In traditional operations research education, models are often presented in isolated chapter formats: linear programming, graph theory, dynamic programming, queuing theory, and inventory control each form their own separate systems. While learners may master solution procedures, they frequently struggle to truly understand the formation mechanisms of optimal solutions and the systemic logic underlying decision-making. This fragmented approach creates significant gaps between theoretical knowledge and practical application.

The OR-LabX (Operations Research Laboratory Experiment) system addresses these challenges through a unified WebApp platform that reconstructs classical operations research models into a "visualizable, interactive, and interpretable" integrated framework. Through deep integration of modeling, solving, simulation, and AI analysis, abstract formulas transform into dynamic processes. Users can observe algorithm paths and state evolution while understanding system stability and decision sensitivity through parameter variations—achieving a cognitive leap from "knowing how to calculate" to "understanding and deciding."

Technical Architecture: A Trinity Structure

The OR-LabX technical architecture embodies a three-in-one structure: Visual Modeling Language × Dynamic Optimization Engine × AI Decision Interpretation.

Visual Modeling Language

This component transforms real-world problems into intuitive, operable mathematical models, making abstract structures concrete and accessible. Instead of staring at equations on paper, users interact with visual representations that bring mathematical concepts to life.

Dynamic Optimization Engine

Responsible for executing various algorithms and iterative processes, this engine presents the paths and mechanisms of optimal solution generation. Users don't just see the final answer—they witness the journey of how the algorithm arrives at that solution, step by step.

AI Decision Interpretation Layer

This layer provides semantic analysis and strategic interpretation of results, revealing the decision logic and systemic significance behind the models. It answers not just "what is the optimal solution" but "why is this optimal" and "what does this mean for real-world decisions."

These three components work synergistically, achieving a complete closed loop from modeling through solving to understanding and decision-making.

Comprehensive Laboratory Overview

The OR-LabX system encompasses seventeen distinct laboratory modules, each designed to illuminate specific aspects of operations research while contributing to a cohesive learning experience.

1. Simplex Method Laboratory

Core Focus: Linear Programming Fundamentals

This laboratory transforms the tedious simplex tableau calculations into a "spatial path visualization" process. Through dynamic display of the objective function's vertex iteration trajectory on a convex polyhedral feasible region, users can intuitively observe how the optimal solution gradually approaches along boundary edges.

Educational Value: This interactive experience helps students deeply understand the essential nature of linear programming: it's not merely algebraic matrix transformation, but a geometric evolution process of "searching for the optimal vertex along the gradient direction" within a constrained space.

Learning Outcome: Students transition from mechanically performing row operations to genuinely comprehending the geometric interpretation of linear optimization.

2. Duality Problem Laboratory

Core Focus: Primal-Dual Symmetry and Economic Interpretation

This laboratory focuses on the profound symmetrical aesthetics between primal and dual problems. Through linked display of resource constraints and shadow prices (dual prices), users can observe in real-time how subtle fluctuations in constraint conditions reflect in the value assessment of dual variables.

Educational Value: The experiment guides students to understand the core economic principle of "constraints as value," re-examining resource scarcity from the shadow price perspective. This achieves a cognitive leap from pure resource allocation calculation to deep resource pricing logic.

Real-World Application: Understanding shadow prices enables managers to make informed decisions about resource acquisition, pricing strategies, and capacity expansion.

3. Sensitivity Analysis Laboratory

Core Focus: Solution Stability and Parameter Robustness

Through parameter slider adjustment and real-time result linkage, this laboratory dynamically presents the stability boundaries of optimal solutions when objective coefficients or resource limits change. Users can intuitively see when the "optimal basis" becomes invalid and identify which parameters the model is most sensitive to.

Educational Value: This is not merely mathematical parameter discussion—it's decision pressure testing simulation. It helps users understand that models provide not just a "static optimal solution" but an "effective range" guaranteeing decision robustness.

Practical Significance: In real-world scenarios, parameters rarely remain fixed. Understanding sensitivity prepares decision-makers for uncertainty and variability.

4. Transportation Problem Laboratory

Core Focus: Logistics and Distribution Optimization

By constructing supply-demand balance matrices and logistics path networks, this laboratory dynamically presents the evolution path of transportation solutions from initial basic feasible solutions (such as the Northwest Corner Method) to final optimal solutions.

