A core industrial & systems engineering reference covering Lean and the eight wastes, Six Sigma and DMAIC, OEE, takt and line balancing, Little's Law and the Theory of Constraints, statistical process control, process capability, inventory models (EOQ/EPQ/ROP), queuing theory, CPM/PERT project scheduling, forecasting, and a master formula quick-reference. Use the contents to jump to a section, or Prev / Next to read through. Every section is rendered on the page so it is fully searchable.
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1. Lean Manufacturing & the 8 Wastes (DOWNTIME)
Lean manufacturing, derived from the Toyota Production System, defines value from the customer's perspective and relentlessly removes everything that does not add value. Anything the customer would not willingly pay for is waste (muda). The classic seven wastes were expanded to eight, remembered by the acronym DOWNTIME: Defects (rework and scrap), Overproduction (making more or sooner than needed — considered the worst waste because it hides the others), Waiting (idle time for people or machines), Non-utilized talent (failing to use people's skills and ideas), Transportation (unnecessary movement of material), Inventory (excess raw, WIP, or finished goods that ties up cash and conceals problems), Motion (unnecessary movement of people), and Excess processing (doing more than the customer requires). Lean attacks waste with tools such as 5S workplace organization, standardized work, single-minute exchange of dies (SMED) for rapid changeover, kanban pull systems, value stream mapping, kaizen (continuous improvement) events, and poka-yoke (mistake-proofing). The two pillars of the Toyota house are Just-In-Time (produce only what is needed, when needed, in the amount needed) and Jidoka (build in quality and stop the line on defects). The goal is one-piece flow at the pace of customer demand (takt) with the minimum inventory.
2. Six Sigma & DMAIC
Six Sigma is a data-driven methodology for reducing variation and defects, targeting a process so capable that defects are extremely rare — 3.4 defects per million opportunities (DPMO) when a 1.5-sigma long-term shift is allowed. A six-sigma process has its nearest specification limit six standard deviations from the mean. The core improvement framework is DMAIC: Define (the problem, customer CTQs, project scope and goals), Measure (collect baseline data, validate the measurement system with Gage R&R, compute current sigma level), Analyze (find root causes using fishbone diagrams, the 5 Whys, hypothesis tests, regression, and Pareto analysis), Improve (pilot and implement solutions, often using design of experiments), and Control (lock in gains with control plans, SPC charts, and standardized work). For new product or process design, DMADV (Define-Measure-Analyze-Design-Verify), part of Design for Six Sigma, is used instead. Practitioners are ranked by belts: Yellow, Green, Black, and Master Black Belt, with Champions sponsoring projects. Lean and Six Sigma are commonly combined as Lean Six Sigma — Lean removes waste and improves flow while Six Sigma reduces variation and defects.
3. OEE & the Six Big Losses
Overall Equipment Effectiveness (OEE) measures how well a manufacturing operation utilizes equipment, expressed as the product of three factors: Availability × Performance × Quality. Availability = run time ÷ planned production time, capturing downtime losses. Performance = (ideal cycle time × total count) ÷ run time, capturing speed losses. Quality = good count ÷ total count, capturing defect losses. A world-class OEE is around 85% (typically 90% availability × 95% performance × 99.9% quality); 100% means making only good parts, as fast as possible, with no stop time. OEE quantifies the Six Big Losses grouped under each factor. Availability losses: (1) unplanned stops/breakdowns and (2) planned stops/setup and changeover. Performance losses: (3) small/idling stops and (4) reduced speed/slow cycles. Quality losses: (5) production defects/rejects and (6) startup/reduced-yield losses. TEEP (Total Effective Equipment Performance) extends OEE by also accounting for utilization (scheduled time vs. all calendar time), revealing the true capacity of the asset. OEE is the central metric of Total Productive Maintenance (TPM).
4. Takt Time, Cycle Time & Line Balancing
Takt time is the rate of customer demand: available work time per period ÷ customer demand per period. It is the pace at which one unit must be completed to meet demand exactly — the heartbeat of a lean line. Cycle time is the actual time between consecutive completed units at a process. To meet demand without overtime, every process cycle time must be at or below takt; any process above takt is a bottleneck. Line balancing distributes the total work content across workstations so each station's time is as close to takt as possible without exceeding it, minimizing idle time. The theoretical minimum number of stations = total task time ÷ takt time (rounded up). Line efficiency = total task time ÷ (number of stations × cycle time). Balance delay (idle %) = 1 − efficiency. The challenge is the precedence diagram: tasks must be assigned respecting their required order. Heuristics like Ranked Positional Weight or Longest-Task-Time assign tasks to stations greedily. A well-balanced line has minimal idle time, smooth flow, and station times just under takt, enabling stable one-piece flow.
