Methodology in Operational Research

In Operational Research (OR), a systematic methodology is essential for analyzing and solving problems effectively. This process typically involves three key steps: Observation, Definition, and Formulation.

1. Observation

Definition: Observation involves gathering data and understanding the problem context by closely examining the system under study.

Key Activities:

• Data Collection: Gather relevant quantitative and qualitative data.
• System Analysis: Understand the components and dynamics of the system.
• Identify Issues: Recognize the main problems and constraints affecting the system.

Practical Case: Supply Chain Management

• Scenario: A company wants to improve its supply chain efficiency.
• Activities:
  • Collect data on supplier performance, transportation times, and inventory levels.
  • Analyze the flow of goods from suppliers to customers.
  • Identify bottlenecks and delays in the supply chain.

2. Definition

Definition: In the definition phase, the problem is clearly defined, including the objectives, constraints, and scope.

Key Activities:

• Problem Statement: Clearly articulate the problem to be solved.
• Objectives: Define the goals, such as minimizing costs or maximizing efficiency.
• Constraints: Identify any limitations or restrictions, such as budget or resource availability.

Practical Case: Project Scheduling

• Scenario: A construction company needs to complete a project on time and within budget.
• Activities:
  • Define the objective as minimizing the project duration.
  • Set constraints like budget limits and resource availability.
  • Create a clear problem statement: "Minimize the project duration while staying within budget and ensuring resource availability."

3. Formulation

Definition: Formulation involves developing a mathematical model that represents the problem, using the information gathered in the previous steps.

Key Activities:

• Model Selection: Choose an appropriate modeling approach (e.g., linear programming, simulation).
• Mathematical Representation: Formulate the problem using mathematical expressions.
• Assumptions: Clearly state any assumptions made in the model.

Practical Case: Workforce Scheduling

• Scenario: A hospital needs to schedule nurses to ensure 24/7 coverage.
• Activities:
  • Model Selection: Use linear programming to optimize the schedule.
  • Mathematical Representation:

  \begin{array}{ll}
  \text{Maximize} & Z = \mathbf{c}^T \mathbf{x} \\
  \text{Subject to} & \mathbf{A} \mathbf{x} \leq \mathbf{b} \\
  & \mathbf{x} \geq 0
  \end{array}

• $( Z )$: Total cost
• $( c_i )$: Cost per nurse (i)
• $( x_i )$: Hours worked by nurse (i)
• $( a_{ij} )$: Availability of nurse (i) for shift (j)
• $( b_j )$: Minimum required staff for shift (j)
  • Assumptions: Assume that nurses can work multiple shifts and that their availability is known.

4. Solution

Definition: The solution phase involves solving the formulated model to find the optimal or feasible solution to the problem.

Key Activities:

• Optimization: Use appropriate algorithms or techniques to find the optimal solution (e.g., simplex method for linear programming).
• Feasibility Check: Ensure that the solution meets all constraints and requirements.

Practical Case: Transportation Problem

• Scenario: A company needs to minimize transportation costs from warehouses to retailers.
• Activities:
  • Formulate the problem as a linear programming model.
  • Use the simplex method or software like Excel Solver to find the optimal shipping plan.

Model:

$$ \begin{aligned} \text{Minimize} \quad & Z = \sum_{i=1}^{m} \sum_{j=1}^{n} c_{ij} x_{ij} \ \text{Subject to} \quad & \sum_{j=1}^{n} x_{ij} = a_i \quad \forall i \ & \sum_{i=1}^{m} x_{ij} = b_j \quad \forall j \ & x_{ij} \geq 0 \quad \forall i, j \end{aligned} $$

• $( Z )$: Total transportation cost
• $( c_{ij} )$: Cost of shipping from warehouse $( i )$ to retailer $( j )$
• $( x_{ij} )$: Amount shipped from warehouse $( i )$ to retailer $( j )$
• $( a_i )$: Supply at warehouse $( i )$
• $( b_j )$: Demand at retailer $( j )$

5. Validation

Definition: Validation ensures that the model accurately represents the real-world system and that the solution is feasible and practical.

Key Activities:

• Verification: Check the model for correctness in terms of logic, structure, and calculations.
• Validation: Compare model outputs with real-world data or expert opinions to ensure accuracy.
• Sensitivity Analysis: Assess how changes in parameters affect the solution to ensure robustness.

Practical Case: Project Scheduling

• Scenario: A construction company needs to validate its project schedule.
• Activities:
  • Verify the scheduling model for logical consistency.
  • Validate the schedule by comparing it with historical data or consulting project managers.
  • Perform sensitivity analysis to check the impact of changes in resource availability or project duration.

6. Modifications

Definition: Modifications involve making necessary adjustments to the model based on validation results or changes in real-world conditions.

Key Activities:

• Model Adjustment: Refine the model to correct any inaccuracies or incorporate new data.
• Re-Solution: Solve the modified model to find an updated solution.
• Iterative Improvement: Continuously improve the model through iterations of validation and modification.

Practical Case: Inventory Management

• Scenario: A retailer finds discrepancies between the model predictions and actual inventory levels.
• Activities:
  • Adjust the model parameters to better reflect actual demand patterns.
  • Re-solve the inventory model to find the updated optimal order quantities.
  • Iterate the process until the model accurately predicts inventory needs.

7. Implementation

Definition: Implementation is the process of applying the model's solution in the real-world system to achieve the desired improvements.

Key Activities:

• Action Plan: Develop a detailed plan for implementing the solution.
• Execution: Carry out the plan in the real-world system.
• Monitoring: Continuously monitor the implementation to ensure it meets the objectives and make adjustments as necessary.

