Sustainable Investment Strategy: A Fuzzy Nonlinear Multi-Objective Programming for Taiwan’s Solar Photovoltaic Billboards

Abstract

In Taiwan, large advertising billboards on commercial buildings consume significant energy, exacerbating environmental challenges and straining sustainability efforts. This study explores the potential of rooftop solar photovoltaic systems (SPVS) to power these billboards, offering a dual solution for energy reduction and financial viability. Using a fuzzy nonlinear multi-objective programming approach, the research demonstrates that SPVS investments become profitable by the ninth year (0.7232% return), rising to 5.4463% by the twentieth year, while a 26-day reduction in construction time cuts carbon emissions by 223.11 kg. The innovative Revenue–Time–Cost–Quality–Carbon Emissions (RTCQCE) framework balances economic gains with environmental benefits, leveraging advertising revenue to fund SPVS. This model bridges a research gap by integrating financial and ecological factors, providing a practical tool for sustainable urban development in Taiwan.

1. Introduction

In Taiwan, large neon advertising billboards on commercial rooftops consume considerable energy, presenting a major obstacle to the nation’s environmental protection and sustainable development objectives. Taiwan amended the “Renewable Energy Development Regulations” in June 2021, requiring new, expanded or renovated buildings larger than 992 square meters to install solar photovoltaic systems (SPVS). This regulation tackles excessive energy consumption and pollution, highlighting Taiwan’s dedication to renewable energy and sustainable cities. Ahmed et al. [1] describe SPVS as a clean, efficient energy source with significant potential, fitting the nation’s energy transition needs. The Ministry of the Interior (MOI) and National Development Council (NDC) report that over 80% of Taiwan’s two million buildings sit in urban areas, making rooftops a key resource for SPVS deployment.

Although these billboards provide substantial rental income, the energy demands, especially from neon designs, raise costs and environmental burdens. Switching to energy-efficient LED billboards and adding SPVS can cut electricity use, operating expenses and pollution while increasing profits. Taiwan offers interest subsidies to hasten SPVS investment returns and the NDC projects that using 10% of rooftop space could produce TWD 200 billion yearly. Companies adopting solar-powered billboards gain a stronger sustainability reputation and lower expenses. Omole, Olajiga and Olatunde [2] affirm that SPVS reduces electricity bills and greenhouse gas emissions, while Christopher et al. [3] stress the advantages of rooftop space for renewable energy production. Chen et al. [4] point out that policies boost investment and Liu et al. [5] suggest an economic model to enhance renewable energy use in buildings, merging economic and environmental aims for a sustainable urban future.

Speeding up SPVS installation brings trade-offs between revenue (R), time (T), cost (C), quality (Q) and carbon emissions (CE), as Qureshi et al. [6] note. Quick deployment accelerates the shift to solar energy but raises initial expenses for labor, equipment and logistics, per Saccardo et al. [7]. Kabir, Matin and Amin [8] propose SPVS as a practical choice in resource-rich regions, though high costs and maintenance needs hinder broad use. Financial success depends on managing these elements. Tsuchiya, Swai and Goto [9] explain that investment recovery fails if energy payback time (EPBT) exceeds the panel lifespan. Tawalbeh et al. [10] underline the importance of thorough environmental assessments, covering emissions from construction and waste, to maintain sustainability.

Using solar energy for billboards demands a plan that blends revenue, time, cost, quality and carbon emissions (RTCQCE). This strategy speeds up SPVS installation while keeping quality high and costs low, creating self-sufficient power models that lessen dependence on external supplies. Advertising revenue can support SPVS, reducing payback time. Ali Banihashemi and Khalilzadeh [11] apply fuzzy goal programming to juggle cost, time and quality in construction, while Mathaba and Abo-Al-Ez [12] optimize energy cost and emissions for homes. Yet, these studies often assume linear patterns, missing the nonlinear nature of real systems. This study uses a nonlinear model to reflect these dynamics accurately, filling a decade-long research gap on billboards’ role in cutting carbon emissions.

This study explores three main questions: (1) How can Taiwan’s commercial rooftops effectively apply SPVS for economic and environmental sustainability? (2) How does speeding up SPVS construction affect RTCQCE? (3) How can advertising revenue fund SPVS investments, balancing economic and environmental gains?

To address these, this study: (1) builds a model combining fuzzy theory, nonlinear modeling and multi-objective programming, (2) uses the RTCQCE framework to weigh trade-offs and carbon impacts and (3) applies fuzzy modified internal rate of return (FMIRR) and α-cut defuzzification for accurate return analysis.

This method, which is different from linear methods used in studies like Tsuchiya et al. [9], uses fuzzy numbers (like FMIRR and FWACC) to handle uncertainty. It is based on fuzzy theory from Bellman and Zadeh [13], fuzzy calculations by Dubois and Prade [14], Dong and Shah [15] and α-cut by Bodjanova [16]. Compared to partial multi-objective models like Ali Banihashemi and Khalilzadeh [11], RTCQCE covers five aspects, uniquely tying in billboard revenue, as shown in a Taiwan case. While Kumar et al. [17] examine rooftop solar in India, billboard revenue models remain unaddressed in that work.

The RTCQCE framework excels in its comprehensive, nonlinear blend, tailoring SPVS to Taiwan’s urban setting. It equips building owners, advertisers, policymakers and investors with a solid tool for sustainable project management, turning billboards into assets for both the economy and the environment. Moreover, this paper mainly suggests and tests a new sustainable investment plan that uses solar photovoltaic systems (SPVS) on the roofs of commercial buildings to power advertising billboards. It applies fuzzy nonlinear multi-objective programming and the RTCQCE framework to benefit the economy and the environment. In particular, this paper is the first to consider using advertising revenue to pay for solar photovoltaic systems (SPVS) on commercial roofs while adding a revenue (R) component to the traditional TCQCE framework to create RTCQCE.

Additionally, it employs fuzzy modified internal rate of return (FMIRR) and α-cut defuzzification technology to evaluate investment returns better and speed up construction times, which contrasts with previous studies that focused on warehouse design (Chiang [18]) or logistics (Chiang [19]). Furthermore, this study addresses a missing area in research about how urban billboards can help reduce carbon emissions, offering a helpful tool that can be used in various locations and policy settings, such as quick implementation in Southeast Asia and focusing on quality in Northern Europe. As a result, it provides valuable insights into sustainable urban development worldwide. These contributions go beyond the scope of previous studies, making this paper not just a case study but a vital advance with methodological and applied innovations.

2. Literature Review

2.1. Application of Solar Photovoltaic Systems in Advertising Billboards

Large billboards installed on the roofs of commercial buildings can generate revenue, but they also increase a city’s carbon emissions. How to apply the RTCQCE trade-off model to accelerate the reconstruction of large rooftop billboards to achieve carbon reduction targets and revenue balance is an important issue.

Şirin, Goggins and Hajdukiewicz [20] highlighted that 40% of global energy demand comes from building energy consumption, accounting for 33% of global greenhouse gas (GHG) emissions. Building-integrated photovoltaic (BIPV) systems efficiently generate electricity and heat and can improve overall building energy performance when installed within a building facade. The research method adopted the literature search method. The key is understanding material selection and the optimal placement of the measurement system within the building to maximize the effectiveness of the technology. BIPV systems will play an important role in the development of sustainable and net-zero energy buildings.

Cuce et al. [21] pointed out that green energy buildings can reduce air conditioning costs through building envelopes. An experiment to install phase change materials (PCMs) on the roof envelope saved $6.29 in air conditioning costs and reduced carbon dioxide emissions by 300.55 kg per year. The payback period is 3.13 years.

Seyedabadi, Eicker and Karimi [22] evaluated the role of green roofs in reducing building energy consumption and studied the overall impact on the building’s carbon footprint. Based on the instrumental monitoring method, the results showed that Sedum serrata, Catharanthus roseus and Vinca roseus reduced annual energy consumption by 8.5%, 8.0% and 7.1%, respectively. It significantly reduced the building’s CO₂ emissions by 28.16, 26.48 and 23.44 kg/m², respectively, demonstrating the practical benefits of green roofs in reducing carbon footprints.

Buildings can reduce their energy consumption in two ways, as discussed in the literature above. One is to improve the building materials mentioned in the literature or use solar shading devices to reduce energy use and carbon emissions. Another is to increase a building’s energy harvesting by using SPVS to absorb heat and convert it into electricity while reducing carbon emissions. Therefore, installing SPVS can also block sunlight and reduce energy consumption, making it feasible to install SPVS to increase green energy.

Ma and Yuan [23] emphasized that solar energy is a viable renewable resource. Auxiliary energy storage in hybrid renewable energy systems using solar photovoltaic systems can increase the energy autonomy of sustainable buildings. The authors evaluated models of two off-grid energy storage structures using particle swarm optimization techniques. These structures include photovoltaic/battery systems and off-grid photovoltaic/hydrogen energy systems. The conclusion is that rooftop photovoltaic and battery systems have low costs, making them the main power source in rural areas.

Kokchang [24] studied the economic feasibility of solar photovoltaic (PV) and battery energy storage system (BESS) for electric vehicle charging stations in a Thai university. This study conducted a sensitivity analysis on the actual measurement method. The results show that installing solar PV charging stations at current subsidy rates provides the greatest economic benefits. However, installing BESS for peak shaving is the least profitable due to its high cost. Sensitivity analysis also reveals that reducing the BESS cost will boost the project’s IRR.