Interactive Features: Users can manually or automatically execute the stepping-stone method and closed-loop method, intuitively perceiving the optimization logic of capacity allocation.

Educational Value: The experiment explains through visualization how to reduce total costs by eliminating "inefficient circular transportation," enabling users to deeply grasp the global cost minimization mechanism in logistics scheduling.

5. Assignment Problem Laboratory

Core Focus: Optimal Matching and Allocation

Centered on the Hungarian algorithm, this laboratory transforms complex person-task matching processes into intuitive matrix transformation visualization. Through displaying each step of "zero element" covering lines and matrix reduction operations, users understand how to find independent zero elements through equivalent transformations in the cost matrix.

Educational Value: This interactive design makes abstract discrete combinatorial optimization easy to observe, helping students master the optimal allocation logic of one-to-one correspondence relationships under limited resources.

Applications: From assigning workers to tasks, matching students to schools, or allocating machines to jobs—assignment problems appear throughout organizational management.

6. Shortest Path Laboratory

Core Focus: Network Navigation and Route Optimization

Based on classic Dijkstra and Floyd algorithms, this laboratory uses dynamic graph structures and node label update processes to display in real-time the generation trajectory of optimal paths from origin to destination.

Visualization: Users can observe how the algorithm performs "sector search" or "dynamic induction" within the network, transforming path optimization from abstract pseudocode into a dynamic construction process.

Educational Value: The experiment emphasizes the influence of weight distribution on path selection, serving as an important introductory tool for understanding complex network navigation and topology optimization.

7. Maximum Flow Laboratory

Core Focus: Network Capacity and Bottleneck Analysis

Based on Edmonds-Karp or the double-labeling method, this laboratory completely presents the entire process of "label search → augmenting path → residual update." Using pipe thickness to represent flow, users can intuitively observe how flow gradually advances through the network, where backflow occurs, and how bottlenecks (minimum cuts) ultimately form.

Educational Value: The experiment aims to help students understand the limits of network carrying capacity, mastering the application essence of fluid models in bandwidth allocation, traffic planning, and other real-world scenarios.

8. Minimum Spanning Tree Laboratory

Core Focus: Network Connectivity with Minimal Cost

Through step-by-step execution of Prim and Kruskal algorithms, this laboratory dynamically displays how a network achieves connectivity of all nodes with minimum total edge weight. Users can observe in real-time how the algorithm avoids cycles and greedily selects the shortest edges.

Educational Value: This helps understand the mathematical logic of "local selection constructing global optimality" in structural optimization. The experiment is particularly suitable for explaining cost control issues in infrastructure construction (such as power grids, optical cable laying), making topology principles concrete.

9. Network Planning Laboratory

Core Focus: Project Management and Critical Path Analysis

This laboratory transforms project management (PERT/CPM) problems into clear directed time-scale network diagrams. By calculating time parameters for each operation, the system automatically highlights the "critical path," enabling users to intuitively identify which tasks are project bottlenecks and which have slack time.

Educational Value: The experiment conveys the balancing art between schedule and resource optimization, helping managers master the core strategies of achieving progress prediction and schedule optimization through node control in complex engineering projects.

10. Decision Tree Laboratory

Core Focus: Multi-Stage Decision Making Under Uncertainty

Through multi-level probability branches and expected return calculations, this laboratory visually displays multi-stage decision paths in uncertain environments. Users can manually input probabilities and profit/loss values for different options, observing how decision trees prune suboptimal branches through "backward induction."

Educational Value: The experiment not only demonstrates mathematical calculations but also conveys the logic of risk preference and rational choice, serving as the core teaching framework for understanding expected value-driven decision mechanisms and handling probabilistic prediction problems.

11. Markov Decision Laboratory

Core Focus: Dynamic State Transitions and Policy Optimization

This laboratory constructs a dynamic state transition environment, demonstrating the formation mechanism of long-term optimal strategies through real-time display of state distribution evolution and policy iteration processes. Users can adjust transition probability matrices and observe how the system moves from chaotic states toward stable distributions.

Educational Value: This laboratory serves as an excellent window for understanding the underlying frameworks of modern reinforcement learning, stochastic processes, and intelligent control systems, making abstract stochastic mathematics clear and tangible.

