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Meta-heuristic algorithms proved to find optimal solutions for combinatorial problems in many domains. Nevertheless, the efficiency of these algorithms highly depends on their parameter settings. In fact, finding appropriate settings of the algorithm’s parameters is considered to be a nontrivial task and is usually set manually to values that are known to give reasonable performance. In this paper, Ant Colony Optimization with Parametric Analysis (ACO-PA) is developed to overcome this drawback. The main feature of the ACO-PA is the ability of deciding the appropriate parameter values within the predefined parameter variations. Besides, a new approach which enables the pheromone information value to be proportional to the heuristic information value is introduced. The effectiveness of the proposed algorithm is investigated through the application of the algorithm to the construction site layout problems taken from the state-of-art. Results show that the ACO-PA can reduce transportation cost up to 16.8% compared to the site layouts generated by Genetic Algorithms and basic ACO. Moreover, the effects of parameter settings on the generated solutions are investigated.

Lately, mathematical optimization models [

One of the key points for getting high-quality solutions by ACO is the parameters and the main drawback is the absence of a general strategy for choosing values for these parameters. In fact, finding appropriate settings of an algorithm’s parameters is considered to be a non-trivial task. ACO algorithm parameters are usually set manually to values that are known to give reasonable performance. These parameter values do not vary from one problem domain instance to another and are fixed throughout the entire run. A better approach for the determination of these parameter values is a fine tuning process which generally results in better solutions; however, in many instances, only several values are tested for parameter tuning and are fixed throughout the entire run [

In this paper, Ant Colony Optimization with Parametric Analysis (ACO-PA) is proposed to overcome this drawback, and, thus, to improve the solution. The main feature of the ACO-PA is the ability of deciding the appropriate parameter values within the predefined parameter variations. The proposed algorithm is applied to the construction site layout problem. An objective function which is related to the total of personnel movement and the distance between facilities is adopted for reducing the transportation time. Next, the effects of parameter settings on the results are investigated. The remaining of the paper is organized as follows. The next section focuses on the ant colony optimization algorithms. In Section 3, application of the proposed algorithm to construction site layout is introduced. Next, benchmark case studies, numerical analysis and conclusions are presented.

In the application process of ACO to optimization problems, a pheromone value

There is an ongoing interest on parameter tuning in the ACO literature. In general methods for parameter settings can be divided into three main approaches. Method of pre-scheduled variations of parameters propose to decrease or increase linearly over the run of the algorithm. In adaptative approaches, some parameters are modified according to some rules that changes the behavior of the algorithm. λ-branching factor [

A construction site layout problem is the allocation of a number of predetermined facilities to a number of predetermined locations. A well-planned site layout is vital for any construction project, since space is one of the most important resources at a construction site. Furthermore, it has significant impact on construction cost and time, especially in the case of large projects in which traveling between facilities can be time consuming. Spatial problems may also occur during the setup of temporary facilities due to the limited amount of available space. This task is typically carried out by project managers or decision makers based on trial-and-error, experience, and intuition. However, as not all of the options are evaluated in these decision making processes, the outcome might be an improper layout, which can result in increased transportation time and costs. Possible alternatives for any construction site layout increases exponentially with the number of facilities.

The problem could be modeled as a quadratic assignment problem, in which equal numbers of facilities and locations exist. If the number of locations is more than that of facilities, dummy facilities should be added to make the numbers equal. In this study, it is assumed that there are equal numbers of facilities as well as locations, namely equal-area facility layout problem, and each of the predetermined location is available for accommodating the largest facility. The predetermined locations are represented as rectangles. The objective of construction site layout is to minimize the total traveling distance of personnel trips between facilities.

The total distance is defined as Equation (1) and Equation (2)

Subject to

where TD is the total traveling distance; n is the number of facilities; δ_{xi} is the permutation matrix variable (=1 if the facilities is assigned to site i); f_{ij} is the frequencies of trips made by construction personnel between facilities i and j; and d_{ij} is the distance between locations i and j. If the two locations are next to each other, then the distance is calculated as the distance between the centers of the two locations. If there is more than one path linking the two locations, then the shorter path is defined as the distance between two locations.

Main aspects of the ACO-PA algorithm are heuristic information, solution construction, pheromone updating rule, local search and termination rule. Steps established for the proposed algorithm are as follows.

Step 1. Heuristic information: Heuristic information is defined in accordance with the problem characteristics, which enable the ACO-PA algorithm to be more applicable in solving any combinatorial problem.