Burhan and Shahzad [25] pointed out that solar energy has the greatest potential. It is a compact and efficient concentrated photovoltaic (CPV) hydrogen system suitable for rooftop applications in urban areas. We experimentally measured a compact CPV hydrogen system. The results explain a CPV efficiency of 28% and a solar-to-hydrogen (STH) efficiency of 18%, suitable for rooftop operation in tropical Singapore.

The above studies show that SPVS energy has enormous potential as a renewable energy source. Utilizing rooftop SPVS is becoming increasingly important and practical due to their ability to generate reliable, sustainable and cost-effective renewable energy. Accelerating the construction of rooftop SPVS can quickly obtain clean electricity to meet the power consumption needs of advertising billboards.

Zakeri [26] studied the economic feasibility of using rooftop solar PV systems with energy storage for residential consumers in the UK. The study analyzes the impact of various policy scenarios on system profitability by combining a cost optimization model with a national power system model. The findings suggest that replacing the feed-in tariff for PV with a self-consumption incentive could provide an attractive return on the investment for household batteries, equivalent to a 70% capital subsidy. Selecting the optimal storage capacity and dynamic pricing strategy is critical to maximizing the profitability of PV battery energy storage systems.

Magni, Marchioni and Baschieri [27] explored the impact of incorporating financial variables into a solar PV system investment decision model. The researchers used a decision model to analyze the impact of different payment policies and financing combinations on project value. The results suggest that firm decisions regarding payment policy and financing mix can favorably affect the value of solar projects, and that understanding the interplay between operational variables and financial decisions is critical to making informed investment decisions.

Kumar et al. [17] proposed a promising market model for rooftop solar (RTS) photovoltaic systems in the commercial and industrial sectors in India. The model utilizes a discounted cash flow (DCF) approach to evaluate the feasibility of various RTS market approaches. It focuses on a typical 15 story commercial high-rise building in India with a maximum demand of 180 kVA. Its centralized model has a minimum electricity cost of 3.39/kWh and a payback period of 5.5 years. When federal subsidies are factored in, the prosumer model has an even lower cost of 2.06/kWh, with a payback period as low as 3.3 years. Notably, these models achieved grid parity under all tariff structures, meaning rooftop solar is economically competitive with conventional grid electricity.

Bensaha et al. [28] studied the cost-effectiveness of photovoltaic systems for houses in remote areas of the Ghardaïa region. The study analyzed the main issues from both an energy and financial perspective, finding that short battery life and low conventional electricity prices were the main obstacles. The study questions the profitability of initial investments and recommends subsidizing photovoltaic technology to replace conventional electricity in order to increase returns and rationalize the use of non-renewable resources.

While rooftop SPVS has promising environmental and social benefits, the financial viability of these projects remains critical for their widespread adoption. The above studies highlight the importance of financial considerations and strategic decisions to ensure a successful rooftop SPVS investment. RTCQCE has a financial and carbon emissions assessment framework that allows for a comprehensive assessment of the financial feasibility of rooftop SPVS projects. It addresses the issue of billboard electricity consumption by prioritizing financial viability, thereby accelerating the deployment of SPVS and realizing the full potential of these renewable energy sources.

Research overwhelmingly supports SPVS as a win-win solution. The study demonstrated the environmental benefits while also highlighting their financial viability. Rooftop solar can achieve grid parity, making it cost-competitive with conventional electricity and offering attractive payback periods (Kokchang [24]; Kumar et al. [17]). Ma and Yuan [23] demonstrated the potential of SPVS combined with battery energy storage to achieve energy independence for buildings. Studies by Zakeri [26] and Magni et al. [27] highlight the importance of strategic financial planning for maximizing profitability.

The RTCQCE model integrates financial and carbon emissions assessments, providing an ideal framework for optimizing the transition to sustainable billboards. Additionally, SPVS has the potential to meet the energy needs of the remaining billboards, thereby reducing overall energy consumption and establishing a sustainable source of electricity for the billboards. It ensured a successful approach by considering the financial aspects.

By combining RTCQCE’s revenue optimization strategy with rooftop SPVS by replacing large advertising billboards, Taiwan’s commercial buildings can achieve a win-win in terms of economic and environmental sustainability. The environmental impact of electricity consumption by billboards is a pressing issue.

2.2. Application of Fuzzy Nonlinear Multi-Objective Programming Model

As the world pays more and more attention to sustainable development and energy efficiency, energy consumption and carbon emissions in the building sector have become a focus of research. Especially in commercial buildings, large advertising billboards on rooftops can bring economic benefits but also impose a burden on the environment due to high energy consumption and carbon footprint. In recent years, advanced decision-making methods, such as fuzzy logic and multi-objective programming, have been widely used to address such challenges and provide solutions that balance economic benefits and environmental responsibilities. This literature review talks about how similar studies have used fuzzy decision models and multi-objective optimization methods to look into how SPVS can be used to save energy in buildings, how cost-effective it is, and how it can be used to come up with new ways to deal with the energy needs of billboard ads.

Ziemba and Szaja [29] analyzed the PV panels available on the Polish market, using a fuzzy approach to achieve their practical objectives. That’s why they made a fuzzy decision model for evaluating photovoltaic panels using MCDM and the fuzzy preference ranking organizational approach for better evaluation (NEAT F-PROMETHEE). The results show that the fuzzy numbers of the fuzzy model can capture more data than the crisp numbers, which makes the fuzzy model have excellent reliability.

A study by Eti et al. [30] used a fuzzy TOPSIS-DEMATEL model (ranking technique with similarity and decision laboratory) to look into how cost-effective solar panels are in hospitals. Adopting solar panels can significantly reduce a hospital’s energy dependency and electricity costs. However, the high initial investment cost is the second most critical factor for TOP-DEMATEL. It highlights the impact of cost-effectiveness as well as power generation benefits.

Auza et al. [31] looked at all 129 studies that were published between 2011 and 2023 to find out about new trends and approaches in the objective function of renewable energy building costs in multi-objective optimization (MOO) problems. The most common original indicators are cost targets based on total costs, which include both life cycle costs and total costs, as well as reclassified energy import costs. Meanwhile, payback periods are more common in less developed countries. Many studies adopt the MOO approach to achieve the investments needed in building energy efficiency to meet the 1.5 °C target by 2050.

The studies reviewed above adopted fuzzy and multi-objective optimization-based decision-making methods to achieve building energy efficiency goals. A particular focus is on exploring the use of renewable energy in hospitals and the cost-effectiveness of renewable energy buildings.

Adedokun and Egbelakin [32] introduced a new fuzzy nonlinear multi-objective programming model (FNMOPM) to analyze decision-making in construction projects. The model’s ability to handle multiple objectives and uncertainty proved valuable in addressing the complexity inherent in such projects. The results of the study highlighted the significant impact of risk factors, primarily cost and time overruns, on project success. The researchers took a nonlinear approach to address these challenges, revealing the complex interdependence among the various factors that influence project outcomes.

Elshenawy et al. [33] studied new clean renewable energy and efficient, economical and environmentally friendly energy management technologies. The first phase involved the use of a demand response program (DRP). Next, the distribution network operator (DNO) needs to find the best multi-objective (MO) load to lower energy losses. They can do this by using both TOPSIS (ranking by similarity of ideal solutions) and EHO (elephant herd optimization) techniques together. The results showed that total energy losses were reduced by 38.13%, total power generation costs were reduced by 9.468% and carbon emissions were reduced by 5.9%.

When faced with large advertising billboards on the rooftops of commercial buildings, in addition to consulting the FNMOPM assessment of Adedokun and Egbelakin [32], the primary consideration is their potential revenue generation. However, the high cost of these billboards can be attributed to increased energy consumption and carbon footprint. Therefore, this study used the RTCQCE framework to comprehensively assess the situation. By applying the FNMOPM approach, the RTCQ architecture of SPVS on the rooftop of a commercial building is utilized, and its effectiveness in solving the CE problem of advertising billboards is demonstrated.

This study fills in a gap in the research by suggesting a multifaceted method that uses fuzzy models to handle uncertain environments, nonlinear models to capture changes in those environments and multi-objective programming to find a fair solution. This innovative approach mitigates the building’s carbon footprint, paving the way for a greener future and transforming the roof into a symbol of economic sustainability through an optimized revenue stream for the RTCQCE, and environmental responsibility through the SPVS.

3. Methodology

3.1. Research Structure

The reliance on energy-intensive lighting for advertising billboards significantly adds to a commercial building’s carbon footprint. LED advertising billboards also do not entirely have carbon emissions. I recommend the accelerated construction of rooftop SPVS to provide an independent power supply for advertising billboards. Integrating SPVS systems into commercial buildings reduces advertising billboards’ reliance on grid electricity, consequently minimizing greenhouse gas emissions. However, a comprehensive revenue analysis guides the acceleration of SPVS installation. Therefore, Figure 1, representing the RTCQCE framework for rooftop SPVS, presents an approach to accelerating commercial building advertising billboards toward carbon emissions mitigation and a sustainable green future.

