FE Industrial & Systems cheat sheet

Central Limit Theorem: for independent samples of size n ≥ 30, the sample mean is approximately normal with mean μ and standard error σ/√n regardless of the population's shape.

Confidence interval for a mean with σ known: x̄ ± z(α/2)·σ/√n; substitute t(α/2, n−1) when σ is estimated from the sample.

Test statistic for a mean: z = (x̄ − μ0)/(σ/√n); for a two-sided test reject H0 when |z| > z(α/2).

Poisson probability: P(X = k) = e^(−λ)·λ^k / k!, where λ is the expected count per interval; mean and variance both equal λ.

Binomial: P(X = k) = C(n,k)·p^k·(1−p)^(n−k), with mean np and variance np(1−p).

Exponential interarrival times: f(t) = λe^(−λt), mean = 1/λ, variance = 1/λ².

Bayes' theorem: P(A|B) = P(B|A)P(A) / [P(B|A)P(A) + P(B|A′)P(A′)].

Combinations C(n,r) = n!/[r!(n−r)!] count unordered selections; permutations nPr = n!/(n−r)! count ordered arrangements.

Least-squares regression: slope b1 = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)² and intercept b0 = ȳ − b1·x̄.

Single-payment factors: F = P(1+i)^n and P = F(1+i)^(−n), written (F/P, i, n) and (P/F, i, n).

Uniform series present worth: (P/A, i, n) = [(1+i)^n − 1]/[i(1+i)^n]; the capital recovery factor (A/P) is its reciprocal.

Uniform series future worth: (F/A, i, n) = [(1+i)^n − 1]/i; the sinking fund factor is (A/F, i, n) = i/[(1+i)^n − 1].

Capitalized cost for a perpetual project: CC = initial cost + A/i, where A is the equivalent uniform annual cost.

Decision rule for independent projects: accept when NPW ≥ 0 or when IRR ≥ MARR; for mutually exclusive alternatives choose the largest positive NPW or the highest incremental IRR.

Straight-line depreciation: D = (first cost − salvage value)/N per year; book value after n years = first cost − n·D.

Break-even quantity Q* = fixed cost/(unit price − unit variable cost); the denominator is the contribution margin per unit.

Real interest rate: i_real = (1 + i_market)/(1 + f) − 1, where f is the inflation rate.

Benefit-cost ratio = equivalent benefits / equivalent costs; a ratio ≥ 1 indicates an economically acceptable public project.

Fundamental canon: engineers shall hold paramount the safety, health, and welfare of the public in performing professional duties.

Seal rule: engineers shall not seal or sign plans or documents not prepared by them or under their direct supervisory control.

Competence rule: engineers shall perform services only in areas of their competence and shall not misrepresent their qualifications or experience.

Conflict of interest: engineers shall disclose known or potential conflicts and shall not accept compensation from more than one party for the same service.

Confidentiality: information obtained in professional practice shall not be disclosed without consent except where required by law or code.

Bribery and gifts: engineers shall not offer, give, solicit, or accept anything of value to influence the judgment of others on a project.

Public statements: engineers shall issue statements only in an objective and truthful manner and shall identify the client when commenting on their work.

Violations: an engineer with knowledge of a code violation shall report it in writing to the proper authority.

Contingency fees: engineers shall not accept compensation contingent on decisions by a public board or commission that affect public safety.

Little's Law: WIP = throughput × flow time; valid for any steady-state system, so shortening flow time cuts WIP at constant throughput.

Takt time = available production time per period / customer demand per period; it sets the required pace of the line.

OEE = availability × performance × quality, where each factor is an actual-over-ideal ratio; world-class is often cited near 85%.

Seven wastes of lean: overproduction, waiting, transportation, overprocessing, inventory, motion, and defects.

Machines required = ceiling of (demand × time per unit)/(available time per machine).

SMED shortens changeover by converting internal setup steps (machine stopped) into external steps (machine running).

Cellular manufacturing applies group technology to form part families produced on dedicated cells, reducing setup and material handling.

Six Sigma improvement sequence: Define, Measure, Analyze, Improve, Control (DMAIC).

Job-shop sequencing: SPT minimizes average flow time and average WIP; EDD minimizes maximum lateness; Johnson's rule solves the two-machine flow shop.

EOQ = √(2DS/H), with D = annual demand, S = cost per order or setup, and H = annual holding cost per unit.

Total annual inventory cost = (D/Q)S + (Q/2)H; at the EOQ, ordering cost equals holding cost.

Reorder point = demand rate × lead time (same time units) plus safety stock when demand or lead time is uncertain.

MRP net requirement = gross requirement − scheduled receipts − on-hand inventory + safety stock.

MRP processing order: explode the MPS through the bill of materials, net requirements, apply lot sizing, then offset by lead time.

Aggregate planning options: chase (vary workforce or output), level (use inventory and backlogs), and mixed; select the lowest total-cost plan.

Learning curve: Y_x = a·x^b with b = log(learning rate)/log 2; an 80% curve gives b ≈ −0.322.

Safety stock for a target cycle service level = z × σ of demand during lead time.

Johnson's rule: repeatedly take the shortest remaining processing time, scheduling that job first if it lies on machine 1 and last if on machine 2.

Systematic Layout Planning (Muther): analyze flow and activity relationships (A, E, I, O, U, X) → relationship diagram → space requirements → block layout.