12. Dynamic Programming Laboratory

Core Focus: Multi-Stage Optimization and Optimal Substructure

Through stage division, state definition, and state transition diagrams, this laboratory breaks down complex global optimization problems into a series of linked sub-problem solutions. Users can observe how the "principle of optimality" functions at each step of recursion or iteration, understanding the memorized search mechanism of "optimal substructure."

Educational Value: Through visualization of classic cases (such as knapsack problems or path selection), the experiment transforms originally difficult-to-understand state transition equations into intuitive table filling and path backtracking.

13. Nash Equilibrium Laboratory

Core Focus: Strategic Interaction and Stable Outcomes

Through game payoff matrices and best response dynamic analysis, this laboratory displays the formation process of stable strategy combinations in multi-agent interactions. Users can simulate different players' psychological expectations and strategy choices, observing how the system converges to Nash equilibrium points.

Educational Value: The experiment aims to reveal the dialectical relationship between competition and cooperation in non-cooperative games, enabling users to understand how individual rationality leads to group equilibrium in group decision-making environments—serving as an intuitive courseware for game theory learning.

14. Queuing System Laboratory

Core Focus: Service System Design and Capacity Planning

Based on discrete event simulation, this laboratory simulates random arrival and service processes, dynamically displaying fluctuations in queue length, waiting time, and server utilization rates. Users can adjust λ (arrival rate) and μ (service rate), intuitively observing when the system becomes congested and when it maintains efficient operation.

Educational Value: The experiment reveals the trade-off mechanism between "efficiency" and "cost" in service systems, helping users master the design principles and capacity planning strategies of random service networks.

15. Inventory Model Laboratory

Core Focus: Supply Chain Optimization and Cost Balancing

This laboratory integrates various classic inventory control models (such as EOQ—Economic Order Quantity), displaying inventory level sawtooth changes over time through parameter driving. Users can observe how ordering costs, holding costs, and shortage losses jointly influence the total cost curve.

Educational Value: Through visual simulation, the experiment enables users to understand the overall operating mechanism of inventory systems, mastering how to find optimal order quantities between supply fluctuations and demand pressures, achieving minimization of capital occupation at the supply chain endpoint.

16. (s,S) Inventory Management Laboratory

Core Focus: Uncertain Environment Inventory Control

Through dynamic time series simulation, this laboratory displays the entire replenishment process executing the (s,S) strategy under random demand. When inventory drops to threshold s, automatic replenishment to upper limit S is triggered. Users can observe the impact of strategy parameters on shortage rates and turnover rates.

Educational Value: Through high-frequency simulation runs, the experiment helps users intuitively understand the art of inventory control in uncertain environments, mastering practical strategies for preventing supply chain risks and optimizing working capital.

17. Unconstrained Optimization Laboratory

Core Focus: Continuous Space Search and Convergence

Through three-dimensional function surfaces and contour maps, this laboratory dynamically displays the iterative search paths of various algorithms such as gradient descent and Newton's method. Users can observe in real-time the influence of initial point selection on convergence speed and how algorithms search for extreme points in complex terrain.

Educational Value: The experiment aims to reveal the search mechanism of optimal solutions in continuous space, serving as the most fundamental mathematical engine for understanding parameter optimization in deep learning, nonlinear fitting, and other modern computational science fields.

Seven-Stage Progressive Learning System

The OR-LabX system organizes these laboratories into a coherent seven-stage learning pathway:

Stage 1: Basic Modeling and Deterministic Optimization (Green Level)

Laboratories: Simplex Method | Duality Problem | Sensitivity Analysis

Core Focus: Linear Programming

Learning begins with "how to model and solve for optimal solutions." Through simplex path visualization, students understand solution generation mechanisms; through duality problems, they comprehend resource value (shadow prices); through sensitivity analysis, they understand solution stability and parameter perturbation effects.

Cognitive Upgrade: From "problem description" → "mathematical modeling" → "optimal solution interpretation"

Stage 2: Continuous Optimization and Iterative Thinking (Yellow Level)

Laboratory: Unconstrained Optimization

Core Focus: Nonlinear Programming

Breaking through linear structures, entering continuous optimization space. Understanding gradient descent, convergence paths, and local optima, establishing the cognition of "approaching optimality through iteration."

