Optimal Positioning of Mobile Cranes on Construction Sites Using Nonlinear Programming with Discontinuous Derivatives

Abstract

Mobile cranes represent conventional construction machinery that is indispensable for the erection of most prefabricated buildings, especially those containing heavy components. However, it is also common knowledge that the engagement of these machines has a significant influence on the environment, various social aspects of the construction process, and its economic benefits. Optimal positioning of the mobile crane on the construction site, primarily driven by the contractor’s interest to perform assembly operations with expensive machinery as effectively as possible, considerably reduces not only the costs of engaging such a machine but indirectly also its negative impacts on construction sustainability. This paper discusses an exact nonlinear model for the optimization task. The optimization model consists of a cost objective function that is subject to various duration and positioning constraints for the mobile crane, including bounds on its degrees of freedom of movement and stop positions. Because the model formulation includes discontinuous and non-smooth expressions, nonlinear programming with discontinuous derivatives (DNLP) was employed to ensure the optimal solution was reached. The model provides the mobile crane operator with exact key information that enables the complete and optimal assembly of the building structure under consideration. Additionally, the information gained on the optimal distribution of the mobile crane rental period to assembly operations allows for a detailed duration analysis of the entire process of building structure erection, which can be used for its further improvement. An application example is given in this study to demonstrate the advantages of the proposed approach.

1. Introduction

Contemporary construction of prefabricated buildings, particularly those with heavy components, is commonly supported by the use of mobile cranes to perform the necessary lifting. Compared to most tower cranes, mobile cranes allow additional flexibility when performing erection operations across the construction site as they can be moved more conveniently closer to assembly locations. However, it is also widely recognized that the use of these machines considerably affects construction sustainability.

In terms of environmental impacts, mobile crane operations powered by an internal combustion engine generate significant carbon emissions. One of the important goals for achieving environmental sustainability in construction is to minimize transportation needs and thus lower greenhouse gas emissions by reducing energy consumption. From a social viewpoint of sustainability, the efficient management of mobile cranes is a crucial task for keeping public facilities intact, as well as ensuring the safety and health of people on the construction site and its immediate vicinity. With regard to the economic benefits of construction, the high rental and operating costs of mobile cranes strongly encourage their well-planned use.

The lifting plan is the basis for employing a mobile crane on a construction site. It significantly contributes to effective risk management [1] associated with the use of mobile cranes in construction. However, the preparation of such a plan involves the detailed determination of all necessary lifting, including the identification of a clash-free space for each uplift and a configuration of the mobile crane that accommodates all lifts, bearing in mind its multiple degrees of freedom of movement, which is a challenging and time-consuming task when using conventional analytical approaches. To overcome these inconveniences and speed up the generation of the most advantageous lifting plans for mobile cranes, various optimization approaches have been proposed in the literature.

Through a screening of relevant databases, it was found that published studies have addressed various optimization challenges of engaging mobile cranes on construction sites, such as path planning, positioning, selection, simulation and visualization of operations, as well as the concurrent use of multiple cranes [2]. Current research is also being conducted on the robotization of lifting processes on construction sites using mobile cranes with automatic operation abilities [3,4]. A recent in-depth review of automated lift planning methods for mobile cranes is available in reference [5].

It turned out that heuristic methods were the most often employed optimization approaches, among which the ant colony algorithm [6], A-star algorithm [7], genetic algorithm [8], and neural network [2] stand out in terms of frequency. Even though heuristic methods can regularly assure that the obtained solution is near-optimal, they have been found to solve a large part of the optimization tasks related to mobile cranes in a time-efficient manner.

Mathematical programming methods are stricter in terms of what they may produce as outputs than other approaches, but they can deliver an exact optimal solution to the problem posed. Because optimization problems involving mobile cranes often require discontinuous and non-smooth terms in model formulation, which most conventional exact mathematical programming algorithms cannot handle, this seems to have resulted in less attention being paid to the aforementioned research field. Nevertheless, there are related published works.

For instance, a specially developed integer programming algorithm implemented in an interactive planning environment for heavy lifting using mobile cranes was introduced in reference [9]. An autonomous mobile crane system considering obstacle recognition and optimal path planning using the branch-and-bound method was proposed in reference [10]. The optimization of heavy lifting plans for mobile cranes assigned to industrial construction sites using Dijkstra’s algorithm was presented in reference [11]. In a broader context, it is also possible to point out research that was devoted to the optimal planning of tower crane usage on construction sites with the support of exact mathematical programming techniques [12,13,14,15,16,17,18], although these were mostly based on linear discrete model formulations.

This study discusses an exact nonlinear model for the optimal positioning of a mobile crane on a construction site. The optimization model consists of a cost objective function subject to various duration and positioning constraints for the mobile crane, including bounds on its degrees of freedom of movement and stop positions. Because the model formulation includes discontinuous and non-smooth expressions, nonlinear programming with discontinuous derivatives (DNLP) was employed to assure the optimal solution was reached.

To the best of our knowledge, such an approach to the optimal positioning of mobile cranes on construction sites has not been proposed in the literature. Thus, the continuation of this paper initially introduces the optimization problem formulation of DNLP, presents the model formulation for the optimal positioning of mobile cranes on construction sites, shows the advantages of the considered approach through an application example, and finally discusses the findings and provides conclusions with recommendations for further research.

