A Fuzzy Mathematical Decision Model for Residential Rooftop PV Adoption: The HEART Approach

M24-1 2nd KOCAELI SCIENCE CONGRESS (KOSC-2025) Kocaeli University, Faculty of Arts and Sciences November 19-21, 2025, İzmit, Kocaeli, Türkiye A Fuzzy Mathematical Decision Model for Residential Rooftop PV Adoption: The HEART Approach Kaan Deveci 1Department of Energy, Science and Technology, Turkish-German University, Istanbul, Türkiye Corresponding author: [email protected] ORCID IDs: First Author: 0000-0003-0301-2296 DOI : 10.5281/zenodo.18024207 Abstract The decreasing cost of photovoltaic (PV) technologies, increasing electricity prices, and evolving environmental concerns have led homeowners to show greater interest in rooftop energy systems. However, adoption decisions depend on multiple economic, technical, and behavioral factors. This study develops a mathematical decision framework to model household investment behavior under uncertainty for Turkey by integrating intuitionistic fuzzy sets with the Hypervolume-based Evaluation and Ranking Technique (HEART). The uncertainty in household preferences is represented through linguistic evaluations provided by decision makers of the study, which are translated into intuitionistic fuzzy sets. The alternatives of decision framework are defined as investing in rooftop PV, rooftop PV with battery storage, and a no-investment baseline. The multi-criteria analysis revealed that no investment case is the most preferable decision under current Turkish conditions. This is predominantly driven by the high capital recovery risk, upfront cost burden, and significant procedural complexity associated with active investment alternatives. The findings strongly indicate that betterdesigned tariff regimes, targeted financing, and streamlined bureaucratic procedures are necessary to overcome the current barriers and accelerate widespread household adoption. Keywords: Investment decision modeling, rooftop photovoltaics adoption, multi-criteria decision making. 1. Introduction and Motivation Global electricity costs have risen markedly over the past decade, amplifying household exposure to price volatility and motivating interest in demand-side measures. At the same time, overall energy needs continue to grow with electrification trends (cooling, appliances, EV readiness), increasing the salience of reliable and affordable supply at the customer side. Technological progress has driven substantial cost declines in renewables, with photovoltaics (PV) experiencing some of the steepest learning rates among clean technologies. As module and balance-of-system costs have fallen, PV adoption has expanded worldwide, especially where retail tariffs and policy instruments translate technical potential into household value. M24-2 2nd Kocaeli Science Congress, November 19-21, 2025 Turkey shares the global decline in PV technology costs; however, household uptake has not matched international peers due to the prevailing retail pricing context—predominantly a single-rate (flat) tariff—and the way tariff and netting rules shape on-site value. Under flat pricing, the arbitrage value of behind-the-meter storage is limited, and PV savings hinge on self-consumption alignment rather than price spreads. Consequently, many households remain on the fence despite improving hardware affordability, particularly when upfront costs, perceived procedural frictions, and behavioral factors are considered alongside tariff design. These elements together create a mixed picture for the average homeowner. Homeowner investors considering investments in solar PV panels face a three-way choice: PV only, PV with battery, or no investment. The research goal is building an investment decision model of a compact decision view that remains transparent under uncertainty. Expert evaluations are expressed through decision makers’ linguistic terms (e.g., very low…very high), by which uncertainty is explicitly modeled via intuitionistic fuzzy sets. The contribution is twofold. First, the Turkish homeowner’s PV decision is framed as a multi-criteria problem with a clear set of nationally relevant criteria. Second, a solution under uncertainty and missing information is demonstrated using the HEART procedure. 