Figure 1’s core involves accelerating investment in rooftop solar photovoltaic systems (SPVS) to provide a sustainable source of electricity for billboards, thereby achieving both environmental protection and economic benefits. In particular, faster investment can provide long-term electricity for billboards and eventually find a balance between rental income and carbon reduction. This can be done by speeding up construction schedules, reducing costs, maintaining quality, and implementing carbon reduction strategies, thereby accelerating the realization of a green future.

3.2. Case Propositions

Installing SPVS power generation on the roof for advertising billboards is necessary. However, adding SPVS equipment to already constructed buildings can significantly reduce carbon emissions. This study presents a series of propositions as follows, each one a key to unlocking the mutual complexity of RTCQCE:

Proposition 1.

SPVS investments have the potential to generate favorable long-term revenues.

Proposition 2.

Shortening the construction period will increase the cost of SPVS.

Proposition 3.

Shortening the construction period compromises quality.

Proposition 4.

As construction progresses, we must maintain quality above safety standards, as it will increase costs.

Proposition 5.

Accelerating the construction period could make a significant impact on reducing carbon emissions.

Proposition 6.

Using SPVS on commercial building rooftops to power advertising billboards is a niche market that requires investment.

Undoubtedly, the most crucial of the six propositions are financial revenue and shortened construction time. These two factors can significantly impact cost, quality and carbon emissions. Getting green and clean energy more quickly and in large amounts could be achieved by speeding up the shortening of commercial buildings during the SPVS construction period of the object. This would be a big step toward sustainable energy and construction.

3.3. Case Methodology

Based on the literature review, this study found that the evaluation model needs to be comprehensive and closer to the actual situation. The researcher proposes an innovative model that integrates fuzzy theory to address uncertain environments, a nonlinear model that better responds to real-world fluctuations, and multi-objective programming to balance advertising billboards’ energy consumption and carbon emissions. This comprehensive model, is based on the fuzzy theory of Bellman and Zadeh [13], with fuzzy calculations referring to the works of Dubois and Prade [14] and Dong and Shah [15], along with the nonlinear multi-objective programming model by Zimmermann [34] and Huang [35] and the α-cut method based on the work of Bodjanova [16]. Therefore, this study integrates the above techniques into an effective evaluation model.

In fuzzy theory, the α-cut method is a technique for ranking fuzzy numbers. Fuzzy numbers are sets of real numbers, with each element assigned a membership degree indicating the extent of its affiliation with the set. The membership function rigorously formalizes this concept, providing a numerical degree of membership for each element within the fuzzy set. A fuzzy number is specifically characterized by two membership functions:$f^{a_{FiL}}$ representing the left membership function and $f^{a_{FiR}}$ representing the right membership function.

Assuming the inverse functions $g^{a_{FiL}}$ and $g^{a_{FiR}}$ exist for $f^{a_{FiL}}$ and $f^{a_{FiR}}$, respectively, the article defines the left integral value, $I^L\left(a_{Fi}\right)$, and the right integral value, $I^L\left(a_{Fi}\right)$, of $a_i$ as specified in Equations (2) and (3), presented by Dubois and Prade [14], Dong and Shah [15] and Bodjanova [16]. Additionally, the mathematical operations on two positive fuzzy numbers, A and B, can be expressed using fuzzy arithmetic as follows: for fuzzy addition, (A$\oplus$B)^α = [A_l^α + B_l^α, A_u^α + B_u^α]; for fuzzy subtraction, (A$\ominus$B)^α = [A_l^α − B_u^α, A_u^α − B_l^α]; for fuzzy multiplication, (A$\otimes$B)^α = [A_l^αB_l^α, A_u^αB_u^α]; and for fuzzy division, (A$\oslash$B)^α = [A_l^α/B_u^α, A_u^α/B_l^α].

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

(1)

$$I^L\left(a_{Fi}\right)={\displaystyle \frac{1}{2}}\underset{k\to \infty }{\mathit{lim}}\left\{{\displaystyle \sum }_{j=1}^k\left[{g_{{\alpha }_{Fi}}}^L\left({\alpha }_{Fj}\right)+{g_{{\alpha }_{Fi}}}^L\left({\alpha }_{Fj-1}\right)\right]\Delta {\alpha }_{Fj}\right\}$$

(2)

$$I^R\left(a_{Fi}\right)={\displaystyle \frac{1}{2}}\underset{k\to \infty }{\mathit{lim}}\left\{{\displaystyle \sum }_{j=1}^k\left[{g_{{\alpha }_{Fi}}}^R\left({\alpha }_{Fj}\right)+{g_{{\alpha }_{Fi}}}^R\left({\alpha }_{Fj-1}\right)\right]\Delta {\alpha }_{Fj}\right\}$$

(3)

Firstly, this study aims to enhance the accuracy of assessing the relative profitability of SPVS. Employ the modified internal rate of return (MIRR) to compare the investment revenue. MIRR (Equation (5)) focuses on cash flow returns and reinvestment rates to calculate expected rates of return. It can also avoid the multiple interest rates and conflicting investment problems that may arise with IRR. Join the fuzzy MIRR (FMIRR) (Equation (6)) framework more aligned with real-world situations to solve investment evaluation issues by Sampaio Filho, Vellasco and Tanscheit [36]. Additionally, FMIRR’s cost will be used as the fuzzy weighted average cost of capital (FWACC) to assess a company’s cost. The cost of all capital sources (including equity, preferred stock, bonds, etc.) affects market values.

$$FWACC=W_d\otimes F{K}_d\otimes \left(1\ominus FT\right)\oplus W_s\otimes F{K}_s$$

(4)

$$C={\displaystyle \sum }_{t=0}^n{\displaystyle \frac{CO{F}_t}{{\left(1+k\right)}^t}}={\displaystyle \frac{TV}{{\left(1+MIRR\right)}^n}}={\displaystyle \frac{{{\displaystyle \sum }}_{t=0}^{n}CI{F}_t{\left(k+1\right)}^{n-t}}{{\left(1+MIRR\right)}^n}}$$

(5)

$$FC={\displaystyle \frac{FTV}{{\left(1\oplus FMIRR\right)}^n}}={\displaystyle \frac{{{\displaystyle \sum }}_{t=0}^{n}FCI{F}_t{\left(1+FWACC\right)}^{n-t}}{{\left(1\oplus FMIRR\right)}^n}}$$

(6)

The definitions for symbols are as follows:

$W_d$: it is debt ratio.

$W_s$: it is common equity ratio.

$COF$: it is cash outflow.

$CIF$: it is cash inflow.

${FK}_d$: it is fuzzy cost of debt.

${FK}_s$: it is fuzzy cost of common equity.

$FT$: it is fuzzy margin tax.

$FC$: it is fuzzy present value.

$FTV$: it is fuzzy terminal value.

$FCIF$: it is fuzzy cash inflow.

$WACC$: it is weighted average cost of capital.

$FWACC$: it is fuzzy weighted average cost of capital.

$FMIRR$: it is fuzzy modified internal rate of return.

n: it is the total number of periods over which the investment occurs.

Next to building upon the foundation of fuzzy nonlinear multi-objective programming established by Zimmermann [34] and Huang [35], this process involves several key steps. Firstly, it defines constant values denoted by $a_{Fi}$. Secondly, it calculates various cost components: time cost $(C_{T,Fi}$), direct fuzzy cost ($C_{Fi}$) and fuzzy quality cost ($C_{Q,Fi}$). Additionally, it determines the fuzzy direct carbon emissions ($C{E}_{T,Fi}$). These factors contribute to the inherent trade-off between time, cost, quality and carbon emissions in a multi-objective setting. Equations are in the (7)–(11).

$$C_{T,Fi}=a_{Fi}\times {(t_{TRi})}^2+b_{Fi}$$

(7)

$$a_{Fi}={\displaystyle \frac{c_{regular,Fi}-c_{shortened,Fi}}{t_{regular,Fi}^2-t_{shortened,Fi}^2}}$$

(8)

$$b_{Fi}={\displaystyle \frac{c_{shortened,Fi}\times t_{regular,Fi}^2-c_{regular,Fi}\times t_{shortened,Fi}^2}{t_{regular,Fi}^2-t_{shortened,Fi}^2}}$$

(9)

$$Q_{T,Fi}=q_{shortened,Fi}+{\displaystyle \frac{q_{shortened,Fi}-q_{regular,Fi}}{t_{shortened,Fi}-t_{regular,Fi}}}\times \left(t_{TRi}-t_{shortened,i}\right)$$

(10)

$$C_{Q,Fi}=\left[\left({\displaystyle \frac{\Delta C_{Q_{shortened,Fi}}-\Delta C_{Q_{regular,Fi}}}{t_{shortened, Fi}-t_{regular, Fi}}}\right)\times \left(t_{TRi}-t_{regular,i}\right)+\Delta C_{Q_{shortened,Fi}}\right]\times \left(q_{Fi}-Q_{T,Fi}\right)$$

(11)

The definitions of the symbols are as follows:

$Fi:$ this represents the number of each fuzzy project.

$t_{TRi}:$ this is a variable of time, representing a specific value of time.

$q_{Fi}$: the quality ratio of each fuzzy project.

$c_{Fi}$: the cost of each fuzzy project.

