Tower Crane Layout Planning: Multi-Optimal Solutions Algorithm

Abstract

Effective tower crane layout planning is essential for the success of construction projects. Traditional optimization algorithms, which often provide a single optimal solution, may not always reveal the global optimum, leaving room for doubt. This paper introduces the competitor algorithm, a novel multi-optimal solution approach inspired by the competitive learning paradigm within classroom settings. This algorithm is designed to provide users with a diverse set of competitive solutions, while avoiding falling into local optima. This strategic diversification ensures that users are equipped with a comprehensive range of options, empowering them to make confident, informed decisions. Furthermore, we have streamlined the positioning range for tower cranes, transitioning from a two-dimensional plane to a one-dimensional segmented line, thus eliminating the need to explore extensive, non-competitive regions. The competitor algorithm’s performance was validated through practical application, showcasing both its stability and optimization prowess, thereby confirming its reliable utility in real-world scenarios.

1. Introduction

The layout planning of tower cranes is regarded as one of the most critical technical issues in construction projects, because it significantly impacts the project’s cost, schedule, and safety [1,2,3]. The traditional method for layout planning of tower cranes involves engineers selecting the most suitable layout through repeated comparison and deliberation. This process requires consideration of numerous factors, such as the balanced distribution of work among cranes, crane blind zone ratios, and distances between cranes, which relies heavily on the engineers’ experience [4]. This task is vital, challenging, intricate, and prone to errors, highlighting the necessity of leveraging computer technology to solidify and automate this expertise.

Leveraging advancements in computer technology, researchers have in recent years engaged in broad and in-depth exploration of optimal solutions for tower crane layout planning [5,6]. A myriad of optimization algorithms have been explored, ranging from genetic algorithms [7,8,9], to bee algorithms [10,11], ant colony optimization [12], simulated annealing [13,14], firefly algorithms [15], particle swarm optimization [16,17], BIM and GIS [18,19], the integrated mathematical optimization approach [20], and to the application of GAN neural networks [4]. Yet, there has been a consistent emphasis on identifying singular optimal solutions, leaving users to wonder if further refinement is possible [21]. To users, algorithms frequently appear as a black box [22], even for those with a thorough understanding of the algorithms’ processes.

In the realm of tower crane layout planning, researchers have historically pursued a range of distinct optimization goals. Riga et al. [23,24,25,26,27] have focused on the spatial relationship between tower cranes and storage yards to curtail transportation costs within the site. Zhou et al. [28] have factored in the travel speed of trailers. Younes et al. have taken into account potential conflicts between operational tower cranes when calculating the time and cost of their operation cycles [6]. Zhang et al. have integrated the rental cost of tower cranes, lifting capacity utilization, and moments to tower cranes into a unified cost optimization objective [29]. Lu et al. [8] have also considered the impact of hoisting efficiency on transportation costs. Khodabandelu et al. [30] have examined the influence of crane operations on the layout. Previous research has typically focused on partial costs as optimization objectives.

On the other hand, prior research predominantly considered the selectable range of tower cranes as a two-dimensional plane [4,31], with coordinate accuracy typically set at 1 × 1 m or larger [16,32]. Such approaches have resulted in suboptimal precision in identifying the best solutions and slower computational speeds.

To address the aforementioned issues, this paper introduces a multi-optimal solution algorithm inspired by the competitive learning paradigm within a classroom: the competitor algorithm. This algorithm is designed to present users with all viable competitive options, enabling them to make confident decisions through a comprehensive comparison of multiple solutions. The optimization objective focuses on the total cost associated with tower crane layout planning, facilitating more scientifically based decision-making processes.

Furthermore, this paper enhances the optimization by transforming the tower crane positioning range from a two-dimensional plane to a one-dimensional polyline, thereby eliminating the exploration of redundant areas and substantially improving both computational efficiency and accuracy.

In studies comparing the competitor algorithm and genetic algorithm using real-world cases, the competitor algorithm exhibited exceptional performance stability in minimizing the total cost, with optimal solutions from ten independent runs consistently converging to approximately CNY 2.41 M with an exceptionally low standard deviation of 0.01. In contrast, the genetic algorithm not only yielded a higher minimum cost of CNY 2.48 M, but also demonstrated considerable variability across its ten runs, with a substantial standard deviation of 11.40. These empirical results convincingly validate both the superior stability and optimization capabilities of the competitor algorithm.

