The Road to Sustainable Logistics: Using the Fuzzy Nonlinear Multi-Objective Optimization Model to Build Photovoltaic Stations in Taiwan’s Logistics Centers

Abstract

In Taiwan, numerous company logistics centers have embraced installing solar photovoltaic power stations (SPPSs) on their rooftops. The primary objective of this study is to expedite the generation of green electricity for sale, bolstering the logistics center’s income and enhancing its environmental, social, and governance (ESG) profile. How can we secure solar photovoltaic power station (SPPS) projects with expedited construction timelines, reduced investment costs, and heightened quality aligned with the long-term ESG objectives? The study applies the critical path method (CPM) to determine the item’s path. Next, the mothed leverages Zimmermann’s mathematical models for nonlinear multi-objectives and Yager’s fuzzy sets to enhance project efficiency, minimizing completion time and cost while maximizing the quality ratio. Subsequently, the project uses Liou and Wang’s defuzzification values and incorporates Dong’s fuzzy to accelerate calculations. In this case, Project HP’s item J, the construction time is reduced from 24.3 to 3.2 days, ensuring that construction quality meets an 85% standard. Item J necessitates expanding the fuzzy cost interval (4549.90, 15,416.65, 26,283.41) (it refers to a scope of possible costs). It becomes evident that construction time plays a pivotal role in controlling costs. For Project HP’s item H, the unit time quality decision ranges from TWD 238,000 to 240,000, to turn into a cost interval of TWD 215,100, 239,000, and 262,900. Consequently, cost transformation transitions from an active to a more passive role, with quality and construction time becoming the driving components. This study uses a fuzzy nonlinear multi-objective model to guide the decision analysis of SPPSs within logistics centers. This strategy enables decision-makers to streamline logistics center operations, ensuring time, cost, and quality (TCQ) alignment during SPPS installation, thereby advancing ESG sustainability goals.

1. Introduction

Taiwan advocates using solar photovoltaic green power generation and encourages installing solar stations on the company’s logistics center rooftops [1]. The carbon footprint generated during the construction life cycle phase is the assembly, machine power, and construction of fortifications by Chang [2]. All construction projects will consume fossil fuels and temporary electricity, thus emitting carbon and causing environmental loads. Chang pointed out that shortening the construction project time reduces carbon emissions. Therefore, the enterprises use their own logistics center’s rooftop area to shorten the time for building solar photovoltaic power stations (SPPSs). These advantages include reducing carbon emissions, producing clean energy, and improving the company’s ESG (environmental, social, and governance) image.

However, Taiwan’s policy is not the only one to propose that Taiwan’s companies build SPPSs on roofs at logistics centers. Thailand also expands and executes strategies regarding implementing SPPS rooftops [3]. Logistics center decision-makers must implement solar construction at low costs while meeting quality requirements and ensuring rapid completion of the building to reduce carbon emissions [4]. Therefore, when building SPPS, planning and calculating the quality, cost, and construction period (how to reduce carbon emission) in advance is essential.

This study aimed to streamline construction time while maximizing quality and minimizing costs when developing SPPSs. As part of the cost–benefit analysis, the logistics center’s decision-making process utilizes the critical path method (CPM) to determine the longest project path that can reduce execution time. Tsao et al. [5] adopt the technique of compressing activities to curtail project duration, albeit at an increasing expense. However, it is known from the explanation by Chang that carbon emissions can be reduced.

After the above explanation, this study focuses on shortening building hours to reduce carbon emissions and achieve ESG goals. Wang, Lai, and Shi [6] pointed out that problems arise in uncertain or fuzzy situations, and the goal of decision-makers is to find the best solution and solve multi-objective issues when information is incomplete or imprecise. The uncertainty that Al-Zarrad and Fonseca [7] discuss cannot be eradicated through any scheduling or estimation techniques. As such, there is a need for a model that can accurately represent real-world uncertainty to address time–cost–quality trade-off problems. This research focuses on developing a fuzzy nonlinear multi-objective model to optimize project scheduling by considering the changing activity costs associated with regular and expedited construction timelines (called “crash time”, which means the construction time becomes short).

The research method program design differs from other designs. It uses α-cut for defuzzification sorting. This sorting allows decision-makers to use interval values to make judgments. It provides more space to alleviate uncertainty. As proposed by Jana and Chakraborty [8], the fuzzy α-cut calculation could offer an effective tool for dealing with problems containing fuzzy information and help decision-makers make better choices when facing uncertainty.

The short-term objective of the fuzzy nonlinear multi-objective model is to expedite the operation of SPPSs and shorten the construction period to reduce carbon. However, the long-term ESG goals can significantly contribute to environmental protection through SPPS production of clean energy in the future. This study employs a fuzzy nonlinear multi-objective model to facilitate the construction of SPPSs, thereby allowing us to understand inherent uncertainties and constraints. It ultimately offers solutions for decision-makers at logistics centers to navigate uncertain environments. In the context of the ESG principles, the research methodology is a tool for shortening the decision analysis time. This study strategically utilizes these short-term (project completion period) tools to realize the long-term (clean energy contract execution period 20 years with Taiwan Power Company) ESG goals.

2. Literature Review

Due to Taiwan’s impending carbon tax in 2025, numerous companies have strategized establishing SPPSs in their in-house logistics centers to make carbon neutrality. This allows companies to sell clean energy for profit and enhances the image of the logistics centers regarding ESG goals. How can these logistics centers effectively employ project compression management techniques to shorten construction time when building stations to reduce carbon emissions?

Monghasemi et al. [9] proposed a multi-criteria decision-making model that optimizes TCQ trade-offs in construction projects. This approach promotes an economy that flourishes while protecting the environment.

Moreover, Ittmann [10] highlighted that integrating a solar photovoltaic power generation system on logistics center rooftops significantly reduces environmental impact, thereby boosting competitiveness. Investing in SPPSs signifies active participation in environmental and social initiatives, which improves a company’s ESG image. Carter and Rogers [11] suggest that solar photovoltaic power secured long-term economic benefits through ecological conservation and social progress, ultimately providing a competitive advantage.

However, it is known from the literature [9,10,11] that green electricity from solar photovoltaic power stations can reduce environmental impact. Moreover, the TCQ trade-off problem in the project is optimized to obtain the most suitable solution. However, it did not point out the essence and far-reaching significance of the problem. The shortening of the construction time in TCQ is mainly to reduce carbon emissions and achieve an ESG goal. Therefore, shortening the construction time on each item belongs to the sustainability content of ESG. This study focuses on the failure of scholars to shorten the construction time to establish ESG goals.

