a. including DPU, DPMO, FTY, RTY Cycle Time; deriving these metrics

including DPU, DPMO, FTY, RTY Cycle Time; deriving these metrics Introduction This article explains how to understand and derive the key process performance metrics: - Defects per Unit (DPU) - Defects per Million Opportunities (DPMO) - First Time Yield (FTY) - Rolled Throughput Yield (RTY) - Cycle Time It focuses on definitions, correct formulas, interpretation, and practical calculation steps, including how these metrics relate to each other and to process defects and opportunities. --- Foundations: Units, Defects, Opportunities Units, Defects, and Defective Units Before deriving the metrics, distinguish three basic concepts: - Unit One item that goes through the process and is judged acceptable or not. - Examples: one order, one component, one claim, one report. - Defect A specific nonconformance to a requirement. A single unit can contain multiple defects. - Examples: missing signature, scratch on a surface, wrong field completed. - Defective unit A unit that has one or more defects such that it does not meet acceptance criteria. - Defective units may be: - Scrapped - Reworked - Returned - Repaired Important relationship: - One unit → can have 0, 1, 2, … defects - If a unit has ≥ 1 defect → it is defective (by definition in most quality systems) Defect Opportunities To compare processes fairly, defects are related to opportunities: - Defect opportunity A meaningful chance for a defect to occur on one unit, based on requirements. - Example: A customer invoice has: - 3 critical fields (name, address, total) - 2 supporting fields (due date, PO number) If all 5 fields are considered meaningful and independently defect-prone: - Opportunities per unit = 5 - Total opportunities for a sample: - Total opportunities = (Units) × (Opportunities per unit) Opportunity definitions must be: - Explicit - Stable over time - Limited to meaningful failure modes, not every trivial check --- Defects per Unit (DPU) Concept and Formula DPU measures the average number of defects per unit, regardless of how many units are defective or how severe the defects are. - Definition - DPU = (Total number of defects) ÷ (Total number of units inspected) - Formula - DPU = D / U Where: - D = total defects found - U = total units inspected Example Calculation and Interpretation - Suppose: - 500 invoices (units) are checked - 160 defects are found across all invoices - Calculation: - DPU = 160 / 500 = 0.32 defects per unit - Interpretation: - On average, each invoice has 0.32 defects. - This does not tell how many invoices are defective, only the average defect count per invoice. Relationship to Defective Units If the defect rate is low and defects per unit follow a Poisson-like distribution, an approximate relationship is: - Probability(unit has no defects) ≈ e^(−DPU) - Probability(unit has at least one defect) ≈ 1 − e^(−DPU) This relationship is fundamental for connecting DPU with yield-based metrics like FTY and RTY (especially under simplifying assumptions). --- Defects per Million Opportunities (DPMO) Concept and Formula DPMO scales defects to a standard base of one million opportunities, allowing comparison across processes with different numbers of opportunities per unit. - Definition - DPMO = (Defects / Opportunities) scaled to one million - Core formula - DPMO = (D / (U × O)) × 1,000,000 Where: - D = total defects - U = total units - O = opportunities per unit Example Calculation A process produces 2,000 circuit boards: - Given: - U = 2,000 boards - O = 10 opportunities per board - D = 48 total defects found - Steps: - Total opportunities = U × O = 2,000 × 10 = 20,000 - DPMO = (48 / 20,000) × 1,000,000 - DPMO = 0.0024 × 1,000,000 = 2,400 DPMO Interpreting DPMO - DPMO expresses defect frequency per opportunity, normalized to 1,000,000. - A low DPMO value indicates: - Fewer defects per opportunity - Better process quality, given the defined opportunities - Comparing across processes: - A process with 1,000 DPMO has fewer defects per opportunity than one with 5,000 DPMO, assuming opportunity definitions are equally rigorous. Link between DPU and DPMO If each unit has O opportunities: - DPU = D / U - Defects per opportunity = D / (U × O) - Therefore: - DPMO = (DPU / O) × 1,000,000 Or rearranged: - DPU = (DPMO × O) / 1,000,000 This relationship is useful when defects are tracked at the opportunity level but calculations must be presented at the unit level. --- First Time Yield (FTY) Concept and Formula First Time Yield (FTY) is the proportion of units that pass a step or an entire process without any rework or repair. - Definition at a single step - FTY(step) = (Units that leave the step “good on first pass”) ÷ (Units that entered the step) Units failing first time and needing rework do not count in the numerator for FTY, even if they are later fixed. Single-Step Example A testing station receives 1,000 units: - 920 pass on first attempt - 80 fail and require rework (some eventually pass, some are scrapped) - FTY(test) = 920 / 1,000 = 0.92 (92%) The FTY reflects “right first time” performance, independent of later recoveries. Multi-Step FTY vs. Defect-Based Yield If only one step exists and units are either passed or rejected (no rework), then: - FTY(step) ≈ “yield” of that step When rework is allowed: - FTY measures initial performance - A different metric (like final yield or RTY) captures the cumulative effect, including rework --- Rolled Throughput Yield (RTY) Concept Rolled Throughput Yield (RTY) measures the probability that a unit passes all steps in a process without a single defect and without rework. - RTY is the product of FTYs across all process steps, under the assumption that: - Each step’s first-pass performance is independent - FTY is measured per step with rework excluded from the numerator Core Formula For a process with n steps: - RTY = FTY₁ × FTY₂ × ... × FTYₙ Where: - FTYᵢ = First Time Yield at step i RTY is always ≤ each individual FTY, and decreases as the number of steps or their defect rates increase. Multi-Step Example Consider a three-step