2. Optimization Problem Formulation

The optimization problem of DNLP can be generally expressed as follows:

	$\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e} \mathrm{o}\mathrm{r} \mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}z=f\left(\mathit{x}\right)$
	subject to:
	$\mathit{h}\left(\mathit{x}\right)=\mathbf{0}$			(DNLP)
	$\mathit{g}\left(\mathit{x}\right)\le \mathbf{0}$
	$\mathit{x}\in \mathit{X}=\{\mathit{x}\mid \mathit{x}\in {\mathit{R}}^{\mathit{n}},{\mathit{x}}^{\mathbf{L}\mathbf{O}}\le \mathit{x}\le {\mathit{x}}^{\mathbf{U}\mathbf{P}}\}$

where $z$ represents the objective variable, $f\left(\mathit{x}\right)$ defines the objective function, $\mathit{x}$ denotes the vector of variables that are continuous real numbers determined in n-dimensional space ${\mathit{R}}^{\mathit{n}}$ and specified by compact set $\mathit{X}$, $\mathit{h}\left(\mathit{x}\right)$ and $\mathit{g}\left(\mathit{x}\right)$ stand for (non)linear functions of equality and inequality constraints, and ${\mathit{x}}^{\mathbf{L}\mathbf{O}}$ and ${\mathit{x}}^{\mathbf{U}\mathbf{P}}$ indicate vectors of lower and upper bounds on the variables, respectively.

The above optimization problem formulation appears to be the same as that for conventional nonlinear programming. Namely, the continuous variables can be included in the linear or nonlinear terms of the objective function and constraints. However, the DNLP allows discontinuous and non-smooth expressions in the optimization problem formulation, such as functions that compute the maximum/minimum of the argument or those that calculate the absolute value of the term. The said features will be incorporated in the model formulation for optimal positioning of the mobile crane on the construction site.

3. Optimization Model Formulation

3.1. Objective Function

Optimal positioning of the mobile crane on the construction site can be performed based on various relevant criteria. However, the minimization of the total cost of mobile crane usage can probably be recognized as an optimization criterion that is commonly applied in construction projects [19]. In this vein, the objective function was added to the optimization model in form of:

$$\mathrm{Minimize}c=c_r t+c_e$$

(1)

where $c$ defines the total cost of mobile crane usage, $c_r$ denotes the rental cost of mobile crane, including any discount, $t$ determines the duration of mobile crane rental, and $c_e$ stands for the extra cost of the mobile crane independent of the rental duration. The above objective function was subjected to various duration and positioning constraints for the mobile crane, including bounds on its degrees of freedom of movement and stop positions.

3.2. Duration Constraints

The mobile crane must be engaged in the assembly process on the construction site from the attachment of the first component to its lifting device until the last component is installed in the building. Accordingly, the necessary duration of mobile crane rental was included in the optimization model with the following expression:

$$t=\sum _{i\in I}\left(t_p+t_{a_i}+t_{f_i}+t_{c_i}\right)$$

(2)

where $i\in I$ stands for the set of building components that should be assembled using the mobile crane, $t_p$ represents the duration of preparing the mobile crane for operation, $t_{a_i}$ determines the duration of verification and positioning of component assemblies, $t_{f_i}$ indicates the duration of the component’s fitment, and $t_{c_i}$ denotes the duration of the component assembly cycle.

The duration of the component assembly cycle was determined as follows:

$$t_{c_i}=t_{t_i}+t_{w_i}\hspace{1em}i\in I$$

(3)

where $t_{t_i}$ is the duration of the mobile crane chassis’ travel between stops prior to transporting the component and $t_{w_i}$ is the duration of the work procedure with the component itself.

The duration of the mobile crane chassis’ travel between stops prior to component transport was defined in the following form:

$$t_{t_i}=\frac{d_{m_i}}{v_m}+t_s+t_d\hspace{1em}i\in S$$

(4)

where $d_{m_i}$ denote the mobile crane chassis travel distance between stops prior to component transport, $v_m$ represents the mobile crane chassis travel velocity, and $t_s$ and $t_d$ determine the durations of the mobile crane setup and dismantling, respectively. It is necessary to emphasize at this point that the above term is active only for those components requiring mobile crane movement to the next stop, i.e., $i\in S$. For all other components, $t_{t_i}=0$.

The duration of the work procedure with the component was covered using the following formula:

$$t_{w_i}=t_{l_i}+t_{r_i}+t_{u_i}+t_{e_i}\hspace{1em}i\in I$$

(5)

where $t_{l_i}$ represents the duration of the component’s hooking to the mobile crane lifting device, $t_{r_i}$ determines the duration of the component’s rise from the supply location to the demand location, $t_{u_i}$ denotes the duration of the component’s temporary setup and unhooking from the mobile crane lifting device, and $t_{e_i}$ represents the duration of the movement of an empty mobile crane lifting device from the demand location to the next supply location.

The rise of the component attached to the mobile crane lifting device from the supply location to the demand location may be achieved through four operations, i.e., turning the boom horizontally, adjusting the boom length, adapting the boom’s vertical angle, and modifying the winch length. The duration of the component’s rise from the supply location to the demand location was thus determined using the following equation:

$$t_{r_i}=\max \left(t_{h_i}, t_{v_i}\right)+\gamma \min \left(t_{h_i}, t_{v_i}\right)\hspace{1em}i\in I$$

(6)

where $t_{h_i}$ and $t_{v_i}$ indicate the durations of the component’s horizontal and vertical movements, respectively, while $\gamma$ represents the coefficient with which the probability that both mentioned movements will be executed at the same time is considered.

