}$ $M_4$ 13|20 20|22 20|23 13 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 7,
$A_{18}$: 8, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 13;$A_{29}$ $M_1$ 15 20 18 15 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 7,
$A_{18}$: 8, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 13, $A_{29}$: 15;$A_{5}$ $M_4$ 4|8|
15|2020|20|
20|2116|20|
27|324 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$:7,
$A_{18}$: 8, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 13, $A_{29}$: 15,
$A_{5}$: 4;$A_{7}$ $M_2$ 5|8|
13|16|1920|21|
23|21|2216|19|
24|28|305 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 13, $A_{29}$: 15,
$A_{5}$: 4, $A_{7}$: 5;$A_{17}$ $M_1$ 8|10|18 20|20|20 16|18|26 8 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 13, $A_{29}$: 15,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8;$A_{19}$ $M_4$ 9|15|20 20|20|22 16|22|27 9 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 13, $A_{29}$: 15,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 9;$A_{22}$ $M_1$ 11|18 20|21 16|23 11 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 13, $A_{29}$: 15,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 9,
$A_{22}$: 11;$A_{20}$ $M_4$ 14|15|20 21|20|22 16|17|22 15 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 13, $A_{29}$: 15,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 9,
$A_{22}$: 11, $A_{20}$: 15;$A_{4}$ $M_2$ 2|8|9|
13|16|1920|23|22|
23|21|2213|19|20|
24|28|302 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 13, $A_{29}$: 15,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 9,
$A_{22}$: 11, $A_{20}$: 15, $A_{4}$: 2;
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表 4 本文算法调度产品$A$过程(续表 3)
Table 4 Scheduling the product $A$ by the algorithm proposed (continued Table 3)
工序号 加工设备 准调度时间点 试调度方案加工总时间 后续工序静态调度时间 确定调度时间点 当前方案中每个工序调度时间点 $A_{8}$ $M_1$ 5|9|10|
14|1820|22|20|20|
2013|17|18|
22|265 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 13, $A_{29}$: 15,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 9,
$A_{22}$: 11, $A_{20}$: 15, $A_{4}$: 2, $A_{8}$: 5;$A_{14}$ $M_4$ 7|8|11|
15|17|2026|20|20|
22|22|2413|14|17|
21|23|268 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 14, $A_{29}$: 17,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 12,
$A_{22}$: 14, $A_{20}$: 17, $A_{4}$: 2, $A_{8}$: 5,
$A_{14}$: 8;$A_{27}$ $M_2$ 12|13|
16|1921|22|
20|2114|15|
18|2116 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 14, $A_{29}$: 17,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 12,
$A_{22}$: 14, $A_{20}$: 17, $A_{4}$: 2, $A_{8}$: 5,
$A_{14}$: 8, $A_{27}$: 16;$A_{12}$ $M_1$ 5|7|
9|10|
17|2020|20|
21|20|
21|2113|15|
17|18|
25|285 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 14, $A_{29}$: 17,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 12,
$A_{22}$: 14, $A_{20}$: 17, $A_{4}$: 2, $A_{8}$: 6,
$A_{14}$: 8, $A_{27}$: 16, $A_{12}$: 5;$A_{31}$ $M_2$ 6|8|
9|13|
16|18|1922|21|
20|21|
21|21|2013|15|
16|20|
23|25|
269 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 14, $A_{29}$: 17,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 12,
$A_{22}$: 14, $A_{20}$: 17, $A_{4}$: 2, $A_{8}$: 6,
$A_{14}$: 8, $A_{27}$: 16, $A_{12}$: 5, $A_{31}$: 9;$A_{24}$ $M_1$ 10|17|
2020|24|
2416|23|
2610 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16, $A_{25}$: 18,
$A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5, $A_{13}$: 8,
$A_{16}$: 10, $A_{23}$: 14, $A_{29}$: 17, $A_{5}$: 4,
$A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 12, $A_{22}$: 14,
$A_{20}$: 17, $A_{4}$: 2, $A_{8}$: 6, $A_{14}$: 8, $A_{27}$: 16,
$A_{12}$: 5, $A_{31}$: 9, $A_{24}$: 10;
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表 5 本文算法调度产品$A$过程(续表 4)
Table 5 Scheduling the product $A$ by the algorithm proposed (continued Table 4)
工序号 加工设备 准调度时间点 试调度方案加工总时间 后续工序静态调度时间 确定调度时间点 当前方案中每个工序调度时间点 $A_{30}$ $M_3$ 14|18 20|20 16|20 14 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 5, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 14, $A_{29}$: 17,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 12,
$A_{22}$: 14, $A_{20}$: 17, $A_{4}$: 2, $A_{8}$: 6,
$A_{14}$: 8, $A_{27}$: 16, $A_{12}$: 5, $A_{31}$: 9,
$A_{24}$: 10, $A_{30}$: 14;$A_{6}$ $M_3$ 5|7|
10|13|
16|1820|20|
21|20|
21|206|8|
11|14|
17|195 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 6, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 14, $A_{29}$: 17,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 12,
$A_{22}$: 14, $A_{20}$: 17, $A_{4}$: 2, $A_{8}$: 6,
$A_{14}$: 8, $A_{27}$: 16, $A_{12}$: 5, $A_{31}$: 9,
$A_{24}$: 10, $A_{30}$: 14, $A_{6}$: 5;$A_{32}$ $M_3$ 13|16|18 20|21|20 14|17|19 13 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 6, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 14, $A_{29}$: 17,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 12,
$A_{22}$: 14, $A_{20}$: 17, $A_{4}$: 2, $A_{8}$: 6,
$A_{14}$: 8, $A_{27}$: 16, $A_{12}$: 5, $A_{31}$: 9,
$A_{24}$: 10, $A_{30}$: 14, $A_{6}$: 5,
$A_{32}$: 13;$A_{33}$ $M_4$ 16|19|
2020|21|
2117|20|
2116 $A_{1}$: 0, $A_{2}$: 2, $A_{10}$: 6, $A_{11}$: 8,
$A_{18}$: 9, $A_{15}$: 10, $A_{21}$: 13, $A_{26}$: 16,
$A_{25}$: 18, $A_{28}$: 19, $A_{3}$: 2, $A_{9}$: 5,
$A_{13}$: 8, $A_{16}$: 10, $A_{23}$: 14, $A_{29}$: 17,
$A_{5}$: 4, $A_{7}$: 5, $A_{17}$: 8, $A_{19}$: 12,
$A_{22}$: 14, $A_{20}$: 17, $A_{4}$: 2, $A_{8}$: 6,
$A_{14}$: 8, $A_{27}$: 16, $A_{12}$: 5, $A_{31}$: 9,
$A_{24}$: 10, $A_{30}$: 14, $A_{6}$: 5, $A_{32}$: 13,
$A_{33}$: 16;
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