2. Materials and Methods

2.1. The Competitor Algorithm

2.1.1. Inspiration

The competitor algorithm draws inspiration from the competitive class paradigm, where every student is driven to achieve superior results. In each examination, a first-place winner emerges as the exemplar for all other students’ studies. However, the learning approach of the first-place winner may not be universally applicable. Therefore, every student should seek out the learning method that is most appropriate and fitting for them.

2.1.2. Mathematical Model

This section will describe the process of competitive learning in a class, with explanations in parentheses that incorporate common algorithmic terms to illustrate the mathematical model of the competitor algorithm. The primary objective of this paper was to minimize the total cost, thereby implying that a lower student grade in this section would be preferable.

Each student (individual) is a member of the class (population). Upon matriculation, each student is granted an initial score. In a learning atmosphere class, all students aspire to achieve superior grades. All students are cognizant of the top performer’s score (best score), yet remain uninformed about the study method (solution) that led to that achievement. Everyone is exploring learning methods that suit them in the hopes of achieving better scores.

Top students (top individuals) are those select few in the class who attain the highest scores. We designate the highest score of these top students as the ‘good score’. The single-subject score for these top students is often comparable or even superior to that of the best individual. Consequently, they may be inclined to explore various learning methods for a specific subject (dimension) at a small scale. If exploration results in an improvement in score, the learning method will be updated. If exploration reveals a drop in score instead, the previous learning method will be restored.

Within the classroom, a certain group of students is labeled as ‘penalized students’ (penalized individuals). These students have breached specific rules (constraints), and as a result, they are subjected to severe penalties (assigned a penalty score). Intervention by the school or public security bureau is then triggered, motivating them to turn over a new leaf (regenerate).

Furthermore, numerous average-performing students (average individuals) within the class exhibit grades that fall between a good score line and a penalty score line. These students may not yet have discovered an optimal learning approach for themselves, leading them to experiment with various subject methods at a larger scale. This process of exploration and experimentation persists until their grades either dip below the good score threshold or climb above the penalty score boundary.

The aforementioned outlines the population renewal mechanism within the competitor algorithm, as depicted in the pink box of Figure 1.

After each exam, students will have a period of time to explore their own new learning methods and participate in the next exam with their new learning methods until graduation. As the number of exam attempts (generations) accumulates, students will uncover the most optimal approaches to learning. The above is the entire process of the competitor algorithm, as shown in Figure 1.

The adaptive step size [33] is incorporated into this algorithm. The iteration step size is determined by multiplying the random number, which ranges from −10 to 10, with the individual score and then dividing it by the best score. This step size is proportional to the individual score; thus, a larger step size is applied when the individual score is high, while a smaller step size is employed when the individual score approaches the best value.

It becomes evident that the competitor algorithm lacks a mechanism to encourage individuals to gravitate towards the top individual. While such a convergence mechanism is prevalent in optimization algorithms and can markedly enhance the rate of convergence, it also poses the risk of steering the algorithm towards local optima rather than the global optimum.

In the competitor algorithm, individuals are categorized into three types, each serving a different function. Top individuals perform well and take small exploratory steps in each iteration, updating their solution only if there is improvement. This ensures that top individuals do not become penalized or average individuals. Their performance steadily improves, helping them find the local optima in their vicinity. Penalized individuals have all violated constraints and are regenerated randomly, which helps to discover more competitive feasible solutions. Average individuals have not violated constraints but their performance is suboptimal. Taking large exploratory steps might help them achieve better results, as they may be near non-competitive local optima, where small steps may prevent them from escaping.

2.2. Case Study

This section presents a case study where the competitor algorithm is applied and is juxtaposed with a genetic algorithm. Each of the two algorithms is run ten times, meticulously capturing data for runtime, best score, and the best individual in each iteration. All experiments were performed on a single computer to ensure uniformity in the execution environment.