The heuristic information (e) is given by Equation (3)

The heuristic desirability (η) is defined as Equation (4)

The vector d_{j} represents the sum of the distances from location j to all other locations, which mean that the lower the value of d_{j} is, the more the location is closer to the center. The vector f_{i} presents the sum of flows from facility i to all other facilities, where the higher the value of f_{i} shows the more important the facility is. This formulation reveals the importance of facilities with higher flows and locations having less distance potentials; as the total traveling distance cost is directly proportional to the distances and flows between facilities.

Step 2. Solution construction: At each construction step, an ant k places an unassigned facility i to an available location with a probability of

where τ_{ij} is the pheromone trail at iteration t. It should be noted that α and β are the parameters that determine the relative influence of the pheromone and heuristic information, respectively.

The assignment sequence of the facilities is generally decided according to the flow rate in an ascending or vice versa orders. However, keeping the same assignment sequence, especially on small-dimensional problems, causes continual increase in pheromone information values and that the algorithm cannot generate different solutions. To avoid this early convergence to a specific solution, the ACO-PA uses the first assignment sequence of the facilities in a non-decreasing order of the flow f_{i}. After determining heuristic and pheromone information, facilities are randomly sorted to construct solutions.

Step 3. Pheromone updating: The pheromone trail update is applied after a complete solution is constructed. Pheromone trails of all couplings (i, j) are updated in accordance with Equation (6)

is the amount of pheromone added by ant k is given by Equation (7)

where

Step 4. Local search: Local search procedure is added to the algorithm to avoid finding a local optimum solution. Although there are several different local search strategies, one of them (2-opt) has been found as the most promising one, which was also selected for the developed algorithm. Local search starts from an initial solution and tries to improve the solution by local chances. If a layout with less objective function value could be found by swapping two distinct nodes in the current assignment, then the current solution is replaced by that assignment’ and the search continues until no better solution is found.

Step 5. Parameter analysis: In the proposed model, parameter values are defined within the prescheduled parameter variations. The algorithm also enables to set maximum and minimum limits for parameter values and steps. The algorithm changes one parameter value at a time and evaluates the solution according to the objective function. Therefore, the algorithm has the ability of generating more layout options and deciding the optimal solution among them.

Step 6. Termination rule: ACO algorithm can be terminated in several ways, such as, constructing solutions for a maximum number of iterations, running the algorithm for a stipulated time or until a stagnation situation is encountered in which all ants converge into the same solution. In the proposed model, the algorithm runs for a certain number of iteration to avoid stagnation in early stages. The flowchart of the proposed algorithm is illustrated in

The proposed algorithm was coded in C++ and executed on IntelCore2 Duo processor at 2.66 GHz and 4 Gb of RAM. A comparison with previously proposed algorithms is conducted to demonstrate the effectiveness of the developed algorithm. Three case studies were selected from the literature [

In case 1 (C1), it is assumed that each of the predetermined location is available for accommodating every facility [

The frequencies of trips (in one day) between facilities are as listed in

The distances of the available locations are as listed in

Site facilities | Abbreviations |
---|---|

Site office | SO |

False work shop | FS |

Labor residence | LR |

Storeroom 1 | S1 |

Storeroom 2 | S2 |

Carpentry workshop | CW |

Reinforcement steel workshop | RW |

Side gate | SG |

Electrical, water and other utilities control room | UR |

Concrete batch workshop | BW |

Warehouse | W |

SO | FS | LR | S1 | S2 | CW | RW | SG | UR | BW | W | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|

SO | 0 | 5 | 2 | 2 | 1 | 1 | 4 | 1 | 2 | 9 | 1 | |

FS | 5 | 0 | 2 | 5 | 1 | 2 | 7 | 8 | 2 | 3 | 8 | |

LR | 2 | 2 | 0 | 7 | 4 | 4 | 9 | 4 | 5 | 6 | 5 | |

S1 | 2 | 5 | 7 | 0 | 8 | 7 | 8 | 1 | 8 | 5 | 1 | |

S2 | 1 | 1 | 4 | 8 | 0 | 3 | 4 | 1 | 3 | 3 | 6 | |

F= | CW | 1 | 2 | 4 | 7 | 3 | 0 | 5 | 8 | 4 | 7 | 5 |

RW | 4 | 7 | 9 | 8 | 4 | 5 | 0 | 7 | 6 | 3 | 2 | |

SG | 1 | 8 | 4 | 1 | 1 | 8 | 7 | 0 | 9 | 4 | 8 | |

UR | 2 | 2 | 5 | 8 | 3 | 4 | 6 | 9 | 0 | 5 | 3 | |

BW | 9 | 3 | 6 | 5 | 3 | 7 | 3 | 4 | 5 | 0 | 5 | |

W | 1 | 8 | 5 | 1 | 6 | 5 | 2 | 8 | 3 | 5 | 0 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0 | 15 | 25 | 33 | 40 | 42 | 47 | 55 | 35 | 30 | 20 | |

