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This paper proposes a heuristic algorithm, called list-based squeezing branch and bound algorithm, for solving a machine-fixed, machining-assembly flowshop scheduling problem to minimize makespan. The machine-fixed, machining-assembly flowshop consists of some parallel two-machine flow lines at a machining stage and one robot at an assembly stage. Since an optimal schedule for this problem is not always a permutation schedule, the proposed algorithm first finds a promising permutation schedule, and then searches better non-permutation schedules near the promising permutation schedule in an enumerative manner by elaborating a branching procedure in a branch and bound algorithm. The results of numerical experiments show that the proposed algorithm can efficiently provide an optimal or a near-optimal schedule with high accuracy such as mean relative error being less than 0.2% and the maximum relative error being at most 3%.

Recently manufacturers face to more competitive situation, because of shorten product life cycles and diversifi- cation of products. Flexible Manufacturing Cell (FMC) has attracted a lot of attention as a production system to cope with a multi-product, small-lot production efficiently in such a situation. The FMC usually consists of two stages: A machining stage with some parallel machines (or flow lines) and an assembly stage with a few robots. Sun et al. [

This paper deals with the machine-fixed MAFS problem with

This paper proposes a kind of hybrid heuristic algorithms, incorporating a local search procedure into the squeezing B & B [

This paper deals with a machine-fixed MAFS model with the following conditions:

・ A machining stage consists of

・ Each of

・ Any assembly operation for each job cannot be started until machining operations for all parts of each job have been completed.

・ Machining time of

・ Setup time is independent of job sequence and included in each processing time.

・ Transfer time between machines is negligible.

・ All jobs are ready at time zero, and no job can be split or preempted.

・ No machine can process more than one operation at a time, and all machines are always available during a scheduling period.

The scheduling criterion is to minimize makespan

It has proved that the best permutation schedule is not always optimal to this scheduling problem [

Since the MAFS problem treated in this paper is NP-complete, we propose an efficient heuristic method, called “List-Based Squeezing Branch and Bound Algorithm (LSQ)”. The LSQ is a B & B-based local search algorithm elaborated for improving the efficiency of the squeezing B & B [

The squeezing B & B is a heuristic method which aims at obtaining a near-optimal schedule as close to the optimum as possible within a given computation time. In the squeezing B & B, parent nodes to be branched at branching level

After

The procedure is terminated when the branching process reaches the bottom level of the search tree, and then the best schedule is selected from among the schedules obtained at the bottom level as the solution by the squeezing B & B.

Since the squeezing B & B does not implement any backtracking, the time complexity of the squeezing B & B can be controlled by the

This branching procedure is called “list-based squeezing” and the B & B-based parallel search algorithm us- ing the list-based squeezing is called list-based squeezing Branch and Bound algorithm (LSQ). In the same way as the squeezing B & B, the basic procedure of the LSQ is terminated when the branching process reaches the bottom level of the search tree, and then the best schedule obtained at the bottom level is selected as the solution. The quality of the solution can be improved by implementing the LSQ iteratively according to the new job-list which is renewed by the current best schedule with a better value of the performance measure than that of the current job-list.

Since the job-list in the LSQ is corresponding to an initial solution in a general local search procedure, the LSQ can be also considered as a kind of local search algorithms which searches neighborhood of an initial schedule in an enumerative manner according to the lower bound like a branch and bound algorithm and the size of neighborhood is restricted by the value of

For the MAFS problem of this paper, the LSQ is applied as a two-phase heuristic search algorithm. In the first phase, a promising permutation schedule is searched according to a job-list and then better non-permutation schedules are searched according to both some job-lists for non-permutation scheduling and the best permuta- tion schedule obtained in the first phase. In both phases, the LSQ is implemented iteratively.

A job-list used in the LSQ is an initial schedule for searching better schedules. In the LSQ, the job-list is ob- tained first by using any promising heuristic method and the neighborhood is searched in an enumerative man- ner by employing a restricted branching procedure according to the job-list. The following four heuristic me- thods are proposed for obtaining a job-list for permutation scheduling to the MAFS problem.

1) Find a machining flow line

inal MAFS problem. By applying Johnson’s algorithm [

2) Construct

3) Find a machining flow line

4) For each

ing

ing a lower bound

Select a schedule with the minimum makespan for the original MAFS problem from among the set of sche- dules generated by the above four heuristic methods and set the schedule as the job-list for permutation sche- duling, denoted by

The job-lists for non-permutation scheduling are obtained as follows:

1) Consider

2) Adopt the best permutation schedule obtained in the first phase to a job-list for non-permutation scheduling, resulting in a job-list

These two kinds of job-lists are used for selecting some unscheduled jobs to be branched in the non-permu- tation scheduling phase. Select first

In the permutation scheduling phase of the proposed algorithm, the ordinary branching rule which generate nodes for the

Furthermore, another branching rule is also proposed for more effective non-permutation scheduling. In this branching rule, find first a bottleneck machining flow line

These two kinds of branching rules for non-permutation scheduling are illustrated in

Since the LSQ selects parent nodes to be branched according to the minimum lower bound rule, introducing the tight lower bound is important for getting a better performance. It is, however, very hard to define a tight lower bound directly for the MAFS problem, because a set of unscheduled jobs for each machining flow line is not always the same as these of the other lines in the non-permutation scheduling phase. Therefore, we adopt the following lower bound of a partial schedule

where

Note that

where,

The basic algorithm of the LSQ with bottleneck line search for the non-permutation scheduling for this MAFS problem is presented as follows:

Step 1: Select a squeezing pattern and specify the values of

Step 2: Find a schedule with minimum makespan from among the schedules generated by using four heuris- tics described in 3.2. Set the job sequence of the schedule as the job-list for permutation scheduling