Practical Case: Workforce Scheduling

• Scenario: A hospital needs to implement a new nurse scheduling system.
• Activities:
  • Develop an action plan detailing the new schedule and how it will be communicated to the staff.
  • Execute the plan by rolling out the new schedule.
  • Monitor the new schedule's performance and gather feedback from nurses to make any necessary adjustments.

---

Variables, Decisions, and Parameters in Operational Research

In Operational Research (OR), understanding the different components of a model is crucial. These components include variables, decisions (controlled variables), and parameters (uncontrollable variables).

Variables

Definition: Variables are elements that can change and take on different values within a model. They are typically categorized into decision variables and parameters.

1. Decision Variables (Controlled)

Definition: Decision variables are those variables that can be controlled or decided upon by the decision-maker. These are the variables we seek to optimize in OR models.

Characteristics:

• Control: Decision-makers have direct control over these variables.
• Objective: They are part of the objective function we aim to optimize.
• Examples: Quantity of products to produce, hours of labor to allocate, routes to take.

Practical Case: Production Planning

• Scenario: A factory needs to decide how many units of each product to produce to maximize profit.
• Decision Variables:
  • $( x_1 )$: Number of units of Product 1 to produce.
  • $( x_2 )$: Number of units of Product 2 to produce.

Model:

$$ \begin{array}{ll} \text{Maximize} & Z = p_1 x_1 + p_2 x_2 \ \text{Subject to} & a_{11} x_1 + a_{12} x_2 \leq b_1 \ & a_{21} x_1 + a_{22} x_2 \leq b_2 \ & x_1, x_2 \geq 0 \end{array} $$

• $( Z )$: Total profit
• $( p_1, p_2 )$: Profit per unit of Product 1 and Product 2
• $( a_{ij} )$: Resource consumption coefficients
• $( b_j )$: Available amount of resource ( j )

2. Parameters (Uncontrollable)

Definition: Parameters are elements that cannot be controlled by the decision-maker. They are constants in the model that represent external conditions or inputs.

Characteristics:

• Fixed Values: Parameters are given and fixed within the context of the model.
• External Influence: They represent factors outside the control of the decision-maker.
• Examples: Market demand, production capacity, resource availability.

Practical Case: Inventory Management

• Scenario: A retailer needs to manage inventory levels considering market demand and holding costs.
• Parameters:
  • $( D )$: Market demand for the product.
  • $( H )$: Holding cost per unit of inventory.
  • $( C )$: Cost per unit of product.

Model:

$$ \begin{array}{ll} \text{Minimize} & TC = \frac{D}{Q} S + \frac{Q}{2} H \ \text{Where} & Q = \sqrt{\frac{2DS}{H}} \end{array} $$

• $( TC )$: Total cost
• $( Q )$: Order quantity
• $( S )$: Ordering cost

Variables de Desempeño (Performance Variables)

Definition: Performance variables are metrics that allow us to evaluate the extent to which desired objectives have been achieved. They provide a way to measure the effectiveness and efficiency of a system or process in reaching its goals.

Characteristics of Performance Variables

• Quantifiable: They should be measurable using numerical data.
• Relevant: They must be directly related to the objectives of the system or process.
• Actionable: They should provide insights that can lead to actionable decisions.
• Timely: They should be measured and reported in a timely manner to allow for prompt decision-making.

Types of Performance Variables

1. Key Performance Indicators (KPIs)

Definition: Specific, quantifiable measures that gauge the performance of a system or process in relation to its objectives.

• Examples:
  • Customer satisfaction score
  • Return on investment (ROI)
  • Order fulfillment rate

2. Efficiency Metrics

Definition: Measures that evaluate how well resources are utilized to achieve objectives.

• Examples:
  • Cycle time: The time taken to complete a process.
  • Resource utilization rate: The percentage of resources used compared to what is available.

3. Effectiveness Metrics

Definition: Measures that assess how well the outcomes align with the desired objectives.

• Examples:
  • Quality of output: The number of defects per unit produced.
  • Achievement of targets: The percentage of objectives met.

Practical Case: Manufacturing Process

Scenario

A manufacturing company wants to improve its production process and ensure it meets customer demand efficiently and effectively.

Performance Variables:

1. Production Rate:
   • Definition: The number of units produced per hour.
   • Objective: Ensure the production rate meets customer demand.
2. Defect Rate:
   • Definition: The number of defective units per batch produced.
   • Objective: Maintain a high quality of products with minimal defects.
3. On-time Delivery:
   • Definition: The percentage of orders delivered on or before the promised date.
   • Objective: Improve customer satisfaction by ensuring timely deliveries.
4. Cost per Unit:
   • Definition: The total cost incurred to produce one unit.
   • Objective: Minimize production costs to increase profitability.

Example Metrics:

$$ \begin{aligned} \text{Production Rate} &= \frac{\text{Total Units Produced}}{\text{Total Hours Worked}} \ \text{Defect Rate} &= \frac{\text{Number of Defective Units}}{\text{Total Units Produced}} \times 100 \ \text{On-time Delivery} &= \frac{\text{Number of On-time Deliveries}}{\text{Total Deliveries}} \times 100 \ \text{Cost per Unit} &= \frac{\text{Total Production Cost}}{\text{Total Units Produced}} \end{aligned} $$

Conclusion

Understanding the roles of variables and parameters in OR models is essential for effective problem-solving. Decision variables are controlled by the decision-maker and are central to the optimization process, while parameters are uncontrollable and represent external conditions.