An Improved Metaheuristic Algorithm for On-site Workshop Availability Cost Problem

We define a new binary decision variable, $y^{\prime}_{ijkt}$. It equals $1$ if activity $i\in J$ with the execution mode $j\in m_{i}$ occupies the OSW $k\in K$ at the period $t=\{0,1,\ldots,T\}$, otherwise it equals $0$. Accordingly, we replace constraint (3) with constraints (11)-(13).

Where $\alpha$ is an arbitrary constant parameter such that $0<\alpha<1$.

We defined a new decision variable, $R^{\prime}_{kt}$, as the availability level of OSW $k\in K$ at period $t=\{0,1,\ldots,T\}$. Then, we replace the constraint (4) with constraints (14)-(17).

Where $M$ is a big constant parameter, such that $0\leq R_{k}\leq M,\forall k\in K$.

Finally, we reformulate the adopted nonlinear MIP (3)-(LABEL:Eq10) to a linear MIP (3), (2), (5)-(17), (LABEL:Eqx3).

The adopted MIP model for MOSWACP has $T\bar{K}m^{\prime}$ decision variables and $n^{\prime}+T\bar{K}m^{\prime}$ constraints. $\bar{K}$, $m^{\prime}$, and $n^{\prime}$ are the total number of available OSWs, the total number of execution modes, and the total number of precedent activities, respectively. MOSWACP extends the Multi-Mode Resource Investment Problem with tardiness (MRIPT). Since MRIPT is NP-hard (Gerhards, 2020), and MOSWACP can be reduced to MRIPT, the MOSWACP is NP-hard, too. The existing commercial solvers cannot solve a large-scale MOSWACP within a reasonable time. Thus, we developed the metaheuristic algorithm, Electron Radar Search Algorithm (ERSA), to generate solutions for large-scale MOSWACPs.

We built a $2n+3\bar{K}$-dimensional vector to represent the solution space for the MOSWACP problem as

$\{S_{1},\ldots,S_{n},SW_{1},\ldots,SW_{\bar{K}},FW_{1},\ldots,FW_{\bar{K}},M_{1},\ldots,M_{n},R_{1},\ldots,R_{\bar{K}}\}$.

where $n$ denotes the total number of activities, and $\bar{K}$ represents total OSWs. The solution representation in the proposed ERSA is a ($2n+3\bar{K}$)-dimensional vector, in which $\{S_{1},\ldots,S_{n}\}$, $\{SW_{1},\ldots,SW_{\bar{K}}\}$, $\{FW_{1},\ldots,FW_{\bar{K}}\}$, $\{M_{1},\ldots,M_{n}\}$, and $\{R_{1},\ldots,R_{\bar{K}}\}$ are the activities’ start time, the installation time of the OSWs, the dismantling duration of OSWs, the execution mode selected for activities, and the level of OSWs’ availability, in corresponding order.

The initial population set must constitute a feasible solution for MOSWACP. First, we find $R_{k},k\in K$, through the allocation of a randomly generated integer that exceeds $\max_{i\in J,j\in m_{i}}\{r_{ijk}\}$. Second, we find $M_{i}(i\in J)$ by selecting a random integer from set $m_{i}$, which is the execution mode of the activity $i$. Third, we use Alg. 2 to determine the initiation time of the activities. Finally, $SW_{k}$ and $FW_{k}$, $k\in K$ are determined based on the activities that require OSW $k\in K$ to begin, as Eq. (19).

ERSA employs an improvement-based local search to create the neighboring solution by practical crossover and mutation operators. Then, it is improved by applying three efficient improvement rules represented in Table 2. The inputs and outputs of the crossover operator are $2$ solutions, each of which known as parent and offspring solutions. We presume $Sol_{1}$ and $Sol_{2}$ are parent solutions as follows.

$Sol_{2}:\{S^{2}_{1},\ldots,S^{2}_{n},SW^{2}_{1},\ldots,SW^{2}_{\bar{K}},FW^{2}_{1},\ldots,FW^{2}_{\bar{K}},M^{2}_{1},\ldots,M^{2}_{n},R^{2}_{1},\ldots,R^{2}_{\bar{K}}\}$.

We determine a new point based on the crossover point of a single-point crossover, $C_{p}$. The crossover then takes place at this point, producing the offspring solutions, $Sol_{3}$ and $Sol_{4}$, as described below.

