_{1}

^{*}

This paper presents a novel hybrid metaheuristic GA-VNS matching genetic algorithm (GA) and variable neighborhood search (VNS) to the dynamic facility layout problem (DFLP). The DFLP is a well-known NP hard problem which aims at assigning a set of facilities to a set of locations over a time planning horizon so that the total cost including material handling cost and re-arrangement cost is minimized. The proposed hybrid approach in this paper elegantly integrates the exploitation ability of VNS and exploration ability of GA. To examine the performance of the proposed hybrid approach, a set of instance problems have been used from the literature. As demonstrated in the results, the GA-VNS is mighty of attaining high quality solution. Compared with some state-of-the-art algorithms, our proposed hybrid approach is competitive.

The DFLP is the determination of the most efficient arrangement of a number of facilities on the plant floor over multiple periods so that the sum of material handling cost and re-arrangement cost is minimized. The DFLP is an extension of static facility layout problem (SFLP) by considering changes in material flow between facilities over time planning periods. The main issue with DFLP is the existence of time period that makes it more complex than SFLP. Historically, most of the research conducted on facility layout problem has been focused on SFLP type; however nowadays there is a necessity of considering dynamic condition due to demand of flexibility and rapid changes on manufacturing systems. For instance, high tech industries such as electronics and software developments need to be designed under a dynamic environment rather than static due to changes on their product design and functionalities. Therefore the static layout is unreliable and economically inefficient for such industries.

In the DFLP, time period can be expressed in terms of week, month or even year. Solution of SFLP consists of a single layout while it is a series of layout (layout map) with each layout is associated with a particular period for the DFLP [

・ If material handling costs are extremely larger than re-arrangement costs, then DFLP can be solved as a series of SFLP. In fact, the optimum layout associated with first period can be attained by solving the SFLP using data for the first period and the optimum solution for the second period can be similarly attained by solving the SFLP using data for the second time period and so on.

・ If re-arrangement cost is extremely larger than handling costs, so the handling cost can be simply ignored and the problem can be solved as a series of SFLP.

So the DFLP deals with selecting a set of layout for each period and then deciding whether a facility needs to be re-arranged in the next period or remains fixed. It is clear that the layout configuration tends to be fixed if the re-arrangement cost is high while there are more tendencies toward changing layout plan over the time horizon if the re-arrangement cost is low. The assumptions of DFLP are defined as follows: the flow between facilities is dynamic and deterministic; facilities and locations are all equal size; the distances between facilities are predetermined [

The DFLP was initially addressed by Rosenblatt [^{T} possible layout maps. It is easy to see that the complexity of DFLP is much larger than SFLP and hence, is computationally intractable. Therefore the results achieved by decent exact methods such as branch and bound are modest. This is the reason there is intense motivation to implement heuristics to solve DFLP. Thereby the major portion of the literature is dedicated to review heuristics approaches.

Lacksonen and Enscore [

The mathematical model of DFLP presented by McKendall et al. [

s.t.

Notations which are used to describe the DFLP model are defined below:

No. Article/author | Solution approach | Problem type |
---|---|---|

1 Proposed hybrid | Hybrid Genetic Algorithm and Variable Neighborhood Search | DFLP |

2 Rosenblatt [ | Dynamic Programming | DFLP |

3 McKendall et al. [ | Simulated Annealing Heuristics | DFLP |

4 McKendall and Shang [ | Hybrid Ant Systems | DFLP |

5 Lacksonen and Enscore [ | Dynamic Programming | DFLP |

6 Balakrishnan et al. [ | Hybrid Genetic Algorithm | DFLP |

7 Baykasoglu and Gindy [ | Simulated Annealing | DFLP |

8Chen [ | Ant Colony Optimization | DFLP |

9 McKendall and Hakobyan [ | Boundary Search Heuristic and Tabu Search | DFLP with unequalsize |

10 Dunker et al. [ | Hybrid Genetic Algorithm and Dynamic programming | DFLP with unequal size |

11 Dunker et al. [ | Genetic Algorithm | SFLP with unequal size |

12 Guan and Lin. [ | Hybrid Shortest Path and Simulated Annealing | DFLP with dynamic environment |

13 Ulutas and Kulturel-Konak [ | Artificial Immune System | DFLP with unequal size |

14 Komarudin and Wong [ | Ant Colony Optimization | DFLP with unequal size |

15 Ripon et al. [ | Adaptive Variable Neighborhood Search | Multiple objective SFLP with unequal size |

16 Kulturel-Konak [ | Probabilistic Tabu Search | SFLP with unequal size |

17 Baykasoglu et al. [ | Ant Colony Optimization | DFLP with budget constraint |

18 Hosseini et al. [ | Hybrid imperialist competitive algorithm | DFLP |

The first term in objective function (1) represents the material handling costs, and the second term is used to obtain the re-arrangement costs. Constraint set (2) guarantees that each location is assigned to only one facility at each time period. Constraint set (3) guarantees that exactly one facility is selected and assigned to each location at each time period. Constraint set (4) adds the re-arrangement cost if a facility re-arranges from its location to a new location within two consecutive time periods. Finally, binary restrictions on the decision variables are presented in (5) and (6).

