## 2 Background  
### Fuzzy Reward System  
In the domain of reinforcement learning, environmental feedback manifests as reward signals, indicating the efficacy of an agent's actions in specific states. To optimize learning efficiency, it is imperative for the reward function to be defined with clarity and precision, incorporating parameters that directly influence the agent's decision-making process and the dynamics of their interactions [24]. The accuracy of the reward function is crucial, as it significantly influences the computational complexity required. Furthermore, multi-criteria decision making in this context often demands a flexible approach to defining optimality [25]. Central to this discussion is the FRS, a computational construct grounded in fuzzy logic for robust information processing and decision making. This system comprises four integral elements: a fuzzifier, a knowledge base (encompassing rule bases or databases), a fuzzy inference engine, and a defuzzifier [26]. Among FS, the two prevalent types are Mamdani-type FS [27] and Sugeno-type FS [28], distinguished by their methodologies in deriving precise outputs from fuzzy inputs--Mamdani through defuzzification and Sugeno via weighted averages. In Mamdani-type FS, the formulation of fuzzy rules incorporates fuzzy linguistic values for both conditions and consequences, ensuring a nuanced interpretation of data inputs and outputs. This precision has led to the widespread adoption of the Mamdani-type FS as the de facto standard in the field. Central to its architecture are four pivotal components: a fuzzification module for converting crisp inputs into fuzzy sets, a fuzzy inference engine for processing fuzzy logic, a knowledge base (KB) which is the nucleus of the system housing the fuzzy rules and data, and a defuzzification module for translating fuzzy conclusions back into precise outputs. The KB is an amalgamation of a rule base (RB) and a data base (DB), where the DB is characterized by scaling functions and membership functions (MFs) delineating the fuzzy sets, and the RB entails an array of IF-THEN fuzzy rules. An exemplar rule in the Mamdani FS might be articulated as  
\[x_{1}\text{ {is} }A_{1}^{i}\text{ {and} }x_{2}\text{ {is} }A_{2}^{i}\dots\text{ {and} }x_{n}\text{ {is} }A_{n}^{i}\text{ {then} }y\text{ {is} }C^{i} \tag{1}\]  
In the framework of the Mamdani FS, the intricate network of fuzzy rules is indexed using \(i\) ranging from 1 to \(N\), with \(N\) signifying the aggregate count of these rules. The system's input fuzzy sets are denoted as \(A_{n}\), while \(X=(x_{1},x_{2},\dots,x_{n})\) and \(y\) embody the linguistic variables for input and output within the Mamdani framework. The term 'and' functions as a fuzzy conjunction operator, orchestrating the interplay between rules. For each rule, \(A_{n}^{i}\) and \(C^{i}(i=1,2,\dots,N)\) represent the fuzzy sets pertaining to the \(i\)th input variable and the output under the \(n\)th rule, correspondingly. In this context, [PERSON]'s fuzzy sets are articulated through MFs, with triangular fuzzy numbers [29] and trapezoidal fuzzy numbers [30] being the predominant choices. Completely overlapping triangular MFs represent a specific category of triangular fuzzy numbers. In this case, each membership function constitutes a triangular fuzzy number, where the two vertices at the base of each triangle precisely coincide with the midpoints of the base edges of the adjacent triangles. This is illustrated in Figure 1.  
When representing a triangular fuzzy number using three parameters for its left, middle, and right points, the number of required parameters for a fuzzy variable with m MFs is \(m-2\). It is important to note that in the context of completely overlapping triangular MFs, these parameters are not equal and must satisfy specific mathematical relationships. For instance, as illustrated in Figure 1, for a fuzzy variable V with five MFs, the number of parameters is three (namely \(a,b,c\)), adhering to the relationship \(0<a<b<c<1\).