Operations Management Formulas and Key Definitions

Posted on Apr 28, 2026 in Mathematics

Waiting Line Management

Definitions and Formulas

  • Arrival rate (λ): Average number of customers arriving per unit time.
    λ = 1 / E(A); where E(A) is the average interarrival time.
  • Service rate (μ): Average number of customers served per unit time per server.
    μ = 1 / E(S); where E(S) is the average service time.
  • Utilization (ρ): Fraction of system capacity being used. This value must be less than 1.
    ρ = λ / (cμ); where c is the number of servers.
  • Coefficient of variation (CV): Measures variability.
    Cₐ = σₐ / E(A), Cₛ = σₛ / E(S); where σ is the standard deviation.
  • Average number in system (L): Total customers in the system.
    L = λW; where W is the average time in the system.
  • Average number in queue (Lq): Customers waiting for service.
    Lq = λWq; where Wq is the waiting time.
  • Average time in system (W):
    W = Wq + 1/μ
  • Average number relationship:
    L = Lq + (λ / μ)

M/M/1 Model Formulas

  • Utilization: ρ = λ / μ
  • Probability system is empty (P0): P0 = 1 − ρ
  • Probability of n customers (Pn): Pn = (1 − ρ)ρⁿ
  • Average number in queue (Lq): Lq = ρ² / (1 − ρ)
  • Average number in system (L): L = ρ / (1 − ρ)
  • Waiting time (Wq): Wq = Lq / λ
  • Time in system (W): W = 1 / (μ − λ)

M/M/c Model Formulas

  • Probability system is empty (P0):
    P0 = 1 / [ Σ from n=0 to c−1 of ( (λ/μ)^n / n! ) + ( (λ/μ)^c / (c! (1−ρ)) ) ]
  • Average number in queue (Lq):
    Lq = [ P0 (λ/μ)^c ρ ] / [ c! (1−ρ)² ]
  • Waiting time: Wq = Lq / λ; Time in system: W = Wq + 1/μ

PK Approximation (General Case)

Lq ≈ [ ρ² (Cₐ² + Cₛ²) ] / [ 2 (1 − ρ) ] × (1 / c)

Quality Management

Definitions and Formulas

  • Specification limits (USL, LSL): The acceptable range of output.
  • Process mean (μ): The average output.
  • Standard deviation (σ): The variation in the process.
  • Process capability ratio (Cp): Measures the spread.
    Cp = (USL − LSL) / (6σ)
  • Process capability index (Cpk): Measures spread and centering.
    Cpk = minimum of [ (USL − μ) / (3σ), (μ − LSL) / (3σ) ]
  • Defect rate:
    Defect rate = number defective / total

Control Charts

  • X-bar control chart (Mean Chart):
    UCL = x̄ + z(σ / √n)
    LCL = x̄ − z(σ / √n)
    where x̄ is the sample mean and n is the sample size.
  • P-chart (Proportion Defective):
    UCL = p̄ + z √[ p̄(1 − p̄) / n ]
    LCL = p̄ − z √[ p̄(1 − p̄) / n ]
    where p̄ is the average defect rate.
  • C-chart (Number of Defects):
    UCL = c̄ + z √c̄
    LCL = c̄ − z √c̄
    where c̄ is the average number of defects.

Inventory Management

Definitions and Formulas

  • Demand (D): Units required per year.
  • Ordering cost (S): Cost per order.
  • Holding cost (H): Cost per unit per year.
  • Unit cost (P): Cost per unit.
  • Economic Order Quantity (EOQ): Optimal order size.
    Q* = √( 2DS / H )
  • Annual ordering cost: (D / Q)S
  • Annual holding cost: (Q / 2)H
  • Total cost (TC): TC = (D / Q)S + (Q / 2)H + PD
  • Number of orders per year: D / Q
  • Time between orders: Q / D

Reorder Point and Safety Stock

  • Reorder point (ROP): ROP = dL
    where d = D / working days and L = lead time.
  • Reorder point with uncertainty: ROP = μLT + zσLT
    where μLT = dL and σLT = σd √L.
  • Safety stock (SS): SS = zσLT
  • Total cost with safety stock:
    TC = (D / Q)S + (Q / 2 + SS)H + PD

Newsvendor Model

  • Underage cost (Cu): Cu = p − c
  • Overage cost (Co): Co = c − b
    where p = selling price, c = cost, and b = salvage value.
  • Critical fractile (CR): CR = Cu / (Cu + Co)
  • Optimal order quantity (Q*): Q* = μ + zσ

Forecasting and Supply Chain Management

Definitions and Formulas

  • Actual demand (A): The observed value.
  • Forecast (F): The predicted value.
  • Forecast error: Error = A − F
  • Mean Absolute Deviation (MAD): Average absolute error.
    MAD = ( Σ |A − F| ) / n
  • Mean Squared Error (MSE): Average squared error.
    MSE = ( Σ (A − F)² ) / n
  • Tracking signal (TS): Measures bias.
    TS = ( Σ (A − F) ) / MAD

Forecasting Methods and Concepts

  • Time series components:
    • Trend: Long-term movement.
    • Seasonality: Repeating patterns.
    • Random variation: Noise.
  • Moving average forecast (n periods):
    F = (A₁ + A₂ + … + Aₙ) / n
  • Weighted moving average:
    F = w₁A₁ + w₂A₂ + … + wₙAₙ; where Σw = 1.
  • Exponential smoothing:
    Fₜ₊₁ = αAₜ + (1 − α)Fₜ; where α is the smoothing constant (0 to 1).
  • Trend-adjusted forecast: Fₜ = level + trend
  • Bullwhip effect: A concept where demand variability increases as one moves upstream in the supply chain.