$Sol_{1}:\{S^{1}_{1},\ldots,S^{1}_{n},SW^{1}_{1},\ldots,SW^{1}_{\bar{K}}\otimes,FW^{1}_{1},\ldots,FW^{1}_{\bar{K}},M^{1}_{1},\ldots,M^{1}_{n},R^{1}_{1},\ldots,R^{1}_{\bar{K}}\}$

$Sol_{2}:\{S^{2}_{1},\ldots,S^{2}_{n},SW^{2}_{1},\ldots,SW^{2}_{\bar{K}}\otimes,FW^{2}_{1},\ldots,FW^{2}_{\bar{K}},M^{2}_{1},\ldots,M^{2}_{n},R^{2}_{1},\ldots,R^{2}_{\bar{K}}\}$

$\downarrow$

$Sol_{3}:\{S^{1}_{1},\ldots,S^{1}_{n},SW^{1}_{1},\ldots,SW^{1}_{\bar{K}}\otimes,FW^{2}_{1},\ldots,FW^{2}_{\bar{K}},M^{2}_{1},\ldots,M^{2}_{n},R^{2}_{1},\ldots,R^{2}_{\bar{K}}\}$

$Sol_{4}:\{S^{2}_{1},\ldots,S^{2}_{n},SW^{2}_{1},\ldots,SW^{2}_{\bar{K}}\otimes,FW^{1}_{1},\ldots,FW^{1}_{\bar{K}},M^{1}_{1},\ldots,M^{1}_{n},R^{1}_{1},\ldots,R^{1}_{\bar{K}}\}$

With the symbol $\otimes$ and $C_{p}=n+\bar{K}$, crossover point is positioned between $n+\bar{K}$ and $n+\bar{K}+1$.

Since the offspring solutions might be infeasible, we designed a crossover operator to transform an infeasible solution to a feasible one. If the crossover modify the beginning time of the activities, we call Alg. 2 to repair the infeasible schedule. Additionally, if the crossover alters the OSW schedule, we transform the infeasible schedule into a feasible one by Eq. (19). Moreover, in case the crossover shifts the execution mode of an activity, we update the schedule of the activities and OSW according to the new execution modes by Alg. 2. Finally, in case the crossover modifies the OSWs’ availability level, we update the schedule of OSWs and activities based off of the new availability levels using Alg. 2.

The mutation operator takes one solution, and creates a mutated solution. The number of bits (elements) mutated in each solution is determined by $r_{m}$. Let’s take the solution, $Sol_{1}$, to mutate as follows.

$Sol_{1}:\{S^{1}_{1},\ldots,S^{1}_{n},SW^{1}_{1},\ldots,SW^{1}_{\bar{K}},FW^{1}_{1},\ldots,FW^{1}_{\bar{K}},M^{1}_{1},\ldots,M^{1}_{n},R^{1}_{1},\ldots,R^{1}_{\bar{K}}\}$

We mutate the elements $n$ and $n+\bar{K}$, such that $r_{m}=2$. The new solution becomes as follows.

$\{S^{1}_{1},\ldots,\mathbf{S^{\prime 1}_{n}},SW^{1}_{1},\ldots,\mathbf{SW^{\prime 1}_{\bar{K}}},FW^{1}_{1},\ldots,FW^{1}_{\bar{K}},M^{1}_{1},\ldots,M^{1}_{n},R^{1}_{1},\ldots,R^{1}_{\bar{K}}\}$

Where $S^{\prime 1}_{n}$ and $SW^{\prime 1}_{\bar{K}}$ are the new elements.

The solution may become infeasible by mutation. In that case, we use Alg. 2 to convert infeasible solutions into feasible solutions. If the mutation changes the activity schedule, we convert the activity’s start times to satisfy precedence relationships and the space capacity constraints. Also, if the mutation operator changes the installation or dismantling times of the OSW, we modify the corresponding installation or dismantling times to maintain the feasibility of the activities schedule. Likewise, if the mutation operator modifies the availability level of an OSW, we modify the corresponding OSW’s availability level to maintain the feasibility of the schedule and space capacity. Finally, if the mutation operator shifts the execution modes, we switch that activity to support the feasibility of activities’ schedules and space capacity. As the fitness function involves the cost of OSWs utilization, we calculate total usage cost by $\sum_{k\in K}R_{k}C_{k}$.

The proposed ERSA in this study incorporates three MOSWACP-specific improvement operators, as shown earlier in Table 2.

