Operations Management: Queueing and Forecasting
Single-Server Queue Model Principles
A key assumption for the single-server queue model is that the average interarrival time (time between arrivals) must be greater than the average process time.
Utilization and Probability Formulas
Utilization (ρ) represents the percentage of time a server is busy. It is calculated as:
ρ = λ (arrival rate) / μ (service rate).
The probability of having n customers in the system is defined as: Pn = (1 – ρ)ρn.
Queue Formation and Capacity
In a scenario where the average arrival rate equals the average service rate and the Coefficient of Variation (CV) of both rates is 1.00, a line will form and gradually increase over time. This happens because the line length can never go below zero, whereas it can exceed zero in various periods.
True or False: A queue cannot form if average server capacity exceeds average demand.
Answer: False.
The formula for the average number of customers in a line is: Lq = ρ² / (1 – ρ).
Forecasting Methods and Data Volatility
The Delphi method is used for emerging technologies or nascent products by consulting top experts in the field, though it can be slow and inconclusive.
Smoothing and Moving Averages
- An exponential smoothing constant (α) of 1 emphasizes the most recent actual demand figure.
- A moving average forecast corresponds to a naïve forecast when the period taken is one.
- To perform single exponential smoothing, one requires the current month’s actual sales, forecasted sales, and an alpha smoothing constant.
Data volatility consists of two primary elements: Variability and Uncertainty. Averaging forecasts from different methods generally results in greater accuracy relative to a single method.
Error Measurement and Accuracy
To present the magnitude, direction (bias), and confidence interval of a forecast error, the following sequence of measures is best: MAD, Mean Error (or MPE/CFE), and RMSE.
The Mean Absolute Percentage Error (MAPE) can be misleading when actual demand in a series occasionally approaches zero. Furthermore, the larger the R² value in a linear regression, the greater the explanatory power of the independent variables.
Inventory Management and EOQ Analysis
Using the LIFO (Last-In, First-Out) system for costing inventory in an environment of rising input costs will likely cause reported profits to decline.
Economic Order Quantity (EOQ)
The EOQ formula is: EOQ = √[(2 × D × S) / H], where:
- D: Annual demand
- S: Average cost of placing an order
- H: Average cost of holding one unit for one year
If annual demand increases ninefold while all other values remain constant, the EOQ will increase by three times.
Holding Costs and Stock Levels
Inventory holding costs include scrap, obsolescence, and spoilage costs. Spoilage is particularly relevant for perishable goods like bread. High stock-out costs relative to holding costs should lead to recommendations for larger orders. Phantom inventory occurs when physical stock differs from the records in the computer system.