2.2.1. Project Description

The project includes two structures: Building A, with a height of 24 m and ground dimensions of 90 by 126 m split across four floors, and Building B, which reaches a height of 14 m and features ground dimensions of 80 by 110 m over two floors. In addition to these structures, the surrounding area features a 30-story residential tower, a worker’s camp, a high-voltage power line, and a fire pool. The layout adopts a planar coordinate system with the southwest corner of Building A as its origin, and the project overview is illustrated in Figure 2A.

The total duration of this project spanned 18 months, with the tower crane usage accounting for 10 months. It was determined that the maximum single-layer lifting area of a single tower crane in this project should not exceed 5000 m². Consequently, it was deemed appropriate to employ 3 tower cranes for Building A and 2 for Building B. The optional models of the tower crane were TC6010, TC6513, and TC7013, with lifting radii of 60 m, 65 m, and 70 m, respectively. Consequently, the layout planning for each crane in this case involved 10 variables, representing the model selection and positioning for the five cranes. The variable costs associated with these models were CNY 36 k, CNY 50 k, and CNY 53 k, respectively, and their fixed costs stood at CNY 89 k, CNY 110 k, and CNY 146 k, respectively.

The cost of a crash shed per meter for high-voltage line anti-fall measures was CNY 5 k, and the cost of a crash shed per square meter for living area anti-fall measures was CNY 30 k. When the linear dimensions of a single hoisting blind zone for a tower crane exceeded 10 m in length or width, a handling fee of CNY 1 k per square meter was applied. Conversely, when these dimensions were less than 10 m, it was customary for workers to forgo the handling fee, and there was no requirement for additional mechanical equipment.

To configure the restricted zone, the residential building’s profile was expanded by 2 m, and the high-voltage power line was realigned 3 m to the north. This restricted zone is represented by the orange lines in Figure 2B. An intersection between the crane’s lifting range and this restricted zone would indicate a breach of the safety distance or an actual collision. Given that the worker camp was merely 6 m high, it did not necessitate extension as it did not interfere with the tower crane operations.

2.2.2. Positioning Range of Tower Crane

It is recommended that tower cranes be positioned as close to the building as feasible in order to enhance their efficiency and utility [34]. Consequently, for structures with a convex polygonal or near-convex polygonal outline, extending the perimeter by 2 m [35] creates a linear boundary that defines the positioning range for the tower crane, thereby eliminating the need to explore extensive, non-productive areas. The designated positioning range for the tower crane, depicted in Figure 2B, is outlined by a multi-segment line. By simplifying the positioning range to one dimension, we can represent the specific location of the tower crane with a single parameter: the tower position.

2.2.3. Objective Function

The objective of this study was to minimize the total cost (TC), which is expressed as:

TC = C₁ + C₂ + C₃ + C₄ + C₅ + C₆

(1)

where:

C₁: rental fees;

C₂: operator wages;

C₃: foundation cost;

C₄: blind zone handling fee;

C₅: installation and removal fee;

C₆: crash shed cost.

Included in the total cost are expenses such as rental fees, operator wages, foundation cost, and costs associated with its installation and removal; these are collectively termed crane usage fees and tend to vary primarily with the model of the crane when the duration of use is fixed. On the other hand, the potential costs for blind zone handling and crash shed measures vary with the position of the tower crane.

When strategizing the layout planning for tower cranes, while the overall cost is a critical consideration, it is equally vital to monitor five reference indicators that should remain within reasonable bounds: crash shed cost, blind zone handling fee, blind zone ratio, tower crane coverage ratios, and jib intersection ratio. If a layout’s reference indicator falls outside the acceptable range, further optimization of the layout may be necessary. In our evaluation of layout quality, we refrained from assigning specific weights to individual indicators, opting for a more comprehensive assessment. This decision stems from our belief that a one-size-fits-all weighting system is not suitable for all projects.

2.2.4. Constraints

When there is a possibility of collisions between tower cranes or the gap distances are insufficient, the layout must be considered impractical [36].

Regarding the horizontal collision constraints for tower cranes, it is crucial to ensure that their lifting ranges do not overlap with the expanded residential tower. Furthermore, the horizontal separation between any two tower cranes must exceed the sum of jib length plus the required safety clearance.