The first improvement operator, denoted as $IR_{1}$, focuses on altering the activities’ mode of execution to achieve two primary goals: (i) Reduce the project’s makespan by parallelizing activities, and (ii) Optimize the utilization of OSWs and avoid the physical space violation.

Let’s take an example of a construction site, shown in Fig. 4, with $5$ OSWs, in which the activity $1$ utilizes the OSWs $\{1,4,5\}$, activity $2$ utilizes $\{1,2,3\}$, and activity $3$ utilizes $\{3,4\}$. The improvement operator, $IR_{1}$, switches the activity’s execution mode $1$ from mode $1$ to mode $2$. In this case, the second execution mode requires activity $1$ to take relatively more space in workshop $4$, while not demanding a space in OSW $5$, as illustrated in Fig. 5. Consequently, we can execute activity $4$ concurrent with activities $1$, $2$, and $3$ due to the precedence relationships outlined in Fig. 4. Activity $4$ occupies workshop $6$ only so that our adjustment does not violate the physical space capacity. In this example, we can reduce the project duration by effectively parallelizing $4$ activities without violating the constraints.

The second improvement operator, denoted as $IR_{2}$, rearranges the scheduling of activities based on their float times and reduces the project’s makespan, consistent with precedence relationships and the area’s constrained capacity. To illustrate the improvement operator $IR_{2}$, let’s take the scenario depicted in Fig. 4 as an example, where the duration of activities $1-4$ are $3$, $4$, $3$, and $6$ periods, respectively. Due to the limited area at the construction site, activity $4$ cannot commence until, say, one of the activities $1-3$ is complete. A feasible schedule for this project is shown in Fig. 6. In this example, activity $1$ is not a direct successor of $2$ or $3$. In such cases, the improvement operator, $IR_{2}$, shifts the activity’s start time $1$ from $0$ to $3$; see Fig. 7. This rule reduces the overall project duration by $33\%$; see Fig. 6, and Fig. 7.

The improvement operator, $IR_{3}$, optimizes the size or availability level of OSWs to incorporate additional workshops on the site. Let’s take the construction site depicted in Fig. 4 as an example. The improvement operator, $IR_{3}$, reduces OSWs’ availability level and maintains the demanded physical space of activities in them. By resizing the workshop(s), we add space to the site and install an additional OSW, as Workshop $6$ in Fig. 8. This adjustment facilitates the concurrent execution of more activities and ultimately improves the project schedules and allocation of resources efficiently.

In this section, we explain how we involve novel parameters used in the literature to generate diverse instances of MOSWACP. The parameters for MOSWACP closely resemble those of the well-known MMRCPSP (Drexl and Gruenewald, 1993). To create MOSWACP instances, we involve $C_{k}$ and $r_{jik}$ in MMRCPSP library²²2http://www.om-db.wi.tum.de/psplib/getdata_mm.html, which represent the usage cost of the OSW, $k$, and the space occupied by activity, $j$, with execution mode, $i$, respectively. We selected the instances, C15, C21, J10, J12, J14, J16, J18, J20, M2, and R3, from this library. We evaluate how the problem responds to the parameter variations using $16$ different instances of J10. Subsequently, we incrementally range the parameter $r_{jik}$ from $0$ to $13$, with other parameters fixed. Similarly, incrementally, we set the parameter $C_{k}$ to several different values from $\{10,20,30,50,100\}$ with other parameters fixed. Finally, we generate MOSWACP instances of various sizes, 34, 84, 536, 101, 117, 45, 93, 64, 89, and 113, for the datasets C15, C21, J10, J12, J14, J16, J18, J20, M2, and R3.

We design experiments over the instances of dataset C15 to obtain a set of values for MOSWACP parameters and optimal parameters for the proposed ERSA using response surface methodology (RSM). The RSM is a statistical and mathematical technique widely employed in experimental design and optimization processes (Khuri and Mukhopadhyay, 2010). It is a powerful tool for understanding complex relationships between input factors and the response of a system. The RSM simultaneously explores the optimal settings for multiple parameters, leading to improved efficiency and performance of a given process or system (Campatelli et al., 2014). In this study, we input the parameters of ERSA and the instances of C15 to the RSM and determine optimal parameters according to the objective function values.

