Building projects with time–cost–quality–environment trade-off optimization using adaptive selection slime mold algorithm

Abstract

Every project manager deals with various challenges, and almost all tasks have backup plans to ensure efficient success. Therefore, it is essential to manage resources, notably in terms of time, cost, quality, and environmental impact, and this needs to be thoroughly shown. As a result, the adaptive selection slime mold algorithm (ASSMA) is proposed for repetitive projects due to multiple concurrent instances. It is made by merging the tournament selection (TS) method and the slime mold algorithm (SMA) model. The new model’s capabilities are demonstrated using a case study of a rural water pipeline project, and the outcomes of the ASSMA are contrasted with those of the data envelopment analysis (DEA) approach utilized by the previous researcher. Consequently, the ASSMA technique is an effective optimization matching method that can help project managers select the best strategy for a given activity. This study is anticipated to expand significantly and outperform other models by utilizing quality performance metrics.

Introduction

Economic recovery and the construction industry have improved over the years in Vietnam following the fourth wave of the COVID-19 pandemic, with a deeper understanding of the significance of building through implementing projects and allocating funds. Planning must be used to overcome challenges caused by disruptions due to force majeure, because the COVID-19 pandemic’s effect in the future is unpredictable. Thus, it is essential to grasp the potential of the entire construction industry recovering and growing in the ensuing years (Table 1).

Table 1.

Pseudocode of the ASSMA

Algorithm: pseudocode of the ASSMA

Begin Inputs: N, d, T, $\delta$, LB, UBS Outputs: XGB and fGB Initialization: for the first iteration of time t = 1, randomly initialize the slime mold $X_i=(x_i^1,x_i^2,...,x_i^d),\forall i\in [1,N]$ within the search boundary of UB and LB While $t\le T$ Utilizing Eq. (7) determine the N slime mold fitness values f(X) Using Eq. (12), sort the fitness value Update the best fitness fLB using Eq. (13) and best individual XLB using Eq. (14) Update the worst fitness fLW using Eq. (15) Update the best fitness fGB and best individual XGB Update the weight W using Eq. (11) Adjust b and c using Eq. (16), (17) For each slime mold, N, where i = 1 Produce arbitrary numbers r1 and r2 Produce the threshold value pi using Eq. (9) Analyze the location of Xni using Eq. (18), Eq. (19), Eq. (20) Analyze the new slime mold’s fitness value f(Xni) If SortedFitness (Temp(i)) < Best [temp(i) = (Randi(size(sortedFitness,2),1), N)] (Tournament Selection strategy) Best = SortedFitness(temp(i)); Index = temp(i); End End Choice = index; End Next iteration t = t + 1 End Return: Worldwide best-solutions arena XGB

The elements affecting the project are always challenging for managers to overcome. The success of a project is consistently determined by the three factors of time, cost, and quality. However, a number of other things may also have an impact. The fourth consideration in this article's concurrent four-factor optimization of a construction project is the environmental impact. Because the majority of conflicting factors cannot be coordinated concurrently to complete a project, it is imperative to optimize time, cost, quality, and environmental effect.

The environmental component, which must be taken into account in the construction industry to implement environmental protection inside and outside the project's borders, is usually overlooked in modern society. Recently, projects that minimize pollution have attracted the eye of many project managers. The external environment, climate, and even the land at the building site are all negatively impacted by the waste of resources. To ease the burden on the industry, Vietnam has also passed a lot of environmental legislation. In their discussion on environmental protection, Fergusson and Langford (2006) concentrated on eliminating the discomfort produced by man-made pollution and the pain it causes others.

Additionally, Wang et al. (2019) highlighted environmental protection strategies because this issue has been neglected and imprecisely quantified. Project managers frequently disregard external issues. This article emphasizes the significance of environmental influences in construction projects.