The duration of the component’s horizontal movement was governed using the following expression:

$$t_{h_i}=\max \left(t_{{\rho }_i}, t_{{\theta }_i}\right)+\beta \min \left(t_{{\rho }_i}, t_{{\theta }_i}\right)\hspace{1em}i\in I$$

(7)

where $t_{{\rho }_i}$ and $t_{{\theta }_i}$ denote the durations of the component’s horizontal movement in the radial and tangential directions, respectively, while $\beta$ is the coefficient by which the probability that both said movements will be performed simultaneously is included in the optimization model.

In the presented Equations (6) and (7), both $\beta$ and $\gamma$ are coefficients spanning from zero to one, which indicate the probability that the mobile crane operator will perform several operations simultaneously. When these coefficients are equal to one, it is assumed that the operations will be executed sequentially, while a value of zero signifies that the operations will be performed concurrently.

The duration of the component’s horizontal movement in the radial direction was defined as follows:

$$t_{{\rho }_i}=t_{{\alpha }_i}+t_{{\delta }_i}\hspace{1em}i\in I$$

(8)

where $t_{{\alpha }_i}$ and $t_{{\delta }_i}$ represent the durations required to rotate and extend the mobile crane boom horizontally in the radial direction to transport the component, respectively.

The duration required to rotate the mobile crane boom horizontally in the radial direction for component transport was established using the following expression:

$$t_{{\alpha }_i}=\frac{\left|{\alpha }_i-{\alpha }_{i-1}\right|}{v_{\alpha }}\hspace{1em}i\in I$$

(9)

where ${\alpha }_i$ indicates the vertical angle of the mobile crane boom when installing the component and $v_{\alpha }$ specifies the angular velocity of the mobile crane boom lift.

The duration necessary to extend the mobile crane boom horizontally in the radial direction for component transfer was determined using the following expression:

$$t_{{\delta }_i}=\frac{t_b \left|l_{b_i}-l_{b_{i-1}}\right|}{\delta }\hspace{1em}i\in I$$

(10)

where $t_b$ denotes the duration required to reach the maximum mobile crane boom extension, $l_{b_i}$ stands for the mobile crane boom length when installing the component, and $\delta$ represents the maximum mobile crane boom extension.

The duration of the component’s horizontal movement in the tangential direction was defined as follows:

$$t_{{\theta }_i}=\frac{{\theta }_i}{v_{\theta }}\hspace{1em}i\in I$$

(11)

where ${\theta }_i$ signifies the horizontal angle between the supply and demand locations for the component transport and $v_{\theta }$ implies the mobile crane slew velocity.

The duration of the component’s vertical movement was determined using the following formula:

$$t_{v_i}=\frac{\left|z_{v_i}\right|}{v_v}\hspace{1em}i\in I$$

(12)

where $z_{v_i}$ is the total vertical change in the mobile crane lifting device position for transporting the component with a winch, while $v_v$ denotes the velocity of vertical lifting with the mobile crane winch.

3.3. Positioning Constraints

Generally, the task of the mobile crane placed at the stop position ${\mathit{p}}_{\mathit{c}}=\left(x_{c_j},y_{c_j},z_{c_j}\right)$, $j\in J\left(i\right)$, is to transport the component, $i\in I$, from its supply location ${\mathit{p}}_{\mathit{s}}=\left(x_{s_i},y_{s_i},z_{s_i}\right)$ to the point ${\mathit{p}}_{\mathit{d}}=\left(x_{d_i},y_{d_i},z_{d_i}\right)$, where it is demanded. Based on this, the horizontal distances were established using the following expressions:

$$l_i=\sqrt{{\left(x_{d_i}-x_{s_i}\right)}^2+{\left(y_{d_i}-y_{s_i}\right)}^2}\hspace{1em}i\in I$$

(13)

$${l_s}_i=\sqrt{{\left(x_{s_i}-x_{c_j}\right)}^2+{\left(y_{s_i}-y_{c_j}\right)}^2}\hspace{1em}i\in I\hspace{1em}j\in J\left(i\right)$$

(14)

$$l_{d_i}=\sqrt{{\left(x_{d_i}-x_{c_j}\right)}^2+{\left(y_{d_i}-y_{c_j}\right)}^2}\hspace{1em}i\in I\hspace{1em}j\in J\left(i\right)$$

(15)

where $l_i$, $l_{s_i}$ and $l_{d_i}$ denote horizontal distances for the component transport between the supply and demand locations, the supply location and the mobile crane, and the demand location and the mobile crane, respectively.

The total vertical change in the mobile crane lifting device position for transporting the component with a winch was considered as follows:

$$z_{v_i}=z_{{\alpha }_i}+z_i\hspace{1em}i\in I$$

(16)

where $z_{{\alpha }_i}$ stands for the vertical change in the mobile crane lifting device position due to the boom angle change when transporting the component and $z_i$ indicates the height difference between the demand and supply locations of the component.