2 | 15 | 0 | 10 | 18 | 25 | 27 | 32 | 42 | 50 | 45 | 35 | |

3 | 25 | 10 | 0 | 8 | 15 | 17 | 22 | 32 | 52 | 55 | 45 | |

4 | 33 | 18 | 8 | 0 | 7 | 9 | 14 | 24 | 44 | 49 | 53 | |

5 | 40 | 25 | 15 | 7 | 0 | 2 | 7 | 17 | 37 | 42 | 52 | |

D= | 6 | 42 | 27 | 17 | 9 | 2 | 0 | 5 | 15 | 35 | 40 | 50 |

7 | 47 | 32 | 22 | 14 | 7 | 5 | 0 | 10 | 30 | 35 | 40 | |

8 | 55 | 42 | 32 | 24 | 17 | 15 | 10 | 0 | 20 | 25 | 35 | |

9 | 35 | 50 | 52 | 44 | 37 | 35 | 30 | 20 | 0 | 5 | 15 | |

10 | 30 | 45 | 55 | 49 | 42 | 40 | 35 | 25 | 5 | 0 | 10 | |

11 | 20 | 35 | 45 | 53 | 52 | 50 | 40 | 35 | 15 | 10 | 0 |

The construction site, where two permanent buildings are to be constructed, is shown in

In case 2 (C2), it is assumed that side gate and main gate are assigned to locations 1 and 10, respectively [

In case 3 (C3), it is assumed that site office, labor residence concrete batch shop cannot be allocated to the relatively smaller locations 7 and 8 [

In the first solution construction, which is used to define the initial pheromone values, parameters were set to α = 0.1, β = 2 and ρ = 0.1 where α, β, and ρ are the parameters that show relative influence of pheromone trail, heuristic information, and pheromone trail evaporation rate, respectively. Number of ants was set to 10 and Q is taken as the value of pheromone trail evaporation rate ρ. The parameter settings were taken from the literature [

A comparison between the ACO-PA and the best-known results in the literature for the benchmark case studies are shown in

Facility | SO | FS | LR | S1 | S2 | CW | RW | SG | UR | BW | W | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Location | [ | C1 | 8 | 6 | 9 | 7 | 4 | 3 | 5 | 10 | 1 | 11 | 2 |

ACO-PA | 1 | 10 | 4 | 5 | 3 | 8 | 6 | 9 | 7 | 2 | 11 | ||

[ | C2 | 11 | 5 | 8 | 7 | 2 | 9 | 3 | 1 | 6 | 4 | 10 | |

ACO-PA | 9 | 11 | 6 | 5 | 8 | 2 | 4 | 1 | 3 | 7 | 10 | ||

[ | C3 | 11 | 5 | 9 | 7 | 2 | 8 | 3 | 1 | 6 | 4 | 10 | |

ACO-PA | 11 | 9 | 6 | 5 | 8 | 3 | 7 | 1 | 4 | 2 | 10 |

In C1, none of the facilities were assigned to the locations that were proposed by Gharaie et al. [

In C2, it can be seen that the ACO-PA met all the constraints, which were to assign side gate and warehouse to locations 1 and 10, respectively. Apart from these two facilities, none of the facilities were assigned to the same locations that were proposed by Li et al. [

In C3, side gate and warehouse had the priority to be assigned as their locations were predetermined in the problem. Next, site office, labor residence, and concrete batch workshop were assigned as they had area constraints. According to the generated site layout by ACO-PA, all the constraints were met and total travel distance was reduced by 16.8% compared to the literature.

With respect to computational processing time, the optimal solution was found in 1.15 seconds on Intel Core 2 Duo processor at 2.66 GHz and 4 Gb of RAM. A comparison of computational time cannot be provided due to the lack of related information in the studies taken from the literature.