5. Little's Law & Theory of Constraints
Little's Law is a fundamental relationship in any stable queuing system: WIP = Throughput × Lead Time (often written L = λW, where L is average number in system, λ is arrival/throughput rate, and W is average time in system). It holds regardless of the arrival distribution or service order, making it one of the most useful laws in operations. Rearranged, Lead Time = WIP ÷ Throughput — so to cut lead time at constant throughput, you must cut WIP. This is the mathematical justification for limiting work-in-process in Lean and Kanban. The Theory of Constraints (TOC), from Goldratt's The Goal, states that every system has at least one constraint (bottleneck) limiting its throughput, and overall performance is governed by that constraint. The Five Focusing Steps are: (1) Identify the constraint, (2) Exploit it (get the most from it without major investment), (3) Subordinate everything else to the constraint's pace, (4) Elevate the constraint (add capacity if still limiting), and (5) Repeat — do not let inertia create a new constraint. Drum-Buffer-Rope schedules the whole system to the drumbeat of the constraint. The key insight: an hour lost at the bottleneck is an hour lost for the entire system; an hour saved at a non-bottleneck is a mirage.
6. Statistical Process Control (X̄-R, p-charts, Western Electric rules)
Statistical Process Control (SPC) uses control charts to distinguish common-cause variation (inherent, random, in-control) from special-cause variation (assignable, signals a process change). Control limits are set at ±3 standard deviations from the centerline, capturing 99.73% of common-cause variation. For variable (measured) data in subgroups, the X̄ and R charts are used together: the X̄ chart monitors the process mean with limits X̄̄ ± A2·R̄, and the R chart monitors within-subgroup spread with limits D4·R̄ and D3·R̄, where A2, D3, D4 are constants depending on subgroup size n. Always check the R chart first — if spread is unstable, the X̄ limits are meaningless. For attribute (count) data, the p-chart tracks the fraction defective, the np-chart the number defective, the c-chart counts of defects per unit, and the u-chart defects per unit for variable sample sizes. The Western Electric (Nelson) rules detect special causes beyond a single point outside 3σ: e.g., 2 of 3 points beyond 2σ, 4 of 5 beyond 1σ, 8 consecutive points on one side of the centerline, or 6 in a row trending. SPC is about monitoring a process over time; capability (Cpk) compares it to specifications.
7. Process Capability (Cp / Cpk / Pp / Ppk)
Process capability indices compare the voice of the process (its variation) to the voice of the customer (specification limits). Cp = (USL − LSL) ÷ 6σ measures potential capability — how the spread fits within the tolerance — but ignores centering. Cpk = min[(USL − μ)/3σ, (μ − LSL)/3σ] measures actual capability, penalizing a process that is off-center; Cpk ≤ Cp always, and they are equal only when the process is perfectly centered. A common rule of thumb: Cpk ≥ 1.33 is capable (4σ), ≥ 1.67 is good, and a Six Sigma process has Cpk = 2.0. Cp/Cpk use the short-term (within-subgroup) sigma estimated from R̄/d2, reflecting inherent capability. Pp and Ppk are the performance indices using the long-term (overall) sample standard deviation s, which includes shift and drift over time — Pp = (USL − LSL)/6s and Ppk = min[(USL − μ)/3s, (μ − LSL)/3s]. The gap between Cpk and Ppk reveals how much the process drifts over time. A capability study requires the process to be in statistical control first (stable via SPC) and the data to be approximately normal; otherwise the indices are not trustworthy. Cpk relates to defect rate: Cpk = 1.0 ≈ 2,700 ppm; Cpk = 1.33 ≈ 63 ppm; Cpk = 2.0 ≈ 0.002 ppm.