2. Mathematical Framework / Material and Methods Fuzzy logic is central to defining and modeling uncertainty because it accommodates human linguistic judgments and approximate reasoning[1]. Yet ordinary fuzzy sets, which use only a single membership degree, can be insufficient for rating criteria since they do not separately capture counterevidence or residual doubt in expert assessments. To overcome this limitation, extended formulations are used. Intuitionistic fuzzy sets (IFS), introduced by Atanassov, represent each evaluation with both a membership and a non-membership degree, and interpret the remainder as hesitation [2]. In this way, support, opposition, and uncertainty are all recorded explicitly, leading to clearer and more reliable decision inputs. Previously, a new multi criteria decision making method, HEART is proposed to as a new multi criteria decision making method with IFS [3]. The solution procedure and flowchart of HEART method is given below. Step 1: Collect the Decision Maker (DM) evaluations on alternatives (𝐴󰆻, ), importance of each criterion (𝑤), and DM expertise for each criterion (𝑒) where 𝑙 (𝑙∈{1,2,…,𝑞}) corresponds to the number of DM,𝑖 (𝑖∈{1,2,…𝑛}) represents the number of alternatives, and 𝑗 (𝑗∈{1,2,…𝑚}) represents the number of criteria. Same notations also will be used in the below equations. Step 2: Aggregate DM evaluations on alternatives (𝐴󰆻, ) defined for each criterion (𝑒) by using IFA and obtain the aggregated evaluation matrix (𝐴󰆻) as given in Eq. 1. 𝐴󰆻 =𝐼𝐹𝐴𝐴󰆻, ,…;𝑒,…=󰇧∑𝜇 𝜇   ∑𝜇   ,∑𝜐 𝜇   ∑𝜇   󰇨 (1) Step 3: Aggregate DM evaluations on criteria importance (𝑤) and obtain the matrix of aggregated criteria importance (𝑤) as below. 𝑤=𝐼𝐹𝐴𝑤,…;𝑒,…=󰇧∑𝜇 𝜇   ∑𝜇   ,∑𝜐 𝜇   ∑𝜇   󰇨 (2) Step 4: Normalize the aggregated evaluation matrix (𝐴󰆻) by using Eq. 3 as follows [4]. 𝐴󰆻 =𝜇,𝜐 𝑁  =󰇫𝜇,𝜐, if j∈B 𝜐,𝜇, if j∈C (3) here B and C represent the set of benefit and cost criteria, respectively. M24-3 2nd Kocaeli Science Congress, November 19-21, 2025 Step 5: Calculate the weighted normalized evaluation matrix (𝑟 =𝑁  ⊗𝑤) by using Eq. 4 as follows. 𝑟 =(𝑥,𝜇 (𝑥)𝜇 (𝑥),𝜐 (𝑥)+𝜐 (𝑥)−𝜐 (𝑥)𝜐 (𝑥)|𝑥∈𝑈) (4) Step 6: Using weighted normalized evaluation matrix (𝑟) generate decision spaces (𝑈 and 𝑈) for each alternative and calculate the HV metric for each decision spaces (𝐻𝑉 and 𝐻𝑉) with respect to reference point (rj). Calculate HVnet. 𝑈 ={𝜇,𝜇,…,𝜇},𝑈 ={𝜐,𝜐,…,𝜐} (5) 𝐻𝑉, =󰇻𝑈, −𝑟󰇻   (6) 𝐻𝑉, =󰇻𝑈, −𝑟󰇻   (7) 𝐻𝑉, =𝐻𝑉, −𝐻𝑉, (8) Step 7: Rank the alternatives based on the HVnet values. C Start Obtain DM evaluations for alternatives and criteria Calculate the aggregated evaluation matrix Calculate the aggregated criteria matrix Calculate weighted aggregated evaluation matrix Calculate hyper-volume of each alternative w.r.t. R for Uµ and Uν. Calculate HVnet in hypervolumes obtained for Uµ and Uν of each alternative. Rank the alternatives based on the values of HVnet Normalize the weighted aggregated evaluation matrix Fig.1: The flowchart of HEART. 3. Problem Definition Household adoption of rooftop solar PV has been rising, driven primarily by declining technology costs and sustained increases in retail electricity prices. Homeowners face three mutually exclusive alternatives for meeting their electricity needs: invest in PV only (A1), invest in PV with battery storage (A2), or not invest and continue purchasing all electricity need from the grid (A3). The decision is evaluated across six criteria selected for nationwide relevance: capital recovery risk (costtype), upfront financial burden (cost-type), grid price sensitivity (benefit type), self-consumption potential (benefit type), behavioral and social influence (benefit type), and procedural complexity (cost type). Within this study, two decision makers (DMs) provide linguistic assessments that encode M24-4 2nd Kocaeli Science Congress, November 19-21, 2025 uncertainty to alternatives and criteria, as given in Table 1 and Table 2, respectively. These assessments are mapped to intuitionistic fuzzy sets and aggregated using HEART (Hypervolumebased Evaluation and Ranking Technique) to perform the MCDM, yielding a robust ranking of the three alternatives under Turkish conditions. A brief explanation of the criterion set used to rank the available energy alternatives is presented below. C1. Capital Recovery Risk: The uncertainty and temporal risk associated with recouping invested capital are reflected. Longer and more volatile recovery horizons are understood to increase risk (e.g., tariff volatility, weak self-consumption, financing terms). The no-investment alternative is evaluated as very low risk because no capital is exposed. C2. Upfront Financial Burden (cost-type): The immediate budget strain due to acquisition and installation is captured, with grants/discounts considered within the net upfront. Higher upfront amounts are understood to reduce affordability. The no-investment alternative is evaluated as zerolow burden. C3. Grid Price Sensitivity: This measures the household's vulnerability to fluctuations in retail electricity tariffs and potential price hikes. Alternatives with high self-sufficiency insulate the user from market volatility, whereas the "no investment" option leaves the household fully exposed (the worst value) to grid pricing dynamics and future tariff increases. C4. Self Consumption Potential: This criterion gauges the degree to which a household can meet its energy demand without relying on the external grid. A combination of PV and battery storage offers the highest level of independence and security of supply, while the "no investment" case represents zero autonomy (the worst value) and total reliance on grid availability. C5. Behavioral and social influence: Covers peer effects, environmental attitudes, and perceived technology risk. Visible neighborhood adoptions and pro-environmental values shift the decision toward investing. C6. Procedural Complexity (cost-type):The administrative and operational burden required for implementation is captured, including permitting, grid-interconnection procedures, metering changes, and on-site installation logistics. Lower complexity is preferred. The no-investment alternative is evaluated as very low complexity, whereas active investment options are evaluated according to current regulatory and technical steps. Table 1. Linguistic labels which are used to evaluate alternatives. Linguistic Term IF Value Extremely High (EH) ( 1 0 ) Very very high (VVH) (0.9 0.1) Very high (VH) (0.8 0.10) High (H) (0.7 0.2) Moderately high (MH) (0.6 0.3) Fair (F) (0.5 0.4) Moderately low (ML) (0.4 0.5) Low (L) (0.25 0.6) Very low (VL) (0.1 0.75) Very very low (VVL) (0.1 0.9) Table 2. Linguistic labels which are used to evaluate the importance of criteria. Linguistic Term IF Value Very important (VI) ( 0.9 0.10 ) Important (I) (0.75 0.2) Medium (M) (0.5 0.45) Unimportant (UI) (0.35 0.6) Very unimportant (VUI) (0.1 0.9) M24-5 2nd Kocaeli Science Congress, November 19-21, 2025 3. Results and Discussion DM evaluations on each alternative and criteria are given in Table 3 and Table 4, respectively. The aggregated DM evaluations on alternatives and criteria set are given in Table 5 and Table 6, respectively. Table 3: DM Evaluations on alternatives. A1 A2 A3 DM1 DM2 DM1 DM2 DM1 DM2 C1 VVH MH F VH (0 1) (0 1) C2 ML MH VVH VH (0 1) (0 1) C3 G G G G VVH VVH C4 H L VH F (0 1) (0 1) C5 H MH H MH L F C6 G G G G (1 0) (1 0) Table 4: DM evaluations on criteria set. C1 C2 C3 C4 C5 C6 DM1 I VI VI M M VI DM2 I I VI M M VI Table 5: Aggregated DM evaluations