supply chain resilience.

The orchestration of modern healthcare supply chains exists in a state of perpetual tension between clinical necessity and economic constraint. For life-critical, perishable pharmaceuticals, ranging from volatile radiophar-maceuticals used in oncology to cold-chain dependent mRNA vaccines, the inventory management problem is far more than a mathematical exercise in cost minimization. It is an existential mandate. In this domain, the traditional Economic Order Quantity (EOQ) logic falters because the "cost" of a stock-out is not merely a lost sale or a back order penalty; it is a permanent, irreversible clinical failure. This creates what we term the Perishability-Service Paradox: the systemic requirement for near-perfect service levels in an environment defined by rapid product decay and profound epistemic uncertainty.

These systems’ vulnerability was exposed by the global disturbances of the early 2020s, showing that past information is not always a good guide to new challenges. The growing complexity of world logistics, according to Rezaei [9], requires a shift to "Best-Worst" Methodologies and more flexible preference modeling in dealing with multi-criteria disruptions. But even these sophisticated models have trouble when the information is uncertain, or "hesitant." In a hospital setting, a procurement officer’s estimate of a demand surge is rarely a single number or even a stable probability distribution. Instead, it is a collection of fragmented intuitions experts often oscillate between several possible membership degrees based on conflicting epidemiological reports.

This psychological and mathematical "hesitation" is the frontier of current uncertainty modeling. The foundational work on fuzzy sets by Zadeh [13] provided the initial language for vagueness, but it was Atanassov [1] who introduced Intuitionistic Fuzzy Sets (IFS) to account for the "non-membership" or hesitation margin. While IFS has been applied extensively to medical supply chains notably by Perez and Gomez [8] in the context of hospital resource allocation the framework is fundamentally limited by the linear constraint µ + ν ≤ 1. In high-pressure medical crises, the sum of an expert’s confidence (µ) and their skepticism (ν) often exceeds this linear boundary, requiring the "mathematical breathing room" provided by the Pythagorean membership space.

The introduction of Pythagorean Fuzzy Sets (PFS) by Yager [12] revolutionized this by expanding the constraint to µ + ν ≤ 1. This quadratic expansion allows for a more nuanced representation of "human-like" uncertainty, particularly in risk-averse environments like emergency medicine. Recent studies, such as those by Peng and Selig [7], have explored information measures for PFS, yet these models remain "single-valued" in their assessment of membership. They do not account for the hesitancy among multiple possible values that a group of medical experts might provide.

Furthermore, the perishability aspect introduces a non-linear decay function into the inventory state, a concept explored by Mirzazadeh and Gholami [6] in their work on green supply chains. While their research highlights the impact of deterioration on total cost, the intersection of Pythagorean membership and Hesitant elements remains a significant research gap. If a pharmacist hesitates between a 60% and 70% membership level for a deterioration rate, traditional models force a choice or an average, thereby losing the "richness" of the uncertainty. The necessity for our proposed Pythagorean Hesitant Fuzzy (PHF) model is further underscored by the limitations identified in contemporary literature. As noted by Behera [2], while triangular fuzzy models effectively capture general vagueness, future research must evolve toward hybrid uncertainty methods, such as hesitant fuzzy sets, to better accommodate real-life supply chain complexities. Our work directly answers this call by expanding the decision-making manifold into a quadratic hesitant space.

This paper bridges these disparate threads. We propose a completely novel Pythagorean Hesitant Fuzzy (PHF) inventory model. Our contribution is three-fold: (i) we mathematically define the PHF membership functions for demand and deterioration; (ii) we derive a non-linear objective function that minimizes the total fuzzy expected cost while maximizing a newly defined "Clinical Resilience Index"; and (iii) we provide a rigorous theoretical proof for the existence of an optimal solution under deep hesitation. By treating hesitation not as "statistical noise" but as a primary decision variable, we offer hospital administrators a robust tool that aligns mathematical precision with the fluid reality of healthcare crises.

The academic field of inventory theory has been transformed, from the inflexible deterministic models of mid-20th century, to the modern era that is focused on soft-computing models. Nevertheless, when we look at the status quo of “uncertainty modeling” in medicine, there is still a wide gap between mathematical beauty and clinical reality.