With respect to the vertical dimension constraints, since both Buildings A and B had maximum heights of 24 m, there was considerable flexibility for the placement of tower cranes at various heights. Therefore, it was unnecessary within the algorithm to apply restrictions concerning the height of the tower cranes.

In cases where the planned layout of a tower crane violates the set constraints, a punitive cost [37,38] of up to CNY 1 billion is imposed. This threshold, well beyond any feasible layout’s total cost, is termed the ‘penalty score line’.

An excessive number of constraints can significantly reduce the convergence speed of the algorithm [39]. Therefore, it is not advisable to set non-essential issues as constraints.

2.2.5. Algorithm Iteration Process

In this case, the competitor algorithm and the genetic algorithm both comprised a population of 1000 individuals, each evolving over 500 generations. Within the genetic algorithm, the process of roulette wheel selection [40] is employed to identify 500 pairs of parent individuals per generation, which then undergo crossover [41] to yield 500 pairs of offspring. Moreover, there is a 5% probability that any given individual in the population will be randomly reinitialized as a new individual [42].

3. Results

In the ten trials of the competitor algorithm, the optimal solutions yielded total costs ranging from CNY 2.4101 M to 2.4105 M, with an average value of CNY 2.4103 M and a standard deviation of 0.11. The computational time spanned from 3610 to 4044 s, averaging 3788 s with a standard deviation of 135.51. For further details, please refer to the Supplementary Materials.

The outcomes of the ten trials were strikingly similar, with the total cost differing by a maximum of CNY 400, a discrepancy deemed insignificant. The positioning of the tower cranes was nearly identical. All layouts featured the same model composition: four 60 m boom tower cranes and one 65 m boom tower crane, with the latter being located at either Tower 4 or Tower 5. Upon meticulous comparison of the ten layouts, the layouts were found to be strikingly similar or exhibiting mirror symmetry, prompting us to regard them as effectively a single layout, as illustrated in Figure 3, Layout 1.

In the ten trials of the genetic algorithm, the optimal solutions yielded total costs ranging from CNY 2.4781 M to CNY 2.8364 M, with an average value of CNY 2.5907 M and a standard deviation of 11.40. The computational time spanned from 5527 to 6285 s, averaging 5828 s with a standard deviation of 195.30. Upon visual comparison of the layout diagrams, the 10 layouts could be streamlined into 7 unique layouts. For further details, please refer to the Supplementary Materials.

Based on the averaged results from ten trials of both algorithms, as illustrated in Figure 4 and

Table 1. Comparison of competitor algorithm (CA) and genetic algorithm (GA).

Algorithm	Best Score			Runtime		Convergence
	Minimum (M CNY)	Avg. (M CNY)	Std. Dev.	Avg. (s)	Std. Dev.	Avg. Generation	Avg. Time (s)
CA	2.4101	2.4103	0.01	3788	135.51	276	2091
GA	2.4781	2.5709	11.40	5828	195.30	31	361

, distinct convergence patterns emerged. The competitor algorithm achieved optimal solution convergence after 276 generations, requiring an average computational time of 2091 s. Within this algorithm, the proportion of penalized individuals stabilized at approximately 78% at about generation 30. In contrast, the genetic algorithm exhibited more rapid convergence of its optimal solution, reaching stability at generation 31 with a significantly shorter computational time of 361 s. However, its convergence value exceeded that of the competitor algorithm by 7%. Similarly, the proportion of penalized individuals in the genetic algorithm demonstrated rapid initial descent, converging to approximately 5% by generation 30.

For our in-depth analysis, we selected the outcome from the fourth execution of the competitor algorithm, which exhibited the lowest total cost. Within this trial, the minimum total cost achieved was CNY 2.401 million, with a group of 63 individuals showing total costs that were within a close margin of CNY 0.2 million from this minimum. Subsequently, we consolidated similar and mirrored layouts, and the four resulting layouts, as depicted in

Table 2. Comparison of top four tower crane layouts.