Building on previous discussions, scholars have paid attention to solutions to construction problems. Singh [12] addressed the multi-objective project scheduling issue under resource constraints, employing rule prioritization and the analytical hierarchy process (AHP) method. Birjandi and Mousavi [13] examined the multi-route resource-constrained project scheduling problem (RCPSP) in construction projects. Their article proposed a fuzzy mixed-integer nonlinear programming (MINLP) model under uncertainty to minimize project costs. Kannimuthu et al. [14] researched optimizing TCQ in multi-mode resource-constrained scheduling to utilize binary integer programming models (in binary problems, each variable can only take on 0 or 1; this may represent selecting or rejecting an option or turning on or off switches, and the objective function has form minimized $\mathrm{Z}={\sum }_{j=1}^n{c}_j{x}_j$), perform multi-objective optimization, and identify Pareto optimal solutions. The results showed that costs can be reduced by increasing the construction period, and the quality can only be improved by rising costs. Ballesteros-Perez et al. [15] proposed nonlinear theoretical models assuming collaborative or non-collaborative resources. Their article used a genetic algorithm (GA) on an application example. The results solve discrete, continuous, deterministic, and stochastic situations. Afruzi, Aghaie, and Najafi [16] conducted a study on the robust resource-constrained multi-project scheduling problem (RRCMPSP). They utilized a scenario relaxation algorithm to derive the optimal solution for the RRCMPSP, focusing on maximizing the weighted difference between the project’s completion time and its assigned deadline.

The above papers show that many scholars used different methods to resolve construction problems. However, in the literature [14,15,16], among the various solution methods, the articles did not show that these problems often existed in uncertain environments. There was a frequent need for fuzzy regarding the environmental coefficients and decision parameters in project management decisions. Bellman and Ladeh [17] introduced a fuzzy decision-making method for fuzzy problems. Fuzzy decision-making represents the intersection of goals and constraints. The point in space where the membership function of fuzzy decision-making reaches its maximum value was defined as maximizing decision-making. Hence, fuzzy theory can construct fuzzy decision-making in fuzzy environments.

Eydi, Farughi, and Abdi [18] studied the balance between time, cost, and quality in projects that have been conducted. Methods have been proposed to reduce project duration and expenses while enhancing quality. By applying models and a hybrid approach that combines the fuzzy AHP strategy and the VIKOR method, both multi-criteria decision-making methods and non-dominated solutions have been identified. This hybrid approach can significantly assist in choosing the most suitable solution.

Akrami et al. [19] researched goal programming for the project TCQ trade-off. The article proposed a grey model to approximate the activity mode’s TCQ parameters to address the problem. The results of this model offer a framework for decision-makers to achieve an acceptable time frame with minimal cost and loss of quality. Thapar, Singh, and Pandey [20] resolved a polynomial geometric optimization problem using max–min fuzzy relational equations (FRE). After solving optimization problems, a single optimal solution was determined. Deep et al. [21] proposed an interactive approach-based method for solving multi-objective optimization problems. The proposed method provided a solution for linear and nonlinear multi-objective optimization problems modeled in a fuzzy or crisp environment. The proposed method considers constraints at a different α-cut (α ϵ [0, 1] to both left and right reference functions of $\mathsf{\alpha }-\mathrm{c}\mathrm{u}\mathrm{t}=\frac{x-a}{b-a} and \frac{x-b}{c-b}$) of the fuzzy parameter. Li [22] proposed a multi-objective train scheduling model that incorporated fuzziness through linear and nonlinear fuzzy membership functions to reduce energy costs, carbon emissions, and total passenger time.

From the literature [18,19,20,21,22] in the previous paragraph, it is evident that the choice of method is closely tied to the nature of the problem. Different problems necessitate different solutions. Fuzzy linear multi-objective decision-making method: This method is employed for issues that involve multiple objectives and uncertainty. Fuzzy AHP: This is used for decision-making problems with multiple criteria. TOPSIS and VIKOR: When items must be ranked relatively quickly. Gray model: This model is utilized to minimize the amount of switching, thereby improving the reliability of the switching system. Maximum and minimum fuzzy relational equations: These equations play a crucial role in solving problems related to fuzzy relational equations or inequality systems. Fuzzy α-cut: This method is suitable for analyzing fuzzy issues and can be used for sorting.

The problem under study in this context exhibits fuzzy uncertainty and is nonlinear. It also involves multi-criteria decision-making and fuzzy values must be sorted. Given the many demands, this study primarily employs the fuzzy nonlinear multi-objective decision-making method. Next, suitable for analyzing fuzzy problems and sorting fuzzy values, α-cut is used in the defuzzification method that best adapts to this situation.

Hashemi and Mousavi [23] explored project management processes to meet objectives, introducing a novel mathematical model that minimizes total cost and completion time while maximizing project management decision quality. A linearization technique was presented, focusing on variable change and piecewise linearization, transforming the nonlinear function into a linear programming model and representing fuzzy set theory and fuzzy mathematical programming to accommodate parameters and variables under uncertainty situations. The model resolves conflict in a fully fuzzy time–cost–quality project management model. Furtado and Sola [24] used the MCDM (multi-criteria decision-making) method and the fuzzy COPRAS (complex proportional assessment) method to solve the problems regarding selecting SPPS sites with conflicting energy project standards. The fuzzy COPRAS method already solves location problems and deals with uncertainty, complexity, and ambiguity problems. Miraj and Berawi [25] chose the best alternative for photovoltaic systems, the selection of which remains a complex problem. Their study proposed MCDM, considering the best–worst model (BWM) and VIKOR (MCDM analysis method) to find suitable photovoltaic alternatives. The result showed that the best scenario was a complete photovoltaic installation into the existing system. However, policymakers favored hybrid options due to their low power generation, with non-renewable energy as the primary energy source. Farsijani and Moradi [26] studied risk control and risk assessment in the electricity market. In Iran’s fuzzy environment, high-risk factors were used by the grey ANP (analytic network process) method. Ultimately, they used the three life cycle stages to examine solar power, demonstrating an increase in the profitability of the renewable energy cycle. Malemnganbi and Shimray [27] conducted a study on selecting optimal solar power plant (SPP) sites. The article presents a detailed analysis of the optimal ranking of SPP sites using the analytical hierarchy process (AHP) of MCDM, a multiple-layer perceptron neural network trained with the backpropagation (MLP-BP) algorithm, and a genetic algorithm (MLP-GA). The study considered three SPP sites in India, demonstrating that the MLP-GA outperformed the MLP-BP and AHP. The MLP-GA could rank the power plant sites precisely. It found that the MLP neural network trained by the GA exhibited superior efficiency in accurately classifying and identifying potential areas for installing solar power stations.

The literature mentioned above [23,24,25,26,27] shows that the primary focus of building SPPS is site selection, with particular consideration for factors such as sunshine duration and energy quality. This is the pursuit of maximizing clean energy output. The challenge of this study is to minimize SPPS’s construction time while maintaining high quality and low cost to reduce the carbon emissions of the construction process. This study complements the ESG’s carbon emission issues of existing SPPS construction.