$t_{shortened,Fi}$: duration time of each fuzzy project with shortened status.

$t_{regular,Fi}$: duration of each fuzzy project with regular status.

$c_{shortened,Fi}$: direct cost of the shortened working status of each fuzzy project.

$c_{regular, Fi}$: direct cost of the regular work status of each fuzzy project.

$q_{shortened,Fi}$: quality ratio of each fuzzy project in the shortened status.

$q_{regular,Fi}$: quality ratio of the regular status of each fuzzy project.

$C_{T,Fi}$: calculate each fuzzy project’s cost/time ratio.

$C_{Q,Fi}$: calculate each fuzzy project’s cost/quality ratio.

$Q_{T,Fi}$: calculate each fuzzy project’s quality/time ratio.

$\Delta C_{Q_{shortened,Fi}}$: unit cost of the quality ratio variation for each fuzzy project with shortened status.

$\Delta C_{Q_{regular,Fi}}$: unit cost of the quality ratio variation for each fuzzy project with regular status.

This paper introduces a new project model to solve many consumption problems. Since decision-making will face multi-objective programming, nonlinear situations and fuzzy environmental changes, using fuzzy theory to analyze interval values is more in line with decision-making needs. However, the construction of solar equipment will involve accelerating SPVS completion time, reducing SPVS cost and optimizing SPVS quality. Moreover, supplying the generated electricity to advertising billboards, reducing external power purchases and adopting an internal power supply model will also be needed. Part of the advertising revenue will be paid to the power provided by SPVS as electricity revenue, and the investment payback period of SPVS will be calculated simultaneously. Finally, calculate how much carbon emissions the SPVS power supply can reduce for advertising billboards. To correctly complete the establishment of the model, this study used Excel 2019 for preliminary calculations, then used Python 3.11 for fuzzy program data processing and finally put the results processed by Python 3.11 into Excel 2019 to integrate the results and form a table.

4. Sample Problem and Results

This paper introduces a time approach for decision-oriented delivery project administration. The approach employs fuzzy theory, α-cut defuzzification, nonlinearity, mathematical programming and multi-objective programming to address the correlations between TCQCE and carbon emissions mitigation of ESG. This research provides a comprehensive technique that enables the decision-maker to convert fuzzy quantities into exact values.

4.1. Case Introduction

The actual case is on the SPVS and advertising billboard installed on the rooftop of SASSWOOD Corporation, a 30 story commercial building in Taipei, Taiwan.

The SASSWOOD Corporation commercial building’s rooftop area is 3892.46 m² and has 1283 solar panels. The total investment for this project amounted to TWD −25,660,000 (the expenditure is a negative sign), and the power generation capacity stands at 474.71 kWh. Taiwan Power Company purchases the generated power at TWD 4.0970 kWh. If it sells solar power to Taiwan Power Company, it will get TWD 2,645,046 yearly. This income can be used to offset the advertising revenue paid for the leased billboards and to determine SPVS’s return on investment. However, for the construction SPVS in capital financing, according to Taiwan’s 10-year corporate bond announcement, the coupon rate is approximately 1.98% to 2.05% (Taipei Exchange bond market). The bank’s loan interest cost rate is 3.07% (Taiwan’s Bank SinoPac loan interest rate in 2024). The interest tax saving rate is at 25%. The quality of all projects must remain above 90% to maintain product durability.

Next, according to information from the Ministry of Economic Affairs Industrial Development Administration, the power consumption of one square meter of outdoor advertising billboard is about 34.7 Wh. The power consumption of a giant outdoor advertising billboard of about 50 square meters is 17,350 Wh/10 h, 6332.75 kWh/year and 3128.38 metric tons/CO₂e/year. The estimated carbon emissions per kilowatt-hour of electricity generated in Taiwan are 0.495 kg.

There are two construction methods for the SPVS project. There is the dry cement construction method and the wet cement construction method. The dry cement construction method does not require cement slurry and typically uses prefabricated materials for assembly. It has a shorter construction time and produces little dust and waste but lacks wet construction’s structural stability and durability. The wet construction method uses cement slurry to fix the structure, providing higher strength and durability. According to the different construction methods used to set up Figure 2.

There are two different ways to build SPVS in Figure 2. One is dry cement construction within Projects 1 and 2, and the other, Project 3, is wet cement construction. Regardless of the method chosen, it is possible to complete the construction of the solar photovoltaic system. The main purpose is to show two feasible construction options rather than to compare the pros and cons. Dry construction may be faster and easier, while wet construction may provide a more stable structure. The method chosen depends on project requirements, environmental conditions and cost considerations.

This study provides practical data for the SPVS project and organizes conventional data and shortened data under the framework of time, cost and quality, as shown in

Table 1. The data related to the SPVS project.

Project	Time (Days)		Cost (TWD)		Quality (%)		Quality Cost (TWD/%)
	Regular	Shortened	Regular	Shortened	Regular	Shortened	Regular	Shortened
1	78	52	25,660,000	27,712,800	90	85	3655	6270
2	78	53	25,660,000	27,633,846	90	86	3655	6169
3	83	53	26,686,400	28,821,312	95	92	3384	5911

. This shortened approach significantly reduced the overall project duration and provided multiple benefits. Step 1: compare the number of days that can be shortened and the cost expenditure of three projects with the same construction method and different construction methods. Step 2: quality indicators show a slight decrease in the shortened method. Although the differences are relatively small, keeping high qualities is still important for organizations seeking to accelerate project completion. This responsibility falls on decision makers because different construction methods have different effects on shortening working hours and costs. The faster the completion, the earlier the rental income can be obtained, so the increased costs and reduced quality will be related to the rental income.

As per the data in

Table 2. Determine the impact of shortened time on cost and quality slopes.

Project	Time (Days)		Cost (TWD)		Quality (%)		Shortened Time (Days) (7) = (1) − (2)	Unit Time Cost (TWD/Day) [((4) − (3))/(7)]	Unit Time Quality (%/Day) [((6) − (5))/(7)]
	Regular (1)	Shortened (2)	Regular (3)	Shortened (4)	Regular (5)	Shortened (6)
1	78	52	25,660,000	27,712,800	90	85	26	78,953.85	−0.19
2	78	58	25,660,000	27,633,846	90	86	20	78,953.84	−0.20
3	83	53	26,686,400	28,821,312	95	92	30	71,163.73	−0.10

, the shortened project execution method has significantly impacted the project schedules of Project 1, Project 2 and Project 3, shortening them by 26, 25 and 30 days, respectively. However, this acceleration also comes with increased costs. This shows that the cost of shortening the project schedule is that for every day of the construction period reduced, an additional cost of TWD 78,953.85/day will be paid. Project 2 costs 78,953.84 TWD/day and Project 3 costs 71,163.73 TWD/day.

Table 2 also highlights the trade-off between project speed and quality. The unit quality slopes of Projects 1, 2 and 3 are negative: −0.19%, −0.16% and −0.10%. This means the quality decreases by 0.19%, 0.16% and 0.10% daily; therefore, a certain level of quality is compromised in the pursuit of quick project completion.

4.2. To RTCQCE from FNMOPM

Table 3. FWACC data.

Wd (%)	FKd (%)	Ws (%)	FKs (%)	FT (%)	FWACC (%)
0	–	100	[1.9800, 2.0148, 2.0500]	[25, 25, 25]	[1.9800, 2.0148, 2.0500]
10	[2.7630, 3.0597, 3.3770]	90	[1.7820, 1.8133, 1.8450]	[25, 25, 25]	[1.8110, 1.8615, 1.9138]
20	[2.4867, 2.7538, 3.0393]	80	[1.5840, 1.6118, 1.6400]	[25, 25, 25]	[1.6402, 1.7025, 1.7679]
30	[2.2104, 2.4478, 2.7016]	70	[1.3860, 1.4104, 1.4350]	[25, 25, 25]	[1.4675, 1.5380, 1.6124]
40	[1.9341, 2.1418, 2.3639]	60	[1.1880, 1.2089, 1.2300]	[25, 25, 25]	[1.2930, 1.3679, 1.4472]
50	[1.6578, 1.8358, 2.0262]	50	[1.3860, 1.4104, 1.4350]	[25, 25, 25]	[1.3147, 1.3936, 1.4773]
60	[1.9341, 2.1418, 2.3639]	40	[1.3860, 1.6118, 1.6400]	[25, 25, 25]	[1.4247, 1.6086, 1.7198]
70	[1.9341, 2.4478, 2.7016]	30	[1.7820, 1.8133, 1.8450]	[25, 25, 25]	[1.5500, 1.8291, 1.9718]

presents the crucial fuzzy weighted average cost of capital (FWACC) data. These data are not just numbers but the backbone of this study’s financial analysis for the SPVS. The FWACC, established in the FMIRR equation, is a cost to our decision-making process. The calculation of FWACC considers the source of investment funds of the project SPVS, be it its own funds or bank loans. Zimmermann [34] and Huang [35] provided a powerful tool for dealing with uncertainty and multi-objective decision-making problems, identified the membership function, which was ${\mu }_{{\tilde{a}}_l}\left(c_i\right)$, the fuzzy set of $\widetilde{{\mathrm{a}}_i}$ defined as $\left(\tilde{{{\mathrm{c}}_l}_{\alpha }}\right)=\left\{c_i|{\mu }_{\widetilde{c_l}}\left(c_i\right)\ge \alpha \right\}$. This interval was the set of all $\widetilde{(c_l}{)}_{\alpha }=\left[{(c_i)}_{\alpha }^L, {(c_i)}_{\alpha }^U\right]$ valued with possibility $a_i.$ Within this model, the fuzzy $\widetilde{{\mathrm{a}}_i}$ for each operation $\widetilde{(c_i}{)}_{\alpha }=\left[{(c_i)}_{\alpha }^L,{(c_i)}_{\alpha }^M, {(c_i)}_{\alpha }^U\right]$ is modeled as a triangular fuzzy number. Project SPVS’s fuzzy number $\widetilde{(c_{SPVP}}{)}_{\alpha }=\left[{(c_{SPVP})}_{\alpha }^L,{(c_{SPVP})}_{\alpha }^M, {(c_{SPVP})}_{\alpha }^U\right]$ serves as an illustration. The fuzzy numbers will be calculated cube root of ${(c_{SPVP})}_{\alpha }^M$ is after average of ${(c_{SPVP})}_{\alpha }^L \mathrm{and} {(c_{SPVP})}_{\alpha }^U$ within 10% upper or lower bounds.