Layout	Cost (CNY k)			Blind Zone Ratio	Coverage Ratios	Jib Intersection Ratio
	Total	Crash Shed	Blind Zone Handling
1	2410	4.059	0	2.3%	29%, 33%, 48%, 67%, 60%	68%
2	2561	4.385	312	3.0%	30%, 36%, 45%, 60%, 59%	60%
3	2571	4.028	0	1.7%	29%, 33%, 48%, 53%, 53%	32%
4	2585	18.238	0	1.8%	56%, 29%, 30%, 60%, 67%	68%

and illustrated in Figure 3, were collectively designated as the top four layouts. We undertook a meticulous analysis of the four layouts to assess the relative advantages and disadvantages of these individuals. Individuals with a total cost exceeding CNY 2.601 million were excluded from our analysis due to the substantial divergence from the best score.

Overall, the four layouts presented no significant principled concerns, and none of the layouts interfered with high-voltage lines. However, each layout slightly encroached on the worker camp area, which incurred a crash shed cost across all layouts. The proportion of blind spots across all four layouts fell within a relatively moderate range of 1.7% to 3%. Only Layout 2 was subject to blind zone handling fees. The overlap of the tower cranes’ operational ranges varied from 32% to 68%, with the highest overlap of 68% observed between two tower cranes in Building B in both Layout One and Layout Four. This level of intersection is deemed acceptable.

Our analysis confirms that Layout One is an exceptional layout. It featured the lowest total cost, recorded at CNY 2.41 million. It achieved exceptional cost-efficiency in crash shed costs, with its crash shed cost merely CNY 31 above that of Layout 3, the most cost-effective in this respect. Layout 1 was free from blind zone handling fees and presented a moderate blind zone ratio of 2.3%. Tower 1 and 2 had coverage ratios near 33%, and Tower 3 had ratios of 48%. In practice, overlapping areas should mainly be assigned to Towers 1 and 2. Prioritizing Tower 3 in crane navigation optimized workload distribution and boosted the overall efficiency. Towers 4 and 5 should share the load for Building B evenly. In essence, Layout 1 not only led in cost-efficiency but also excelled in reference indicators, rendering it an exemplary layout choice.

The following analysis evaluates the comparative advantages and disadvantages of Layouts 2 through 4 with respect to Layout 1. Layout 2’s primary modification involved downgrading Tower 4 from model 6513 to 6010. Although this adjustment achieved a reduction of CNY 161 k in crane usage fees, it substantially increased the blind zone, resulting in additional handling fees of CNY 312 k and consequently elevating the total cost by CNY 150 k. Layout 3 was distinguished by two key modifications: upgrading Building B’s crane from model 6010 to 6513 and adopting a north-south arrangement. While these changes effectively reduced the crane intersection ratio to 32%, the increased crane usage costs of CNY 161 k resulted in an equivalent rise in total expenses. Layout 4’s implementation, featuring a model 6513 for Tower 1, led to a total cost increase of CNY 175 k, comprising additional usage fees of CNY 161 k and a crash shed fee of CNY 14 k.

Based on the comprehensive analysis above, Layouts 2 through 4 exhibited fewer competitive advantages compared to Layout 1, while other layouts, with total costs exceeding that of Layout 1 by more than CNY 200 k, merit even less consideration. Consequently, Layout 1 conclusively emerged as the optimal layout.

4. Discussion

The top four layouts encompass all the potentially optimal layouts we have envisioned. A meticulous evaluation of these layouts validated that Layout 1 is the superior solution. Moreover, the consistent convergence of the competitor algorithm to layouts equivalent to Layout 1 across all ten trials in the case study substantially reinforced our confidence in identifying Layout 1 as the optimal solution. This consistent performance also attests to the robustness of the competitor algorithm, demonstrating its reliability in solving complex layout planning problems like tower crane placement.

The adoption of Tower 1, 2, and 3 configuration from Layout 3 to Layout 1 would yield a cost reduction of CNY 31 in total expenses. This indicates that the competitor algorithm might not have discovered the true solution within 500 generations. Nevertheless, the CNY 31 discrepancy is inconsequential within the scope of the case study, and there was no substantive distinction between Layout 1 and the true solution. As such, we can regard Layout 1 as the optimal solution. For practical purposes, a minimal fine-tuning of Layout 1 would suffice for its application.