Optimization development between objectives is utilizing many original or hybrid algorithms or creating more robust ones. Time and cost optimization was used to create the initial foundation (Feng et al., 1997). These methods have been widely employed by researchers to address the three optimization issues of time, cost, and quality (El-Razek et al., 2010) and time, cost and safety (Afshar & Zolfaghar Dolabi, 2014). Numerous studies have shown major evidence using algorithmic models for concurrent three-factor optimization. The successful implementation of time, cost, quality, and safety optimization raises performance to unprecedented heights (Sharma & Trivedi, 2020), creating a variety of distinctive results. Additionally, Pareto’s capacity for convergence is more varied. This study applies the concurrent four-factor optimization of time, cost, quality, and environmental impact in an infrastructure project (ASSMA). However, this issue has encountered numerous challenges, since the complexity rises and the search space’s aims change. Slime mold algorithm (SMA) was presented by Li et al. (2020) to optimize the algorithm’s operations. To identify the best solution possible, the SMA's primary function is used most effectively during the exploration and exploitation phases. However, SMA’s potential for exploration and exploitation is constrained because it is guided by two randomly selected search agents who choose a path to proceed in and then modify it later to look for the best outcomes. We suggest employing the tournament selection (TS) approach to overcome the SMA’s limitations (Yang and Li 2010) to enhance convergence from random to best candidate selection. This combination helps improve the original method, reduce the algorithm’s risks, help find results quickly, and provide superior convergence through the Pareto front.

Systematic issues are solved using the ASSMA technique. Even while some issues cannot be resolved at random, it is at least possible to provide solutions that fall within the algorithm’s acceptable bounds. Algorithms are created to solve problems repeatedly, not just once. ASSMA does a fantastic job at solving a number of issues, including: (1) finding the “shortest path” when there is no effective way to do; (2) processing a lot of data; (3) using the same steps at each time; and (4) processing many elements simultaneously.

The ASSMA capabilities are fully utilized in this study to facilitate the analysis of time, cost, quality, and environmental impact. The objective is to demonstrate that the recommended algorithm outperforms data envelopment analysis at solving the problem, achieves rapid convergence without sacrificing variety, and outperforms DEA in comparison.

Literature review

In 2020, Li et al. (2020) released the SMA with many new and improved features. Moreover, Mostafa et al. (2020) found a new strategy for SMA to optimize solar power. To address the issue of urban water demand, Zubaidi et al. (2020) employed ANN along with SMA. The SMA and whale optimization were combined by Abdel-Basset et al. (2020) to address the chest X-ray picture segmentation challenge. Wazery et al. (2021) also studied the SMA combined with K-nearest applied in healthcare. Yu et al. (2021) endorsed the SMA on quantum rotation gates, Houssein et al. (2021) presented hybrid SMA with a differential evolution algorithm to solve the comprehensive optimization problem. Both Liu et al. (2021) and Houssein et al. (2022) employed the SMA to solve the multiobjective problem. Liu et al. (2021) used the SMA to calculate the parameters of the photovoltaic models. The SMA has not been combined with many different methods or widely applied to construction management, especially for crucial objective optimization.

The optimization problem is profoundly interesting to many authors and readers. Forming two, three, and four objectives are always complex in the concurrent optimization problem. Afshar et al. (2009) analyzed a new multicolony ant algorithm which was used to solve the time–cost multiobjective optimization problem. Both Tiwari and Johari (2015) and Zahraie and Tavakolan (2009) made the case that time and cost are two important considerations in building. Babu's method was used by Khang and Myint (1999) in the construction of a cement plant, and they confirmed its efficacy. The time–cost trade-off was expanded to include time–cost–quality (El-Rayes & Kandil, 2005), time–cost–safety trade-off optimization models (Luong et al., 2018), and more. The GA model was utilized by Liu et al. (2020) to address the issues of time, cost, and quality. Additionally, other evolutionary hybridization methods have been utilized successfully to address the time–cost–quality problem (Mungle et al., 2013; Tran et al., 2015; Zhang et al., 2014). The four-objective optimization phase, which includes time, cost, resources, and the environment (Panwar & Jha, 2019); time, cost, quality, and safety (Sharma & Trivedi, 2020); and time, cost, resources, and cash flow (Pamal and Manoj 2020), has been considered to be the best (Elbeltagi et al., 2016). From the above bases, the desire for multiobjective optimization is increasing, and the full use of the algorithm’s capabilities brings success to the entire project. This is the preliminary stage and the premise for further development and expansion of research directions and solutions to solve new difficulties in the four-objective optimizations from concurrent optimization with two to three objectives.