Beyond that, the short-term goal is to address multi-objective decision-making for construction. The long-term goal is to build SPPSs on the logistics center’s rooftops to solve the impact of traditional power generation models on climate and pollution, increase clean energy power generation, and promote environmental sustainability to achieve the ESG mission.

Wang et al. [28] used building roof data, optimal tilt angle, maximum solar radiation calculation, and GIS to estimate the potential of photovoltaics on old residential rooftops in Nanjing. They found these could meet 17.7–20% of residential electricity demand under three photovoltaic performance ratios (PR) scenarios. The carbon reduction potential of rooftop photovoltaics during their lifecycle reached 13,912,874.12 t (PR = 0.85), 13,094,469.76 t (PR = 0.8), and 12,276,065.4 t (PR = 0.75). However, the economic potential result showed that rooftop photovoltaics could not produce economic benefits when the NPV value was less than 0.

Lee et al. [29] studied zero-energy building operations (ZEBO) for ESG goals. The ZEBO policy of the solar power generation system is formulated considering environmental impact and social relations. In contrast, the formulation of the solar power generation system is based on an ESG operating strategy with the execution of data-driven power generation forecasts.

Durgapal [30] showed that climate change and pollution have created a new normal for natural disasters. India’s adoption of solar photovoltaic systems is a key focus for ESG goals. India has the potential to significantly reduce carbon emissions without compromising its economic growth. India’s solar energy target is 100 GW, including 40 GW for rooftop installations. Each additional megawatt of solar energy is equivalent to planting 49,000 teak trees, saving 31,000 tons of CO₂.

Toba et al. [31] reported that renewable energy technologies are harnessed to meet energy needs, achieve societal objectives, and reach climate change goals. The adoption of solar photovoltaic (PV) technology for ESG goals is being considered on a large scale in Southeast and East Asia. Their study suggests that integrating ESG goals into business strategies is feasible and can foster business expansion and sustainable development.

Liao et al. [32] developed a recycling strategy for end-of-life photovoltaic modules, creating silicon–carbon composite anode materials. The W–Si-rM@G material, used as a lithium-ion battery anode, showed an initial discharge capacity of 1770 mA h g⁻¹ and maintained 913 mA h g⁻¹ after 200 cycles. The economic analysis confirmed the feasibility of this approach.

Most of the ESG literature [28,29,30,31,32] on solar power generation focuses on selecting regions for developing solar power generation, the carbon reduction benefits of solar power generation, the development of zero-carbon buildings, or the recycling of materials after solar power generation. No literature discussed shortening the construction of SPPSs to reduce carbon emissions and achieve ESG goals.

Therefore, based on the findings of the above literature, multi-objective programming and fuzzy sets (including defuzzification) can be used to provide resolutions for uncertain nonlinear problems. This paper constructs the following research methodology to solve uncertain nonlinear issues for shortening the construction of SPPSs to reduce carbon emissions and increase clean energy. It is also an excellent way to develop sustainable renewable energy.

3. Methodology

3.1. Case Assumptions

Based on the provided context, there is a conflict within the time–cost–quality (TCQ) triangle in construction projects. The research hypotheses could be formulated as follows:

Hypothesis 1.

A decrease in construction time leads to increased costs.

Hypothesis 2.

A decrease in cost results in an increase in construction time.

Hypothesis 3.

An improvement in quality leads to increased construction time.

The foundations of the hypotheses are that shortening the construction time can reduce carbon emissions, and more solar photovoltaic stations can be built within a fixed period when the construction time is shortened, thereby increasing clean energy.

3.2. Case Methodology

This study formulated a mathematical nonlinear calculation model that minimizes the project’s direct costs, considering that a shorter activity duration results in higher costs. The subsequent step involves integrating the nonlinear mathematical model with fuzzy numbers and employing a defuzzification method to evaluate the project’s decision-making direction. Before implementing the fuzzy nonlinear multi-objective project, it is essential to determine the longest construction project path using the CPM mode to understand the implications of shortening the construction period. The following is the fuzzy nonlinear multi-objective model process:

$$\begin{gathered} Min{S}_{finish} \\ \left(\mathrm{This}\mathrm{minimizes}\mathrm{the}\mathrm{scheduling}\mathrm{time}\mathrm{of}\mathrm{the}\mathrm{crash}\mathrm{status}\mathrm{of}\mathrm{items}\right). \end{gathered}$$

(1)

$$\begin{gathered} Min\sum _{i=1}^m{C}_i \\ (\mathrm{This}\mathrm{minimizes}\mathrm{the}\mathrm{scheduling}\mathrm{cost}\mathrm{of}\mathrm{the}\mathrm{crash}\mathrm{status}\mathrm{of}\mathrm{items}). \end{gathered}$$

(2)

$$\begin{gathered} Max\frac{{\sum }_{i=1}^m{Q}_i}{n} \\ (\mathrm{This}\mathrm{maximizes}\mathrm{the}\mathrm{scheduling}\mathrm{quality}\mathrm{of}\mathrm{the}\mathrm{crash}\mathrm{status}\mathrm{of}\mathrm{items}). \end{gathered}$$

(3)

The restrictions are:

$$\begin{gathered} s_i+t_i+{Lag}_{i-(i+1)}\le s_{i+1} \\ \left(\mathrm{This}\mathrm{is}\mathrm{the}\mathrm{finish-to-start}\mathrm{relationship}\mathrm{between}\mathrm{item}i\mathrm{and}\mathrm{item}j\right). \end{gathered}$$

(4)

$$\begin{gathered} s_i+{Lag}_{i-(i+1)}\le s_{i+1} \\ \left(\mathrm{This}\mathrm{is}\mathrm{the}\mathrm{start-to-start}\mathrm{relationship}\mathrm{between}\mathrm{item}i\mathrm{and}\mathrm{item}j\right). \end{gathered}$$

(5)

$$\begin{gathered} s_i+{Lag}_{ij}\le s_j+t_j \\ \left(\mathrm{This}\mathrm{is}\mathrm{the}\mathrm{start-to-finish}\mathrm{relationship}\mathrm{between}\mathrm{item}i\mathrm{and}\mathrm{item}j\right). \end{gathered}$$

(6)

$$\begin{gathered} s_i+t_i+{Lag}_{ij}\le s_j+t_j \\ \left(\mathrm{This}\mathrm{is}\mathrm{the}\mathrm{finish-to-finish}\mathrm{relationship}\mathrm{between}\mathrm{item}i\mathrm{and}\mathrm{item}j\right). \end{gathered}$$

(7)

$$\begin{gathered} s_i\ge 0 \\ \left(\mathrm{This}\mathrm{is}\mathrm{the}\mathrm{non-negative}\mathrm{bounds}\mathrm{on}\mathrm{the}\mathrm{start}\mathrm{time}\mathrm{of}\mathrm{each}\mathrm{item}\right). \end{gathered}$$

(8)

$$\begin{gathered} t_i\in \mathbb{R} \\ \left(\mathrm{The}\mathrm{start}\mathrm{time}\mathrm{of}\mathrm{each}\mathrm{item}\mathrm{is}\mathrm{a}\mathrm{positive}\mathrm{integer}\right). \end{gathered}$$

(9)

$$t_{crash,i}\le t_i\le t_{normal,i}$$

(10)

$$q_{crash,i}\le q_i\le 100\%$$

(11)

$$c_{normal, i}\le c_i\le c_{crash, i}$$

(12)

The explanation for the meaning of each equation follows.