Step 3: Set

Step 4: Generate

Step 5: Select first

Step 6: Calculate the lower bound for each child node by using Equation (1). If there exists nodes whose lower bounds are larger than

Step 7: If

Step 8: Determine the number of nodes to be selected, that is

Step 9: Select the

Step 10: Set the incumbent best schedule as

Step 11: Set

Step 12: Set the job sequence of the schedule obtained by applying Johnson’s algorithm for each machining flow line

Step13: Find a machining flow line of which completion time of the last job is the latest among

Step 14: Set

Step 15: If

Step 16: Select first

Step 17: Generate a child node for each parent node by sequencing the job sequenced at the

Step 18: Calculate the lower bound for each child node. If there exists nodes whose lower bounds are larger than

Step 19: If

Step 20: Determine the number of nodes to be selected

Step 21: Select the

Step 22: Set the makespan of the schedule

Step 23: Set

Step 24: The schedule

In this algorithm, the Steps 1-11 present the permutation scheduling procedure and Steps 12-24 present the non- permutation scheduling procedure.

For the case that all lines search is adopted in the non-permutation scheduling phase, Steps 13, 15 and 17 are removed from the above algorithm and Step 16 is replaced by the following Step 16’.

Step 16’: Select first

To evaluate the performance of the proposed algorithm, numerical experiments are implemented under the fol- lowing conditions.

One hundred instances are generated for each combination of

・ LSQ(a): The LSQ with

・ LSQ(b): The LSQ with

・ LSQ(c): The LSQ with

・ LSQ(d): The LSQ with

All algorithms are coded in C-language and run it on a personal computer with CPU of Phenom II X6 3.20 GHz.

Results of numerical experiments are summarized in

(N, L) | LSQ(a) | LSQ(b) | LSQ(c) | LSQ(d) | Proposed LSQ | |
---|---|---|---|---|---|---|

(10, 2) | ta | 0.39 | 0.41 | 0.63 | 0.64 | 0.18 |

na | 1.43 | 1.72 | 1.76 | 1.57 | 1.20 | |

m | 6.26 | 6.26 | 4.73 | 4.76 | 2.98 | |

p | 73 | 76 | 64 | 59 | 85 | |

(30, 5) | ta | 0.02 | 0.03 | 0.12 | 0.04 | 0.01 |

na | 0.27 | 0.33 | 0.54 | 0.33 | 0.18 | |

m | 0.55 | 0.80 | 2.15 | 0.82 | 0.30 | |

p | 91 | 92 | 77 | 87 | 93 |

N | 10 | 15 | 20 | 30 | ||||
---|---|---|---|---|---|---|---|---|

Method | B & B | LSQ | B & B | LSQ | B & B | LSQ | B & B | LSQ |

ta | 0.00 | 0.18 | 0.10 | 0.08 | 0.15 | 0.04 | 0.40 | 0.01 |

na | 0.00 | 1.20 | 1.07 | 0.53 | 0.78 | 0.47 | 1.34 | 0.31 |

m | 0.00 | 2.98 | 3.57 | 1.52 | 2.91 | 1.09 | 4.34 | 0.43 |

p | 100 | 85 | 91 | 84 | 81 | 92 | 70 | 96 |

N | 10 | 15 | 20 | 30 | ||||
---|---|---|---|---|---|---|---|---|

Method | B & B | LSQ | B & B | LSQ | B & B | LSQ | B & B | LSQ |

ta | 0.02 | 0.13 | 0.07 | 0.10 | 0.27 | 0.06 | 0.80 | 0.01 |

na | 0.42 | 0.95 | 0.77 | 0.96 | 1.30 | 0.81 | 1.61 | 0.16 |

m | 0.96 | 2.87 | 2.44 | 2.05 | 4.29 | 1.95 | 6.37 | 0.40 |

p | 96 | 86 | 91 | 90 | 79 | 93 | 50 | 95 |

N | 10 | 15 | 20 | 30 | ||||
---|---|---|---|---|---|---|---|---|

Method | B & B | LSQ | B & B | LSQ | B & B | LSQ | B & B | LSQ |

ta | 0.03 | 0.16 | 0.08 | 0.11 | 0.22 | 0.04 | 0.86 | 0.01 |

na | 0.80 | 0.68 | 0.69 | 1.08 | 1.03 | 0.37 | 1.65 | 0.18 |

m | 1.33 | 1.91 | 1.79 | 1.74 | 4.29 | 1.06 | 3.66 | 0.30 |

p | 96 | 77 | 88 | 90 | 79 | 88 | 48 | 93 |

(or best) schedule. The “best” schedule means the best of all schedules obtained by all heuristic algorithms and the one-hour-truncated branch and bound algorithm, and this term is used when the branch and bound algorithm cannot provide the “optimal” schedule within one hour. The notation “

Tables 2-4 show the experimental results of the proposed method and the one-hour-truncated branch and bound algorithm [

In this paper, a branch-and-bound-based heuristic algorithm, called “list-based squeezing branch and bound al- gorithm (LSQ)” is proposed for solving a machine-fixed, machining-assembly flowshop (MAFS) scheduling problem with L parallel two-machine flow lines at the machining stage and one assembly robot at the assembly stage. Since an optimal schedule to minimize makespan for this MAFS problem is not always a permutation schedule, two-phase search is implemented by using the LSQ. The first phase provides a promising permutation schedule and the second phase searches better non-permutation schedules near the permutation schedule. Results of numerical experiments show that the proposed LSQ efficiently provides optimal or near-optimal schedules with total average relative error is less than 0.2% and the maximum relative error is at most 3%.