Time Series Analysis, Forecasting & Operations Management Concepts

Posted on Apr 4, 2025 in Mathematics

Time Series Components

A time series is a set of values of a variable measured at successive points in time.

Key Components

  • Trend: The gradual increase or decrease of a time series over a longer period.
  • Seasonality: Regular patterns of variability within certain time periods, such as a year, week, etc.
  • Cycle: Any regular pattern above and below the trend line lasting more than one year.
  • Random Variations: Fluctuations caused by chance events and/or unusual situations.

Time Series Forecasting Methods

Smoothing Methods

Used for stable time series and short-term (ST) forecasting.

  • Simple Moving Average (SMA): Calculates the average of a specific number of the most recent observations. For example, if SMA = 3, the first forecast starts using values from periods 1, 2, and 3 to predict period 4.
  • Weighted Moving Average (WMA): Averages recent observations, giving more weight to the values closest to the forecasted period.
  • Simple Exponential Smoothing (SES): A forecasting method where the forecast for the next period is a weighted average of the actual value from the current period and the forecast from the current period. Formula: F(t) = α * X(t-1) + (1-α) * F(t-1), where α is the smoothing constant.

Decomposition Methods

Difficulty in isolating components: Trend (very easy), Seasonality (easy), Cycle (difficult), Random (impossible).

  • Simple Linear Regression: Used to model the trend component. Formula: Ft = A0 + A1 * t. Factors A0 (intercept) and A1 (slope) are determined by minimizing the sum of squared errors. Useful for understanding the general trend, identifying issues, providing a base for forecasting, and removing biases.
  • Seasonal Index:
    1. Model the trend (e.g., using Linear Regression) for past periods.
    2. Calculate the seasonal ratio for past periods: (Actual Value - Forecasted Trend Value) / Forecasted Trend Value or Actual Value / Forecasted Trend Value depending on the model (additive vs multiplicative).
    3. Determine the average seasonal index for each period (e.g., average index for all Januarys, Februarys, etc.).
    4. Establish forecasts by combining the models (e.g., Additive: Forecast = Trend + Seasonality or Multiplicative: Forecast = Trend * Seasonality. Example using Linear Regression (LR) trend and Seasonal Index (SI) for period p: Forecast = LR(t) + SI(p) (additive) or Forecast = LR(t) * SI(p) (multiplicative)).

Evaluating Forecast Quality

  • Mean Squared Error (MSE): The average of the squared differences between forecasted values (Ft) and actual values (Xt). Formula: MSE = Average[(Ft - Xt)^2]. A smaller MSE indicates better forecast accuracy.
  • Average Margin of Error: Typically represented by the Standard Error of the forecast, which is the square root of the MSE (sqrt(MSE)).

Probability in Operations Management

Probabilities are used in operations management for:

  • Determining service levels
  • Resource planning
  • Inventory management
  • Infrastructure planning and organization

Normal Distribution

A continuous probability distribution. Often used with functions like NORMDIST (to find probability) and NORMINV (to find a variable’s value for a given probability) in spreadsheet software.

Poisson Distribution

A discrete probability distribution expressing the probability of a given number of events occurring in a fixed interval of time or space. Parameter: λ (lambda) = average number of occurrences in an interval.

Examples (assuming λ = 15):

  • P(X = 20) = POISSON.DIST(20, 15, FALSE) ≈ 4.2%
  • P(X < 20) = P(X ≤ 19) = POISSON.DIST(19, 15, TRUE) ≈ 87.5%
  • P(X ≤ 20) = POISSON.DIST(20, 15, TRUE) ≈ 91.7%
  • P(X > 20) = 1 – P(X ≤ 20) = 1 – POISSON.DIST(20, 15, TRUE) ≈ 8.3%
  • P(X ≥ 20) = 1 – P(X ≤ 19) = 1 – POISSON.DIST(19, 15, TRUE) ≈ 12.5%
  • P(15 ≤ X ≤ 20) = P(X ≤ 20) – P(X ≤ 14) = POISSON.DIST(20, 15, TRUE) - POISSON.DIST(14, 15, TRUE)

Exponential Distribution

A continuous probability distribution describing the time between events in a Poisson point process. Parameter: λ (lambda) = average number of occurrences per time unit (rate).

Process Management Fundamentals

A process is a sequence of tasks that use resources (human, machines, tools) and materials to produce and deliver a product or service to a customer.

Lean Six Sigma Methodology (DMAIC)

A structured approach to process improvement:

  1. Define: Define the problem, project goals, and customer deliverables.
  2. Measure: Measure process performance.
  3. Analyse: Analyse the process to determine root causes of variation and defects.
  4. Improve: Improve the process by eliminating defects.
  5. Control: Control future process performance.

Define (D) Phase

1. Process Modeling (What?)

Visualizing the process:

  • Flowchart
  • Swimlanes Flowchart (Cross-functional flowchart)
  • SIPOC Diagram (Supplier, Input, Process, Output, Customer)

2. Voice of the Customer (VOC) (Why?)

Understanding customer needs and feedback:

  • Identifying customer needs: Market research, customer surveys, focus groups.
  • Tracking customer voice: Monitoring satisfaction, managing complaints, analyzing sales performance.

Measure (M) Phase

1. Process Analysis (How many/much/long?)

  • Capacity: The maximum output rate. Calculated based on available resources and their processing times (e.g., Number of Resources / Processing Time per Unit).
  • Process Time / Cycle Time: The total time required to complete all sequential tasks in a process.
  • Bottleneck: The task or activity that limits the overall process capacity.
  • Balance Demand with Process Capacity:
    • If Workload > Capacity: Process is under-capacity.
    • If Capacity > Workload: Process is over-capacity.
  • Adjusting Task/Activity Capacity: Capacity can be modified by changing the number of resources assigned, altering the duration of the task (e.g., through improvements), or redesigning the process (e.g., merging or splitting tasks).