The vertical change in the mobile crane lifting device position due to the boom angle change when transporting the component was considered using the following expression:

$$z_{{\alpha }_i}=l_{b_i} \left(\sin {\alpha }_i-\sin {\alpha }_{i-1}\right)\hspace{1em}i\in I$$

(17)

while the height difference between the demand and supply locations of the component was determined by employing the following formula:

$$z_i=z_{d_i}-z_{s_i}\hspace{1em}i\in I$$

(18)

It is necessary to emphasize at this point that the optimization model was developed for mobile cranes with a telescopic boom, which are most frequently used in construction practice. In this way, the relations that apply between the horizontal distances of different locations of the supply, demand, and mobile crane, the length and vertical angle of the mobile crane boom when installing the component, and the horizontal angle between the supply and demand locations for the component were included in the optimization model with the following two equations:

$${\alpha }_i=\arccos \frac{l_{d_i}}{l_{b_i}}\hspace{1em}i\in I$$

(19)

$${\theta }_i=2\pi -\arccos \frac{{l_{d_i}}^2+{l_{s_i}}^2-{l_i}^2}{2 {l_{d_i}}^2 {l_{s_i}}^2}\hspace{1em}i\in I$$

(20)

where the penultimate expression presented above considers that the boom length does not change unless the component to be transported is not outside the crane’s currently set operating radius, while the latter term works under the assumption that there are obstacles present between the supply and demand locations when the mobile crane is operating.

3.4. Bounds on Degrees of Freedom of Movement and Stop Positions

The degrees of freedom of the mobile crane movement on the construction site were defined within the optimization model by $\mathit{d}\mathit{o}\mathit{f}=\left({\alpha }_i,{\theta }_i,d_{m_i},l_{b_i},z_{v_i}\right)$ and bounded using the following expression:

$${\mathit{d}\mathit{o}\mathit{f}}^{\mathbf{L}\mathbf{O}}\le \mathit{d}\mathit{o}\mathit{f}\le {\mathit{d}\mathit{o}\mathit{f}}^{\mathbf{U}\mathbf{P}}$$

(21)

where ${\mathit{d}\mathit{o}\mathit{f}}^{\mathbf{L}\mathbf{O}}$ and ${\mathit{d}\mathit{o}\mathit{f}}^{\mathbf{U}\mathbf{P}}$ designate the lower and upper bounds on the degrees of freedom of the mobile crane movement, respectively.

It should be noted here that the determination of bounds on $d_{m_i}$ may be performed according to the size of the construction site area that is prepared beforehand for the mobile crane movement and the maneuvering space available for its operation, while this determination for ${\alpha }_i$, ${\theta }_i$, $l_{b_i}$ and $z_{v_i}$ can be implemented based on the technical specifications of the mobile crane manufacturer. The degrees of freedom of the mobile crane movement are presented in Figure 1.

Similarly, the feasible area for determining the mobile crane stop positions on the construction site was bounded within the optimization model by the following conditional inequality:

$${{\mathit{p}}_{\mathit{c}}}^{\mathbf{L}\mathbf{O}}\le {\mathit{p}}_{\mathit{c}}\le {{\mathit{p}}_{\mathit{c}}}^{\mathbf{U}\mathbf{P}}$$

(22)

where ${{\mathit{p}}_{\mathit{c}}}^{\mathbf{L}\mathbf{O}}$ and ${{\mathit{p}}_{\mathit{c}}}^{\mathbf{U}\mathbf{P}}$ denote the lower and upper bounds on the mobile crane stop positions, respectively. The bounds presented above can be concretely set based on the size of the construction site area that is ready for mobile crane movement and the room at disposal for its maneuvers. The optimization model notations are given in the Appendix A.

The optimization model discussed above was applied to a real case in the following section to demonstrate its advantages for optimal mobile crane positioning on a construction site.

4. Application Example

4.1. General Information

The construction under consideration represents the extension of a production and storage facility [20] in Gornja Radgona, Slovenia. Figure 2 presents the construction site location with a limited area of plot 401 for mobile crane operations. In addition, the construction site is also characterized by the fact that when performing assembly work, it is necessary to pay attention to the existing buildings in the immediate vicinity and the one to which the extension is connected; there is also an active transport route right next to the construction site. There are no height barriers in the construction site area, so a special height control for the mobile crane boom is not required.

Figure 3 shows the facility extension to be built. It is backed by a ground floor steel structure with a gable roof in a repeating pattern, which is to be assembled using a mobile crane (Figure 4). The extension is positioned on level ground right next to the existing production and storage facility.

4.2. Input Data

The building structure to be assembled using a mobile crane consists of 49 components, i.e. $i\in I=\left\{1, 2,\dots ,49\right\}$. The coordinate system origin and the labels of the component mounting locations are given in Figure 5.

The set of mobile crane stops $j\in J\left(i\right)$ was determined according to the guidelines given in reference [21]. Accordingly, it was estimated that two stops of the mobile crane are necessary. The first stop $\left(j=1\right)$ of the mobile crane is determined for the assembly of components 1 to 28, while the second one $\left(j=2\right)$ covers the installation of components 29 to 49.

The input data of the optimization model include the values of scalar and indexed parameters. Scalar parameters incorporate cost parameters, parameters of operation durations, probability parameters, parameters from mobile crane technical specifications, and limiting parameters for crane positioning, movements, and maneuvers. In this application example, the input values of the parameters were set as presented in

Table 1. Input values of scalar parameters.