on alternatives. Criteria A1 A2 A3 C1 (0.75 0.2) (0.65, 0.25) (0, 1) C2 (0.5, 0.4) (0.85, 0.1) (0, 1) C3 (0.7, 0.2) (0.7, 0.2) (0.9, 0.1) C4 (0.475, 0.4) (0.65, 0.25) (0, 1) C5 (0.65, 0.25) (0.65, 0.25) (0.375, 0.5) C6 (0.7, 0.2) (0.7, 0.2) (1, 0) Table 6: Aggregated criteria importance. Criteria 𝑤   C1 (0.75 0.2) C2 (0.825 0.15) C3 ( 0.9 0.1 ) C4 (0.5 0.45) C5 (0.5 0.45) C6 ( 0.9 0.1 ) Once the aggregated matrices are obtained, the normalized aggregated DM evaluation and the weighted normalized aggregated DM evaluation matrices are obtained as given in Table 7 and Table 8, respectively. M24-6 2nd Kocaeli Science Congress, November 19-21, 2025 Table 7. Normalized aggregated DM evaluation matrix. A1 A2 A3 C1 (0.2,0.75) (0.25,0.65) (1,0) C2 (0.4,0.5) (0.1,0.85) (1,0) C3 (0.7,0.2) (0.7,0.2) (0.9,0.1) C4 (0.475,0.4) (0.65,0.25) (0,1) C5 (0.65,0.25) (0.65,0.25) (0.375,0.5) C6 (0.7,0.2) (0.7,0.2) (1,0) Table 8. Weighted normalized aggregated DM evaluation matrix. A1 A2 A3 C1 (0.15,0.8) (0.1875,0.72) (0.75,0.2) C2 (0.33,0.575) (0.0825,0.8725) (0.825,0.15) C3 (0.63,0.28) (0.63,0.28) (0.81,0.19) C4 (0.2375,0.67) (0.325,0.5875) (0,1) C5 (0.325,0.5875) (0.325,0.5875) (0.1875,0.725) C6 (0.63,0.28) (0.63,0.28) (0.9,0.1) Finally, the hypervolume values of the decision spaces are obtained as in Table 9. And the ranking order is obtained as A3≻A1≻A2, where A1 corresponds to PV only option, A2 is PV with battery option, and A3 corresponds to no investment. This ordering is explained by heightened capitalrecovery risk and upfront burden for the investment alternatives, together with limited tariff-induced demand under the current flat pricing. Consequently, PV-only was evaluated above PV+battery, while the no-investment option remained preferred overall. Table 9. HV values for decision spaces of the available alternatives. A1 A2 A3 𝐻𝑉  6.663 5.997 13.043 𝐻𝑉  12.314 13.298 6.232 𝐻𝑉  -5.659 -7.302 6.810 5. Conclusions A rooftop PV investment model was generated as a multi-criteria decision under uncertainty and solved with an intuitionistic fuzzy HEART procedure. Three alternatives are selected, investing in PV, investing in PV and battery, and no investment, against six nationally relevant criteria using linguistic inputs from two decision makers. The final ranking was A3 ≻ A1 ≻ A2. This ordering was driven by higher capital recovery risk and upfront burden for investments, limited tariff-induced demand under the current single-rate structure, modest self-consumption alignment, and greater procedural complexity for PV options. The outcome aligns with Turkey’s present diffusion pattern, where rooftop PV remains far from widespread household adoption. Policy levers were indicated: better-designed tariff regimes (e.g., netting or TOU spreads), targeted financing to lower upfront costs, and streamlined procedures to cut complexity. Such changes are expected to narrow the gap to A3, with PV only improving first and PV with battery becoming competitive under stronger TOU spreads or storage incentives. The future work can expand DM panels, segment weights, and financing detail, and track outcomes under evolving tariffs and incentives. M24-7 2nd Kocaeli Science Congress, November 19-21, 2025 References [1] Zadeh, Lotfi A. "Fuzzy logic and approximate reasoning: In memory of Grigore Moisil." Synthese, 30.3 (1975): 407-428. [2] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. vol. 20 (1) (1986) 87–96, https://doi.org/10.1016/S0165-0114(86)80034-3. [3] Deveci, Kaan, and Önder Güler. "Ranking intuitionistic fuzzy sets with hypervolume-based approach: An application for multi-criteria assessment of energy alternatives." Applied Soft Computing, 150 (2024): 111038. [4] Wang, Weize, and Xinwang Liu. "The multi-attribute decision making method based on intervalvalued intuitionistic fuzzy Einstein hybrid weighted geometric operator." Computers & Mathematics with Applications, 66.10 (2013): 1845-1856.