Operational Research: Understanding Systems through Modelization

Operational Research (OR) involves analyzing complex systems to help make better decisions. The first step in OR is to understand the system, which we achieve through modelization. A model is a simplified representation of reality used to understand, analyze, and improve systems.

Types of Models

1. Physical Models

Definition: These are tangible, three-dimensional representations of the system.

• Example: A scale model of a building.
• Practical Case: In urban planning, physical models of city blocks are used to visualize the impact of new construction on traffic flow and pedestrian movement.

2. Iconic Models

Definition: These are visual representations that resemble the system but on a different scale.

• Example: Maps and blueprints.
• Practical Case: In logistics, an iconic model like a network diagram of a supply chain helps visualize the flow of goods from suppliers to customers.

3. Analog Models

Definition: These use different physical phenomena to represent the system.

• Example: An electrical circuit used to simulate the flow of traffic.
• Practical Case: Hydraulic models that simulate water flow through a dam to study flood control measures.

4. Symbolic Models

Definition: These use symbols and mathematical expressions to represent the system.

• Example: Equations, graphs, and computer programs.
• Practical Case: In project management, a symbolic model like a Gantt chart helps plan and monitor project timelines and resources.

Why Modelization is Important

• Simplifies Complexity: By reducing a complex system to its essential components, models make it easier to understand and analyze.
• Predictive Power: Models allow us to simulate different scenarios and predict outcomes, helping in decision-making.
• Cost-Effective: Testing different strategies on models is much cheaper and safer than in real life.

Engaging with Practical Cases

Case Study 1: Transportation Problem

• Scenario: A company needs to minimize the cost of shipping goods from multiple warehouses to multiple retailers.
• Model Used: Symbolic Model using linear programming.
• Approach: Develop a cost matrix, formulate the objective function and constraints, and use optimization software to find the best shipping plan.

Case Study 2: Inventory Management

• Scenario: A retailer wants to optimize inventory levels to balance storage costs and stockout risks.
• Model Used: Symbolic Model using the Economic Order Quantity (EOQ) formula.
• Formula:

EOQ = \sqrt{\frac{2DS}{H}}

• $( D )$: Demand rate
• $( S )$: Order cost per order
• $( H )$: Holding cost per unit per year
• Approach: Calculate EOQ to determine the optimal order size that minimizes total inventory costs.

Games and Activities for Better Understanding

Game 1: Model Building Challenge

• Objective: Build different types of models for a given real-world system (e.g., a local park).
• Rules: Students work in groups to create physical, iconic, analog, and symbolic models. Each group presents their models and explains how they would use them for system analysis.

Activity 2: Simulation Exercises

• Objective: Use simulation software to model and solve a real-life problem (e.g., traffic light optimization at an intersection).
• Approach: Students input data, run simulations, and analyze the results to propose improvements.

Conclusion

Understanding and using different types of models is crucial in Operational Research. Each type of model has its strengths and applications, and practical cases help illustrate their importance in solving real-world problems.

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Deterministic and Stochastic Models in Operational Research

In Operational Research (OR), understanding whether a model is deterministic or stochastic is crucial for accurately representing and analyzing systems.

Deterministic Models

Definition: Deterministic models are those where the outcomes are precisely determined through known relationships among states and events, without any random elements.

Characteristics:

• Predictability: Given the initial conditions, the future state of the system can be predicted exactly.
• No randomness: All inputs and processes are known and fixed.
• Examples: Linear programming, certain scheduling problems, and network flow models.

Practical Case: Production Scheduling

• Scenario: A factory needs to schedule production to meet demand orders.
• Approach: Use a deterministic linear programming model where demand, production capacity, and costs are known and fixed.
• Model:

  \text{Minimize } Z = \sum_{i=1}^n c_i x_i

  \text{Subject to:}

  \sum_{i=1}^n a_{ij} x_i \leq b_j \quad \text{for } j = 1, 2, \ldots, m

• $( Z )$: Total cost
• $( c_i )$: Cost per unit of product (i)
• $( x_i )$: Quantity of product (i) to produce
• $( a_{ij} )$: Resource requirement of product (i) for resource (j)
• $( b_j )$: Available amount of resource (j)

Stochastic Models

Definition: Stochastic models incorporate random variables and processes, acknowledging that uncertainty and variability are inherent in most real-world systems.

Characteristics:

• Uncertainty: The model includes elements of randomness and unpredictability.
• Probabilistic outcomes: Outcomes are described by probability distributions.
• Examples: Inventory models with uncertain demand, queuing models, and Monte Carlo simulations.

Practical Case: Inventory Management with Uncertain Demand

• Scenario: A retailer needs to manage inventory levels when demand is uncertain.
• Approach: Use a stochastic model like the Newsvendor model, which incorporates probabilistic demand.
• Model:

  \text{Expected Profit} = \int_{0}^{Q} (p \cdot x - c \cdot Q) f(x) \, dx + \int_{Q}^{\infty} (p \cdot Q - c \cdot Q - h \cdot (x - Q)) f(x) \, dx

• $( Q )$: Order quantity
• $( p )$: Selling price per unit
• $( c )$: Cost per unit
• $( h )$: Holding cost per unit for unsold inventory
• $( f(x) )$: Probability density function of demand $(x)$

Key Differences

Aspect	Deterministic Models	Stochastic Models
Predictability	Outcomes are precisely determined	Outcomes are probabilistic
Inputs	Fixed and known	Include random variables and distributions
Applications	Situations with certainty	Situations with uncertainty and variability

Conclusion

Understanding the nature of the system you are analyzing is crucial in choosing the right type of model. Deterministic models are suitable for systems with predictable and fixed behaviors, while stochastic models are essential for systems where uncertainty and randomness play significant roles.