Let’s consider a DFLP instance with 6 facilities and 3 time periods (T = 1, 2, 3) as depicted in

The solution to this specific instance can be represented by a vector with size of (1 × 18) as represented in

In general, the solution of DFLP problem with N facilities and T time periods can be represented as in

GA is a stochastic optimization technique which is inspired from the biology and evolution process. GA starts with a set of solutions, generated either randomly or using a given heuristicis referred to as the population, and each individual in the population is called a chromosome. The size of the initial population depends on the complexity of problem. A GA with a smaller population is faster but the same time the risk of premature convergence increases. Once the initial population has been generated, the chromosomes are then evaluated by means of fitness function. Parent chromosomes are then selected based on their fitness values to generate new chromosome (child). There are different techniques for selecting parent chromosomes such as roulette wheel selection, tournament selection, and reward-based selection; however the roulette wheel selection is the most commonly used technique. In roulette wheel selection technique [

・ Single point crossover

The proposed single point crossover shame applied in this paper works with single parent instead of two parents. This way of crossover produces feasible offspring and there is no need of checking for solution feasibility. It is also easier to implement. Following are the steps of the proposed single crossover point.

1) Select a parent

2) Select a time period randomly t, such that

3) Select a random number n, such that

4) If

5) Facilities from n to N in parent

6) Similarly, facilities from 1 to (

An illustrative example of single point crossover is depicted in

・ Two exchange (swap) mutation

A cyclic order mutation scheme applied in this paper to modify the locations of facilities in a cyclic order as represented in

1) Generate a random time period T such that

2) Choose two facilities

3) Swap facilities

Variable neighborhood search (VNS) is a recent stochastic local search algorithm proposed by Mladenovic and Hansen [

To begin with the VNS, we need to define a set of neighborhood structure

Different types of neighborhood structure can be used for the VNS. However, we used three neighborhoods; two exchanges

・ Two exchanges

The two exchange neighborhood structure used in the VNS is functionally same as two exchange (swap) mutation addressed in Section 4.

・ Insertion

Step 1: Generate a random time period t such that

Step 2: For the selected time period t, choose two facilities _{1} and

Step 3: If a < b, then insert facility

Example 3: Considering the same instance, the first time period is randomly selected. According to

Insertion operation is depicted in

・ Three exchanges

Step 1: Generate a random time period t such that

Step 2: For the selected time period t, choose three facilities

Step 3: Exchange facility

Example 2: Consider example above, three facilities 4, 2 and 6 are randomly selected, then exchange facility

4 with facility 2, facility 2 with facility 6 and finally facility 6 with facility 2, as shown in

Note that the size of neighborhood

A variety of stopping criteria can be used to terminate the running of VNS, such as maximum allowed elapsed time, maximum number of iterations, etc. In this paper, the VNS continues for the number of iterations since the last improvement reaches at the given maximum of

With study of recent works on hybrid metaheurisitcs, it has found that hybrid VNS has dramatically effect on balancing between exploitation and exploration in combinatorial optimization problems [

The main properties of the hybrid GA-VNS are as follows:

Property 1: Generating the random initial solution.

The hybrid approach begins with a population produced randomly.

Property 2: Diversification using GA

It is proven that GA is high capable of shuffling the solution space to prevent search stagnation, but the same time fails to intensify the search toward promising regions of the search space. Nevertheless, hybridization of GA with some suitable local search techniques may overcome the shortcoming feature of each individual algorithm.

Property 3: Intensification using VNS local search.

The concept borrowed by the GA-VNS is that utilizing different neighborhood structures could prevent of getting trapped into local optimal and could also promotes expanding the search scope.

The driving parameter of proposed hybrid GA-VNS is hybridizing coefficient (HC). HC parameter addresses the percentage of population that is assigned to GA. The proposed hybrid with HC = 1 will be converted to pure GA. It means that the whole population will be evolved with GA, while HC = 0 will be resulted in pure VNS. To take the advantages of features of both GA-VNS, it is clear that HC must be set to some value between 0 and 1 (

where

To compare the efficiency of the proposed algorithm against some existing algorithms; SAI, SAII [

As it can be seen from

where

It is necessary to analyze the computational costs of GA-VNS against the other algorithms but it is hard to have a fair comparison due to utilizing different operating systems, programming procedure, and etc. However, the average CPU time for problem P02 with six facilities, 5 time periods and problem P17 with 15 facilities and 5 time periods were 876 s and 1795 s respectively.

Problem | Problem No. | SAI [ | SAII [ | Hybrid ACO [ | GA-VNS |
---|---|---|---|---|---|

N T | Best Solution | Best Solution | Best Solution | Best Solution | |

6 5 | P02 | 104,834 | 104,834 | 104,834 | 104,834 |

6 5 | P04 | 106,399 | 106,399 | 106,399 | 106,399 |

6 5 | P06 | 103,985 | 103,985 | 103,985 | 103,985 |

6 5 | P08 | 103,771 | 103,771 | 103,771 | 103,771 |

15 5 | P17 | 480,453 | 480,496 | 480,453 | 480,637 |

15 5 | P20 | 484,405 | 484,414 | 484,446 | 483,312 |

15 5 | P23 | 487,232 | 486,779 | 486,853 | 485,729 |

15 5 | P24 | 491,034 | 490,812 | 491,016 | 490,952 |

In this paper, a new hybrid GA-VNS was introduced to deal with the DFLP. In the proposed hybrid algorithm, the population was splitted into two parts; one part was assigned into the GA and the other part was assigned to the VNS. The percentage of splitting population was adjusted by a parameter called hybridizing coefficient (HC) through the course of run. Extensive computational experiments were performed to identify the most suitable parameters setting. The performance of the GA-VNS was compared against SAI, SAII, and hybrid ACO in terms of the quality of solutions. The results reveal that the hybrid GA-VNS gives good or even better solutions in most of the cases.

Md SanuwarUddin, (2015) Hybrid Genetic Algorithm and Variable Neighborhood Search for Dynamic Facility Layout Problem. Open Journal of Optimization,04,156-167. doi: 10.4236/ojop.2015.44015