To minimize the scheduling time and cost of the crash status of engineering items, according to the item cost function proposed by Ghodsi et al. [33], the sum of each item’s time cost ($C_{T,i}$), direct cost $\left(C_i\right)$, and quality cost ($C_{Q,i}$) is calculated. Diao et al. [34] first proposed the trade-off problem of combining time cost with quality. The functional relationship between time and cost transforms into a curve function relationship. The mathematical model proposed by Zimmermann [35] makes ${\mathrm{a}}_i$ the one with a definite value, and the time cost ${(C}_{T,i}$) is thus determined by the following calculation:

$$C_{T,i}={\mathrm{a}}_i\times (t_i{)}^2+b_i$$

(13)

$${\mathrm{a}}_i=\frac{c_{crash,i}-c_{normal,i}}{t_{crash,i}^2-t_{normal,i}^2}$$

(14)

$$b_i=\frac{{c_{normal,i}\times t}_{crash,i}^2-c_{crash,i}{\times t}_{normal,i}^2}{t_{crash,i}^2-t_{normal,i}^2}$$

(15)

$$Q_{T,i}=q_{normal,i}+\frac{q_{normal,i}-q_{crash,i}}{t_{normal,i}-t_{crash,i}}\times \left(t_i-t_{normal,i}\right)$$

(16)

$$C_{Q,i}=\left. [(\frac{∆C_{Q_{normal, i}-}∆C_{Q_{crash,i}}}{t_{normal, i}-t_{crash, i }})\times \left(t_i-t_{crash,i}\right)+∆C_{Q_{normal,i}}\right]\times \left(q_{i-}{Q}_{T,i}\right)$$

(17)

The definitions for the above symbols are as follows:

• $i$ is the number of each item in the project, i = A, B, …, m;
• $m$ is the total number of items in the project;
• $s_i$ is the start time of each item in the project;
• $t_i$ is the duration of each item in the project;
• ${Lag}_{i-(i+1)}$ is the necessary time interval between the project’s two items, i and (i +1);
• $q_i$ is the quality ratio of each item in the project;
• $c_i$ is the cost amount of each item in the project;
• $t_{normal,i}$ is the duration time of each item in the normal state of the project;
• $t_{crash,i}$ is the duration of each item in the crash status of the project;
• $c_{normal, i}$ is the direct cost of the normal working status of each item in the project;
• $c_{crash, i}$ is the direct cost of the crash work status of each item in the project;
• $q_{normal,i}$ is the quality ratio of each item in the normal state of the project;
• $q_{crash,i}$ is the quality ratio of the crash status of each item in the project;
• $C_{Q,i}$ calculates the cost of quality ratio for each item in the project;
• $Q_{T,i}$ calculates the quality of time ratio for each item in the project;
• $∆C_{Q_{normal, i}}$ is the unit cost of the quality ratio variation of each item in the normal state;
• $∆C_{Q_{crash, i}}$ is the unit cost of the quality ratio variation of each item in the crash state.

The defuzzification methods encompass the following approaches: (1) utilizing the center of gravity method to locate the center of gravity within the shadow area resulting from inference; (2) employing the height method to establish the height of the consequent part based on the fitness of the antecedent part of the rule (in contrast, the center of gravity method necessitates integration for resolution); (3) adopting α-cut defuzzification, which also requires integration for resolution (however, it facilitates the ranking of solutions when dealing with fuzzy numbers; hence, the α-cut defuzzification method is advantageous to this study’s required solution ranking framework) [36].

The α-cut method is used to sort fuzzy numbers [37] (a fuzzy number extends a regular natural number by a range of values with individual numbers between 0 and 1, known as the membership function) [38]. In this method, let $a_1, a_2,\dots ,a_i$ be I fuzzy numbers, and the left and right membership functions of the fuzzy number $a_i$ are $f^{a_{iL}}$ and $f^{a_{iR}}$; suppose that $g^{a_{iL}}$ and $g^{a_{iR}}$ are the inverse functions of $f^{a_{iL}}$ and $f^{a_{iR}}$, respectively; and define the left integral value $I^L\left(a_i\right)$ and right integral value $I^L\left(a_i\right)$ of $a_i$ (Equations (19) and (20)) [39]. The mathematical manipulations involving two positive fuzzy numbers A and B can be represented as fuzzy addition: (A⊕B)^α = [A_l^α + B_l^α, A_u^α + B_u^α]; fuzzy subtraction: (AΘB)^α = [A_l^α − B_u^α, A_u^α − B_l^α]; fuzzy multiplication: (A$\otimes$B)^α = [A_l^αB_l^α, A_u^αB_u^α]; and fuzzy division:(AØB)^α = [A_l^α/B_u^α, A_u^α/B_l^α] [40]:

$$(a_i{)}_{\alpha }=\left. [(a_i{)}_{\alpha }^L, (a_i{)}_{\alpha }^U\right]$$

(18)

$$I^L\left(a_i\right)=\frac{1}{2}\underset{k\to \mathrm{\infty }}{\lim }\left\{\sum _{j=1}^k\left. [{g_{ai}}^L\left({\alpha }_j\right)+{g_{ai}}^L\left({\alpha }_{j-1}\right)\right]∆{\alpha }_j\right\}$$

(19)

$$I^R\left(a_i\right)=\frac{1}{2}\underset{k\to \mathrm{\infty }}{\lim }\left\{\sum _{j=1}^k\left. [{g_{ai}}^R\left({\alpha }_j\right)+{g_{ai}}^R\left({\alpha }_{j-1}\right)\right]∆{\alpha }_j\right\}$$

(20)

In this research, the proposed model offers a comprehensive analysis for scheduling planning considering project deadlines. The model considers the execution stages to achieve a balance of each item’s TCQ content and aims to construct an optimal compressed schedule plan while estimating the project cost. The objectives of the model encompass “minimizing completion time”, “minimizing cost”, and “maximizing quality ratio”.

In summary, numerous studies suggest that each method is apt for problem-solving. This study amalgamates the accuracy methods proposed by Zimmermann [35], Yager [38], Liou, and Wang [39] with Dong [40] to address nonlinear trade-off problems. This consolidated research approach enables decision-makers in the company’s logistics center to make quick decisions and understand the disparity between the minimum and maximum values.

4. Sample Problem and Results

This article uses various functions to establish relationships among these features by considering the decision maker’s risk attitude. It applies the α-cut method to transform the fuzzy project duration and direct costs into decision values.