We illustrate the ranges and optimal values of the parameters obtained by ERSA in Fig. 9. Since the output of RSM for the parameters, $N$, $E^{0}_{n}$, $CV$, and $M$ were float numbers, we rounded them to the nearest integer. Then, we found the best values for the parameters to minimize the objective function as $N=60$, $\beta=0.5$, $E^{0}_{n}=1500$, $CV=50$, $M=3600$, and $r=0.2$. Also, Fig. 1 (see A) shows the behavior of the objective function value, $OF$ (represented on the y-axis), for different values of the parameters, $N$, $\beta$, $E^{0}_{n}$, $CV$, $r$, and $M$ (represented on the x-axis). This figure shows that the objective function value negatively correlates with $M$ and $E^{0}_{n}$ but has no meaningful relationship with the remaining parameters. The objective function value in these experiments was the average of the objective function value of the first $10$ instances of the C15 dataset returned by ERSA.

In this section, we solve MOSWACP at different sizes by the exact commercial solver, CPLEX, and illustrate computational results in Table 3.

Due to the execution time limit, we set it to one hour ($T=3600$ seconds), and left few instances from each dataset unsolved to optimality. As shown in the table, we solved all instances of the datasets C15 and J10 in about $18$ and $1$ minutes, respectively. Also, the MIP solver efficiently solves the instances of datasets C21, J12, J14, and J16 by obtaining optimal solutions in more than $90\%$ of the instances. However, solving the instances of dataset J20 is challenging for CPLEX since only less than $60\%$ of cases are solved optimally by CPLEX. On average, $88\%$ of all instances are solved optimally by CPLEX in about $12$ minutes.

In this section, we solve MOSWACP instances of different sizes using our proposed ERSA, equipped with improvement rules. Also, we compare the performance of the proposed ERSA with SA, GA, and PSO, as summarized in Table 4. We set the execution time limit to $1$ minute in these experiments and obtain the optimal value of parameters by RSM (Dean et al., 2017). We put the parameters of GA to $N=50,p_{c}=0.3,p_{m}=0.2,r_{m}=0.2,C_{p}=0.2$, where $N$ is the population size, $p_{c}$ is the crossover probability, $p_{m}$ is the mutation probability, $r_{m}$ is the mutation rate, and $C_{p}$ is the crossover point. Also, we found optimal values for the SA parameters as $T_{max}=1000,T_{min}=0.01,\alpha=0.98,N=20$, where $\alpha$ is the cooling rate, and $N$ is the number of iterations at each temperature. We also set the parameters of PSO to $n_{s}=20$, $M=100$, and $c_{1}=c_{2}=2$, where $n_{s}$ denotes the size of the swarm, $M$ is the maximum number of iterations. Also, $c_{1}$ and $c_{2}$ are the learning factors.

In Table 4, we illustrate $5$ different versions of ERSA equipped with a different set of improvement rules involved: (i) ERSA with no improvement rule, $ERSA_{0}$; (ii) ERSA with all improvement rules except $IR_{3}$, $ERSA_{1}$; (iii) ERSA with all improvement rules except $IR_{2}$, $ERSA_{2}$; (iv) ERSA with all improvement rules except $IR_{1}$, $ERSA_{3}$; and (v) ERSA with all improvement rules, $ERSA_{4}$. Also, we compare metaheuristic solvers according to three different criteria: (i) $N^{*}$, the number of instances solved to optimality; (ii) $R^{*}$, the average of relative percentage between the best-found solutions, $f$ (returned by metaheuristics), and the optimal solutions, $f^{*}$ (returned by MIP solver), where $R^{*}=\frac{f-f^{*}}{f^{*}}*100$; and (iii) $T^{*}$, the average execution time. As shown in Table 4, our proposed ERSA with the proposed improvement rules performs better than existing metaheuristic solvers in terms of the number of instances solved to optimality and the average gap between the best-found and optimal solutions. Moreover, the designed improvement rules are superior to the solvers regarding solution quality and closeness to the optimal solution. We compare the proposed ERSA with the existing metaheuristic solvers in terms of the number of instances solved to optimality, $N^{*}$, as well as the average of relative percentage between the best-found solution and the optimal solution, $R^{*}$, in Fig. 10. The result reveals that the proposed ERSA bundled with improvement rules, $ERSA_{4}$, has a better performance compared to the existing metaheuristic solvers.