Evolution of Fuzzy Medical Inventory Models. The initial foray into fuzzy inventory logic was largely defined by the application of triangular and trapezoidal fuzzy numbers to represent demand. Although such models for example, the classic model of Zadeh [13] enabled us to speak of “vagueness,” they could not capture the hesitation prevailing in human expert judgment. This was partly resolved by Atanassov [1] through Intuitionistic Fuzzy Sets (IFS), which introduced a “margin of hesitation” by separating membership and non-membership.

In the specific context of healthcare, Sharma and Gupta [10] recently demonstrated the efficacy of IFS in managing vaccine cold-chains. Their work significantly reduced wastage by modeling temperature fluctuations as intuitionistic variables. Yet, their approach remains tethered to the linear constraint µ + ν ≤ 1. When dealing with high-stakes emergency drug shortages, the "degree of rejection" or skepticism regarding a supply lead time often consumes a larger portion of the decision-space than a linear model allows. This is the precise point where the Pythagorean logic of Yager [12] becomes a mechanical necessity rather than a theoretical luxury.

The Rise of Pythagorean and Hesitant Frameworks. The leap from Intuitionistic to Pythagorean Fuzzy Sets (PFS) offered researchers a "quadratic expansion" of the uncertainty domain. Kumar and Singh [4] utilized PFS to optimize multi-objective pharmaceutical supply chains, proving that PFS captures risk profiles that IFS simply misses. However, a subtle but critical limitation persists in the work of Kumar and others: they assume that for any given parameter, there is a single Pythagorean pair.

In reality, a panel of hospital board members or a group of epidemiologists rarely agrees on a single value. They provide a set of possible values, reflecting a deep, collective hesitancy. Torra [11] introduced Hesitant Fuzzy Sets (HFS) to capture this multi-valued reality, and very recently, Li and Zhao [5] integrated HFS with Type-2 fuzzy sets for medical diagnosis. While Li’s work is groundbreaking in its handling of expert disagreement, it has not yet been translated into the temporal and physical constraints of inventory systems involving perishable goods.

The evolution of inventory modeling is no longer confined to the corporate warehouse. Recent advancements have seen the migration of inventory logic into public service sectors where "demand" represents critical societal needs. For instance, Behera and Mohanta [3] developed a stochastic inventory framework for crime resolution, treating case backlogs as "inventory" and prioritizing resources based on severity.

This cross-sectoral application provides a powerful precedent for our work. Just as Behera and Mohanta argue that law enforcement requires adaptive resource allocation under stochastic pressure, we contend that healthcare systems require a similarvyet more mathematically flexible approach. By moving from their stochastic model to our Pythagorean Hesitant Fuzzy framework, we take their concept of "prioritization under backlog" and apply it to the "clinical risk of expiration" in medical supply chains.

Recent studies have begun to bridge the gap between environmental sustainability and fuzzy uncertainty. For instance, Behera [2] introduced a fuzzy inventory framework that incorporates carbon emission penalties and perishable shelf life using triangular fuzzy numbers[cite: 401, 402]. While their model successfully utilizes graded mean integration to provide actionable sustainable decisions, it is primarily restricted to a single-valued fuzzy space.

By ignoring the "hesitant" nature of Pythagorean variables, current models force a "false consensus" among experts, which leads to fragile inventory policies. Our work is positioned to dismantle this consensus-bias. We propose a Pythagorean Hesitant Fuzzy (PHF) model that allows experts to be both "quadratically uncertain" and "hesitant among multiple values," providing the most psychologically accurate and mathematically robust representation of healthcare supply chains to date. In order to offer a better structural view of the positioning of our work within the state-of-the-art, we have distilled a comparison in Table 1. This contrasts the linear decision spaces restricted in current healthcare models and signifies the vital role of our quadratic multi-valued method. Debunking the “illusory agreement” imposed by single-valued models, we introduce a Pythagorean Hesitant Fuzzy (PHF) model that allows experts to articulate their quadratic uncertainty while hesitating amongst a set of values, providing the most psychologically realistic as well as the most mathematically potent representation of the supply chain in healthcare to date.

In order to build a model of emergency medical logistics that reflects its dynamic nature, we have to make a step beyond the "point-estimate" misconception. In this section, we define the architecture of our Pythagorean Hesitant Fuzzy (PHF) inventory system. We assume a scenario where a high-value medical commodity (e.g., a specialized immunotherapy drug) is subject to constant deterioration and hesitant demand patterns.