4.1. Case Introduction

This study follows the case of Ghodsi et al. [33] to conduct analysis. Data (as shown in

Table 1. Case data.

Item	Prerequisites	Time (Days)		Quality (%)		Cost (TWD)		Variable Cost per Unit of Quality (TWD/%)
		Normal	Crash	Normal	Crash	Normal	Crash	Normal	Crash
A		21.3	18.5	82	46	359,000	365,000	1600	7060
B		23.3	11.4	79	48	390,000	408,000	1010	3850
C	A	7.6	5.7	85	48	96,000	97,000	460	1970
D	B	23.3	5	76	37	349,000	392,000	1110	4720
E	B	17.7	7.8	80	45	273,000	293,000	680	2640
F	C	7	5.9	79	31	81,000	82,000	210	840
G	D, E	10.6	7.8	83	47	117,000	119,000	440	1580
H	D, E	16	13.9	83	26	238,000	240,000	780	2770
I	F	24.2	15.2	77	34	461,000	475,000	1350	5440
J	G, H	24.3	3.2	80	27	300,000	326,000	1450	6190
K	I, J	8.2	6.4	80	28	128,000	130,000	470	2090
L	J	24.5	20.8	82	50	296,000	300,000	1050	3810

) were computed using Lingo 20, Excel, and Python fuzzy programs. This study utilized Lingo 20 to assist in computing the solution. Then, input the calculated results into Excel and the Excel execution results into the Python fuzzy program to calculate.

The network and operation time are used to find the start time point and operation time of each operation from front to back. For example, the relationship between items A and C is finish–start. The time constraint equation is $S_A+t_A\le S_c$, the start time of item A is $S_A=0$, and the operation time is $t_A=21.3$ days. That is, the start time $S_C$ of item C must be greater than or equal to 21.3 days, as shown in

Table 2. The updated start time for item C.

Item	A	B	C	D	E	F	G	H	I	J	K	L
Prerequisites			A	B	B	C	D, E	D, E	F	G, H	I, J	J
Start time $S_i$	0	0	* 21.3	0	0	0	0	0	0	0	0	0
Time $t_i$	21.3	23.2	7.6	23.2	17.7	7	10.6	1.60	24.2	24.3	8.2	24.5

The schedule for each operation in the project and the cost are shown in Table 3. Furthermore, it identifies the project path as B → D → H → J → L (as indicated in *).

.

The CPM is according to the preoperational items in the above tables, as shown in Figure 1.

The path items of the project are organized, as shown in

Table 3. Project data clarifying the normal schedule of the environment.

Item	Prerequisites	Start Time ${\mathit{S}}_{\mathit{i}}$	Time (Days) ${\mathit{t}}_{\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l},\mathit{i}}$	Quality (%) ${\mathit{q}}_{\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l},\mathit{i}}$	Cost (TWD) ${\mathit{c}}_{\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l},\mathit{i}}$
A		0	21.3	97	359,000
* B		0	* 23.2	94	390,000
C	A	21.3	7.6	100	96,000
* D	* B	23.2	* 23.2	91	349,000
E	B	23.2	17.7	95	273,000
F	C	28.9	7	94	81,000
G	D, E	46.5	10.6	98	117,000
* H	* D, E	46.5	* 16.0	98	238,000
I	F	35.9	24.2	92	461,000
* J	* G, H	62.5	* 24.3	95	300,000
K	I, J	86.8	8.2	95	128,000
* L	* J	86.8	* 24.5	97	296,000

Note: The * was the project path.

(presented as in *), and in

Table 4. The calculation of each item’s crash time, unit time cost slope, and unit time quality slope.

Item	Time (Days)		Quality (%)		Cost (TWD)		Crash Time (Days) (7) = (1) − (2)	Unit Time Cost (TWD/Day) [((5) − (6))/(7)]	Unit Time Quality (%/Day) [((3) − (4))/(7)]
	Normal (1)	Crash (2)	Normal (3)	Crash (4)	Normal (5)	Crash (6)
A	21.3	18.5	97	86	359,000	365,000	2.8	(2142.86)	3.93
* B	* 23.2	11.4	94	88	390,000	408,000	11.8	(1525.42)	0.51
C	7.6	5.7	100	88	96,000	97,000	1.9	(526.32)	6.32
* D	* 23.2	5	91	77	349,000	392,000	18.2	(2362.64)	0.77
E	17.7	7.8	95	85	273,000	293,000	9.9	(2020.20)	1.01
F	7	5.9	94	71	81,000	82,000	1.1	(909.09)	20.91
G	10.6	7.8	98	87	117,000	119,000	2.8	(714.29)	3.93
* H	* 16	* 13.9	98	66	238,000	240,000	* 2.1	(952.38)	15.24
I	24.2	15.2	92	74	461,000	475,000	9	(1555.56)	2.00
* J	* 24.3	* 3.2	95	57	300,000	326,000	21.1	(1232.23)	1.80
K	8.2	6.4	95	68	128,000	130,000	1.8	(1111.11)	15.00
* L	* 24.5	* 20.8	97	90	296,000	300,000	3.7	(1081.08)	1.89

, each item’s crash time, unit cost slope, and unit quality slope are calculated. The critical path items are B, D, H, J, and L—called item H to name Project HP because the unit time cost is the smallest. The number of days available for crashing is 2.1 days (presented as in *). The total time is 111.2 days (23.2 + 23.2 + 16 + 24.3 + 24.5), the average quality is 95% ((94 + 91 + 98 + 95 + 97)/5), and the total cost is TWD 1,573,000 (390,000 + 349,000 + 238,000 + 300,000 + 296,000) (presented as in *).

4.2. A Fuzzy Nonlinear Multi-Objective Method for TCQ Decision-Making under Uncertain Crash States

Zimmermann [35] demonstrated that for the time cost slope, the membership function is ${\mu }_{{\tilde{a}}_l}\left(c_i\right)$, and the fuzzy set of $\widetilde{{\mathrm{a}}_l}$ is defined as $\left(\stackrel{~}{{{\mathrm{c}}_l}_{\alpha }}\right)=\left\{c_i|{\mu }_{\widetilde{c_l}}\left(c_i\right)\ge \alpha \right\}$. This interval contains all ${\left(\widetilde{c_l}\right)}_{\alpha }=\left. [(c_i{)}_{\alpha }^L, (c_i{)}_{\alpha }^U\right]$ values under the possibility $,{\mathrm{a}}_i$.

