A Novel Framework for Multi-Agent Multi-Criteria Decision-Making: Mathematical Modeling and Rastrigin-Based Benchmarking

Multi-Criteria Decision-Making (MCDM) serves as an essential methodology for evaluating, selecting, or ranking alternatives in complex scenarios where conflicting objectives and diverse criteria must be systematically balanced

As decision-making complexity escalates in group settings, Group Decision-Making (GDM) extends the MCDM paradigm to scenarios where a group of decision-makers (DMs) collaborates to reach a consensus

In contrast, Multi-Agent Systems (MAS) provide a decentralized framework in which autonomous agents make independent decisions while interacting within a shared environment

Notably, there is no universally superior MCDM technique or algorithm, as the appropriateness of a method depends on the specific decision situation. While almost all methodologies in MCDM share common steps like problem organization and decision matrix construction, they differ in how they synthesize information

The proposed Multi-Agent Multi-Criteria Decision-Making (MAS-MCDM) framework is a novel approach that addresses the challenges of scalability and global optimality in decentralized decision-making systems. It enables agents to rank alternatives based on local criteria while simultaneously negotiating system-wide constraints. Four models are introduced: 

First, a full enumeration strategy exhaustively evaluates all combinations to guarantee global optimality, although it is computationally infeasible for large-scale problems. 

Second, a decomposition approach enables independent agent evaluations, improving scalability at the potential cost of global optimality. 

Third, an iterative refinement mechanism enables agents to adjust their decisions based on system-level feedback, thereby enhancing feasibility under resource constraints. 

Finally, an extended iterative framework incorporates explicit inter-agent interaction terms to model cooperative synergies and competitive conflicts among agents.

To validate the framework, we adapt the Rastrigin function

The remainder of the paper is organized as follows. Section 2 details the methodology, including mathematical models and workflows for each approach. Section 3 validates the framework experimentally using the discretized Rastrigin function, with a focus on resource constraints and inter-agent dynamics. Section 4 concludes with a discussion of strengths, limitations, and future research directions. 

2.1. Problem Setup

We define the following parameters:

For all following approaches, the decision variable

If there are

The preference scores represent the normalized ranks of options that are calculated and ranked by the TMF algorithm. Based on the criteria for each agent, the TMF gives the value of the preference for each option.

2.2. Approach 1: Full Enumeration of

This approach systematically valuates all possible combinations of options for N agents, each with M options. The total number of combinations is

2.2.1. Mathematical Model

Agent-level Objective (

System-Level Objective (

Objective Function:

Combine agent objectives and system-level objectives:

where:

Ensure

Constraints:

Single Option Per Agent: Each agent selects exactly one option:

System Constraints: The system imposes constraints such as:

Resource limits:

Set of Solutions: Each solution

Optimal Solution: The optimal solution is:

2.2.2. Strengths and Limitations of the Approach

This approach guarantees the globally optimal solution by systematically evaluating all

2.3. Approach 2: Decomposition into Agent Contributions

This approach reduces the computational complexity of multi-agent decision-making by evaluating each agent’s decision independently and approximating the system-level objectives as the sum of individual agent contributions. While it may not guarantee the global optimum and does not consider the existence of the resources constraints like the full enumeration approach, it significantly improves scalability and remains practical for larger systems without constraints.

2.3.1. Mathematical Model

2.3.1.1. Agent-Level Objective

Each agent independently evaluates its options:

where:

2.3.1.2. System-Level Objective

Approximate the system-level objective as the sum of agent-level contributions:

2.3.1.3. Constraints

Each agent must select exactly one option:

2.3.1.4. Optimal Solution

The optimal solution selects the best option for each agent based on their individual objectives and system-level contribution:

2.3.2. Step-by-Step workflow

Step 1: Define Agent Contributions

For each agent

Step 2: Apply Multi-Criteria Decision-Making (MCDM) Locally

For each agent

Use TMF to calculate the individual preferences (

Step 3: Generate the ranking

Generate a priority ranking for each agent by combining the individual preferences (

Step 4: Aggregate Agent-Level Decisions

After evaluating each agent's options locally, aggregate the top-ranked choices to approximate the system-level solution.