Mathematical Preliminaries: The PHF Space. A Pythagorean Hesitant Fuzzy Set (PHFS) on a universe X is defined by a set of possible membership degrees (µ) and non-membership degrees (ν) such that for every x ∈ X, the square-sum of the maxima of these sets does not exceed unity. Let P ˜ be a PHFE (Pythagorean Hesitant Fuzzy Element):

P ˜ = {⟨µ , µ , . . . , µ ⟩, ⟨ν , ν , . . . , ν ⟩} (1)

1 2 k 1 2 m

Subject to:

max µi + min νj ≤ 1 (2)

i=1...k j=1...m

This quadratic boundary is the "mathematical lung" that allows our model to breathe under extreme uncertainty where Intuitionistic Fuzzy Sets (IFS) would collapse.

Assumptions and System Notations. To maintain the structural integrity of the model, we establish the following human-centric assumptions:

• Hesitant Demand (D ˜ ): Demand is not a static number but a PHFE, reflecting the conflicting intuitions of clinical experts.

• Continuous Deterioration (θ): A proportion of the on-hand inventory continuously decays per unit time. We consider θ to be a PHF parameter to model changes in storage temperature.

• Zero Lead Time: In order to isolate the effect of hesitation and perishability, we consider in the main model that replenishment is instantaneous.

• Ethical Shortage Penalty: A shortage is permitted but carries a steep "clinical risk" penalty, Cs, which signifies the human cost associated with running out of stocks.

The Governing Inventory Differential Equation. Over the cycle [0, T], the inventory level I(t) at any time t is reduced by two factors: the reluctant demand D ˜ and the fuzziness deterioration rate θ. The rate of change is governed by:

dI(t) + θI(t) = −D ˜ , 0 ≤ t ≤ T (3) dt Solving this first-order linear differential equation with the boundary condition that inventory hits zero at the end of the cycle (I(T) = 0), we obtain:

P H θ(T −t)

I(t) = e − 1 (4)

The initial order quantity Q, which must satisfy the total demand and the "lost" volume due to decay, is found at t = 0:

P H θT

Q = I(0) = e − 1 (5)

The Objective Function: Minimizing the Total Fuzzy Cost. The Total Inventory Cost per unit time, T C g(T), is a composite of the ordering cost (A), the fuzzy holding cost (Ch), and the fuzzy deterioration cost (Cd).

" Z Z # T C g(T) = A + C I(t)dt + C θI(t)dt (6) T 0 0

Substituting the expression for I(t) and integrating:

A D ˜ (C + θC ) g P H h d θT T C(T) = + e − θT − 1 (7) T θ T This is a non-linear transcendental equation. Because D ˜ , θ, C , and C are all Pythagorean Hesitant Fuzzy

PH h d

Elements, the "cost" is not a value but a fuzzy manifold. Our task in the next section is to defuzzify this manifold using a novel score function to find the crisp optimal cycle time T .

The transition from a fuzzy manifold to a concrete decision requires a rigorous bridge. In this section, we develop the mathematical infrastructure to prove that an optimal, unique cycle time T exists for the Pythagorean Hesitant Fuzzy (PHF) cost function.

Defuzzification via the PHF-Score Function. To handle the multi-valued nature of our parameters, we define a novel Score Function S(P ˜) for a Pythagorean Hesitant Fuzzy Element (PHFE). Given the inherent risk-aversion in medical systems, our score function prioritizes the "quadratic distance" from the ideal membership state. For P ˜ = {µ, ν}, let:

1 X 1 X ˜ 2 2 S(P) = γ − η (8) |h| |g|

γ∈h η∈g

where h is the set of hesitant membership values and g is the set of non-membership values. This score function maps the PHFE into a crisp real-valued space R ∈ [−1, 1], allowing us to treat the fuzzy objective function T C g(T) as a crisp equivalent Ψ(T).

Existence and Uniqueness of the Optimal Policy. We must ensure that the total cost function is not just solvable, but globally convex.

Theorem 1. The crisp equivalent cost function Ψ(T) derived from the Pythagorean Hesitant Fuzzy inventory model is strictly convex with respect to the cycle time T in the domain T ∈ (0, ∞), provided the holding cost Ch and ordering cost A are positive.