The weight changes from 0% to 70% in increments of 10%, representing the proportion of debt (d) and common equity (s), respectively. Under each weight combination, the corresponding fuzzy debt cost (FKd) and fuzzy common equity cost (FKs) are given, and these costs are expressed in the form of triangular fuzzy numbers. The upper and lower bounds of the interest rate are 110% and 90% of the base interest rate. When the bank interest rate is 3.07% ((2.763% + 3.377%)/2 = 3.07%) and the fuzzy cost of common equity (FKs) is 3.0597%, this is calculated by the cube root of the derived value ($\sqrt[3]{2.763\times 3.07\times 3.377}$).

The table also contains a fixed fuzzy tax rate (FT) set to [25%, 25% and 25%]. The last column gives the FWACC calculated under different weight combinations and presented as triangular fuzzy numbers. As seen from the table, the debt ratio plays a significant role in the trend of FWACC, showing a pattern of first decreasing and then growing. This trend reflects the impact of capital structure on the company’s overall capital cost, a crucial insight into financial decision-making. These FWACC data will be used in the FMIRR equation to calculate the annual rate of return. Ultimately, these data will be applied to the FNMOPM to evaluate the TCQ of SPVS and the CE of advertising billboards. This approach provides an essential reference for project SPVS financial decisions while optimizing capital structure and environmental impact.

Based on the FWACC content mentioned above, the next step is performing an α-cut to determine which structure meets the most appropriate weighted capital cost. Therefore, α-cuts are performed from 0 to 1 at intervals of 0.2. The procedure involves calculating lower and upper bounds and aggregating the data to derive the FWACC value. In

Table 4. FWACC using α-cuts defuzzification.

Debt Ratio (%)	α-Cut	FWACC (%)	${\mathbf{I}}^{\mathbf{L}}$(FWACC)	${\mathbf{I}}^{\mathbf{R}}$(FWACC)	$\mathbf{R}$(FWACC)
0	0	[1.98, 2.05]	1.997398	2.032398	4.029797
	0.2	[1.9869, 2.0429]
	0.4	[1.9939, 2.0359]
	0.6	[2.0008, 2.0288]
	0.8	[2.0078, 2.0218]
	1	[2.0147, 2.0147]
10	0	[1.8110, 1.9138]	1.836245	1.887620	3.723865
	0.2	[1.8211, 1.9033]
	0.4	[1.8312, 1.8929]
	0.6	[1.8413, 1.8824]
	0.8	[1.8514, 1.8719]
	1	[1.8615, 1.8615]
20	0	[1.6402, 1.7679]	1.671369	1.735214	3.406584
	0.2	[1.6527, 1.7548]
	0.4	[1.6651, 1.7418]
	0.6	[1.6776, 1.7287]
	0.8	[1.6901, 1.7156]
	1	[1.7025, 1.7025]
30	0	[1.4675, 1.6124]	1.502771	1.575181	3.077952
	0.2	[1.4816, 1.5975]
	0.4	[1.4957, 1.5826]
	0.6	[1.5098, 1.5677]
	0.8	[1.5239, 1.5529]
	1	[1.5380, 1.5380]
40	0	* [1.2930, 1.4472]	1.330450	1.407520	* 2.737970
	0.2	[1.3080, 1.4313]
	0.4	[1.3230, 1.4155]
	0.6	[1.3379, 1.3996]
	0.8	[1.3529, 1.3837]
	1	* [1.3679, 1.3679]
50	0	[1.3147, 1.4773]	1.354146	1.435471	2.789618
	0.2	[1.3305, 1.4606]
	0.4	[1.3463, 1.4438]
	0.6	[1.3620, 1.4271]
	0.8	[1.3778, 1.4104]
	1	[1.3936, 1.3936]
60	0	[1.4247, 1.7198]	1.516647	1.664152	3.180800
	0.2	[1.4615, 1.6975]
	0.4	[1.4983, 1.6753]
	0.6	[1.5350, 1.6530]
	0.8	[1.5718, 1.6308]
	1	[1.6086, 1.6086]
70	0	[1.5500, 1.9718]	1.689542	1.900461	3.590004
	0.2	[1.6058, 1.9433]
	0.4	[1.6616, 1.9147]
	0.6	[1.7175, 1.8862]
	0.8	[1.7733, 1.8576]
	1	[1.8291, 1.8291]

Note: The minimal value of R(FWACC), 2.737970, is denoted by an asterisk (*).

, the FWACC occurs when the debt ratio is 40%, and its interest rate cost will fall between 1.2930% and 1.4472%.

Table 5. Presentation of the revenues generated by the FMIRR.

Year	FMIRR (%)	Year	FMIRR (%)
1	[−89.6574, −89.6919, −89.6574]	11	[2.4755, 2.5863, 2.7701]
2	[−54.2058, −54.2490, −54.1366]	12	[3.1343, 3.2503, 3.4347]
3	[−31.6772, −31.6866, −31.5387]	13	[3.6399, 3.7604, 3.9454]
4	[−18.9670, −18.9451, −18.7812]	14	[4.0316, 4.1561, 4.3415]
5	[−11.3825, −11.3367, −11.1647]	15	[4.3372, 4.4651, 4.6510]
6	[−6.5726, −6.5088, −6.3322]	16	[4.5766, 4.7075, 4.8940]
7	[−3.3668, −3.2890, −3.1098]	17	[4.7645, 4.8983, 5.0852]
8	[−1.1448, −1.0561, −0.8751]	18	[4.9120, 5.0483, 5.2357]
9	[0.4435, 0.5410, 0.7232]	19	[5.0273, 5.1660, 5.3539]
10	[1.6068, 1.7116, 1.8946]	20	[5.1170, 5.2579, 5.4463]

shows the FMIRR values over 20 years. Significant negative FMIRR values were obtained between the first and eighth years, with an extreme low of roughly −89.65% in the first year. In the ninth year, it has a breakeven point of 0.7232%. By the twentieth year, the FMIRR has reached 5.4463%. The consistent growth in FMIRR from the ninth year to the twentieth year reflects rising revenue expectations for increasing interest rates. As shown in FMIRR, SPVS provides the power to support electricity for billboards.

Table 6. The shortening unit of Project SPVS exhibits nonlinear variations characterized by fuzzy numbers.

Project	Time (Days)			Cost (TWD)			Quality (%)
	${({\mathit{c}}_{\mathit{S}\mathit{P}\mathit{V}\mathit{S}})}_{\mathit{\alpha }}^{\mathit{L}}$	${({\mathit{c}}_{\mathit{S}\mathit{P}\mathit{V}\mathit{S}})}_{\mathit{\alpha }}^{\mathit{M}}$	${({\mathit{c}}_{\mathit{S}\mathit{P}\mathit{V}\mathit{S}})}_{\mathit{\alpha }}^{\mathit{U}}$	${({\mathit{c}}_{\mathit{S}\mathit{P}\mathit{V}\mathit{S}})}_{\mathit{\alpha }}^{\mathit{L}}$	${({\mathit{c}}_{\mathit{S}\mathit{P}\mathit{V}\mathit{S}})}_{\mathit{\alpha }}^{\mathit{M}}$	${({\mathit{c}}_{\mathit{S}\mathit{P}\mathit{V}\mathit{S}})}_{\mathit{\alpha }}^{\mathit{U}}$	${({\mathit{c}}_{\mathit{S}\mathit{P}\mathit{V}\mathit{S}})}_{\mathit{\alpha }}^{\mathit{L}}$	${({\mathit{c}}_{\mathit{S}\mathit{P}\mathit{V}\mathit{S}})}_{\mathit{\alpha }}^{\mathit{M}}$	${({\mathit{c}}_{\mathit{S}\mathit{P}\mathit{V}\mathit{S}})}_{\mathit{\alpha }}^{\mathit{U}}$
1	78.000	64.120	52.000	25,660,000.000	26,673,234.532	27,712,800.000	90.000	87.480	85.000
2	78.000	64.690	53.000	25,660,000.000	26,634,733.190	27,633,846.000	90.000	87.980	86.000
3	83.000	66.880	53.000	26,686,400.000	26,668,847.486	28,821,312.000	95.000	93.490	92.000
Project	Unit time cost				Unit time quality
	${(c_{SPVS})}_{\alpha }^L$		${(c_{SPVS})}_{\alpha }^M$		${(c_{SPVS})}_{\alpha }^U$	${(c_{SPVS})}_{\alpha }^L$	${(c_{SPVS})}_{\alpha }^M$	${(c_{SPVS})}_{\alpha }^U$
1	71,058.465		78,689.788		86,849.235	−0.171	−0.189	−0.209
2	71,058.456		78,689.778		86,849.224	−0.180	−0.199	−0.220
3	64,046.700		70,925.722		78,279.300	−0.090	−0.100	−0.110

