Free FE Industrial & Systems practice questions.

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Opportunity cost is the value of the best foregone alternative. If capital is invested in Project A, the opportunity cost is the return that could have been earned in Project B (next best option). This concept underlies MARR selection.

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OSHA 29 CFR 1910.95 (5 dB exchange rate): 90 dBA = 8 hr, 95 dBA = 4 hr, 100 dBA = 2 hr, 105 dBA = 1 hr. Each 5 dB increase halves the permissible duration. Noise dose D = Σ(C_i/T_i) ≤ 1.0. OSHA 5 dB exchange rate (permissible durations): 90 dBA=8 hr, 95 dBA=4 hr, 100 dBA=2 hr, 105 dBA=1 hr, 110 dBA=0.5 hr. NIOSH uses a stricter 3 dB exchange rate with an 85 dBA criterion: 85 dBA=8 hr, 88 dBA=4 hr. Noise dose D = Σ(Cᵢ/Tᵢ) ≤ 1.0; when D>1.0, engineering or administrative controls are required. Engineering controls (barriers, enclosures) are preferred over hearing protection.

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Safety stock = Z × σ_d × √(LT) for variable demand, constant lead time. Z = service level factor (Z=1.65 for 95% service level). Higher service level → more safety stock → more holding cost. Reorder Point = demand during LT + safety stock. Safety stock formula for variable demand: SS = Z × σ_d × √LT. For 95% service level: Z=1.65; 99%: Z=2.33. Reorder Point (ROP) = average demand × LT + SS. Higher service level dramatically increases safety stock (and holding cost) — there's a trade-off between service level and inventory investment.

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∫₀² (3x² + 2x) dx = [x³ + x²]₀² = (8+4) - (0+0) = 12. Power rule: ∫xⁿ dx = x^(n+1)/(n+1) + C. Definite integral: evaluate antiderivative at upper minus lower limit. Verify by differentiation: d/dx(x³ + x²) = 3x² + 2x ✓. Definite integration computes net area under the curve—negative area occurs where f(x) < 0, and positive where f(x) > 0.

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0-1 Knapsack: maximize Σv_i × x_i subject to Σw_i × x_i ≤ W, x_i ∈ {0,1}. v_i = item values, w_i = item weights, W = capacity. NP-hard in general. Dynamic programming gives pseudo-polynomial solution. Applications: resource allocation, capital budgeting, cargo loading. Knapsack is NP-hard in the general case, but a pseudo-polynomial DP solution runs in O(nW) time. Greedy (highest value/weight ratio first) gives optimal solution for the fractional knapsack, but not 0-1 knapsack. Applications beyond packing: portfolio selection (capital budgeting), resource allocation with budget constraints.

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2³ factorial: 3 factors (A, B, C), each at 2 levels (+/-). Total runs = 2³ = 8. Allows estimation of 3 main effects, 3 two-way interactions (AB, AC, BC), and 1 three-way interaction (ABC). Full factorial estimates all effects without confounding. 2ᵏ factorial: k factors at 2 levels (+/-). 2³ = 8 runs. Effects estimated: 3 main effects (A, B, C), 3 two-factor interactions (AB, AC, BC), 1 three-factor interaction (ABC) = 7 effects + intercept. For k=4: 2⁴=16 runs; k=5: 32 runs. Fractional factorial 2^(k-p): 2^(3-1)=4 runs (half-fraction) — used when full factorial is too expensive.

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Little's Law: L = λW relates average inventory in system to throughput rate and average time in system. Holds for any stable system regardless of arrival/service distribution. In manufacturing: WIP = throughput × cycle time. In service: average customers in system = arrival rate × average service time. Universal applicability is its power.

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DMAIC: Define (problem, scope, goals), Measure (baseline performance, data collection), Analyze (root causes), Improve (solutions, pilot), Control (sustain gains, monitoring plan). Used for improving existing processes. Each phase has distinct tools: Define uses project charter and SIPOC; Measure uses capability studies and MSA; Analyze uses fishbone, regression, and hypothesis tests; Improve uses DOE and piloting; Control uses control charts and OCAP. Contrast with DMADV (Define-Measure-Analyze-Design-Verify), used for designing new processes.

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Newton's 2nd Law: ΣF = ma (vector equation). In x, y, z: ΣFx=max, ΣFy=may, ΣFz=maz. This is the fundamental equation of dynamics. For statics: a=0 → ΣF=0. 1st Law: F=0 → a=0. 3rd Law: F_AB = -F_BA. Newton's Second Law is a vector equation—x, y, z components must each balance independently: ΣFx = max, ΣFy = may. For constant acceleration, use kinematic equations: v = v₀ + at and x = v₀t + ½at². Newton's Third Law action-reaction pairs act on different bodies and never cancel each other in the same free-body diagram.

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Axial strain ε = δ/L = 1.5 mm / 2000 mm = 0.00075 (dimensionless). Percent elongation = 0.075%. Strain is unitless (mm/mm or in/in). Strain is dimensionless (mm/mm or in/in) and is often expressed as a percentage or in microstrain (με = 10⁻⁶ strain). True (logarithmic) strain ε = ln(L/L₀) is used for large deformations; engineering strain δ/L₀ is valid in the linear elastic range. Elastic strain is fully recoverable; plastic strain beyond the yield point is permanent.

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Systems engineering: interdisciplinary approach to enable the realization of successful systems through: requirements analysis, functional analysis, design synthesis, verification/validation. Key processes: INCOSE defines in SE Handbook. Considers technical and non-technical aspects across the full lifecycle.

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Value Engineering analyzes functions to achieve required performance at the lowest overall cost. Value = Function/Cost. VE asks 'what does it do?' and 'what does it cost?' to find alternatives that deliver the same function more economically.