Proof : Let Ψ(T) be the defuzzified total cost function:

A S(D ˜ )(C + S(θ)C )

P H h d S(θ)T

Ψ(T) = + e − S(θ)T − 1 (9) T S(θ) T

To prove convexity, we examine the second-order condition 2 > 0. First, we find the first derivative:

dT

˜ S(θ)T S(θ)T dΨ A S(DP H)Φ T S(θ)e − (e − 1) = − + (10)

2 2 2

dT T S(θ) T S(θ)T x

where Φ = Ch + S(θ)Cd. Expanding the exponential term e using its Taylor series repesentation e = 1 + x + + + . . . , we obtain:

2! 3!

2 2 3

A T S(θ)T S(θ) T Ψ(T) = + S(D ˜ )Φ + + + . . . (11) T 2! 3! 4!

By differentiating again with re to T:

d Ψ 2A 2S(θ) 6S(θ) T = + S(D ˜ )Φ + + . . . (12)

2 3 P H

dT T 3! 4!

Since A, T, S(D ˜ ), Φ, S(θ) > 0 for any viable medical inventory system, every term in the second derivative is strictly positive. Consequently, 2 > 0 for all T > 0. This confirms that Ψ(T) is a strictly convex function,

dT dΨ ∗

ensuring that the local minimum found by setting = 0 is the unique global optimal cycle time T . □

dT

Structural Properties of the Optimal Solution. By setting the first derivative to zero, we derive the optimal condition:

2 S(θ) 2S(θ) T A T + + . . . = (13) 3! 4! S(D ˜ )(C + S(θ)C )

PH h d

This expression exposes an important managerial insight: in a Pythagorean Hesitant environment, the optimal cycle length T is the negative power of the "hesitation-weighted" demand. When experts become more “hesitant” and assign higher membership values to demand, the system automatically reduces the replenishment-cycle length to hedge against the risk of running out of stock.

The inherent complexity of Pythagorean Hesitant Fuzzy (PHF) parameters where membership and non-membership are sets of discrete expert intuitions requires a structured computational pipeline. To address the non-linear transcendental equation obtained in Section 4, we present the Pythagorean Hesitant Fuzzy Optimization Algorithm (PHFOA). This approach guarantees that the “hesitation” present in the original input data is maintained during the defuzzification procedure.

The PHFOA Algorithmic Framework. The procedure for determining the optimal cycle time T and the optimal order quantity Q is structured into the following five discrete phases:

Phase 1: Data Aggregation and Normalization: Collect the PHF elements for demand (D ˜ ), de-terioration (θ), and costs (Ch, Cd). Since different experts may provide hesitant sets of varying lengths, we apply a normalization bond by adding the pessimistic value (the minimum membership) to shorter sets until all sets reach equal cardinality, as suggested by Torra [11].

Phase 2: Score Function Mapping: Use the novel quadratic score function S(P ˜) in (9). This is to transform the multi-valued PHF manifolds into crisp representative values with a penalty term on the number of non-members.

Phase 3: Initial Value Estimation: Calculate an initial "Seed" cycle time T0 using the classic deter-ministic EOQ logic with the defuzzified values:

T0 = (14) S(D ˜ )S(C )

P H h S(θ)T

Phase 4: Iterative Refinement: Due to the presence of transcendental terms (e ) in Equation (14) which is the optimality condition, we utilize the Newton-Raphson Iterative Method.Let

dΨ th

f(T) = .The approximation to x at the (n + 1) iteration is given by:

dT

f(Tn) Tn+1 = Tn − (15) f (Tn) The iteration continues until |Tn+1 − Tn| < ϵ, where ϵ = 10 is the convergence tolerance.

Phase 5: Output and Sensitivity Validation: Calculate the optimal order size Q based on Equation (5) and check the stability of the model by sensitivity.

Visual Representation of the Solution Flow. For clarity in guiding hospital IT systems adopting this framework, we illustrate the logical sequence from expert hesitation to inventory decision in Figure 1.

To validate the proposed Pythagorean Hesitant Fuzzy (PHF) model, we present a case study based on the procurement of high-value biologicals in an Intensive Care Unit (ICU) setting. The data reflects the "hesitation" of three senior clinical pharmacists regarding the weekly demand (D ˜) and the deterioration rate (θ) during a seasonal surge in respiratory distress cases.

Data Acquisition and PHF Input Sets. The following parameters were derived based on expert consensus. Note that experts were more or less generous with the numbers of membership and non-membership degrees, reflecting different degrees of “epistemic confidence.”

