## A Parallel Genetic Algorithm for the Set Partitioning Problem (1994)

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Citations: | 69 - 1 self |

### BibTeX

@MISC{Levine94aparallel,

author = {David Levine},

title = {A Parallel Genetic Algorithm for the Set Partitioning Problem},

year = {1994}

}

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### Abstract

In this dissertation we report on our efforts to develop a parallel genetic algorithm and apply it to the solution of the set partitioning problem--a difficult combinatorial optimization problem used by many airlines as a mathematical model for flight crew scheduling. We developed a distributed steady-state genetic algorithm in conjunction with a specialized local search heuristic for solving the set partitioning problem. The genetic algorithm is based on an island model where multiple independent subpopulations each run a steady-state genetic algorithm on their own subpopulation and occasionally fit strings migrate between the subpopulations. Tests on forty real-world set partitioning problems were carried out on up to 128 nodes of an IBM SP1 parallel computer. We found that performance, as measured by the quality of the solution found and the iteration on which it was found, improved as additional subpopulations were added to the computation. With larger numbers of subpopulations the genetic algorithm was regularly able to find the optimal solution to problems having up to a few thousand integer variables. In two cases, high-quality integer feasible solutions were found for problems with 36,699 and 43,749 integer variables, respectively. A notable limitation we found was the difficulty solving problems with many constraints.

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Citation Context ...ximate algorithms which try to find "good" solutions quickly, and exact algorithms which attempt to solve the SPP to optimality. Here we mention some of the more recent methods. See Balas an=-=d Padberg [4]-=- for a survey of older methods. An important approximate approach (as well as the starting point for most exact approaches) is to solve the linear programming (LP) relaxation of the SPP. In the LP rel... |

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Citation Context ... the AIX operating system. The SP1 uses a high-performance switch for connecting the nodes. The SP1 supports the distributed-memory programming model. Our code was written in C and used the Chameleon =-=[15]-=- message-passing library. Chameleon is designed to provide a portable, high-performance message-passing system. Chameleon runs on top of many other message-passing systems, both vendor-specific and th... |

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Citation Context ...f above-average schemata via the strings that migrate. This increases the overall selective pressure since additional reproductive trials are allocated to those strings that are fit enough to migrate =-=[29]-=-. At the same time, the introduction of migrant strings into the local population helps to maintain genetic diversity, since the migrant string arrives from a different subpopulation which has evolved... |

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Citation Context ...ossibilities exist for additional research in this area, including stronger penalty terms (e.g., quadratic), the ranking approach of Powell and Skolnick [24], or the dynamic penalty of Smith and Tate =-=[26]-=- for which we had mixed results [19]. However, for the aa problems, we are less optimistic. Table 7 appears to indicate diminishing returns with respect to the reduction in infeasibilities in these pr... |

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Citation Context ...e approach if the algorithm is terminated "early" with a "good" integer solution. Various bounding strategies have been used, including linear programming and Lagrangian relaxation=-=. Fischer and Kedia [11]-=- use continuous analogs of the greedy and 3 \Gamma opt methods to provide improved lower bounds. Of recent interest is the work of Eckstein [10], who has developed a general-purpose mixed-integer prog... |

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Citation Context ... columns were described as "very large" [22]. Today, problems with hundreds of thousands of columns are "very large," and one benchmark problem has been generated with 837 rows and=-= 12,753,313 columns [6]-=-. Because of the widespread use of the SPP (and often the difficulty of its solution), a number of algorithms have been developed. These can be classified into two types: approximate algorithms which ... |

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Citation Context ...rogramming and Lagrangian relaxation. Fischer and Kedia [11] use continuous analogs of the greedy and 3 \Gamma opt methods to provide improved lower bounds. Of recent interest is the work of Eckstein =-=[10]-=-, who has developed a general-purpose mixed-integer programming system for use on the CM-5 parallel computer and applied it to, among other problems, set partitioning. The most successful approach app... |

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Citation Context ...PP were generated manually. However, airline crew scheduling problems have grown significantly in size and complexity. In 1981 problems with 400 rows and 30,000 columns were described as "very la=-=rge" [22]. Today, p-=-roblems with hundreds of thousands of columns are "very large," and one benchmark problem has been generated with 837 rows and 12,753,313 columns [6]. Because of the widespread use of the SP... |

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Citation Context ...or limits on the amount of work allocated to a particular base not explicitly part of the SPP model. Fourth, genetic algorithms contain a population of possible solutions. As noted by Arabeyre et al. =-=[3], "Th-=-e knowledge of a family of good solutions is far more important than obtaining an isolated optimum." Finally, we believe genetic algorithms have great potential for scaling to take advantage of t... |

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Citation Context ... 0 otherwise. (6) Airline crew scheduling is a very visible and economically significant problem. Estimates of over a billion dollars a year for pilot and flight attendant expenses have been reported =-=[1, 5]-=-. Even a small improvement over existing solutions can have a large economic benefit. At one time, solutions to the SPP were generated manually. However, airline crew scheduling problems have grown si... |

22 |
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Citation Context ...nt years it has been noted that as SPP problems grow in size, fractional solutions occur more frequently, and simply rounding or performing a "small" branch-and-bound tree search may not be =-=effective [2, 5, 13]. Bra-=-nch-and-bound may be viewed as an exact approach if the algorithm runs until an integer solution (if one exists) is proven optimal, or as an approximate approach if the algorithm is terminated "e... |

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Citation Context ...ve the linear programming (LP) relaxation of the SPP. In the LP relaxation, the integrality restriction on x j is relaxed, but the lower and upper bounds of zero and one are kept. A number of authors =-=[5, 13, 22] have note-=-d that for "small" SPP problems the solution to the LP relaxation either is all integer, in which case it is also the optimal integer solution, or has only a few fractional values that are e... |

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Citation Context ...lity is to use an adaptive mutation rate that changes based on the value of some GA statistic such as population diversity or the Hamming distance between two parent strings [30]. Several researchers =-=[7, 28]-=- make the case for a high mutation rate when mutation is separated from crossover, as it is in our implementation. We found that the random choice of columns to add or delete to the current solution t... |