To prove the researchability of the article, the author has also made an in-depth study on algorithms for solving multiobjectives. Specifically, Ngoc-Tri Ngo et al. (2022) presented the meta-model optimization algorithm combined with computer models to solve the energy use of buildings. Son and Lien (2022) introduced a case study of crowd-sourced arbitration using the Rhubarb platform to resolve disputes. Vu-Hong-Son et al. (2022) presented a new hybird artificial intelligent model to optimize the material supply chain save cost for construction contractors. Son and Anh (2021) used game-theoretic method to solve the Nash equilibirum solution. Son and Khoi (2022) optimized the time–cost–quality trade-off problem by using the new model of slime mold algorithm for construction projects. Son and Nguyen Thanh (2022) applied a new artificial intelligence algorithm to solve the design of the water distribution system. Son et al. (2022) optimized the schedule using the matrix method and the dolphin algorithm. Son et al. (2021) presented the optimization algorithm of the dragonfly particle swarm for the cost optimization of construction material. Son and Anh (2021) created a game model for the contractor selection process’ compensation idea, and Son (2021) used a dynamic hybrid bacterium to optimize the layout of the construction site. To improve freight coordination, Son and Khoi (2020) created the African wild dog algorithm. Nguyen et al. (2020) analyzed construction failure factors in the finishing phase of high-rise building projects and Leu et al. (2015) created a new approach to building procurement.

Methodology

According to Li et al. (2020), the development of the ASSMA is based on the modification of the slime approach behavior with a TS decision (Yang and Li 2010), to aid the original algorithm's convergence and locate better candidates.

Tournament selection

The standard TS selects a sample of k people at random from the current populations, and then it selects the competitor with the best fitness. The TS process is composed of two steps of sampling and selection. Since 2, 4, and 7 are popular tournament sizes, N populations of tournaments are needed to produce all of the individuals in the subsequent generation, as shown in Fig. 1.

Fig. 1.

Simulation of tournament selection

When compared to other selection modes, the following characteristics of TS stand out: (i) selection pressure can be modified easily; (ii) it is simple to code; (iii) no prearrangement of populations is necessary; and (iv) it is notably time-consuming (Goldberg & Deb, 1991).

The selection process of TS, which includes sample and selection, therefore requires a large amount of focus on different samples and also different selections, as shown in Harik (1995), Filipović et al. (2000), Luke and Panait (2002), Matsui (1999), and Sokolov and Whitley (2005).

Equation (1) gives the selection probability of a person of rank j in a TS under the implicit premise that the population is completely diverse (each individual has a unique fitness value):

$$N^{-k}({(N-j+1)}^k-{(N-j)}^k).$$ 1

Blickle and Thiele (1994) expanded the selection probability model to describe the selection probability of individuals with the same fitness to predict the expected fitness distribution following TS in a population with a more broad form. They introduced the cumulative fitness distribution and defined the worst person to be placed first. S(fj) denotes the number of individuals with fitness value f_j or worse. The selection probability of the people defined by Eq was then calculated (2):

$${\left(\frac{S(f_j)}{N}\right)}^k-{\left(\frac{S\left(f_{j-1}\right)}{N}\right)}^k.$$ 2

The probability of one individual not being sampled in one tournament is therefore computed by Poli and Langdon (2006) as 1-N1, and the expected number of people not being sampled in any tournament is then determined by Eq. (3):

$$N{\left(\frac{N}{N-1}\right)}^{-ky},$$ 3

where y represents the total number of competitions necessary to create a brand-new generation.

TS is indifferent to population size (the population refers to the same value and so has the same rank), and Xie et al., (2007) proposed a sampling probability model that any program p is sampled to demonstrate this at least once in $y\in \left\{1,\dots ,N\right\}$ tournaments by Eqs. (4) and (5):

$$1-{\left({\left(\frac{N-1}{N}\right)}^N\right)}^{\frac{y}{N}k}$$ 4

$$1-{\left(1-\frac{{\left(\frac{{\sum }_{i=1}^j\left|S_i\right|}{N}\right)}^k-{\left(\frac{{\sum }_{i=1}^{j-1}\left|S_i\right|}{N}\right)}^k}{\left|S_j\right|}\right)}^y,$$ 5

where $\left|S_j\right|$ is the number of programs of the same jth rank.