Also, we compare the fraction of the total instances solved to the optimality of each dataset. As shown in Fig. 11, the cases solved optimally in each dataset are highlighted in blue, and those not solved optimally are shaded in red. The prominent blue areas in these figures demonstrate the enhanced ERSA, incorporating the proposed improvement rules and finding relatively better optimal solutions compared to the existing solvers.

Also, we compare the convergence of $ERSA_{4}$ with the other solvers in Fig. 12 for large instances of datasets C21, J20, M2, and R3, respectively. Even though GA outperforms PSO and SA in finding relatively better solutions on average, our proposed $ERSA_{4}$ generates relatively higher-quality solutions. It searches for neighbor solutions more efficiently for large instances of MOSWACP. These figures illustrate that $ERSA_{4}$ finds high-quality solutions in relatively earlier iterations and searches local solutions more efficiently than other methods.

Finally, we compare $ERSA_{4}$ with the CPLEX solver in the last instance of each dataset. As given in Table 5, the proposed $ERSA_{4}$ reaches optimality in $2$ of datasets. Although the solutions by the CPLEX solver are acquired in one hour ($3600$ seconds), the proposed algorithm found solutions close to CPLEX solutions in approximately one minute ($60$ seconds) on average. Also, the gap, defined as the relative difference between the best-found solution by CPLEX and the proposed $ERSA_{4}$, depicts a faster performance of $ERSA_{4}$ for large instances of datasets compared to the CPLEX.

The computational results demonstrate that our proposed Electron Radar Search Algorithm (ERSA) with all improvement rules outperforms the existing metaheuristics, achieving more instances solved to optimality and maintaining smaller gaps between the best-found solutions and optimal counterparts. Furthermore, the results highlight the effectiveness of the improvement rules compared to other solvers lacking enhancements, such as solution quality and proximity to the optimal solution.

Moreover, the parameter of the spatial capacity of the construction site (denoted by $Q$) is changed to observe its impact on the objective function. The ten instances of dataset C15 were selected, and the average objective function over these instances was reported. Fig. 13 shows the impact of changing the capacity of the construction/production site on the objective function (minimizing the usage cost of OSWs). As observed, the average cost is reducing (improving) by increasing the site’s capacity ($Q$). The reason is that by providing more capacity for a construction site, the OSWs could be installed anytime. So, the associated activities could utilize these OSWs, leading to lower makespan and efficient utilization of OSWs. Also, in lower capacities, the marginal impact is higher; however, after some point, increasing the capacity does not change the usage cost of OSWs. Therefore, the project managers must know that the larger site’s capacity does not constantly improve the costs, and the optimal value of the $Q$ could be found when the costs change drastically.

This section presents the application of the OSWs and their practicability for an actual case study related to a trailer production project. In this case study, we aim to locate the OSWs within the production site (considering the occupied space and their lifetime) to make the project scheduling more efficient and to utilize the resources more effectively, compared to traditional policies. Detailed information on the company is confidential upon their request. Since 2000, the company has expanded production lines, parts making, and assembly to more than 30,000 square meters of land. Data from the case study may be available upon a reasonable request and owner’s permission. The information on project activities, duration, and precedence relations related to this project of the trailer production line is shown in Table 6. Also, the parameters related to the OSWs and storage areas within the manufacturing site are shown in Table 7.

The project duration (deadline) is 50 days, with a total of eight activities (A1-A8) and three On-Site Workshops (W1-W3). The space limitation is two OSWs at a time. Also, OSWs must not exceed their maximum lifetime on-site. The project’s purpose is to minimize total cost, which includes the cost of OSW availability while finding the installation and dismantling time of OSWs and the schedule of activities. Also, in the case study, one execution mode is considered for each activity. Also, the space occupied by each activity is assumed to be the same considering execution mode or OSW. The definition of each activity in the case study is stated as follows:

1. 1.

   A1: Chassis Preparation. This is the initial phase where the trailer chassis (the base frame) is assembled. This involves cutting, welding, and assembling steel beams to create the skeleton of the trailer.

2. 2.

   A2: Axle and Suspension Installation. Once the chassis is ready, the axles and suspension system are installed. These components ensure that the trailer can bear weight and move smoothly. Its predecessor is A1 (Chassis Preparation).

3. 3.

   A3: Wiring and Electrical Setup. During this step, electrical wiring and systems are installed for brake lights, indicators, and internal electrical components. This includes routing cables along the chassis. Its predecessor is A1 (Chassis Preparation).

4. 4.