• Ordering Cost (A): $800 per order (fixed).

• Holding Cost (Ch): $15 per unit/week.

• Shortage/Clinical Risk Cost (Cs): $250 per unit (reflecting the ethical weight of a stock-out). • Deterioration Cost (Cd): $120 per expired unit.

The hesitant parameters are defined in Table 2. We use the "pessimistic extension" method to normalize the lengths of the hesitant sets.

A mathematical model is only as robust as its performance under shifting parameters. In the volatile context of emergency medicine, "fixed" values are a dangerous illusion. In this section, we perform a granular sensitivity analysis to observe how the optimal inventory policy (T , Q ) responds to fluctuations in expert hesitation, deterioration rates, and holding costs.

Impact of Expert Hesitation on Decision Variables. We first examine the "Hesitation Sensitivity." By shifting the membership values in the Pythagorean Hesitant Fuzzy Set (PHFS) for Demand (D ˜), we simulate a scenario where clinical experts become increasingly uncertain or "cautious" regarding an upcoming infection surge.

The results, documented in Table 4, reveal a fascinating non-linear relationship. As the square-sum of membership values increases (indicating higher expected demand with deep hesitation), the model naturally contracts the cycle time T .

The management of life-critical medical inventories is an arena where the cold logic of mathematics meets the warm, often hesitant reality of human clinical judgment. In this research, we have argued that the traditional "certainty-driven" models of the past century are no longer sufficient to navigate the deep epistemic fog of modern healthcare crises. By developing a novel Pythagorean Hesitant Fuzzy (PHF) inventory framework, we have provided a mathematical language that respects the "multi-layered hesitation" of medical experts while maintaining the rigorous optimization required for pharmaceutical stewardship.

Our contribution is defined by three primary achievements. First, we successfully synthesized the quadratic membership space of Pythagorean logic with the multi-valued flexibility of hesitant fuzzy sets, creating a model that "breathes" with the expert’s uncertainty. Second, through a rigorous Taylor-series expansion, we provided a theoretical proof of the model’s convexity, ensuring that hospital administrators can rely on a unique, stable, and globally optimal ordering cycle. Third, our numerical simulations and sensitivity analysis revealed a "Hesitation Buffer" a systemic safety cushion that traditional intuitionistic and deterministic models consistently ignore, often at the cost of clinical resilience.

Practical Significance. For the hospital administrator, the takeaway is clear: "Expert disagreement" is not a noise to be averaged out, but a primary data variable that should dictate the frequency and volume of medical orders. Our model demonstrates that when experts are deeply hesitant, the system must respond with shorter, more frequent replenishment cycles (T ) to mitigate the risk of stock-outs. Furthermore, the high sensitivity of the total cost to the deterioration rate (θ) underscores the urgent need for investment in smart, IoT-enabled cold-chain infrastructure for high-value biologicals.

Limitations and The Path Ahead. Of course, this study is already a giant leap in the right direction, but it does have its limitations. The present model assumes a constant decay rate, while in many biological situations, decay is non-linear and time-varying. It is expected that further research in this directions will be focused on combining Type-2 Hesitant Fuzzy Logic with non-instantaneous deterioration functions, which allows one to model uncertainty at even finer levels.

Additionally, as we move toward an era of decentralized healthcare, the integration of this PHF framework into Blockchain-enabled supply chains offers a promising frontier. Through the utilization of distributed ledger technology to log the “hesitancy scores” of various stakeholders live, we can build a real-time, self-correcting healthcare inventory system that is equally efficient and robust. And through it all, the aim of this work is to help make sure that when the next crisis comes, the supply chain is not a source of failure, but a source of hope.

The authors wish to thank the clinical pharmacists and ICU specialists whose "hesitant" insights provided the empirical heartbeat of this mathematical study.

All authors have read and agreed to the published version of the manuscript.

The authors declare that no external funding, grants, or support were received for the research presented in this paper, including administrative, technical, or in-kind contributions.

The datasets generated and analyzed during the current study, including the Pythagorean Hesitant Fuzzy sets and the parameters for the Newton-Raphson iteration, are included within this article. Detailed computational steps are provided to ensure reproducibility.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

This article does not contain any studies with human participants or animals performed by any of the authors. The clinical data used in the numerical illustration are based on expert-guesstimated limits and simulated scenarios for pharmaceutical logistics validation [2].

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