2.3.3. Strengths and Limitations

This approach significantly improves scalability by reducing the computational complexity from

However, since agents make decisions independently, it does not guarantee a globally optimal solution, as inter-agent dependencies and constraints are not directly accounted for. This can lead to suboptimal system-wide performance, especially in cases where resource constraints or dependencies between agent choices significantly impact the overall objective. Additionally, the approximation of the system-level objective as a sum of individual agent contributions may not always be valid — some global objectives are inherently non-decomposable, meaning that the interactions between agents play a critical role in decision-making.

2.4. Approach 3: Iterative Refinement

The Iterative Refinement approach is a hybrid optimization method that balances computational efficiency and solution quality by iteratively improving agent decisions. Unlike the Full Enumeration approach, which explores all possible combinations, and the Decomposition approach, which assumes independence among agents, Iterative Refinement incorporates feedback between the system-level evaluation and agent-level decisions. This feedback loop progressively adjusts decisions to achieve a better balance between individual preferences and system-wide objectives.

2.4.1. Mathematical Model

2.4.1.1. Objectives

(a) Initial Agent-Level Objective

Each agent computes its initial score for all options j based on a weighted combination of individual preference and system-level contribution like in the Decomposition into Agent Contributions approach:

(b) System-Level Objective

Approximate the system-level objective as the sum of agent-level contributions:

c) Adjusted Agent-Level Objective

Based on feedback from the system-level evaluation, agents adjust their scores by incorporating penalties for violating system constraints:

where:

Key Benefits of saving the previous rank from the previous iteration:

- Smooth Transition Between Iterations: By retaining the

- Better Constraint Handling: The penalties applied in the previous iteration persist in influencing the next iteration, effectively discouraging agents from reverting to previously penalized options.

- Faster Convergence: This memory mechanism reduces oscillations in decision-making, potentially speeding up the convergence to a feasible and optimal solution.

2. Constraints

(a) Individual Agent Constraint

Each agent selects one and only one option:

(b) System-Wide Constraints

System resources must remain within their limits:

3. Solution

The process iteratively refines decisions until a stable, constraint-satisfying solution

2.4.2. Step-by-Step Workflow

Step 1: Initial Solution

Compute Initial Ranks: Each agent evaluates its options using Equation (2.13).

Make Initial Selection: Each agent selects the option

Form Initial Solution: The initial solution

Step 2: System-Level Evaluation

Check Constraints: Evaluate total resource usage for each constraint

Identify Violations: Compute penalties for constraint violations:

Step 3: Feedback and Adjustment

Compute Agent Impact: For each agent, compute its contribution to resource usage:

Adjust Rankings: Penalize options that contribute heavily to constraint violations. The applied penalty will be saved during the iteration:

Re-select Options: Each agent recomputes its rankings using the adjusted scores and selects the best option:

Step 4: Repeat Until Convergence

Convergence Check: Stop the iteration when:

- The system objective

- No agent changes its decision between many iterations: 

- Reach the iteration limit.

Output Final Solution:

The final solution

2.4.3. Strengths and Limitations

The Iterative Refinement approach strikes a balance between computational efficiency and solution quality by iteratively improving agent decisions through a feedback mechanism. By continuously refining choices based on violations of system constraints, it achieves a more adaptive and responsive optimization process. A key advantage of this method is that it reduces computational complexity significantly compared to full enumeration while still capturing partial interdependencies between agents, making it more scalable for larger systems. Additionally, the iterative structure helps mitigate constraint violations over time, leading to a feasible and balanced solution without requiring an exhaustive search of the solution space. This process also allows for parallel updates in decentralized settings, making it applicable to multi-agent systems with distributed decision-making.

However, without proper tuning of penalty weights and feedback mechanisms, the iterative process can oscillate between solutions, causing instability. Another limitation is that this approach assumes that all agents respond rationally and iteratively adjust their decisions based on system feedback, which may not hold in dynamic or adversarial environments where agents have strategic behaviors or incomplete information.