This model assumes that the fuzzy unit time cost slope $\widetilde{{\mathrm{a}}_l}$ of each operation item ${\left(\widetilde{c_l}\right)}_{\alpha }=\left. [(c_i{)}_{\alpha }^L,(c_i{)}_{\alpha }^M, (c_i{)}_{\alpha }^U\right]$ is a triangular fuzzy number, such as the fuzzy number ${\left(\widetilde{c_A}\right)}_{\alpha }=\left. [(c_{HP}{)}_{\alpha }^L,(c_{HP}{)}_{\alpha }^M, {(c}_{HP}{)}_{\alpha }^U\right]$ of Project HP. The $(c_{HP}{)}_{\alpha }^M$ is average and assumes that the upper limit of the cost slope is an increase of 10%, and the lower limit is a decrease of 10%. The following are the solution steps:

• Step 1: According to Table 3, find the main path (Project HP) on the network map.

In Table 3, the critical path items are B, D, H, J, and L. However, the project used item H to name this Project HP.

• Step 2: Calculate the shortened time of each job: shortened time = normal time ($t_{normal,i})$—crash time ($t_{crash,i}$) (as shown in Table 4).
• Step 3: Find the normal cost ($c_{normal,i}$) and crash cost ($c_{crash,i}$) of each job and calculate the cost slope (${∆\mathrm{C}}_i$). The methodology establishes the increased cost variation for each unit time shortened.
• Step 4: Next, establish different degrees of uncertainty attitudes, and using α-cut = 0, 0.2, 0.4, 0.6, 0.8 to 1 of defuzzification, let fuzzy numbers ${\left(\widetilde{c_{HP}}\right)}_{\alpha }$ be definite numbers, as shown in

  Table 5. The uncertainty triangular fuzzy numbers of nonlinear variation of the crashing unit of Project HP.

  Item	Time (Days)				Quality (%)				Cost (TWD)
  	$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{L}}$	$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{M}}$		$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{U}}$	$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{L}}$		$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{M}}$	$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{U}}$	$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{L}}$		$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{M}}$		$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{U}}$
  B	15.57	17.3		19.03	82		91	100	359,100		399,000		438,900
  D	12.69	14.1		15.51	76		84	92	333,450		370,500		407,550
  H	13.46	14.95		16.445	74		82	90	** 215,100		** 239,000		** 262,900
  J	* 12.38	* 13.75		* 15.125	* 68		* 76	* 84	281,700		313,000		344,300
  L	20.39	22.65		24.915	84		94	103	268,200		298,000		327,800
  Item	Unit time cost							Unit time quality
  	$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{L}}$		$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{M}}$			$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{U}}$		$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{L}}$		$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{M}}$		$({\mathit{c}}_{\mathit{i}}{)}_{\mathit{\alpha }}^{\mathit{U}}$
  B	−1373		−1525.42			−1678		0.459		0.51		0.561
  D	−2126		−2362.64			−2599		0.693		0.77		0.847
  H	−857		−952.38			−1048		** 13.716		** 15.24		** 16.764
  J	* −1109		* −1232.23			* −1355		1.62		1.8		1.98
  L	−973		−1081.08			−1189		1.701		1.89		2.079

  Note: The * represented item J’s construction time, which was minimal, and the quality was the lowest. The ** showed item H, which had the lowest cost.

  .

Table 6. The uncertainty triangular fuzzy numbers on defuzzification in shortened time = normal time ($t_{normal,i})$—crash time ($t_{crash,i}$).

α	Item
	B	D	H	J	L
0	[15.57, 19.03]	[12.69, 15.51]	[13.455, 16.445]	[12.375, 15.125]	[20.385, 24.915]
0.2	[15.916, 18.684]	[12.972, 15.228]	[13.754, 16.146]	[12.65, 14.85]	[20.838, 24.462]
0.4	[16.262, 18.338]	[13.254, 15.847]	[14.053, 15.847]	[12.925, 14.575]	[21.291, 24.009]
0.6	[16.608, 17.992]	[14.352, 14.664]	[14.352, 15.548]	[13.2, 14.3]	[21.744, 23.556]
0.8	[16.954, 17.646]	[13.818, 14.382]	[14.651, 15.249]	[13.475, 14.025]	[22.197, 23.103]
1	[17.3, 17.3]	[14.1, 14.1]	[14.95, 14.95]	[13.75, 13.75]	[22.65, 22.65]
Ranking	34.6	28.2	29.9	* 27.5	45.3

shows that the defuzzification ranking is J = 27.5, D = 28.2, H = 29.9, B = 34.6, and L = 45.3. The numerical ranking does not indicate a reduction in construction days. Instead, it illustrates item J, where the reduction in days is minimal, and the unit time cost is comparatively low. In item L, under the defuzzification value, this reduction is the largest, signifying that even in a crash state, the number of crash days remains the highest, and the unit time cost remains the largest (as shown in Table 5, denoted by *).

In the defuzzification ranking (

Table 7. The uncertainty triangular fuzzy numbers on defuzzification in quality ranking values.

α	Item
	B	D	H	J	L
0	[81.9, 100.1]	[75.6, 92.4]	[73.8, 90.2]	[68.4, 83.6]	[84.15, 102.85]
0.2	[83.72, 98.28]	[77.28, 90.72]	[75.44, 88.56]	[69.92, 82.08]	[86.02, 100.98]
0.4	[85.54, 96.46]	[78.96, 89.04]	[77.08, 86.92]	[71.44, 80.56]	[87.89, 99.11]
0.6	[87.36, 94.64]	[80.64, 87.36]	[78.72, 85.28]	[72.96, 79.04]	[89.76, 97.24]
0.8	[89.18, 92.82]	[82.32, 85.68]	[80.36, 83.64]	[74.48, 77.52]	[91.63, 95.37]
1	[91, 91]	[84, 84]	[82, 82]	[76, 76]	[93.5, 93.5]
Ranking	182	168	164	* 152	187

Note: The * showed item J’s unit time cost had the most significant decrease.

), the quality ranking values are J = 152, D = 168, H = 164, B = 182, and L = 187, which means that the quality of item J is the lowest and needs to be strengthened. Nevertheless, for item J, in terms of the unit time cost, the unit time quality is not the most expensive (as shown in Table 6, represented by *).

In the ranking in

Table 8. The uncertainty triangular fuzzy numbers on defuzzification in cost ranking values.

α	Item
	B	D	H	J	L
0	[359,100, 438,900]	[333,450, 407,550]	[215,100, 262,900]	[281,700, 344,300]	[268,200, 327,800]
0.2	[367,080, 430,920]	[340,860, 400,140]	[219,880, 258,120]	[287,960, 338,040]	[274,160, 321,840]
0.4	[375,060, 422,940]	[348,270, 392,730]	[224,660, 253,340]	[294,220, 331,780]	[280,120, 315,880]
0.6	[383,040, 414,960]	[355,680, 385,320]	[229,440, 248,560]	[300,480, 325,520]	[286,080, 309,920]
0.8	[391,020, 406,980]	[363,090, 377,910]	[234,220, 243,780]	[306,740, 319,260]	[292,040, 303,960]
1	[399,000, 399,000]	[370,500, 370,500]	[239,000, 239,000]	[313,000, 313,000]	[298,000, 298,000]
Ranking	798,000	741,000	* 478,000	626,000	596,000

, item H has the lowest value, which means it has the lowest cost (shown in Table 8, indicated by *).