Cognitive Upgrade: From "exact solution" → "search and approximation to optimality"

Stage 3: Resource Allocation and Matching Optimization (Orange Level)

Laboratories: Transportation Problem | Assignment Problem

Core Focus: Allocation Optimization

Applying optimization thinking to "people-goods-tasks" matching problems, understanding the differences between continuous and discrete resource allocation, mastering cost minimization structures.

Cognitive Upgrade: From "single-variable optimization" → "multi-object matching optimization"

Stage 4: Network Structure and System Modeling (Blue Level)

Laboratories: Shortest Path | Maximum Flow | Minimum Spanning Tree | Network Planning

Core Focus: Graphs and Networks

Abstracting complex systems into network structures, understanding path selection, flow allocation, and system connectivity, introducing the time dimension (critical path).

Cognitive Upgrade: From "independent problems" → "structured system modeling"

Stage 5: Dynamic Decision-Making and Multi-Stage Optimization (Purple Level)

Laboratories: Decision Tree | Markov Decision (MDP) | Dynamic Programming

Core Focus: Dynamic Systems

Expanding from single-stage decisions to multi-stage processes, understanding state transitions, policy selection, and long-term returns, achieving global optimality through dynamic programming.

Cognitive Upgrade: From "static optimality" → "time-evolving optimal strategies"

Stage 6: Multi-Agent Games and Strategic Interaction (Brown Level)

Laboratory: Nash Equilibrium

Core Focus: Game Theory

Introducing multiple decision-making agents, understanding the interdependent relationships between strategies, forming equilibrium concepts.

Cognitive Upgrade: From "single-person optimality" → "multi-agent interaction equilibrium"

Stage 7: Stochastic Systems and Operations Optimization (Red Level)

Laboratories: Queuing System | Inventory Management | (s,S) Strategy

Core Focus: Stochastic Processes

Handling randomness in demand and arrivals, analyzing trade-off relationships between service efficiency, waiting times, and inventory costs.

Cognitive Upgrade: From "deterministic systems" → "uncertain system optimization"

AI-Enhanced Decision Support Framework

The OR-LabX system integrates AI capabilities across three progressive layers:

Explanation Layer (Explainable AI)

The explanation layer is primarily responsible for transforming computational results from operations research models into understandable semantic information, giving abstract mathematical outputs intuitive interpretability. For example, conducting path significance analysis on optimal solutions obtained from simplex method or network optimization, providing economic interpretation of shadow prices in dual variables, and explaining the practical impact of constraint changes on results—thereby helping learners understand "why the optimal solution holds" rather than merely "what the optimal solution is."

Reasoning Layer (Reasoning AI)

The reasoning layer is used for systematic comparison and analysis between different strategies and parameter conditions, revealing model behavior patterns through scenario deduction and sensitivity analysis. For example, analyzing cost differences between different resource allocation plans, evaluating the impact of parameter perturbations on optimal solution stability, and exploring performance changes under multiple decision paths—thereby achieving expansion from static results to dynamic reasoning, forming deep understanding of system structure and optimization mechanisms.

Decision Layer (Decision AI)

Building upon analysis from the first two layers, the decision layer further outputs executable decision recommendations oriented toward practical application scenarios, supporting multi-option selection and multi-objective trade-offs. In uncertain or complex constraint environments, this layer can comprehensively integrate optimization results, system stability, and risk factors to generate more practically actionable strategy recommendations, elevating models from "analysis tools" to "decision support systems."

Through this framework, AI completes a progressive leap from "interpreter" to "reasoning engine" to "decision participant," marking operations research's transition from traditional result-solving paradigms to a new stage of intelligent optimization centered on understanding and decision-making.

Unified Methodology: A Consistent Framework

The OR-LabX experimental system follows a unified methodological paradigm in its overall design to ensure consistency and progression in the cognitive path from basic optimization to complex system decision-making. All experimental modules revolve around the core closed loop of "Modeling → Solving → Structural Analysis → Dynamic Evolution → Decision Interpretation."

Modeling Phase

Used to abstract real-world problems into mathematical structures, including linear programming, network structures, queuing systems, or game relationships.