2.5. Approach 4: Iterative Refinement with Inter-agent Interactions

While the previous approaches addressed many fundamental challenges—such as exponential complexity, balancing local vs. global objectives, and adaptive penalty mechanisms to enforce resource constraints—they did not fully capture how agents’ choices interact with each other beyond shared resources — whether they be cooperative synergies, competitive conflicts, or more nuanced relational dynamics. Building on the iterative penalty-based framework of Approach 3, approach 4 closes this gap by incorporating a joint contribution function that quantifies synergy or conflict based on the specific combination of options chosen across agents. In doing so, each agent’s objective is no longer limited to local preference, system-level contribution, and resource penalties; it also accounts for how its choice aligns (or clashes) with the decisions of other agents. As a result, the proposed method supports a wide range of cooperative, competitive, and hybrid scenarios, providing a more realistic and dynamic decision-making framework for multi-agent systems.

2.5.1. Mathematical Model

2.5.1.1. Joint Contribution Function for Inter-agent Interactions

2.5.1.1.1. Motivation

When multiple agents coordinate (or compete), the benefit or cost of an agent

For example:

Cooperative scenario:

Competitive scenario:

Partial synergy:

2.5.1.1.2. Interaction Matrix

In addition to

Hence the total interaction term for agent

2.5.1.1.3. Examples of defining the joint contribution function

We define the Interaction matrix as:

Simple joint function 

A common binary function:

Positive

Negative

Complex joint function

below is a partially rewards (or penalizes) agents

Example: Location-Based Partial Synergy

Suppose each agent’s options correspond to possible locations (or positions in a metric space). Let:

We can design a piecewise function that:

- Rewards pairs of locations that are very close (e.g., good for collaboration).

- Gives a smaller bonus (or neutral effect) if they are moderately distant.

- Penalizes them if they are too far apart (e.g., high communication or transportation cost).

A sample function might look like:

Where:

For intermediate distances

Other Possible Joint contribution function formulations

- Attribute Similarity: Instead of location distance, we could define a function based on the similarity of agents’ skill sets, technology types, or preferences. For instance, if

- Nonlinear or Multi-Attribute: We might combine multiple factors—distance + type alignment + resource overlap—into a single function, weighting each sub-factor differently.

- Time-Dependent: For dynamic environments, let

- Probabilistic: Define expected synergy or conflict, e.g.

2.5.1.2. Refined Objective Function

Agent-Level Objective

At iteration

Local preference (

System-level contribution (

Penalty for resource overuse (adaptive from iteration

Interaction with other agents.

The general objective for agent

Each agent

2.5.2. Step-by-step workflow: Iterative Refinement Process

The algorithm proceeds in iterations 

Step 1: Initialization

- Set an initial decision matrix

- Initialize

-

Step 2: Compute Resource Usage & Penalties

- For each resource

- Determine

- Allocate impacts based on

Step 3: Compute Interaction Terms

For each agent

Step 4: Agent-Level Optimization

Each agent

where

Step 5: Convergence Check

Check if:

- The total system-level objective (or sum of all

- All resource constraints are satisfied (or the penalty remains unchanged).

- No agent changes its decision, i.e.,

If these conditions are met, stop. Otherwise, increment t and repeat from Step 2.

2.5.3. Strengths and Limitations

Approach 4’s main advantage is its ability to capture a wide range of cooperative or competitive inter-agent dynamics by introducing explicit interaction terms beyond simple resource-sharing constraints. It adaptively refines decisions in an iterative manner, allowing for parallel agent updates and offering greater scalability than exhaustive methods, while still preserving a multi-criteria perspective by combining individual preferences, system-level contributions, penalty terms, and interaction factors into a unified objective. However, tuning the additional interaction weights and penalty parameters introduces further complexity and can lead to oscillations if not carefully calibrated. The accuracy of the results relies heavily on defining the joint function. Finally, in highly adversarial or rapidly changing environments, agents may not faithfully follow the iterative penalty updates, and more robust game-theoretic or reinforcement learning approaches may be required for stable and fair outcomes.

The Rastrigin function was selected as a representative benchmark for evaluating the performance of the proposed MAS-MCDM approaches. This function is traditionally used in single-agent global optimization to test an algorithm’s ability to escape local minima. It was used to verify the proposed models for a multi-agent scenario by discretizing it. These discretizing points represent the agent’s options.