shows the accelerated shortening of construction time. The time of Project 1 was reduced from 78 days to 52 days, the cost increased from TWD 25,660,000 to TWD 27,712,800 and the quality dropped from 90% to 85%. The time of Project 2 was shortened from 83 days to 53 days. The cost increased from TWD 26,668,400 to TWD −28,821,312 and the quality decreased from 92% to 95%. The natural environment is undergoing ‘non-linear changes’, which refer to changes that do not follow a straight line or a predictable pattern. Due to the accelerated reduction of working hours, the unit time cost of Project 1 increased from TWD 73,653.63 to TWD 90,323.20, and the unit time quality dropped from −0.18 to −0.22; the unit time cost of Project 2 dropped from TWD 64,046.70 increased to TWD 78,279.30 and the quality per unit time dropped from −0.09 to −0.11. Addressing the significance of multiple objectives and the associated uncertainty factors, a model analysis will be conducted based on this data as an empirical foundation.

Table 7. Defuzzification to shorten the construction period.

α	Project
	1	2	3
0	[78, 52]	[78, 58]	[83, 53]
0.2	[75.224, 54.424]	[75.338, 55.338]	[79.776, 55.776]
0.4	[72.448, 56.848]	[72.676, 57.676]	[76.552, 58.552]
0.6	[69.672, 59.272]	[70.014, 60.014]	[73.328, 61.328]
0.8	[66.896, 61.696]	[67.352, 62.352]	[70.104, 64.104]
1	[64.12, 64.12]	[64.69, 66.88]	[66.88, 66.88]
Ranking	* 129.12	132.69	134.88

Note: The asterisk (*) indicates that Project 1 can achieve better construction time reduction.

displays the outcomes of three projects that utilized fuzzy methods to determine the construction duration. Project 1’s anticipated construction duration spans from 78 to 52 days, with a median of 64.12 days; Project 2’s projected construction timeframe extends from 78 to 53 days, with a normal of 64.69 days and Project 3’s projected construction interval varies from 83 to 53 days, with a median of 66.88 days. Following defuzzification, Project 1 attained a score of 129.12, Project 2 achieved a score of 130.19 and Project 3 recorded a score of 134.88. The reduced score suggests that Project 1’s reduction in the number of days is appropriate. This strategy enables project managers to comprehend the project’s timeline more effectively and make more informed decisions by reducing construction duration.

Table 8. Defuzzification cost ranking analysis.

α	Project
	1	2	3
0	[25,660,000, 27,712,800]	[25,660,000, 27,633,846]	[26,686,400, 28,821,312]
0.2	[25,862,647, 27,504,887]	[25,854,946, 27,434,023]	[26,682,889, 28,390,819]
0.4	[26,065,294, 27,296,974]	[26,049,893, 27,234,200]	[26,679,379, 27,960,326]
0.6	[26,267,941, 27,089,061]	[26,244,839, 27,034,378]	[26,675,868, 27,529,833]
0.8	[26,470,588, 26,881,148]	[26,439,786, 26,834,555]	[26,672,358, 27,099,340]
1	[26,673,235, 26,673,235]	[26,634,733, 26,634,733]	[26,668,847, 26,668,847]
Ranking	53,359,634.532	53,281,656.190	54,422,703.496

shows the defuzzification cost ranking analysis, which evaluates the ranking values of the three projects 1, 2 and 3 under different fuzziness levels as 53,359,634.532, 53,281,656.190 and 54,422,703.496. These results show that Project 2 has the highest cost-effectiveness and the lowest ranking score, followed by Project 1, while Project 3 has the highest cost. This analysis highlights the advantages of Project 2 in minimizing expenses, providing a key reference for decision makers aiming to optimize the SPVS construction process.

There are three projects (Project 1, Project 2 and Project 3) with different levels of blurring.

Table 9. Defuzzification quality ranking analysis.

α	Project
	1	2	3
0	[90, 85]	[90, 86]	[95, 92]
0.2	[89.496, 85.496]	[89.596, 86.396]	[94.698, 92.298]
0.4	[88.992, 85.992]	[89.192, 86.792]	[94.396, 92.596]
0.6	[88.488, 86.488]	[88.788, 87.188]	[94.094, 92.894]
0.8	[87.984, 86.984]	[88.384, 87.584]	[93.792, 93.192]
1	[87.48, 87.48]	[87.98, 87.98]	[93.49, 93.49]
Ranking	174.98	175.98	186.99

displays the deblurred quality ranking analysis, which rates the quality level of each project in terms of a percentage. For Project 1, the ranking score is 174.98. Project 2 ranks at 175.98. Project 3 ranks at 186.99. These findings suggest that Project 3’s higher ranking indicates that it experienced the least decline in quality despite accelerated construction, which is consistent with Chiang’s [18] assertion that quality is fundamental to project success. The table provides valuable insights for decision makers, highlighting that Project 3 is the preferred option when quality is a priority, while Projects 1 and 2 can achieve comparable quality with additional investment.

Table 10. Defuzzification unit time cost ranking analysis.

α	Project
	1	2	3
0	[73,653.63829, 90,323.2]	[71,058.456, 86,849.224]	[64,046.7, 78,279.3]
0.2	[75,290.38581, 88,626.03518]	[72,584.72047, 85,217.33487]	[65,422.35898, 76,808.43898]
0.4	[76,927.13332, 86,928.87035]	[74,110.98494, 83,585.44574]	[66,798.01795, 75,337.57795]
0.6	[78,563.88084, 85,231.70553]	[75,637.24942, 81,953.55662]	[68,173.67693, 73,866.71693]
0.8	[80,200.62836, 83,534.5407]	[77,163.51389, 80,321.66749]	[69,549.3359, 72,395.8559]
1	[81,837.37588, 81,837.37588]	[78,689.77836, 78,689.77836]	[70,924.99488, 70,924.99488]
Ranking	163,825.795	157,643.618	142,087.994

presents the fuzzy unit time cost ranking analysis, which assesses the variations in per-unit time costs for three projects (Project 1, Project 2 and Project 3) across varying levels of fuzziness. The table presents the ambiguous intervals of the unit time cost (in TWD/day) and computes the ranking score to assess cost efficiency over a period. A ranking score of 163,825.795 was achieved for Project 1. Project 2 possesses a ranking score of 157,643.618, whereas Project 3 has a ranking score of 142,087.994. The rankings indicate that Project 1 exhibits the most cost variability across time, signifying increased sensitivity to time compression, whilst Project 3 demonstrates the lowest cost volatility, suggesting enhanced cost stability over the expedited timetable. This approach assists decision makers in comprehending the trade-off between time reduction and expense escalation.

The defuzzified unit time quality ranking analysis for the three projects is shown in

Table 11. Defuzzification unit time quality ranking analysis.

α	Project
	1	2	3
0	[−0.171, −0.209]	[−0.180, −0.220]	[−0.090, −0.110]
0.2	[−0.209, −0.205]	[−0.183, −0.215]	[−0.091, −0.107]
0.4	[−0.178, −0.201]	[−0.187, −0.211]	[−0.093, −0.105]
0.6	[−0.182, −0.197]	[−0.191, −0.207]	[−0.095, −0.103]
0.8	[−0.185, −0.193]	[−0.195, −0.203]	[−0.097, −0.101]
1	[−0.189, −0.189]	[−0.199, −0.199]	[−0.099, −0.099]
Ranking	−0.379	−0.399	−0.199

. It looks at the rate of quality change in unit time (as a percentage per day). Negative numbers indicate a decline in quality resulting from an abbreviated building timeline. The quality ranking score unit time for Project 1 is −0.379. Project 2 possesses a score of −0.399, while Project 3 holds a grade of −0.199. The ranking values indicate that Project 3 exhibits the least quality degradation unit time (−0.199), followed by Project 1 (−0.379) and Project 2 (−0.399), signifying that Project 3 is the least impacted by time compression regarding quality loss. In contrast, Project 2 encountered the most significant decline in quality, indicating a more substantial trade-off when emphasizing speed. This table offers critical insights into the impact of time reduction on quality, assisting decision makers in reconciling speed and quality objectives.