8. Inventory: EOQ, EPQ, ROP, Safety Stock
Inventory models balance the cost of ordering/setup against the cost of holding stock. The Economic Order Quantity (EOQ) minimizes total cost: EOQ = √(2DS/H), where D is annual demand, S is the cost per order, and H is the annual holding cost per unit. At the EOQ, annual ordering cost equals annual holding cost. The Economic Production Quantity (EPQ, or production order quantity) modifies EOQ for finite production rate p relative to demand d: EPQ = √(2DS/H) × √(p/(p−d)), accounting for stock building up gradually rather than arriving all at once. The Reorder Point (ROP) triggers a new order: ROP = d×L + safety stock, where d is demand per unit time and L is lead time. Safety stock buffers against variability in demand and lead time during the lead-time window: SS = z × σ_dLT, where z is the service-level factor (e.g., 1.65 for 95%, 2.33 for 99%) from the standard normal, and σ_dLT is the standard deviation of demand over lead time. Higher service levels require disproportionately more safety stock. ABC analysis (Pareto) classifies items so tight control is focused on the high-value A items. The fixed-order-quantity (Q) system orders a constant amount when stock hits ROP; the fixed-period (P) system reviews and reorders at set intervals.
9. Queuing Theory (M/M/1, M/M/c)
Queuing theory models waiting lines using Kendall notation A/B/c: arrival process / service process / number of servers. M denotes a Markovian (Poisson arrivals, exponential service) process. For M/M/1 (single server, Poisson arrivals at rate λ, exponential service at rate μ): utilization ρ = λ/μ must be < 1 for stability. The average number in the system L = ρ/(1−ρ); average number in queue Lq = ρ²/(1−ρ); average time in system W = 1/(μ−λ); average time waiting in queue Wq = ρ/(μ−λ) = λ/(μ(μ−λ)). Little's Law links them: L = λW and Lq = λWq. A crucial, counterintuitive insight is that waiting time grows nonlinearly and explodes as utilization approaches 100% — the 1/(1−ρ) factor means a system run at 90% utilization has roughly ten times the queue of one at 50%. This is why high-variability systems must keep buffer capacity. For M/M/c (c parallel servers sharing one queue), the Erlang C formula gives the probability an arrival must wait; pooling servers into one queue (M/M/c) dramatically outperforms c separate M/M/1 lines. Variability in arrivals and service inflates waiting (captured by the VUT equation: Wait ≈ Variability × Utilization × Time), so reducing variability shortens queues even without adding capacity.
10. Project Scheduling: CPM / PERT
The Critical Path Method (CPM) and Program Evaluation and Review Technique (PERT) schedule projects modeled as networks of activities with precedence relationships. A forward pass computes the Earliest Start (ES) and Earliest Finish (EF = ES + duration) for each activity; a backward pass computes the Latest Finish (LF) and Latest Start (LS = LF − duration). Total slack (float) = LS − ES = LF − EF; activities with zero slack lie on the critical path — the longest path through the network, which determines the minimum project duration. Any delay to a critical activity delays the whole project. CPM uses single deterministic durations. PERT handles uncertainty with three time estimates per activity: optimistic (a), most likely (m), and pessimistic (b). The expected duration te = (a + 4m + b)/6 (a beta-distribution approximation), and the activity variance σ² = ((b − a)/6)². The project duration variance is the sum of variances of critical-path activities; by the central limit theorem, project completion time is approximately normal, allowing probability statements (e.g., the chance of finishing by a target date via a z-score). Crashing shortens the project by adding resources to critical activities at the lowest cost-per-day slope, until the critical path shifts. Resource leveling smooths resource usage but may extend duration.
11. Forecasting (MA, exponential smoothing, MAD/MAPE)
Forecasting predicts future demand to drive capacity, inventory, and scheduling decisions. Time-series methods assume the future resembles the past. The simple moving average (MA) forecasts the next period as the average of the last n observations — easy but lags trends and weights all periods equally. The weighted moving average assigns larger weights to recent periods. Exponential smoothing forecasts Ft+1 = α·At + (1−α)·Ft, where α (0–1) is the smoothing constant: higher α reacts faster to change but is noisier; it weights past data with exponentially decaying importance. For data with trend, Holt's double exponential smoothing adds a trend component; for trend plus seasonality, Holt-Winters (triple) smoothing adds a seasonal index. Forecast accuracy is measured by the error et = At − Ft. Common metrics: MAD (mean absolute deviation) = Σ|et|/n; MSE (mean squared error) = Σet²/n, which penalizes large errors; MAPE (mean absolute percentage error) = (Σ|et/At|/n)×100%, which is scale-independent and easy to interpret; and bias (mean error or running tracking signal = cumulative error ÷ MAD) to detect consistent over- or under-forecasting. A tracking signal beyond ±4 to ±6 indicates the model is out of control and needs review. Choose the method by data pattern: stationary, trended, or seasonal.