Parameters	Input Values
Cost parameters
Rental cost of mobile crane	$c_r=550 \mathrm{c}.\mathrm{u}./\mathrm{d}\mathrm{a}\mathrm{y}$
Extra cost of mobile crane	$c_e=55 \mathrm{c}.\mathrm{u}.$
Parameters of operations durations
Duration of preparing the mobile crane for operation	$t_p=0.347 \mathrm{h}$
Duration of the mobile crane setup	$t_s=0.167 \mathrm{h}$
Duration of the mobile crane dismantling	$t_d=0.167 \mathrm{h}$
Duration needed to reach the maximum mobile crane boom extension	$t_b=0.067 \mathrm{h}$
Probability parameters
Coefficient of concurrent horizontal radial and tangential movements	$\beta =1.00$
Coefficient of concurrent horizontal and vertical movements	$\gamma =0.25$
Parameters from mobile crane technical specifications
Mobile crane chassis travel velocity	$v_m=\mathrm{78,000} \mathrm{m}/\mathrm{h}$
Mobile crane angular velocity of boom lift	$v_{\alpha }=90 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{h}$
Mobile crane slew velocity	$v_{\theta }=96 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{h}$
Mobile crane winch’s vertical lifting velocity	$v_v=7800 \mathrm{m}/\mathrm{h}$
Minimum vertical angle of the mobile crane boom	${\alpha }^{\mathrm{L}\mathrm{O}}=0.0000 \mathrm{r}\mathrm{a}\mathrm{d}$
Maximum vertical angle of the mobile crane boom	${\alpha }^{\mathrm{U}\mathrm{P}}=1.4486 \mathrm{r}\mathrm{a}\mathrm{d}$
Minimum horizontal slew angle of the mobile crane boom	${\theta }^{\mathrm{L}\mathrm{O}}=0.0000 \mathrm{r}\mathrm{a}\mathrm{d}$
Maximum horizontal slew angle of the mobile crane boom	${\theta }^{\mathrm{U}\mathrm{P}}=2\pi \mathrm{r}\mathrm{a}\mathrm{d}$
Minimum length of the mobile crane boom	${l_b}^{\mathrm{L}\mathrm{O}}=10.200 \mathrm{m}$
Maximum length of the mobile crane boom	${l_b}^{\mathrm{U}\mathrm{P}}=40.000 \mathrm{m}$
Minimum total vertical change in the crane lifting device position	${z_v}^{\mathrm{L}\mathrm{O}}=-40.000 \mathrm{m}$
Maximum total vertical change in the crane lifting device position	${z_v}^{\mathrm{U}\mathrm{P}}=40.000 \mathrm{m}$
Limiting parameters for crane positioning, movements, and maneuvers
Minimum mobile crane chassis travel distance between stops	${d_m}^{\mathrm{L}\mathrm{O}}=0.000 \mathrm{m}$
Maximum mobile crane chassis travel distance between stops	${d_m}^{\mathrm{U}\mathrm{P}}=93.555 \mathrm{m}$
Minimum x-coordinate of the mobile crane at the stop	${x_c}^{\mathrm{L}\mathrm{O}}=0.000 \mathrm{m}$
Maximum x-coordinate of the mobile crane at the stop	${x_c}^{\mathrm{U}\mathrm{P}}=90.000 \mathrm{m}$
Minimum y-coordinate of the mobile crane at the stop	${y_c}^{\mathrm{L}\mathrm{O}}=0.000 \mathrm{m}$
Maximum y-coordinate of the mobile crane at the stop	${y_c}^{\mathrm{U}\mathrm{P}}=19.500 \mathrm{m}$
Minimum z-coordinate of the mobile crane at the stop	${z_c}^{\mathrm{L}\mathrm{O}}=0.000 \mathrm{m}$
Maximum z-coordinate of the mobile crane at the stop	${z_c}^{\mathrm{U}\mathrm{P}}=0.000 \mathrm{m}$

Note: The abbreviation c. u. indicates currency units.

. Here, the cost parameters were established based on information from the company that rents out the mobile crane. An eight-hour workday was planned. The parameters of operations durations were fixed as stated in reference [22], while the probability parameters were specified in accordance with reference [19]. The parameters from the mobile crane technical specifications were provided by its manufacturer. Lastly, the limiting parameters for mobile crane positioning, movements, and maneuvers were entered into the optimization model on the basis of the construction site location with a bounded area of plot 401 (Figure 2 and Figure 5).

Values for the following indexed parameters were determined for each building component $i\in I$: (i) duration of verification and positioning of component assemblies, (ii) duration of the component’s fitment, (iii) duration of the component’s hooking to the crane lifting device, (iv) duration of the component’s unhooking from the crane lifting device, (v) coordinates of the component’s demand location, (vi) coordinates of the component’s supply location, and (vii) crane boom length when installing the component (

Table 2. Input values of indexed parameters.