Mathematical formulation of ASSMA

The following diagram illustrates the location and fitness of N slime molds at the current iteration:

$$X(t)=\left[\begin{array}{c}{x}_1^1{x}_1^2...x_1^d\\ x_2^1{x}_2^2...x_2^d\\ ⋮⋮⋮⋮\\ x_N^1{x}_N^2...x_N^d\\ \end{array}\right]=\left[\begin{array}{c}{X}_1\\ X_2\\ ⋮\\ X_N\\ \end{array}\right],$$ 6

$$f(X)=[f(X_1),f(X_2),...,f(X_N)]\text{.}$$ 7

Using Eq. (8), the slime mold's location in the SMA is updated for the subsequent iteration (t + 1):

$$X_i(t+1)=\left\{\begin{array}{c}{X}_{\text{LB}}(t)+V_b(\text{W}.X_A(t)-X_B(t))r_1\ge \delta \text{and}{r}_2<p_i\\ V_c.X_i(t)r_1\ge \delta \text{and}{r}_2\ge p\\ r\text{and}(\text{UB}-\text{LB})+\text{LB}{r}_1<\delta \\ \end{array}\right),\forall i\in [1,N],$$ 8

where:

• X_LB: represents for the current iteration the local best person,

• X_A and X_B: slime mold from current populations collected at random,

• W: the weight component,

• V_b and V_c: the random velocity,

• r₁ and r₂: random numbers between 0 and 1,

• $\delta$: the slime mold’s fixed 0.03 initialization probability at a random search position.

p_i is the threshold value of the ith slime mold that aids in choosing the best slime mold individual or the slime mold itself for the subsequent iteration, as measured by the following criteria:

$$p_i=tanh\left|f(X_i)-f_{\text{GB}}\right|,\forall i\in [1,N],$$ 9

where:

• f(X_i): the value of fitness ith slime mold X_i,

• f_GB: the global best position’s Eq. (10) global best fitness value X_GB

  $$f_{\text{GB}}=f(X_{\text{GB}}).$$ 10

The weight W for N slime molds in iteration t is defined as follows:

$$W({\text{SortInd}}_f(i))=\left\{\begin{array}{c}1+r\text{and}log\left(\frac{f_{\text{LB}}-f(X_i)}{f_{\text{LB}}-f_{\text{LW}}}+1\right)1\le i\le \frac{N}{2}\\ 1-r\text{and}log\left(\frac{f_{\text{LB}}-f(X_i)}{f_{\text{LB}}-f_{\text{LW}}}+1\right)\frac{N}{2}<i\le N\\ \end{array}\right),$$ 11

where:

• Rand: a random number between 0 and 1,

• f_LB: the local best fitness value,

• f_LW: the local worst fitness value,

• f_LB and f_LW are determined from the fitness value f given in Eq. (7).

Sort the fitness values as follows in ascending order to reduce the issue:

$$\left[{\text{Sort}}_f,{\text{SortInd}}_f\right]=\text{sort}\left(f\right).$$ 12

The best fitness value f_LB and the corresponding best individual X_LB are proposed as follows:

$$f_{\text{LB}}=f\left({\text{Sort}}_f\left(1\right)\right),$$ 13

$$X_{\text{LB}}=X\left({\text{SortInd}}_f\left(1\right)\right).$$ 14

The worst fitness individual value f_LW is proposed as follows:

$$f_{\text{LW}}=f\left({\text{Sort}}_f\left(N\right)\right).$$ 15

V_b and V_c: in Eqs. (16) and (17), a random velocity selected from the continuous uniform distribution in the intervals [−b,b] and [−c,c] was used. b and c for the iteration t were chosen as:

$$b=\text{arc}\text{tan h}\left(-\left(\frac{t}{T}\right)+1\right),$$ 16

$$c=1-\frac{t}{T},$$ 17

where:

• T: maximum iteration.

X_A and X_B are two random slime mold positions, and the ability to solve is not explicitly mentioned in exploration and exploitation. Random pooling can be used to get around this restriction. Therefore, the position of slime mold i (I = 1,2,…, N) is updated according to the principle of Eqs. (8) to (18), (19), and (20):

$$X{n}_i(t)=X_{\text{LB}}(t)+V_b(W.X_{\text{LB}}(t)-X_B(t))\text{if}{r}_1\ge \delta \text{and}{r}_2<p_i.$$ 18

Slime mold search for trajectories guided by X_LB, X_A, and X_B are randomly merged from the search space of N slime molds with velocity V_b. This process aids in the trade-off between exploration and exploitation:

$$X{n}_i(t)=V_c.X_i(t)\text{if}{r}_1\ge \delta \text{and}{r}_2\ge p_i.$$ 19

The algorithm’s position with velocity serves as a compass for the slime mold's trajectory V_c. This action aids in exploitation:

$$X{n}_i(t)=r\text{and}(\text{UB}-\text{LB})+\text{LB}\text{if}{r}_1<\delta .$$ 20

Back in the search area, the slime mold first appears. This action aids in exploring.