According to the unit time cost, items B, D, H, and J produce all negative values. When readers only look at the numerical value, it is gradually declining, indicating that the cost per unit time among various items has a downward trend. The primary purpose is to improve crash work efficiency and minimize losses. Item H has the fastest crash efficiency, but the negative slope of item L is increasing. This means the reason for the crash work has made the efficiency worse during item L (as shown in

Table 9. The uncertainty triangular fuzzy numbers on defuzzification in the unit time cost.

α	Item
	B	D	H	J	L
0	[−1373, −1678]	[−2126, −2599]	[−857, −1048]	[−1109, −1355]	[−973, −1189]
0.2	[−1403.484, −1647.484]	[−2173.328, −2551.728]	[−876.076, −1028.876]	[−1133.646, −1330.446]	[−994.616, −1167.416]
0.4	[−1433.968, −1616.968]	[−2220.656, −2504.456]	[−895.152, −1009.752]	[−1158.292, −1305.892]	[−1016.232, −1145.832]
0.6	[−1464.452, −1586.452]	[−2267.984, −2457.184]	[−914.228, −990.628]	[−1182.938, −1281.338]	[−1037.848, −1124.248]
0.8	[−1494.936, −1555.936]	[−2315.312, −2409.912]	[−933.304, −971.504]	[−1207.584, −1256.784]	[−1059.464, −1102.664]
1	[−1525.42, −1525.42]	[−2362.64, −2362.64]	[−952.38, −952.38]	[−1232.23, −1232.23]	[−1081.08, −1081.08]
Ranking	−2959.388	−3142.388	−4583.296	* −4867.096	−1847.532

Note: The * showed item J’s unit time cost had the most significant decrease.

).

Table 10. The uncertainty triangular fuzzy numbers on defuzzification in unit time quality.

α	Item
	B	D	H	J	L
0	[0.459, 0.561]	[0.693, 0.847]	[13.716, 16.764]	[1.62, 1.98]	[1.701, 2.079]
0.2	[0.561, 0.5508]	[0.7084, 0.8316]	[14.0208, 16.4592]	[1.656, 1.944]	[1.7388, 2.0412]
0.4	[0.4794, 0.5406]	[0.7238, 2.0034]	[14.3256, 16.1544]	[1.692, 1.908]	[1.7766, 2.0034]
0.6	[0.4896, 0.5304]	[0.7392, 0.8008]	[14.6304, 15.8496]	[1.728, 1.872]	[1.8144, 1.9656]
0.8	[0.4998, 0.5202]	[0.7546, 0.7854]	[14.9352, 15.5448]	[1.764, 1.836]	[1.8522, 1.9278]
1	[0.51, 0.51]	[0.77, 0.77]	[15.24, 15.24]	[1.8, 1.8]	[1.89, 1.89]
Ranking	1.02	1.54	* 30.48	3.6	3.78

Note: The * represented item H had a significant quality improvement.

shows the slope value of unit time quality. These values represent the rate of quality variation at different points in time. A positive slope indicates an increase in quality. The quality index of items B to D increased slightly. Item H represents a significant improvement in quality and a substantial increase in the quality index. This means that the critical issues of item H may be solved when the item crashes, resulting in a rapid quality improvement (as shown in Table 10).

For item J of Project PH, the unit time cost is sorted by α-cut, which can save more costs. Its unit time cost value is (−1109, −1232.23, −1355). The corresponding values in the α-cut order of time are (12.38, 13.75, 15.125), which also show the minimum value. However, for quality requirements lower than the minimum level of 85%, an additional TWD (4549.90, 15,416.65, 26,283.41) will be required to increase quality to 85%:

$$\begin{gathered} Q_{T,J}=q_{normal,J}+\frac{q_{normal,J}-q_{crash,J}}{t_{normal,J}-t_{crash,J}}\times \left(t_J-t_{normal,J}\right) \\ =(68,76,84)+\frac{(68,76,84)-57}{24.3-11.4}\times \left(24.2-24.3\right) \\ =\left(68,76,84\right)+\left(-0.85,-1.47,-2.09\right)=\left(67.15,74.53,81.91\%\right) \\ (\mathrm{This}\mathrm{is}\mathrm{the}\mathrm{variation}\mathrm{in}\mathrm{the}\mathrm{time}\mathrm{quality}\mathrm{of}\mathrm{item}\mathrm{J}). \end{gathered}$$

$$\begin{gathered} C_{Q,J}=\left. [\left(\frac{∆C_{Q_{normal,J}-}∆C_{Q_{crash,J}}}{t_{normal,J}-t_{crash,J}}\right)\times \left(t_J-t_{crash,J}\right)+∆C_{Q_{normal,J}}\right]\times \left(q_{J-}{Q}_{T,J}\right) \\ =\left. [\left(\frac{1450-6190}{24.3-3.2}\right)\times \left(24.2-24.3\right)+1450\right]\times \left. [85-(67.15,74.53,81.91)\right] \\ =\left. [\left(-224.64\times -0.1\right)+1450\right]\times (3.09,10.47,17.85) \\ =(4549.90,\mathrm{15,416.65},\mathrm{26,283.41})\mathrm{TWD} \\ (\mathrm{This}\mathrm{is}\mathrm{the}\mathrm{variation}\mathrm{in}\mathrm{the}\mathrm{quality}\mathrm{cost}\mathrm{of}\mathrm{item}\mathrm{J}). \end{gathered}$$

Observing the quality’s slope over the unit time is crucial. Item H displays unit time quality values of (13.716, 15.24, 16.764). Item H exhibits a more pronounced slope in the unit time quality than the other items. The cost will be affected by the number of days. For project H, the number of days will decrease, and the cost range will fall within TWD (215,100, 239,000, 262,900). This is also in line with the research Hypothesis 1 that the number of days will affect the change in cost. On the other hand, Hypothesis 2 is also true. Consequently, achieving efficient construction time results is more attainable for item H.

Table 5 verifies that the quality meets the 85% standard. When the quality of item J increases to 85%, the cost will increase within this range TWD (4549.90, 15,416.65, 26,283.41), which is consistent with the research Hypothesis 3 that the cost increases due to the increase in quality. Therefore, this research initially scrutinized item J’s quality (refer to Table 7), which displayed the smallest value in sorting, then turned attention to item H (refer to Table 10), which displayed the most significant alteration in the unit time quality slope.

5. Discussion

An issue worthy of discussion is how to plan to shorten SPPS building times. This study suggests the building of additional solar power stations within the same timeframe, generating revenue and enhancing the ESG image by selling more clean energy. Renewable energy sources, such as wind and hydropower, could also be considered to further increase the production of clean energy. Moreover, renewable energy sources like wind and hydropower have lower operational costs than traditional ones, making them an economically feasible option for long-term use.