$\mathit{i}\in \mathit{I}$	${\mathit{t}}_{{\mathit{a}}_{\mathit{i}}}\left[\mathbf{h}\right]$	${\mathit{t}}_{{\mathit{f}}_{\mathit{i}}}\left[\mathbf{h}\right]$	${\mathit{t}}_{{\mathit{l}}_{\mathit{i}}}\left[\mathbf{h}\right]$	${\mathit{t}}_{{\mathit{u}}_{\mathit{i}}}\left[\mathbf{h}\right]$	${\mathit{x}}_{{\mathit{d}}_{\mathit{i}}}\left[\mathbf{m}\right]$	${\mathit{x}}_{{\mathit{s}}_{\mathit{i}}}\left[\mathbf{m}\right]$	${\mathit{y}}_{{\mathit{d}}_{\mathit{i}}}\left[\mathbf{m}\right]$	${\mathit{y}}_{{\mathit{s}}_{\mathit{i}}}\left[\mathbf{m}\right]$	${\mathit{z}}_{{\mathit{d}}_{\mathit{i}}}\left[\mathbf{m}\right]$	${\mathit{z}}_{{\mathit{s}}_{\mathit{i}}}\left[\mathbf{m}\right]$	${\mathit{l}}_{{\mathit{b}}_{\mathit{i}}}\left[\mathbf{m}\right]$
1	0.000	0.063	0.025	0.367	0.110	39.400	5.650	10.400	8.200	0.000	40.000
2	0.167	0.063	0.060	0.367	0.160	39.400	2.830	10.400	8.200	0.000	40.000
3	0.167	0.063	0.025	0.367	8.680	39.400	5.650	10.400	8.200	0.000	34.200
4	0.167	0.063	0.060	0.367	8.780	39.400	2.820	10.400	8.200	0.000	34.200
5	0.167	0.063	0.060	0.367	4.390	39.400	5.650	10.400	10.410	0.000	34.200
6	0.167	0.063	0.025	0.367	0.110	39.400	11.650	10.400	8.200	0.000	40.000
7	0.167	0.063	0.060	0.367	0.210	39.400	8.650	10.400	8.200	0.000	40.000
8	0.167	0.063	0.025	0.367	8.680	39.400	11.650	10.400	8.200	0.000	34.200
9	0.167	0.063	0.060	0.367	8.780	39.400	8.650	10.400	8.200	0.000	34.200
10	0.167	0.063	0.060	0.367	4.390	39.400	11.650	10.400	10.410	0.000	34.200
11	0.167	0.063	0.025	0.367	0.110	39.400	17.650	10.400	8.200	0.000	40.000
12	0.167	0.063	0.060	0.367	0.210	39.400	14.650	10.400	8.200	0.000	40.000
13	0.167	0.063	0.025	0.367	8.680	39.400	17.650	10.400	8.200	0.000	34.200
14	0.167	0.063	0.060	0.367	4.390	39.400	17.650	10.400	10.410	0.000	34.200
15	0.167	0.063	0.060	0.367	8.780	39.400	14.650	10.400	8.200	0.000	34.200
16	0.167	0.063	0.025	0.367	26.680	39.400	5.650	10.400	8.200	0.000	34.200
17	0.167	0.063	0.060	0.367	26.570	39.400	2.820	10.400	8.200	0.000	34.200
18	0.167	0.063	0.060	0.367	17.680	39.400	5.650	10.400	10.410	0.000	34.200
19	0.167	0.063	0.025	0.367	26.680	39.400	17.650	10.400	8.200	0.000	34.200
20	0.167	0.063	0.060	0.367	17.680	39.400	17.650	10.400	10.410	0.000	34.200
21	0.167	0.063	0.060	0.367	26.780	39.400	2.820	10.400	8.200	0.000	34.200
22	0.167	0.063	0.025	0.367	44.680	39.400	5.650	10.400	8.200	0.000	34.200
23	0.167	0.063	0.060	0.367	44.680	39.400	2.820	10.400	8.200	0.000	34.200
24	0.167	0.063	0.060	0.367	35.680	39.400	5.650	10.400	10.410	0.000	34.200
25	0.167	0.063	0.025	0.367	44.680	39.400	17.650	10.400	8.200	0.000	34.200
26	0.167	0.063	0.060	0.367	35.680	39.400	17.650	10.400	10.410	0.000	34.200
27	0.167	0.063	0.060	0.367	44.570	39.400	11.210	10.400	8.200	0.000	34.200
28	0.167	0.063	0.060	0.367	44.780	39.400	11.210	10.400	8.200	0.000	34.200
29	0.167	0.063	0.060	0.367	26.570	39.400	11.210	10.400	8.200	0.000	34.200
30	0.167	0.063	0.060	0.367	26.780	39.400	11.210	10.400	8.200	0.000	34.200
31	0.167	0.063	0.060	0.367	17.730	39.400	11.650	10.400	10.410	0.000	34.200
32	0.167	0.063	0.060	0.367	35.680	39.400	11.650	10.400	10.410	0.000	34.200
33	0.167	0.063	0.025	0.367	62.680	39.400	5.650	10.400	8.200	0.000	34.200
34	0.167	0.063	0.060	0.367	62.570	39.400	2.820	10.400	8.200	0.000	34.200
35	0.167	0.063	0.025	0.367	62.680	39.400	17.650	10.400	8.200	0.000	34.200
36	0.167	0.063	0.060	0.367	44.780	39.400	2.820	10.400	8.200	0.000	34.200
37	0.167	0.063	0.060	0.367	53.680	39.400	5.650	10.400	10.410	0.000	34.200
38	0.167	0.063	0.060	0.367	53.680	39.400	17.650	10.400	10.410	0.000	34.200
39	0.167	0.063	0.060	0.367	62.570	39.400	11.210	10.400	8.200	0.000	34.200
40	0.167	0.063	0.060	0.367	53.680	39.400	11.650	10.400	10.410	0.000	34.200
41	0.167	0.063	0.025	0.367	84.780	39.400	5.650	10.400	8.200	0.000	40.000
42	0.167	0.063	0.060	0.367	84.780	39.400	2.820	10.400	8.200	0.000	40.000
43	0.167	0.063	0.060	0.367	73.720	39.400	5.650	10.400	10.410	0.000	34.200
44	0.167	0.063	0.025	0.367	84.780	39.400	11.650	10.400	8.200	0.000	40.000
45	0.167	0.063	0.060	0.367	84.780	39.400	8.650	10.400	8.200	0.000	40.000
46	0.167	0.063	0.060	0.367	73.660	39.400	11.650	10.400	10.410	0.000	34.200
47	0.167	0.063	0.025	0.367	84.020	39.400	17.670	10.400	8.200	0.000	40.000
48	0.167	0.063	0.060	0.367	84.400	39.400	14.660	10.400	8.200	0.000	40.000
49	0.167	0.063	0.060	0.367	73.340	39.400	17.660	10.400	10.410	0.000	34.200