After analysis based on fuzzy theory, this study revealed unique strengths among the three projects. Project 1 performed well in terms of time and cost efficiency, with significant savings potential, as evidenced by its ranking of 129.12 in construction time reduction and 53,359,634.532 in cost. In contrast, Project 2 performs well in terms of quality maintenance, unit time–cost stability, and minimal quality degradation, with a quality ranking of 175.98, a unit time–cost ranking of 157,643.618, and a unit time quality ranking of −0.399. It has been proven that shortening the construction period can effectively reduce costs and CO₂ emissions when quality is prioritized over speed; Project 2 shows less quality loss and lower unit cost impact. Overall, Project 1 was more cost-effective, driven by lower initial costs and efficient time management, resulting in significant cost savings. However, Project 2 provides superior quality results and may justify the higher cost for applications where durability is critical. The choice between these projects depends on whether cost-saving or quality is a priority. Fuzzy analysis of the time–cost–quality (TCQ) framework shows that accelerated construction has a greater impact on cost and quality for Project 1 than for Project 2. The fuzzy numerical algorithm further quantified the cost savings of 42,293,058.58 TWD resulting from the accelerated progress of Project 1. The initial investment costs range from −24,132,589.16 TWD to −35,848,857.80 TWD (negative numbers mean expenditure), but the reasonable investment baseline of 27,712,800 TWD fits with Project 1’s cost profile, which makes it more likely to happen. Therefore, this study uses Project 1 as a case to evaluate fuzzy nonlinear multi-objective programming.

$$\begin{gathered} C_{T,F1}=a_{F1}\times {(t_{TR1})}^2+b_{F1} \\ {\displaystyle \frac{c_{shortened,F1}-c_{regular,F1}}{t_{shortened,F1}^2-t_{regular,F1}^2}}\times {(t_{TR1})}^2+{\displaystyle \frac{c_{regular,F1}\times t_{shortened,F1}^2-c_{shortened,F1}\times t_{regular,F1}^2}{t_{shortened,F1}^2-t_{regular,F1}^2}} \\ =\left(\left[971.74,-605.31,-2186.41\right]\right)\times {77.9}^2 \\ +{\displaystyle \frac{\left(\left[\mathrm{23,094,000.00},\mathrm{25,574,179.96},\mathrm{28,226,000.00}\right]\times ({\left[52,52,52\right]}^2\right)-\left(\left[\mathrm{24,941,520.00},\mathrm{27,620,114.36},\mathrm{30,484,080.00}\right]\times {\left[78,78,78\right]}^2\right)}{({\left[52,52,52\right]}^2-{\left[78,78,78\right]}^2)}} \\ =\left(\left[971.74,-605.31,-2186.41\right]\right)\times {77.9}^2+\left(\left[\mathrm{36,396,144.00},-\mathrm{20,459,343.97},-\mathrm{22,580,800.00}\right]\right) \\ =\left[\mathrm{42,293,058.58},-\mathrm{24,132,589.16},-\mathrm{35,848,857.80}\right]\mathrm{TWD} \\ \left(\mathrm{The}\mathrm{time}\mathrm{costs}\mathrm{associated}\mathrm{with}\mathrm{fuzzy}\mathrm{nonlinear}\mathrm{variations}\mathrm{in}\mathrm{Project}1\right). \end{gathered}$$

(12)

$$\begin{gathered} a_{F1}={\displaystyle \frac{c_{shortened,F1}-c_{regular,F1}}{t_{shortened,1}^2-t_{regular,1}^2}}={\displaystyle \frac{\left(\left[\mathrm{24,941,520.00},\mathrm{27,620,114.36},\mathrm{30,484,080.00}]-[\mathrm{23,094,000.00},\mathrm{25,574,179.96},\mathrm{28,226,000.00}\right]\right)}{({\left[52,52,52\right]}^2-{\left[78,78,78\right]}^2)}} \\ =\left[971.74,-605.31,-2186.41\right] \\ \Delta C_{F1}={\displaystyle \frac{c_{shortened,F1}-c_{regular,F1}}{t_{regular,F1}-t_{shortened,F1}}} \\ ={\displaystyle \frac{\left(\left[\mathrm{24,941,520.00},\mathrm{27,620,114.36},\mathrm{30,484,080.00}\right]-\left[\mathrm{23,094,000.00},\mathrm{25,574,179.96},\mathrm{28,226,000.00}\right]\right)}{\left(\left[78,78,78\right]-\left[52,52,52\right]\right)}} \\ =\left[-\mathrm{126,326.15},\mathrm{78,689.78},\mathrm{284,233.85}\right] \mathrm{TWD} \\ \left(\mathrm{Shortened}\mathrm{unit}\mathrm{time}\mathrm{cost}\mathrm{in}\mathrm{the}\mathrm{fuzzy}\mathrm{equation}\right). \end{gathered}$$

(13)

Next, an evaluation of the quality differences associated with Project 1 was conducted. By calculating the change in mass relative to the change in time and subsequently multiplying it by the original mass, the resulting quality percentage was found to range between 81.05% and 98.91%. This finding indicates that a reduction in project duration significantly influences quality. However, it is imperative to exercise caution to ensure that any reduction in time does not result in quality levels exceeding acceptable thresholds. Quality standards stipulate a minimum requirement of 90%. The lower limit of this range suggests that the quality improvement may not be sufficient to meet the established standards, whereas the upper limit indicates that quality can still conform to the required standards despite a reduction in construction time. Furthermore, a fuzzy nonlinear model is employed to assess the cost variations resulting from changes in quality. Due to the abbreviated construction timeline, the daily costs are projected to fluctuate between −29,433.48 (where a negative value signifies an increase in expenditure) and 35,842.81 (where a positive value denotes cost savings). This analysis reveals that cost variations are influenced by the reduction in time, changes in quality and adherence to quality standards. Potential strategies for realizing cost savings include optimizing resource allocation and enhancing operational efficiency. In summary, this model analysis serves as a valuable reference for decision making among project managers.

$$\begin{gathered} Q_{T,F1}=q_{regular,F1}+{\displaystyle \frac{q_{regular,F1}-q_{shortened,F1}}{t_{regular,F1}-t_{shortened,F1}}}\times \left(t_{TR1}-t_{regular,1}\right) \\ =\left(\left[81.00,89.70,99.00\right]\right)+{\displaystyle \frac{\left(\left[81.00,89.70,99.00]-[76.50,85.00,93.50\right]\right)}{\left(\left[78,78,78\right]-\left[52,52,52\right]\right)}}\times \left(77.9-78\right) \\ =\left(\left[81.00,89.70,99.00\right]\right)+\left(\left[0.05,-0.02,-0.09\right]\right)=\left[81.05,89.68,98.91\right]\% \\ \left(\mathrm{The}\mathrm{effects}\mathrm{of}\mathrm{shortening}\mathrm{construction}\mathrm{time}\mathrm{on}\mathrm{the}\mathrm{quality}\mathrm{of}\mathrm{Project}1\right). \end{gathered}$$

(14)

$$\begin{gathered} C_{Q,F1}=\left({\displaystyle \frac{\Delta C_{Q_{regular,F1}}-\Delta C_{Q_{shortened,F1}}}{t_{regular,1}-t_{shortened,1}}}\right)\times \left(t_{TR1}-t_{shortened,1}\right)+\Delta C_{Q_{regular,F1}} \\ \times \left(q_{standard, 1-}{Q}_{T,F1}\right) \\ =\left({\displaystyle \frac{\left[3289.50, 3642.78, 4020.50]-[5643.00,6249.03,6897.00\right]}{\left(\left[78,78,78]-[52,52,52\right]\right)}}\right)\times \left(84.9-85\right)+ \\ \left[3289.50, 3642.78, 4020.50\right]\times \left(90-\left[81.05,89.68,98.91\right]\right) \\ =\left[13.88,10.02,6.24\right]+\left[-29447.35,-1162.33,35836.57\right] \\ =\left[-29433.48,-1152.31,35842.81\right] \mathrm{TWD} \\ \left(\mathrm{The}\mathrm{cost}\mathrm{impact}\mathrm{of}\mathrm{shortening}\mathrm{construction}\mathrm{time}\mathrm{on}\mathrm{quality}\mathrm{improvement}\mathrm{for}\mathrm{Project}1\right). \end{gathered}$$

(15)

The timeline of Project 1 was shortened by 26 days. The model’s results indicate that the quality enhancement price varied between TWD −29,433.48 and TWD 35,842.81. The daily electricity use was measured at 17.350 kilowatts, resulting in carbon emissions of 0.495 kg per kilowatt. Thus, the cumulative carbon emissions over the 26 days totaled 223.11 kg. The daily cost for quality improvement was TWD 29,433.48, resulting in a cumulative amount of TWD 765,270.48 over 26 days. The cost per ton of carbon is calculated by dividing the entire investment cost by the total carbon emissions. The price per kilogram of carbon increased by TWD 3430.01 due to a reduction in construction time by 26 days and a decrease in carbon emissions by 223.11 kg. Additionally, quality enhancements were realized, resulting in a total decrease of 223.11 kg of carbon emissions per kilogram within the same timeframe. Nevertheless, efficient quality management methods have significantly reduced overall expenses, enabling each ton of carbon to compensate for the initial investment cost of TWD 4176.92.