process: - Step 1: Input 1,000 units, 950 pass first time - FTY₁ = 950 / 1,000 = 0.95 - Step 2: Input 950 units, 900 pass first time - FTY₂ = 900 / 950 ≈ 0.9474 - Step 3: Input 900 units, 855 pass first time - FTY₃ = 855 / 900 = 0.95 - RTY = 0.95 × 0.9474 × 0.95 - RTY ≈ 0.855 (85.5%) Interpretation: - About 85.5% of units are expected to pass all three steps without any rework. Estimating RTY from Defects per Unit If DPU per step is known and defects per unit are assumed to follow a Poisson distribution: - Approximate yield per step: - FTYᵢ ≈ e^(−DPUᵢ) - Then RTY is: - RTY ≈ e^(−(DPU₁ + DPU₂ + ... + DPUₙ)) Equivalently: - Total DPU across all steps: DPU_total = Σ DPUᵢ - RTY ≈ e^(−DPU_total) This relationship is essential when: - Defects are counted per step - You need to estimate the overall probability a unit has zero defects across the process RTY vs. Final Yield - Final Yield: fraction of units leaving the process as acceptable, regardless of rework. - RTY: fraction of units that are defect-free through all steps, without rework. If rework exists: - Final yield ≥ RTY - The gap between them indicates impact of rework and reprocessing. --- Connecting DPU, DPMO, FTY, and RTY From Defects to Yield (Single Step) Given a step with known DPU_step: - Approximate probability a unit has no defects at the step: - FTYstep ≈ e^(−DPUstep) - Approximate probability a unit has at least one defect: - 1 − FTYstep ≈ 1 − e^(−DPUstep) This comes from modeling defects per unit as a Poisson random variable with mean DPU_step. From Defects to RTY (Multiple Steps) Given DPU at each step: - DPU_total = DPU₁ + DPU₂ + ... + DPUₙ - RTY ≈ e^(−DPU_total) So: - If DPU_total = 0.3 defects per unit across all steps: - RTY ≈ e^(−0.3) ≈ 0.7408 (74.08%) Interpretation: - Approximately 74% of units are expected to have zero defects across all steps. From DPMO to Yield If DPMO is known and each unit has O opportunities: - DPU = (DPMO × O) / 1,000,000 - Then yield (probability of zero defects) per unit is approximately: - Yield ≈ e^(−DPU) For processes where “zero defects per unit” is the pass criterion, this yield is equivalent to the probability a unit is non-defective. --- Cycle Time Concept and Types of Time Cycle Time is the total elapsed time for a unit to move from a defined start point to a defined end point of a process. Two related concepts are important: - Processing time Time during which the unit is actively being worked on (value-adding or non–value-adding activity time). - Waiting time Time when the unit is in queue, waiting, or on hold (no active processing). Cycle time includes both: - Cycle Time = Processing Time + Waiting Time Basic Cycle Time Metrics At the level of a single unit: - Single-unit cycle time - CT_unit = End time − Start time Over multiple units: - Average cycle time - CT_avg = (Sum of individual cycle times) / (Number of units) Cycle time can be measured: - Per step - Across the full end-to-end process Relationship to Yield and Defects While cycle time and defect metrics measure different dimensions, they influence each other: - High defect rates and high rework typically: - Increase total processing time and waiting time - Increase cycle time - Improvements in FTY and RTY often: - Reduce the need for rework - Reduce queues and congestion - Shorten overall cycle time A process analysis often examines: - How changes in DPU, DPMO, FTY, and RTY affect cycle time - How long cycle times might hide defect detection (defects discovered late are more costly) --- Deriving and Using These Metrics in Practice Step-by-Step Data Collection To correctly derive DPU, DPMO, FTY, RTY, and cycle time: - Define the process scope - Start point and end point for units - Steps within the process (for multi-step metrics) - Define units - What is counted as a single unit (e.g., one part, one order) - Define opportunities - List defect opportunities per unit - Determine O = opportunities per unit - Collect data - Count units entering and exiting each step - Count defects per unit or per step - Record time stamps at start and end points Deriving the Metrics from Raw Data Given data for a sample: - U = units observed - D = total defects observed - O = opportunities per unit - For each step i: - U_in,ᵢ = units entering - Ufirstpass_out,ᵢ = units that pass first time Compute: - DPU - DPU = D / U - DPMO - DPMO = (D / (U × O)) × 1,000,000 - FTY at each step - FTYᵢ = Ufirstpassout,ᵢ / Uin,ᵢ - RTY - RTY = Π (FTYᵢ) across all steps - Cycle time - For each unit j: CTⱼ = Endtimeⱼ − Starttimeⱼ - Average CT_avg = (Σ CTⱼ) / U Interpretation for Improvement These metrics help answer different questions: - DPU and DPMO: - How frequently do defects occur? - Is the process improving over time? - FTY: - How well does each step perform the first time? - Where is rework most frequent? - RTY: - What portion of units are defect-free end-to-end without rework? - How does adding or removing steps change the chance of a perfect unit? - Cycle Time: - How long does the process take? - Where are the major delays or bottlenecks? Used together, they create a quantitative view of process quality and speed. --- Summary This article has focused on understanding and deriving five tightly related process metrics: - DPU measures average defects per unit. - DPMO normalizes defects per opportunity to one million opportunities, allowing cross-process comparison. - FTY captures the fraction of units that pass a step without rework. - RTY is the product of FTYs across steps, giving the probability a unit passes the entire process without a defect or rework, and links directly to total DPU through an exponential relationship under common assumptions. - Cycle Time measures the total elapsed time for units to move through the process, including waiting and processing time, and is strongly influenced by defects and rework. By defining units and opportunities clearly, collecting consistent defect and time data, and applying the formulas systematically, these metrics can be derived and interpreted to understand and improve process performance in terms of quality and speed.