Notation: $i\in I—$building component, $t_{a_i}—$duration of verification and positioning of component assemblies, $t_{f_i}—$duration of the component’s fitment, $t_{l_i}—$duration of the component’s hooking to the crane lifting device, $t_{u_i}—$duration of the component’s temporary setup and unhooking from the crane lifting device, ${\mathit{p}}_{\mathit{d}}=\left(x_{d_i},y_{d_i},z_{d_i}\right)—$coordinates of the component’s demand location, ${\mathit{p}}_{\mathit{s}}=\left(x_{s_i},y_{s_i},z_{s_i}\right)—$coordinates of the component’s supply location, $l_{b_i}—$crane boom length when installing the component.

).

The values of the durations of the verifications and positioning of component assemblies were set according to reference [22], whereby the value of the mentioned parameter for the first component was equal to zero. The durations for fitment and hooking to and unhooking components from the crane lifting device were also fixed as indicated in reference [22]. It should be noted that the duration of the component’s hooking to the crane lifting device was set differently for columns and beams.

Coordinates of demand locations for components were generated regarding their position in the building structure relative to the coordinate system origin (Figure 4 and Figure 5). The coordinates of supply locations were the same for all components and were determined near the initial mobile crane stop position. Regarding the crane boom length when installing the component, it should be noted that the maximum length was selected for the installation of components positioned on the side edges of the building, while the length of 34.2 m was chosen for all other components.

4.3. Optimization Setup

The objective of the optimization is to determine the optimal mobile crane stop positions that lead to the minimum total cost of its usage on the construction site and allow the complete assembly of the building structure in question. The optimization model discussed in Section 3 was applied here to achieve this goal. A high-level language known as a general algebraic modeling system (GAMS) [23] was employed to create a computationally usable optimization model. The coordinates of the component demand locations were exported from the AutoCAD 2021 software in which the building structure was modeled to an Excel spreadsheet, where each component demand location was given its own tab named after the coordinate label. The rest of the data shown in Table 2 was also generated in the Excel spreadsheet. The input values of the indexed parameters set in the Excel spreadsheet were transferred to the optimization model over the GAMS Data eXchange (GDX) file. The complete optimization procedure is shown in Figure 6.

Section 3 shows that the optimization model is nonlinear but also includes discontinuous and non-smooth expressions, i.e., Equations (6) and (7) with functions that compute the maximum/minimum of the argument set, as well as Equations (9), (10) and (12) with functions that calculate the absolute value of the term. Therefore, DNLP was selected to solve the optimization problem. After tests were previously performed and the six GAMS solvers showed effective optimization ability (see

Table 3. Test results.

Optimization Solver	Objective Value [c. u.]	CPU Time [s]
CONOPT/LINDO/MSNLP/OQNLP	2591.6325	0.156/0.905/9.937/11.934
IPOPT	2596.2831	0.248
SNOPT	2591.7477	0.234

), the task was managed using the CONOPT solver (based on a generalized reduced gradient method, GRG) [24]. Here, the default termination tolerances were selected for the GRG algorithm to perform optimization.

Before running the optimization, the starting mobile crane stop positions were set with ${p_c}_1^0=\left(29.200, 11.000, 0.000\right)$ and ${p_c}_2^0=\left(49.500, 9.800, 0.000\right)$ given in [m], while the initial vertical angle of the mobile crane boom was specified using ${\alpha }_0=1.20 \mathrm{r}\mathrm{a}\mathrm{d}$. It was also preset that the duration of the component’s rise from the supply location to the demand location is equal to the duration of the movement of an empty crane lifting device from the demand location to the next supply location, i.e., $t_{r_i}=t_{e_i}$. The optimization was then performed on a 64-bit operating system using a personal computer with an Intel(R) Xeon(R) W-2255 CPU @ 3.70 GHz, 64 GB of random access memory, and a 512 TB solid-state drive. The total solver time required to obtain the optimal solution was less than a second (Table 3).

4.4. Optimization Results

The optimization led to the minimum total cost of mobile crane usage on the construction site in the amount of 2591.63 c. u. (

Table 4. Results of optimal mobile crane positioning.