Case study

According to a case study involving the rural water pipeline project (Banihashemi & Khalilzadeh, 2020), the ASSMA’s performance is superior to that of the author’s findings. There are two DEA types used in project management. The first type assesses the success of diverse initiatives using a specific set of inputs and outputs. Initiatives that select project portfolios and rank various initiatives prior to investing fall under the second category. Using this model, the mode of execution for each activity is determined by how effectively the various project activity execution modes perform. Figure 2 shows the eight distinct construction scenarios and eight activities that make up the rural water pipeline project, which was the study’s main focus. The project's specific operations are listed in Table 2.

Fig. 2.

Project network

Table 2.

Specific activities method

ID	Activities	Predecessors
1	Equipping the ingot workshop	–
2	Canal lining and drilling	1
3	Spin the pipes	2
4	Channel regression and leveling	1, 2
5	Welding and transfer of pipes to the floor of the canal	3
6	Tubing and steaming	4, 5
7	Testing	6
8	Channel filling	7

To determine the actual project time required, the Gant chart is employed (i.e., zero lead time). The project's overall cost is estimated using the activity's total cost. The project’s overall quality and environmental impact are also determined by the average of each factor in each activity. After that, it is necessary to use the plan selection to determine how to reduce time, money, and environmental effect while retaining the high caliber of the job, as this forms the basis for showing the viability and potential of the ASSMA.

Tables 3 and 4 list the input data along with three resources: laborers (R1), laborers without special skills (R2), and excavators (R3) (R3). The time, expense, quality, and environmental impact of a particular operation are given along with the output statistics. Activity 1 is fixed; however, activities 2 through 7 each have seven unique instances. The project is anticipated to have a total of 823,544 outcomes, which will result in variations in TCQE.

• Input parameters

Table 3.

A project’s data

Activity number	Project activities	EM	Resources			Time	Cost	Quality	Environmental impact
			R1	R2	R3
1	Equipping the ingot workshop	1	0	2	0	2	1000	0.8	0.3
2	Canal lining and drilling	1	0	3	1	14	1260	0.78	0.36
		2	0	7	1	10	1220	0.8	0.44
		3	0	5	2	8	1376	0.83	0.64
		4	0	10	1	7	1022	0.82	0.44
		5	0	7	1	12	1464	0.84	0.52
		6	0	5	2	10	1720	0.87	0.64
		7	0	10	1	10	1460	0.88	0.48
3	Spin the pipes	1	0	2	0	25	400	0.74	0.2
		2	0	5	0	20	800	0.8	0.3
		3	0	7	0	16	896	0.83	0.4
		4	0	10	0	12	960	0.9	0.5
		5	0	5	0	22	880	0.82	0.4
		6	0	7	0	18	1008	0.85	0.5
		7	0	10	0	14	1120	0.92	0.5
4	Channel regression and leveling	1	0	4	0	5	160	0.77	0.27
		2	0	7	0	4	224	0.8	0.4
		3	0	10	0	3	240	0.83	0.47
		4	0	13	0	2	208	0.86	0.53
		5	0	7	0	5	280	0.85	0.4
		6	0	10	0	4	320	0.88	0.53
		7	0	13	0	3	312	0.9	0.53
5	Welding and transfer of pipes to the floor of the canal	1	1	3	1	5	570	0.76	0.45
		2	2	3	1	3	414	0.8	0.6
		3	1	4	1	4	488	0.75	0.45
		4	2	5	1	2	308	0.85	0.6
		5	2	3	1	4	552	0.86	0.6
		6	1	4	1	5	610	0.82	0.45
		7	2	5	1	3	462	0.92	0.6
6	Tubing and steaming operations	1	0	4	0	22	704	0.78	0.3
		2	0	5	0	20	800	0.8	0.4
		3	0	6	0	17	816	0.81	0.5
		4	0	7	0	15	840	0.82	0.6
		5	0	5	0	22	880	0.82	0.4
		6	0	6	0	19	912	0.83	0.6
		7	0	7	0	17	952	0.84	0.6
7	Testing	1	0	1	0	11	880	0.82	0.4
		2	0	2	0	7	112	0.8	0.5
		3	0	3	0	4	96	0.81	0.5
		4	0	4	0	2	64	0.84	0.6
		5	0	2	0	8	128	0.83	0.5
		6	0	3	0	5	120	0.83	0.6
		7	0	4	0	3	96	0.87	0.6
8	Channel filling	1	0	8	0	19	1216	0.82	0.3
		2	0	10	0	15	1200	0.8	0.35
		3	0	6	1	6	684	0.74	0.35
		4	0	8	1	5	650	0.76	0.5
		5	0	10	0	16	1280	0.81	0.3
		6	0	6	1	7	798	0.75	0.44
		7	0	8	1	6	780	0.77	0.44