5. Discussion

The pressing need to lower carbon emissions fuel the overarching goal of environmental sustainability, a challenge calling for innovative management strategies and strong academic frameworks. This study furthers this aim by identifying investment opportunities that harness solar photovoltaic systems (SPVS) to power advertising billboards on commercial building rooftops in Taiwan. Unlike conventional methods relying on grid electricity purchases, this research unveils a pioneering model that uses advertising revenue to fund SPVS deployment. By integrating the time–cost–quality (TCQ) model during construction with carbon emissions considerations, the approach delivers a dual benefit: reducing environmental impact while securing financial viability. This blend of revenue generation and carbon reduction sets the proposed model apart from traditional strategies, establishing it as a progressive solution for sustainable urban development. From an academic point of view, this study is different because it combines fuzzy nonlinear multi-objective programming with the RTCQCE framework. This is different from previous research that usually looks at the environmental and financial aspects separately. Traditional methods, like linear cost–benefit analyses or static lifecycle assessments (Chiang [19]), make things too easy by only looking at one phase at a time or assuming that trends will run in a straight line. On the other hand, this method uses fuzzy theory (Bellman and Zadeh [13]), nonlinear optimization (Zimmermann [34]; Huang [35]) and multi-objective programming to show how revenue, time, cost, quality and carbon emissions change over time and in uncertain situations. Using fuzzy modified internal rate of return (FMIRR) and α-cut defuzzification techniques, the model accurately shows investment returns and trade-offs. It does this better than standard internal rate of return (IRR) methods, which have trouble with different interest rates or situations that are at odds with each other (Sampaio Filho et al. [36]). This holistic approach bridges a key literature gap, offering a flexible tool that aligns with the nonlinear realities of sustainable project management. It advances energy economics by connecting revenue streams to green investments, enriches environmental management with integrated carbon evaluations and elevates construction project management by tackling multi-dimensional trade-offs, paving the way for interdisciplinary exploration. The management implications of this analysis provide actionable guidance for stakeholders, including commercial building owners, advertisers and policymakers. For building owners, adopting SPVS powered by advertising billboards through the RTCQCE framework cuts energy costs and enhances sustainability. Analysis indicates a cost saving of TWD 42,293,058.58 via accelerated construction, plus intangible benefits like a boosted green reputation. To optimize billboard operations, building owners can focus on swift SPVS installation (e.g., reducing construction to 26 days, cutting carbon emissions by 223.11 kg, Proposition 5) while keeping quality above 90% (spanning 81.05% to 98.91%, Proposition 4). Allocating resources strategically, such as adding labor during peak stages, balances cost rises (TWD 3430.01 to TWD 4176.92 per kilogram of carbon) with durability, ensuring long-term gains and market standing. Advertisers collaborating with SPVS-powered billboards can elevate corporate social responsibility profiles, meeting consumer demand for eco-friendly brands. The RTCQCE framework shows profitability from the ninth year (FMIRR of 0.7232%) to the twentieth year (5.4463%, Proposition 1), backing long-term investment choices. Advertisers can enhance billboard management by targeting high-revenue urban sites and negotiating cost-sharing deals with building owners for SPVS setup, maximizing economic and environmental outcomes. This positions advertisers as sustainability pioneers, sharpening competitive advantage. Policymakers, such as the Taiwan Power Company, gain a versatile tool to foster green infrastructure. The model justifies higher procurement prices for SPVS-generated electricity, as Zhu et al. [37] note, leveraging advertising revenue forecasts to support subsidies. Unlike rigid subsidy structures, this adaptable framework adjusts to market changes, spurring private investment. Policymakers can introduce tiered pricing linked to carbon cuts (e.g., rewarding a 223.11 kg reduction with higher rates) and offer tax breaks for projects hitting RTCQCE quality benchmarks (above 90%), expanding low-carbon efforts across Taiwan’s cities. Investors gain clarity from FMIRR’s transparent valuation, validating initial costs (TWD −24,132,589.16 to TWD −35,848,857.80) with defined return schedules. The RTCQCE framework’s versatility lets stakeholders customize strategies: Project 1 shines in cost-effectiveness (daily costs from TWD −29,433.48 to TWD −1152.31, Proposition 4), while Project 2 excels in quality. Decision makers have the ability to select options that align with their priorities, such as cost savings for immediate benefits, durability for long-term value and the optimization of billboard operations for both economic and environmental harmony. This stakeholder-focused approach turns billboards into emblems of sustainability, supporting global economic and ecological aims. Academically, this research expands horizons with fresh angles. In energy economics, it shows advertising revenue funding renewable shifts, questioning sole reliance on subsidies. In environmental management, it embeds carbon emissions into project planning, delivering a unified assessment tool. In construction project management, it pioneers nonlinear trade-off analysis, moving past linear assumptions and sparking interdisciplinary studies. Together, these contributions establish the RTCQCE framework as a vital instrument for sustainable urban development, linking theory to practice.

6. Conclusions

6.1. Research Conclusions

This study investigates the application of solar photovoltaic systems (SPVS) for powering advertising billboards. Utilizing fuzzy theory, the investment feasibility of SPVS is evaluated based on long-term return rates. Additionally, fuzzy nonlinear multi-objective programming is employed to analyze the impacts of construction duration on carbon emissions, quality and cost. Project 1 demonstrates that the return on investment for SPVS becomes profitable in the ninth year, starting with an initial return rate of 0.7232%, which increases to 5.4463% by the twentieth year. The annual positive returns consistently rise from the ninth to the twentieth year, supporting Proposition 1, which posits that investments in SPVS can yield favorable long-term returns.

Furthermore, this study employs a fuzzy nonlinear multi-objective model to investigate the trade-offs between TCQ and carbon emissions to minimize construction time. The research using a fuzzy numerical algorithm indicates that cost savings from reduced construction time amount to TWD 42,293,058.58. Conversely, documented cost expenditures are TWD −24,132,589.16 and TWD −35,848,857.80. Considering the investment amount of TWD −27,712,800 (noting that the negative sign indicates expenditure); the TWD 42,293,058.58 surpasses TWD −24,132,589.16. Consequently, the investment expenditures range from TWD −24,132,589.16 to TWD −35,848,857.80. Proposition 2 suggests that the negative cost figures reflect increased expenses associated with expedited construction timelines.

According to Proposition 3, the quality of construction is anticipated to fall within the range of 81.05% to 98.91% when the construction period is shortened. This finding supports Proposition 3, indicating a potential compromise in quality. Proposition 4 asserts that quality must remain above the 90% safety threshold as construction progresses, with costs escalating daily, falling within the range of TWD −29,433.48 to TWD −1152.31 and TWD 35,842.81. The negative values represent current expenses, indicating an increase in costs. The acceleration of construction is shown to influence the reduction of carbon emissions. Specifically, reducing the construction time to 26 days can decrease carbon emissions by 223.11 kg, with an associated increase in investment costs of TWD 3430.01 per kilogram of carbon. With effective resource allocation and management, it is feasible to offset the original investment of TWD 4176.92 per kilogram of carbon produced, in alignment with Proposition 5.

Regarding Proposition 6, using SPVS to power advertising billboards on the rooftops of commercial buildings is identified as a niche market that merits investment. The scenarios presented in Propositions 1 and 5 further substantiate that Proposition 6 represents a viable niche market with potential investment revenues.

Building on this, the RTCQCE framework’s practical implications extend beyond Taiwan, offering a robust tool for sustainable urban development globally. Its adaptability to diverse geographical and policy contexts enhances its value. In regions with abundant sunlight (e.g., Southeast Asia or the Middle East), the framework can prioritize rapid SPVS deployment to maximize energy yield, adjusting construction timelines and costs to local labor and material availability. In contrast, in temperate zones with less solar potential (e.g., Northern Europe), it can emphasize quality and durability over speed, integrating battery storage to offset lower solar output.

Policy-wise, the framework adjusts to varied incentives: in subsidy-rich markets like Germany, it can leverage feed-in tariffs to shorten payback periods, while in emerging economies with limited subsidies, such as India, it can harness advertising revenue to offset initial costs, as demonstrated in Taiwan. This flexibility enables stakeholders such as building owners, advertisers and policymakers to tailor the RTCQCE model to local conditions, balancing economic returns with environmental goals.

Using a fuzzy nonlinear multi-objective programming model to assess the effects of shortened construction timelines on advertising billboards brings several notable contributions to research. Firstly, it creates a comprehensive framework for decision making within sustainable advertising practices. Secondly, this study fills existing gaps in the literature and enriches the body of research concerning financial investment, time efficiency, cost assessment, quality improvement and carbon emissions reduction. Thirdly, it offers crucial insights for businesses seeking to maximize return on investment while reducing their environmental impact. Overall, the model underscores the importance of incorporating financial and carbon emissions considerations into the decision-making process for constructing advertising billboards. This approach can help businesses make more informed decisions that align with sustainability and profitability goals.

6.2. Research Recommendations

This study provides significant insights into the potential applications of SPVS in the commercial building sector, particularly in the context of powering advertising billboards. The findings indicate that investments in SPVS can yield long-term financial advantages. To further enhance this research, the following recommendations are proposed:

Incorporate policy analysis: explore the impact of government policies, such as feed-in tariffs and tax incentives, on the economic viability and environmental benefits of SPVS installations.

Investigate engineering economics: conduct sensitivity analysis to minimize initial costs and expedite the adoption of solar photovoltaics. Executing these recommendations will improve future research, advance sustainable energy solutions, and facilitate evidence-based decision making in the commercial building sector.