   A4: Fabrication of Body Components. The trailer body (e.g., side panels, floor, and roof) is fabricated and prepped. This involves metal cutting, shaping, and welding to create the trailer’s exterior. Its predecessors are A2 (Axle and Suspension Installation), A3 (Wiring and Electrical Setup).

5. 5.

   A5: Paint and Surface Treatment. The entire trailer chassis and body are primed, painted, and treated with anti-corrosion materials to ensure durability and resistance to weather conditions. Its predecessor is A4 (Fabrication of Body Components).

6. 6.

   A6: Interior Fitting and Insulation. Insulation materials and internal components (like partitions, insulation layers, or specialized fittings) are installed inside the trailer if required (for refrigerated or specialized trailers). Its predecessor is A4 (Fabrication of Body Components).

7. 7.

   A7: Assembly of External Components. This stage involves assembling external features such as doors, loading ramps, handles, and other accessories. Quality checks are also conducted at this stage. Its predecessors are A5 (Paint and Surface Treatment), A6 (Interior Fitting and Insulation).

8. 8.

   A8: Final Quality Control and Testing. The trailer goes through final inspections, quality control checks, and testing (e.g., brake tests, durability tests, etc.) to ensure everything is functioning properly before delivery. Its predecessor is A7 (Assembly of External Components).

After solving the case study with our proposed methodology, the optimal activities’ start and finish times were obtained as given in Table 8. Moreover, considering the spatial constraint (maximum of 2 OSWs installed at any time) and the lifetimes of the OSWs, the optimal schedule for OSW availability is shown in Table 9. According to the results, W1 is installed on Day 1 to handle initial activities like Chassis Preparation (A1) and Axle and Suspension Installation (A2). It also stays on-site until Day 26, covering the Interior Fitting (A6). W2 is installed on Day 16 to handle body fabrication and painting tasks like Fabrication of Body Components (A4) and Paint and Surface Treatment (A5). It remains on-site for 27 days until Day 43, when the Assembly of External Components (A7) is completed. W3 is installed on Day 32 to handle the final stages of assembly and Final Quality Control (A8), being dismantled after Day 50. Its availability is shorter, as it serves only during the final phase. The total usage cost for the OSWs is computed based on their availability times and per-day costs:

Thus, the total OSW usage cost (objective function value) is:

Activities’ optimal start and finish times are provided, ensuring all precedence constraints are respected. The OSWs are optimally scheduled with installation and dismantling times to minimize costs while adhering to spatial constraints. The total usage cost of the OSWs (objective function) is 22,800 USD. This setup ensures that the resources are optimally allocated, minimizing the costs associated with the OSWs.

In the following, we compare the total cost returned by our methodology with the total cost obtained by the traditional policies (based on past experiences). The activity’s start and finish times and OSW’s dismantling and installation time returned by the conventional policy are in Tables 10-11, respectively. Based on these tables, the total usage cost of OSWs when using traditional policy equals $9,000+16,800+8,750=34,550$ USD. Compared with the optimal solution (22,800 USD), the savings with the Optimal Solution equal $34,550-22,800=11,750$ USD. So, the optimal solution is 33.99% cheaper than the traditional approach. The reason is that in the optimal solution, the installation and dismantling of OSWs are timed more efficiently, reducing idle times. On the other hand, in the traditional solution, the OSWs are installed too early and kept for too long, which leads to unnecessarily higher usage costs.

Furthermore, the optimal solution respects the space constraint by installing only two OSWs simultaneously, ensuring minimal overlap and avoiding resource congestion. The solution using traditional policy doesn’t fully account for space optimization. Although it respects the constraint, the inefficiency in scheduling leads to wasted space, as some OSWs remain idle for extended periods. Also, the optimal solution leverages more precise timing of the availability of OSWs, coordinating their use exactly when needed and dismantling them when their role is complete. In contrast, the traditional solution installs OSWs too early and dismantles them too late, resulting in longer usage durations and higher costs.

In addition, in the optimal solution, OSWs are scheduled to avoid idle periods, ensuring they are only used when necessary for the associated activities. The traditional solution, however, leads to extended periods where OSWs are idle, occupying space but not contributing to the project, leading to unnecessary costs. Regarding cost efficiency, the optimal solution minimizes the number of days that OSWs are in use, resulting in a significant cost reduction. In the traditional solution, the usage periods are excessive, inflating the cost by nearly 12,000 USD.