Table 4.

ASSMA parameters

Input	Notation	Value
Number of populations	N	100
Maximum iteration	T	200
Number of decision variables	D	25
$\delta$ parameter	$\delta$	0.03
Lower boundary	LB	−100
Upper boundary	UB	100

Optimization results obtained using the ASSMA

Table 5 shows the convergence results from the ASSMA for time, cost, quality, and environmental effect, which led to the best optimal solutions in the rural water pipeline project. It is crucial for a project manager to understand the best-case scenario to calculate the trade-off between objectives and prevent negative consequences on their project. The project’s minimum completion time is 45 days, and its maximum completion time is 98 days. The project’s minimum and maximum costs are $4308 and $7882, respectively. Its overall quality ranges from 0.77 to 0.86875, and its EI ranges from 0.3225 to 0.53375, as shown in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 3031, 32, 33, and 34. What is even more amazing is that activity number 4 is a non-essential activity for the project according to Gant [0, 1, 2, 3, 5, 6, 7 and 8].

Table 5.

Result Pareto-optimal solutions

No	Pareto-optimal of projects	Time	Cost	Quality	Environmental impact	Gant	Iteration (1/200)
I. BEST TIME
1	1 4 4 1 4 4 4 4	45	5004	0.82	0.47625	1 2 3 5 6 7 8	117
2	1 4 3 2 4 4 4 4	49	5004	0.815	0.48	1 2 3 5 6 7 8	70
3	1 4 4 4 7 4 6 7	50	5392	0.84	0.50125	1 2 3 5 6 7 8	68
4	1 4 3 2 7 4 4 3	51	5192	0.82125	0.46125	1 2 3 5 6 7 8	83
II. BEST COST
1	1 4 1 4 4 1 4 4	65	4356	0.80625	0.43375	1 2 3 5 6 7 8	63
2	1 4 1 1 4 2 4 4	63	4404	0.7975	0.41375	1 2 3 5 6 7 8	87
3	1 4 1 1 4 3 4 3	61	4454	0.7963	0.4075	1 2 3 5 6 7 8	100
4	1 4 1 4 4 4 4 3	59	4526	0.8088	0.4525	1 2 3 5 6 7 8	109
III. BEST QUALITY
1	1 7 7 7 7 7 7 1	68	6618	0.86875	0.48875	1 2 3 5 6 7 8	112
2	1 7 7 7 7 6 7 4	56	6012	0.86	0.51375	1 2 3 5 6 7 8	82
3	1 4 7 7 7 3 7 1	65	6044	0.8575	0.47125	1 2 3 5 6 7 8	65
4	1 7 4 6 7 7 7 7	53	6030	0.8575	0.50625	1 2 3 5 6 7 8	102
IV. BEST ENVIRONMENTAL IMPACT
1	1 1 1 1 3 1 1 5	94	6172	0.78125	0.3225	1 2 3 5 6 7 8	122
2	1 1 1 1 6 1 1 3	85	5698	0.7813	0.32875	1 2 3 5 6 7 8	78
3	1 4 1 1 3 1 1 3	77	5338	0.7775	0.33875	1 2 3 5 6 7 8	88
4	1 1 1 1 3 1 2 3	80	4808	0.77	0.34125	1 2 3 5 6 7 8	118

Fig. 3.

Best TC trade-off

Fig. 4.

Best TQ trade-off

Fig. 5.

Best TEI trade-off

Fig. 6.

Best CT trade-off

Fig. 7.

Best CEI trade-off

Fig. 8.

Best CQ trade-off

Fig. 9.

Best QT trade-off

Fig. 10.

Best QC trade-off

Fig. 11.