3.1. Implementation Overview

3.1.1. Discretized Options

For each agent

The generated points (Set of agent’s options) in this implementation are: 

3.1.2. Preference, Contribution, and Resources

For each agent

Then assign:

Figure 1 - The Rastrigin function with parameter A=10. Point B at x=0 is the continuous global minimum in single-agent scenarios; in multi-agent settings, additional constraints and interactions can shift the solution away from x=0 to A and C points

DOI:10.60797/IRJ.2025.158.94.1

3.1.3. Objectives Weighting

The weights

3.1.4. Constraints

- Resource Constraints: Resource-usage terms derived from the negative Rastrigin values and impose a global limit

- Let the global resource threshold

- Interaction Constraints: Reward-based interactions were incorporated in the implementation of Approach 4, which can steer agents to local minima near

3.1.5. Inter-Agent Interactions:

3.1.5.1. Interaction Setup

We created a scenario where first and fifth agents set as the first group and the other agents set as the second group, each group benefit from matching their values sign (cooperative between agents in the same group). Additionally, cross-group interactions are set as conflict, discouraging all five from picking the same sign (competitive between groups).

Interaction matrix:

3.1.5.1. Normalization

The normalized preferences and contributions onto

3.2. Results and Observations

- Approach 1 (Full Enumeration)

Outcome:

Correctly identifies the global minimum

With Constraints:

When resource usage rules push agents away from

Runtime: Exponential in

Despite guaranteeing the global optimum, the exponential complexity restricts this method to small values of

- Approach 2 (Decomposition)

Outcome:

Each agent individually picks the best discrete point, ignoring synergy or resource constraints. All agents choose

- Approach 3 (Iterative Refinement)

Outcome:

Agents iteratively adjust choices based on feedback from resource penalties until finding the solution where satisfy the constraints.

Resource

This demonstrates that iterative penalty-based methods handle resource constraints effectively without enumerating all solutions.

- Approach 4 (Iterative Refinement with Inter-Agent Interactions)

Outcome:

There are two combinations that solve the problem of constraints and interactions. This approach found a combination that satisfy the constraints with total usage value equals

As shown, incorporating inter-agent synergy and conflict can yield solutions where agent groups naturally self-organize, surpassing purely penalty-based approaches in scenarios with strong interactions.

The Rastrigin experiments validate that each proposed approach can be adapted to MAS-MCDM, handling both simpler unconstrained cases and more intricate interactions/constraints. The results underscore the trade-offs between optimality, scalability, and the capacity to model inter-agent effects — key considerations for multi-agent systems in many practical domains.

In this paper, we have presented a comprehensive framework for MAS MCDM to effectively balance local agent preferences with system-wide objectives and constraints. We introduced and rigorously formulated four distinct approaches — full enumeration, decomposition into agent contributions, iterative refinement, and iterative refinement with inter-agent interactions — each offering a unique set of trade-offs between global optimality and computational scalability. This work systematically demonstrates that while exhaustive search guarantees a global optimum, its exponential complexity renders it impractical for large-scale systems, thereby motivating the need for more scalable methods.

A key contribution of the research is the novel iterative penalty-based refinement strategy. This mechanism dynamically adjusts agent-level decisions based on feedback from resource constraint violations and, in the most advanced approach, inter-agent interactions. By integrating these dynamic adjustments, the framework is capable of converging to robust and feasible solutions even in the presence of complex interdependencies and resource limitations. The incorporation of explicit interaction terms further allows the model to capture cooperative and competitive dynamics, thereby enhancing its applicability to a wide range of real-world scenarios — from tightly coupled, mission-critical systems to decentralized and distributed decision-making. The validation experiments using a discretized version of the Rastrigin function underscore the efficacy of the proposed methods. The experimental results highlight the strengths of each approach and illuminate the inherent trade-offs between achieving optimality and ensuring scalability. In particular, the iterative refinement strategies have shown promise in navigating challenging optimization landscapes. Future work may investigate further enhancements to the interaction models, integrate with reinforcement learning and game-theoretic strategies, and apply these methods to real-world challenges in autonomous systems, resource management, and other areas.