With the case studies, the proposed model is applied to both equal-area and unequal area facility layout where feasible solutions are generated. In equal-area facility layout, each of the predetermined places is capable of accommodating any of the facilities as in C1. On the other hand, in unequal-area facility layout, like in C2 and C3, the number of predetermined places should be equal to or greater than facilities so that no larger facilities are assigned to smaller locations. Objective values of C2 and C3 are slightly higher than the results of C1. This indicates that the unequal-area constraints can affect the layout solution. It is observed that the ACO-PA performed better under constraints. As more constraints are defined, the better objective function values are found.

These parameter values show that the influence of heuristic and pheromone information do not dominate the solution construction regardless of introducing constraints to the problem. Parameter settings used in each case and the objective function values are as in

As can be seen from

In this study, the initial value of pheromones (τ_{0}) is defined as a function of the objective function value of the initial solution, where only heuristic information is used. Therefore, the value τ_{0} is avoided to be too high or too low with this improvement. If the initial value of τ_{0} is too low, then the search is quickly biased towards the first solutions of the ants and this leads to the exploration of inferior zones in the search space. However, if the initial pheromone values are too high, then many iterations are wasted in waiting until enough pheromone evaporation is done and, thus, the pheromone added by ants can start to bias the search [

Furthermore, 3-D surface map indicating the effect of α and ρ variations on cost function of C1 case is given in

Case | Relative influence of heuristic trail (β) | Relative influence of pheromone trail (α) | Pheromone trail evaporation rate (ρ) | Number of ants |
---|---|---|---|---|

C1 | 0.30 | 1.25 | 0.90 | 1.00 |

C2 | 0.00 | 0.00 | 0.35 | 1.00 |

C3 | 0.00 | 0.00 | 0.60 | 1.00 |

function, and it is not possible to observe a tendency (or pattern). Even though this graph indicating the effects of two parameters on one case, other parameter (β) also exhibited the same influence, and conclusions were found to be same in all cases.

In this study, an improved ACO algorithm, ACO-PA, was proposed for the construction site layout problem and was applied to case studies with different constraints. The construction site layout problem is formulated as a quadratic assignment problem with an objective function of minimizing the total transportation distance, and, thus, total transportation cost. The proposed algorithm is applied to three case studies in which feasible solutions for both equal-area and unequal area facility layouts are generated.

The developed algorithm introduced several improvements, which enhances the solution construction process. Firstly, a new approach to define initial pheromone values was integrated to existing ACO frameworks. The first solution was constructed according to only the heuristic information and initial pheromone values were defined using the objective function value of this solution. This approach enabled the initial pheromone values to be proportional to the heuristic information values and that the solution construction was not dominated by the information which had higher values. The ratio between heuristic and pheromone information should prove to be useful to speed the convergence.

Secondly, the developed algorithm is capable of deciding the better parameter settings for α, β and ρ, where α, and β are the parameters that show relative influence of pheromone trail and heuristic information, respectively, and ρ is the pheromone trail evaporation rate. The maximum-minimum limits and steps are defined for each of the parameters. The algorithm changes one parameter value at a time and evaluates the solution according to the objective function. Therefore, the algorithm finds out the proper parameter settings automatically, which leads to the best solution. It should also be noted that the effect of model parameters, namely α, β, and ρ, is noteworthy and the surface map illustrating the effect of these parameters on the variation of the cost function showed us the difficulty of selecting manually.

There are several advantages of applying the ACO-PA algorithm to site layout problems. First, key facilities can be assigned to certain locations. In site layout planning, the project manager would assign facilities one by one to locations that are left over based on experience, trial-and-error, or intuition. In the ACO-PA algorithm, more layout choices can be considered and evaluated in a reasonable time. Secondly, the ACO-PA algorithm uses pheromone and heuristic information which make the algorithm more effective and controllable in the solution of real-life problems. The developed algorithm should enable construction managers and site engineers to plan and optimize the utilization of construction site space in order to minimize the total traveling distance. The developed ACO-PA algorithm has a number of possible extensions which are currently being pursued by the authors. These include developing hybrid systems based on ACO-PA, incorporating multi-objectives in the optimization and extending the research to the dynamic construction site layout planning problem. Future studies may also focus on expanding the application areas of the proposed algorithm to other construction tasks such as equipment routing planning and material storage layout on site.

GulbenCalis,OrhanYuksel, (2015) An Improved Ant Colony Optimization Algorithm for Construction Site Layout Problems. Journal of Building Construction and Planning Research,03,221-232. doi: 10.4236/jbcpr.2015.34022