Best QEI trade-off

Fig. 12.

Best EIT trade-off

Fig. 13.

Best EIC trade-off

Fig. 14.

Best EIQ trade-off

Fig. 15.

Best TCQ trade-off

Fig. 16.

Best TCEI trade-off

Fig. 17.

Best TQEI trade-off

Fig. 18.

Best CQT trade-off

Fig. 19.

Best CEIT trade-off

Fig. 20.

Best CQEI trade-off

Fig. 21.

Best QTC trade-off

Fig. 22.

Best QEIC trade-off

Fig. 23.

Best QTEI trade-off

Fig. 24.

Best EICQ trade-off

Fig. 25.

Best EITQ trade-off

Fig. 26.

Best EICT trade-off

Fig. 27.

Value path for best time

Fig. 28.

Value path for best cost

Fig. 29.

Value path for best quality

Fig. 30.

Value path for best EI

Fig. 31.

2D Column for time, cost, quality, and EI

Fig. 32.

Resource project of R1

Fig. 33.

Resource project of R2

Fig. 34.

Resource project of R3

Figures 3–26 show the best TCQE trade-off in terms of two- and three-dimensional factors.

In Figs. 27–30, the value path graph for the best-optimized TCQE solutions derived from Pareto is displayed. The vertical axis in the path graph indicates that the values have been restricted to the range of [0,1], while the horizontal axis displays the TCQE values, which are acquired values joined to form a straight line. With regard to the various transformations between the straight lines of the factors in the search space, the suggested ASSMA demonstrates how well the search for optimal solutions is done.

Comparing optimization results obtained using the ASSMA with DEA

Table 6 illustrates the compromises between the DEA and the suggested model, demonstrating that the ASSMA's capacity to find optimal solutions is superior to the DEA through the use of a SMA with a characteristic that is evenly and widely distributed in a search space; in addition, it is also combined with the TS method to aid slime mold increase exploration in a new search area and choose the best optima to give the best goals in the model’s capabilities. Utilizing search agents, randomizing candidates with high value, and then utilizing TS to determine the top candidate to update the location of slime mold at time t = 1 result in exploitation. Using different execution mechanisms, the DEA mathematical model enables the measurement of activity performance with respect to desired and undesirable results. To find the most effective order of project implementation procedures, the key innovation of the DEA study is tested along with the effectiveness of each activity’s execution modes. Besides, Sayyid and Mohammad (2020) used the DEA combined with the ideal and anti-ideal methods (Wang & Luo, 2006), using GAMS software to solve the problem. In contrast to the ASSMA, the number of slime molds is increased based on the original method to optimize the capacity to search the output data in a search space after merging with the TS method. The ASSMA has demonstrated the model’s ability to condense data due to the abundance of output data. As a result, the model's searchability has simply produced the anticipated outcomes. The ASSMA used Matlab R2019b primarily to resolve the aforementioned issue. The results showing the ideal accommodation between the DEA and ASSMA are shown in Table 6.

Table 6.

Result comparison between DEA and ASSMA

Sayyid and Mohammad, 2020
EM	Time	Cost	Quality	Environmental impact
DEA
1–4-3–4-2–3-7–2	63	5652	0.823	0.465
1–3-2–3-1–1-2–5	80	6082	0.800	0.407
1–4-3–1-2–3-7–2	63	5604	0.811	0.432
1–4-3–1-4–4-7–2	53	5478	0.827	0.488

Proposed model
EM	Time	Cost	Quality	Environmental impact
ASSMA
1–4–3–2–7–4–4–3	51	5192	0.82125	0.46125
1–4–1–4–4–1–4–4	65	4356	0.80625	0.43375
1–4–1–1–3–1–1–3	77	5338	0.7775	0.33875
1–4–7–7–7–3–7–1	65	6044	0.8575	0.47125

Rural plumbing solely indicates the required number of skilled laborers (R1) for Activity 5 according to the case study. Activity 5 only requires one excavator for each case; however Activity 2 may require as many as three excavators (R3) to complete the assignment. Additionally, the number of workers without a degree is uneven (R2). As a result, Figs. 32–34 show the resource graph of the series of tasks [1, 4, 4, 1, 4, 4, 4] corresponding to R1, R2, and R3 to assess the equipment and human resources employed during the project.