This study uses a fuzzy nonlinear multi-objective method to guide decision-makers to shorten the building time of SPPSs to accelerate electricity production. Does this study propose that the fuzzy nonlinear multi-objective method is better than other methods? Compared with other methods, this study offers a primary way for decision-makers to think about shortening the construction period while maintaining a high-quality standard. This method can measure an interval range to achieve quality requirements, shorten the project’s construction period, and achieve a low-cost result. Hence, this study offers decision-makers an appropriate model. Decision-makers can also choose to utilize this model based on their specific circumstances.

A logistics center strives to shorten project completion time, reduce costs, and maximize quality. The first consideration is quality rather than low-cost production. The general agreement regarding photovoltaic power stations is that one should not compromise quality for a lower cost. Therefore, a prerequisite is the quality, not the cost or construction period. The second consideration is that the construction time will affect construction costs. This is the base principle for Taiwan’s logistics industry to develop solar energy.

In addition to the abovementioned situations, our findings regarding the quality, construction period, and costs are as follows.

Quality corresponds to standardization: A high quality ensures long-term operational reliability. Standardization is a procedure that improves the system quality and reduces failure probability and maintenance costs.

Shortening the construction period corresponds to innovation and efficiency improvement: Innovative technologies and high-efficiency processes can help shorten the construction period. For example, new construction methods, modern engineering designs, and prefabricated components can speed up the construction process and reduce overall construction time.

Cost corresponds to effectiveness and economic feasibility: The cost-effectiveness of green energy lies in the fact that when costs continue to fall, equipment using clean energy becomes more economically viable in the long term.

6. Conclusions

6.1. Research Conclusions

Meeus et al. [41] pointed out that renewable energy requires efficient technologies to bring to market. This study proposes decision analysis for construction project management to shorten the construction time of renewable energy and achieve the goal of building more SPPSs, which will help generate more green energy and fulfill the purpose of environmental sustainability.

Zhu et al. [42] concluded that power projects adopt system thinking and establish a framework for high-quality standards to meet the requirement of sustainability construction elements. This study finds the same conclusion as Zhu et al. that quality must be the primary concern for building a logistics center for SPPS. Regardless of the cost, that quality must remain above 85% standard. In the mathematical model, low cost is not a necessary prerequisite, and shortening construction time is not the most critical consideration. Therefore, the model-established crash time can be analyzed under the quality.

This study uses Zimmermann [35] to propose the mathematical model, the fuzzy method by Yager [38], defuzzification by Liou and Wang [39], and the calculation principle by Dong [40], thereby combining the methods of multiple scholars to conduct fuzzy decision analysis. It is crucial to make effective decisions in uncertain environments. Applying fuzzy nonlinear multi-objective models enables decision-makers to evaluate potential outcomes based on quality tolerance levels. This allows decision-makers to find solutions that are more robust, enabling them to select the best solution based on quality preferences.

In this case, after the α-cut sorting, attention should be directed to the unit time quality of item H, corresponding to the fuzzy cost of item H as TWD (215,100, 239,000, 262,900), which includes the normal cost of TWD 238,000 and the crash cost of TWD 240,000. In item H, the most optimistic case cost is TWD 215,100, and the most pessimistic case cost is TWD 262,900. However, the unit time cost slope is TWD 238,000 to 240,000.

Following this, this study explains item J. After α-cut sorting, it is found that the value of the fuzzy unit time cost is the smallest. The corresponding normal quality of 95% and the crash quality of 57% are of concern. The related fuzzy quality (68, 76, 84) results are obtained because the crash quality is lower than the 85% standard. The construction period dropped from 24.3 to 3.2 days, which is questionable because the crash construction took only 3.2 days to complete. Therefore, the calculation in this study must reach the 85% standard, and the fuzzy cost needs to be increased at TWD (4549.90, 15,416.65, 26,283.41). The results show that under an uncertain environment, decision-makers can calculate the range of costs to be improved. Construction time has been identified as the key factor in controlling costs. Therefore, the cost has become a passive element, and the active components are quality and construction time.

This study uses the same case as Ghodsi et al. [33], focusing on the Pareto optimal solution of comprehensive TCQ. In comparison, this study obtained the interval value of each item’s TCQ under fuzzy calculation. It can better provide decision-makers with the ability to make decisions about building SPPSs and strengthen the goal of achieving sustainable development.

This research introduces a fuzzy nonlinear multi-objective model for project planning, offering critical insights to decision-makers for harmonizing TCQ in managerial applications. The model employs the α-cut method, arming decision-makers with the necessary tools to make informed decisions in uncertain project scenarios. This study’s contribution to construction-related decision-making emphasizes its practicality and relevance in real-world applications. It guides decision-makers in selecting optimal solar photovoltaic station construction solutions for logistics centers.

With Taiwan’s impending carbon tax in 2025, numerous companies have strategized to establish SPPSs in their in-house logistics centers in 2024, aiming for a sustainable reduction in carbon emissions. Therefore, the crash model presented contributes significantly to the ESG principles, concentrating on two key aspects. The construction of many SPPSs’ green power is crucial for realizing ESG goals. As suggested by Dianat et al. [43], these policies should ultimately foster the sustainable development of energy systems.

In augmenting green power, adopting a crash mode approach to expedite the construction of many SPPSs mirrors the company’s long-term values. Lorne and Dilling [44] posit that sustainability can be achieved by aligning with the value created. The study’s primary contribution lies in its effective use of compressed construction time to meet short-term goals while concurrently addressing long-term clean energy issues. Thus, the strategic application of short-term tools to realize long-term ESG goals is underscored. Additionally, this study introduces a decision analysis model for planning SPPSs in logistics centers under fuzzy conditions, offering insights and tools for decision-makers to balance TCQ and select the most effective solution.

6.2. Research Recommendations

Enhance Dynamic Fuzzy Models: Improve the existing fuzzy nonlinear multi-objective model to adapt in real time. Utilize live data and continuous monitoring to adjust fuzzy functions based on project progress. This ensures decision-makers have precise TCQ information in uncertain project scenarios. Real-time monitoring using big data and artificial intelligence is an ideal state. For artificial intelligence, this research is the preliminary content of algorithms.

Carbon Reduction: Expand this research method and add more objectives to evaluate the carbon footprint reduction brought about by the construction of SPPSs. Research innovative technologies and construction practices that align with sustainable development goals and calculate the linkage of SPPSs in logistics centers to carbon emissions. Moreover, models also can help optimize bioenergy production, enhance geothermal systems, and increase the efficiency of hydropower installations.

This research is the basis for the development of future artificial intelligence decision-making models to build solar photovoltaic power plants in logistics centers using fuzzy nonlinear multi-objective models. It can further emphasize the sustainability goal of reducing carbon emissions advocated through the ESG principles. The results of this study will also have a specific influence on our team’s future research on fulfilling carbon reduction goals.