Assembly-line balancing: cycle time C = available time per period / required output; minimum workstations = ceiling(Σ task times / C).

Ranked positional weight heuristic: order tasks by their time plus the times of all following tasks, then assign greedily without exceeding cycle time.

Load-distance score = Σ(load_i × distance_i); the preferred layout minimizes this sum.

Center-of-gravity location: x* = Σw_i·x_i / Σw_i and y* = Σw_i·y_i / Σw_i, with w_i the volume moved to or from site i.

A From-To chart quantifies material flows between departments; a relationship chart is used when qualitative closeness ratings dominate.

ABC classification: class A items are roughly 20% of the items and about 80% of the value, justifying tight control and frequent cycle counts.

Computerized layout tools: CRAFT improves an existing layout by pairwise department exchange; ALDEP and CORELAP construct layouts from relationship ratings.

Cross-docking transfers goods directly from receiving to shipping with little or no storage, reducing inventory and dwell time.

NIOSH lifting equation: RWL = LC × HM × VM × DM × AM × FM × CM; lifting index = load weight / RWL, and LI > 1 indicates elevated risk.

Anthropometric practice: design to the 5th percentile where reach is limiting and to the 95th percentile where clearance is limiting.

Fitts' law: movement time MT = a + b·log2(2D/W); larger targets and shorter distances reduce pointing time.

OSHA noise limits: 90 dBA time-weighted average for an 8-hour day, with an 85 dBA action level; allowable exposure time halves for each 5 dB increase.

Illuminance guidance: about 500 lux for routine office and reading tasks, with substantially higher levels for inspection of small details.

Cognitive limits: short-term memory holds about 7 ± 2 chunks (Miller), and choice reaction time grows with the log2 of the number of alternatives (Hick-Hyman law).

Workstation height for light seated work is approximately seated elbow height; work sustained above shoulder height accelerates shoulder fatigue.

Physiological workload measures include oxygen consumption, heart rate, and metabolic energy expenditure, used to set work-rest cycles.

Cp = (USL − LSL)/(6σ); Cpk = min[(USL − μ)/3σ, (μ − LSL)/3σ], so Cpk penalizes off-center processes.

x̄-chart limits: UCL/LCL = x̄̄ ± A2·R̄; R-chart limits: UCL = D4·R̄ and LCL = D3·R̄, using constants tabulated for subgroup size n.

Out-of-control signals: any point beyond a control limit, or a run of seven consecutive points on one side of the centerline, indicates a special cause.

Taguchi loss function: L(x) = k(x − T)², with k = A0/Δ0², where A0 is the loss at deviation Δ0 from the target T.

OC curve risks: α is the producer's risk of rejecting a lot at the AQL, and β is the consumer's risk of accepting a lot at the LTPD.

Seven QC tools: Pareto chart, cause-and-effect diagram, check sheet, histogram, scatter diagram, control chart, and flow chart.

p-chart limits: p̄ ± 3√[p̄(1 − p̄)/n]; a c-chart tracks the count of defects per inspection unit.

Capability benchmark: Cpk ≥ 1.33 is commonly required for an established process, and Cpk ≥ 1.67 for new or safety-critical processes.

Factorial design: a 2^k experiment estimates k main effects and their interactions using 2^k treatment combinations.

M/M/1 results: ρ = λ/μ, L = ρ/(1−ρ), Lq = ρ²/(1−ρ), W = 1/(μ−λ), and Wq = λ/[μ(μ−λ)].

LP duality: the shadow price of a constraint equals the change in the optimal objective per unit increase in its right-hand side, and is zero when the constraint has slack.

Expected value of perfect information: EVPI = expected value with perfect information − maximum EMV without it.

CPM network: forward pass gives ES and EF, backward pass gives LS and LF, slack = LS − ES, and the critical path has zero slack.

PERT three-point estimate: tE = (a + 4m + b)/6 with variance [(b − a)/6]².

Transportation problem: northwest-corner or least-cost rule gives a starting solution; the stepping-stone or MODI method tests and achieves optimality.

Max-flow min-cut theorem: the maximum flow from source to sink equals the capacity of the minimum cut separating them.

Assignment problem: the Hungarian method finds the minimum-cost one-to-one assignment of jobs to resources.

Decision criteria under uncertainty: maximax (optimistic), maximin (pessimistic), and minimax regret, contrasted with EMV under risk.

Life cycle phases: concept, development, production, utilization and support, and retirement; the V-model pairs each design decomposition level with a corresponding verification level.

Good requirements are unambiguous, verifiable, and traceable to design elements and test cases.

N² (N-squared) diagrams map functional and physical interfaces; the design structure matrix exposes coupling and suggests sequencing.

Earned value: CPI = EV/AC and SPI = EV/PV; CPI < 1 indicates cost overrun and SPI < 1 indicates schedule slippage.

Risk exposure = probability of occurrence × consequence; handling options are avoid, mitigate, transfer, and accept.

Reliability: series R = R1×R2×…×Rn; parallel R = 1 − (1−R1)(1−R2)…(1−Rn); exponential model R(t) = e^(−λt) with MTBF = 1/λ.

Configuration baselines: functional (requirements), allocated (derived requirements), and product (as-built); changes flow through a change control board.

Trade study procedure: define weighted evaluation criteria, score each alternative, run sensitivity analysis, and document the rationale for the selection.