Solving Phase

Obtains basic optimal solutions through methods such as the simplex method, dynamic programming, shortest path algorithms, or MDP recursion.

Structural Analysis Phase

Further reveals the properties of solutions, such as duality relationships, network flow structures, resource matching mechanisms, and policy stability, thereby achieving deepened understanding from results to structures.

Dynamic Evolution Phase

Emphasizes the change process of systems under time and uncertainty, including Markov decisions, inventory fluctuations, queuing waits, and multi-stage scheduling, expanding models from static optimization to dynamic system analysis.

Decision Interpretation Phase

Conducts comprehensive integration and semantic expression of multi-model results, transforming optimization outcomes into understandable decision logic, achieving the leap from "numerical optimal solutions" to "system decision cognition."

This unified methodology runs through all operations research experimental modules, enabling linear programming, network optimization, dynamic programming, game theory, and stochastic system analysis to operate synergistically within the same framework, forming a clear-structured, hierarchically progressive intelligent decision closed-loop system, achieving overall evolution from mathematical modeling to intelligent decision-making.

Real-World Application Mapping

The OR-LabX system extends beyond theoretical learning to practical engineering applications:

Resource Allocation and Production Planning

Linear programming and sensitivity analysis can be used for cost minimization and capacity optimization, helping enterprises achieve optimal resource allocation under multi-constraint conditions; nonlinear optimization further applies to complex constraint and continuous variable systems, such as energy scheduling and parameter optimization problems.

Logistics and Supply Chain Management

Transportation and assignment problems optimize warehouse allocation and distribution paths, achieving optimal matching among people, goods, and tasks; graph and network optimization models (shortest path, maximum flow, and minimum spanning tree) can be used for traffic network design, communication network scheduling, and flow control, improving overall system efficiency from a structural perspective.

Project Management and Engineering Scheduling

Network planning methods identify critical paths and bottleneck links, optimizing the execution efficiency of complex projects from a time dimension.

Dynamic Decision-Making and Intelligent Control

Dynamic programming and Markov decision processes (MDP) are widely applied in inventory management, robot path planning, and intelligent recommendation systems, achieving optimal strategy generation for multi-stage decisions; game theory and Nash equilibrium models can be used for market competition analysis, auction mechanism design, and multi-agent system modeling, depicting the strategic evolution process of "competition-cooperation-equilibrium."

Stochastic Systems and Operations Optimization

Queuing theory analyzes waiting times and resource utilization in service systems, widely applied in customer service systems, hospital queuing, and computer task scheduling; inventory theory (such as (s,S) strategies) is used for supply chain replenishment and inventory cost optimization, achieving stable operation and cost control in demand fluctuation environments.

Conclusion: From Calculation to Comprehension

The OR-LabX experimental system takes multi-stage capability evolution as its main thread, starting from linear programming and nonlinear optimization, gradually expanding to transportation and assignment problems, graph and network optimization, network planning and scheduling, dynamic programming and Markov decisions, game theory and Nash equilibrium, as well as queuing theory and inventory control—constructing an experimental process that progresses layer by layer from "deterministic optimization → structured modeling → dynamic decision-making → stochastic systems → multi-agent games."

Through web interaction and visualization, each experiment transforms abstract mathematical models into operable, observable dynamic processes, enabling learners to understand optimization path formation mechanisms, system structure evolution laws, and strategic equilibrium principles during experiments.

Through AI-pervasive enhancement and unified methodological closed loops, the system enables all experiments to share a consistent framework of "modeling → solving → structural analysis → dynamic evolution → decision interpretation," achieving cross-module knowledge integration and capability transfer.

Ultimately, this system not only strengthens operations research modeling and solving capabilities but also propels learners from "knowing how to calculate optimal solutions" to "understanding system decision logic," forming intelligent optimization and decision-making capability systems oriented toward complex real-world systems.

The essence of operations research is not merely solving mathematically optimal solutions, but more importantly understanding how complex systems gradually evolve and form relatively optimal decision mechanisms under the combined effects of multiple constraints, internal structural relationships, and external uncertainties. It focuses not only on "optimal results" but emphasizes "how the process achieves optimality," revealing the deep logic between resource allocation, system operation, and decision selection through modeling, analysis, and optimization methods, thereby achieving structured understanding and systematic optimization of real-world problems.