Comparing the evaluation indicators of the ASSMA, rather than DEA, with IDMU and ADMU

It is required to demonstrate that the ASSMA is effective based on the quality assessment parameters to evaluate the convergence of Pareto’s search for optimal solutions and variety in the search in the vertical direction of the Pareto front. The essay concluded by suggesting that the qualities given below are sufficient for assessing how effectively the proposed ASSMA performs (Abhilasha and Kumar 2019):

• number of solutions (NS): the number of Pareto optimum front solutions,

• spacing: the variance of the separation of Pareto front solutions,

• mean ideal distances (MID): the Pareto front solution’s rate of convergence from the ideal point,

• spread of nondominant solution (SNS): the variety of solutions provided by the Pareto front,

• quality metric (QM): a variable that aids in evaluating the effectiveness of the Pareto optimum solutions found,

• diversity: extending Pareto-optimal solutions,

• hypervolume (HV): the region of the objective space that solutions occupy,

• epsilon (E): a metric indicating how unsatisfactory a solution set is in comparison to the most well-known Pareto front,

• computional time (CT): the amount of time it takes to create a Pareto-optimal front.

According to Sayyid and Mohammad (2020), they used DEA based on ideal decision-making units (IDMU) to assess IDMU’s greatest possible relative efficacy and DEA based on anti-ideal decision-making units (ADMU) to assess ADMU’s worst possible relative result. Different conclusions could be drawn based on these two unique efficacy assessments. Utilizing a relative index, which provides an evaluation of each DMU’s overall performance and may be used as a standard for comparing and assessing DMU performance values ranging from great to substandard, the two specific effectivenesses are combined.

Thereby, the mathematical model used for the DEA is applied to the basis of effectiveness assessment of two different models, determining the ranking of good and poor performance, from which it is possible to ascertain which activities are predominant and lead to good optimization in finding results, which is one of the intriguing methods for future development. The ASSMA is a model that effectively makes use of exploration and exploitation capabilities to generate an optimal or nearly optimal solution by randomly choosing search agents from the entire population to decide the best search agents’ next course of action. Several international researches indicate that the majority of performance evaluation metrics used to assess the model’s quality are those shown in Table 7. When doing an evaluation, it is advised to use the ASSMA rather than the DEA because it offers a more diverse array of new approaches. Additionally, the two models demonstrate the greatest capacity for adjusting to an uncertain environment.

Table 7.

The evaluation parameters of the ASSMA

Algorithms	NS	Spacing	MID	SNS	QM	Diversity	HypE	E	CT
ASSMA	33	0.38	1.97	81,366	0.95	0.85	0.90	1.25	145

Conclusion

Project managers in the field of project management must make plans to ensure that the project is completed successfully while reducing time, cost, and environmental effect and maintaining a high level of quality. The basis for proving that the project manager needs a lot of experience and knows how to deal with changes arising in the project is that it can be very difficult to achieve the trade-off of factors, and sometimes project managers have to increase the time and pay extra costs for problems that result in an increase or decrease in quality and environmental impact. As a result, they must think very carefully about what is best for the project’s life cycle. The paper uses ASSMA to balance time, cost, quality, and environmental impact to assure optimization. The ASSMA used Pareto to construct a value path graph, a three-dimensional Pareto graph between three components, and a two-dimensional Pareto graph between two factors to build an overall image.

The ASSMA is used to assess the mode of execution for each activity inside the project network. Concurrent tasks are performed, and time, cost, quality, and environmental impact are defined as the input and output data. The results show that the project implementation approach was successfully used, producing the best results with the least amount of time, money, and environmental impact. The DEA with IDMU and ADMU is compared against the SMA with TS to determine which option is the fastest and most efficient. Finding the solution with the least amount of time, money, and environmental impact while keeping the highest degree of quality is the major goal of a construction project.

Directions for future research

The hybrid algorithm model can be implemented in the future with greater improvement using additional research. We broaden the multiobjective optimization model to take into account the critical elements required to deliver the best outcomes for project managers. Construction, logistics, and other industries can leverage multiobjective issues to broaden the model’s application to each new topic. Kaveh (2021) presented the most well-known and efficient metaheuristic algorithms, so the authors will apply or test our studies to compare the effectiveness and superiority with other metaheuristic algorithms.

Author contributions

Both of authors prepared, wrote and reviewed this manuscript

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Data availability

Data available on request from the authors.

Declarations

Conflict of interest

There is no conflict of interest.