In the ten genetic algorithm trials, the minimum total cost achieved was CNY 2.4781 million, yet the outcomes exhibited considerable variability. Despite these trials, the true solution remained elusive. Consequently, if an algorithm yields only one solution, there may be concerns about whether this represents a local optimum. Even upon identifying Layout 1, users might question whether a superior layout exists that could further reduce crash shed costs or if Tower 4 could be substituted with a 6010 model. The competitor algorithm, by comparing various layouts, effectively alleviates such concerns. We consider this comparative capability to be the primary advantage and greatest value of the competitor algorithm.

The genetic algorithm converges significantly faster than the competitor algorithm. This could be attributed to the competitor algorithm’s method of independent random exploration by all individuals, making them more susceptible to penalties, with the proportion of penalized individuals converging to 78%. In contrast, the genetic algorithm uses high-performing individuals to crossbreed and generate new pairs, reducing their susceptibility to penalties, and leading to a lower proportion of penalized individuals converging to just 5%. However, this mechanism also makes the genetic algorithm prone to local optima.

We believe that the generalizability of the competitor algorithm will be excellent. The algorithm only requires setting two parameters: population size and the number of iterations. These parameters affect only the convergence speed and not the convergence result, as long as a convergence result can be generated. This simplicity enhances the potential for widespread adoption of the competitor algorithm. Furthermore, since the algorithm allows all individuals to explore independently, it avoids premature convergence to local optima. We believe that as problem complexity increases, traditional optimization algorithms are more likely to fall into local optima, making the advantages of the competitor algorithm more apparent. Moreover, the algorithm’s distinctive strength lies in its ability to generate a diverse set of competitive solutions, empowering users to make well-informed decisions, thus enhancing its value for widespread adoption.

5. Conclusions

5.1. Summary

Drawing inspiration from competitive learning in class, this paper introduces a novel intelligent optimization algorithm: the competitor algorithm. This algorithm generates multiple optimal solutions, offering users a range of options or inspirations. By conducting a horizontal comparison of multiple solutions, users can confidently select the optimal solution that truly meets their needs.

This study implemented the competitor algorithm in a tower crane layout optimization case study. Through ten trials, the algorithm demonstrated consistent convergence to the vicinity of the optimal solution, with a standard deviation of merely 0.11. These results substantiated the algorithm’s robust stability and superior optimization capabilities.

In this study, we refined the potential range of tower crane placement from a two-dimensional plane to a one-dimensional polyline. This strategic simplification is anticipated to enhance computational accuracy and speed significantly by eliminating the need to explore regions outside the polyline, since no points in these areas are better positioned than those that lie directly on it. By concentrating on the polyline, the search for optimal placement becomes more focused, which could result in either increased computational efficiency or improved accuracy.

This paper also introduces an evaluative framework for tower crane layouts, using total cost as the primary determinant. The framework considers factors such as crash shed cost, blind zone handling fee, blind zone ratio, the tower crane coverage ratios, and the jib intersection ratio as reference indicators. This provides a comprehensive method for evaluating tower crane layouts.

5.2. Limitations

In this paper, we have refined the model for the positioning range of tower cranes by simplifying it from a two-dimensional plane to a one-dimensional polyline, effectively eliminating the need to explore extensive non-competitive zones. This enhancement is anticipated to elevate computational precision and accelerate convergence rates. Though the anticipated improvements are considered obvious, space limitations have precluded a formal demonstration within the scope of this paper.

Excessive blind zones in tower crane layout planning can result in additional costs from manual handling operations, mechanical equipment requirements, and reduced construction efficiency. In this case study, we implemented a penalty coefficient of 1 k CNY per square meter for blind zone handling. A more precise cost assessment could be achieved by quantifying the additional expenses associated with manual handling operations, mechanical equipment utilization, and efficiency losses based on the specific location and dimensions of blind zones. Nevertheless, Layout 1 is anticipated to maintain its position as the optimal solution. Further investigation of this aspect was not pursued due to the page limitations of this study.

6. Patents

This research has resulted in two patent applications that have been accepted by the China National Intellectual Property Administration, with application numbers CN202410286653.4 and CN202410680943.7.