$\mathit{i}\in \mathit{I}$	${\mathit{\alpha }}_{\mathit{i}}\left[\mathbf{r}\mathbf{a}\mathbf{d}\right]$	${\mathit{\theta }}_{\mathit{i}}\left[\mathbf{r}\mathbf{a}\mathbf{d}\right]$	${\mathit{d}}_{{\mathit{m}}_{\mathit{i}}}\left[\mathbf{m}\right]$	${\mathit{z}}_{{\mathit{v}}_{\mathit{i}}}\left[\mathbf{m}\right]$	Minimum Total Cost of Mobile Crane Usage:
1	0.573	3.231	0.000	–7.380	$\min c=2591.63 \mathrm{c}.\mathrm{u}.$
2	0.553	3.313	0.000	7.504
3	0.745	3.276	0.000	13.434	Mobile crane stop positions:
4	0.721	3.386	0.000	7.584	${p_c}_1^{*}=\left(33.406, 10.132, 0.000\right)$${p_c}_2^{*}=\left(49.496, 9.986, 0.000\right)$
5	0.538	3.250	0.000	5.372
6	0.586	3.232	0.000	9.794
7	0.590	3.142	0.000	8.360
8	0.761	3.247	0.000	12.735
9	0.765	3.157	0.000	8.312
10	0.556	3.238	0.000	4.764
11	0.548	3.408	0.000	7.947
12	0.581	3.321	0.000	9.209
13	0.714	3.481	0.000	11.909
14	0.502	3.440	0.000	4.484
15	0.749	3.368	0.000	15.030
16	1.332	3.684	0.000	18.132
17	1.274	3.916	0.000	7.672
18	1.072	3.375	0.000	7.742
19	1.271	4.027	0.000	10.843
20	1.036	3.632	0.000	7.155
21	1.278	3.932	0.000	11.523
22	1.208	5.860	0.000	7.431
23	1.167	5.663	0.000	7.674
24	1.423	5.137	0.000	12.789
25	1.163	5.740	0.000	5.773
26	1.339	5.050	0.000	12.295
27	1.237	6.232	0.000	7.221
28	1.230	6.233	16.090	8.127
29	0.835	6.271	0.000	1.315
30	0.843	6.270	0.000	8.388
31	0.376	6.272	0.000	–2.569
32	1.152	6.204	0.000	29.092
33	1.153	3.418	0.000	8.219
34	1.120	3.602	0.000	7.720
35	1.109	3.709	0.000	8.035
36	1.317	5.253	0.000	10.693
37	1.394	3.904	0.000	10.968
38	1.313	4.254	0.000	9.812	Notation:
39	1.177	3.276	0.000	6.712	$i$	–	building component,
40	1.439	3.561	0.000	12.734	${\alpha }_i$	–	vertical angle of the mobile crane boom
41	0.476	3.223	0.000	−13.107			when installing the component,
42	0.451	3.301	0.000	7.290	${\theta }_i$	–	horizontal angle between the supply and
43	0.768	3.278	0.000	19.257			demand locations for the component,
44	0.489	3.230	0.000	−0.805	$d_{m_i}$	–	mobile crane chassis travel distance
45	0.489	3.145	0.000	8.226			between stops prior to component
46	0.784	3.251	0.000	18.475			transport,
47	0.486	3.402	0.000	−1.357	$z_{v_i}$	–	total vertical change in the crane lifting
48	0.494	3.316	0.000	8.493			device position for transporting the
49	0.749	3.494	0.000	17.476			component with a winch.

).

Table 4 demonstrates the related optimization results, which include (i) optimal mobile crane stop positions, (ii) vertical angles of the mobile crane boom for installing components, (iii) horizontal angles between the supply and demand locations for components, (iv) mobile crane chassis travel distances between stops prior to component transportation, and (v) total vertical changes in the crane lifting device position for transporting components with a winch. Figure 7 shows the determined optimal stopping positions of the mobile crane.

The obtained results showed the main feature of the model, i.e., its capability to provide the mobile crane operator with key information that enables the complete and optimal assembly of the building structure under construction. At this point, it should be emphasized that the DNLP approach provided the exact optimal results actionable on the construction site. The model also yielded the optimal duration of the mobile crane rental of 36.896 hours, which is necessary to perform the assembly of all components, including the optimal duration values of those assembly operations that depend on the mobile crane positioning. The information gained on the optimal distribution of the mobile crane rental period to the assembly operations allowed a detailed duration analysis of the entire process of building structure erection (Figure 8).

For instance, Figure 8 shows that in the case under consideration, almost half of the optimal mobile crane rental period needs to be allocated to temporarily setting up components and unhooking them from the crane lifting device. Such data can be important to the contractor for improving the efficiency of the assembly process, as it indicates which operations it makes sense to focus on when introducing changes in procedures.

5. Discussion and Conclusions

This study presented the optimal positioning of a mobile crane on a construction site using the DNLP approach. The optimization model formulation was discussed in detail, which consisted of the cost objective function subject to various duration and positioning constraints for the mobile crane, including the bounds on its degrees of freedom of movement and stop positions. An application example was given in this paper to demonstrate the advantages of the considered approach.

The computationally usable optimization model was developed using GAMS. The coordinates of the component demand locations were exported from the AutoCAD 2021 software in which the building structure was modeled to an Excel spreadsheet. The input values of the indexed parameters set in the Excel spreadsheet were transferred to the GAMS model over the GDX file. The defined DNLP problem was solved to optimality using the CONOPT algorithm. To the best of our knowledge, such an approach to the optimal positioning of mobile cranes on construction sites has not been proposed in the literature.

Employed to a real case, the DNLP approach displayed the ability to deliver the exact optimal solution for mobile crane positioning on a construction site while consuming the total solver time that is acceptable for practice. Therefore, the mobile crane operator is provided with key information that enables the complete and optimal assembly of the building structure under construction. In addition, the DNLP output information on the optimal distribution of the mobile crane rental period to the assembly operations allowed a detailed duration analysis of the entire process of building structure erection, which can be useful for implementing future improvements.

Despite the fact that the DNLP model focused on the economic aspect of the sustainable engagement of a mobile crane on a construction site, it is reasonable for future research to direct its objectives also to related environmental and social issues. The ability of the DNLP model to operate based on fully exact input and output data opens up space for future studies on its integration with other computer-aided models that have been proven to enhance construction sustainability, such as building information models. Finally, the approach shown requires basic programing skills, so future investigations need to concentrate on developing a more user-friendly system that automatically generates an optimization model, performs optimization in the background, and visualizes the results.