(a) A firm has identified four operations, which are to be conducted in succession for an order to be processed. The tolerance…

(a)(i) Natural tolerance limits

Method: for independent normal times, the sum is also normal; means add and variances add. Taking each given tolerance as the natural tolerance half-width, half-width = 3σ.

σ₁ = 0.6/3 = 0.2 h, σ₂ = 0.6/3 = 0.2 h, σ₃ = 0.8/3 = 0.2667 h, σ₄ = 0.3/3 = 0.1 h.

Total mean time, μ_T = 7 + 5 + 9 + 6 = 27 h.

Total variance, σ_T² = (0.2)² + (0.2)² + (0.8/3)² + (0.1)² = 0.04 + 0.04 + 0.07111 + 0.01 = 0.16111 h².

σ_T = √0.16111 = 0.4014 h.

Natural tolerance half-width = 3σ_T = 3 × 0.4014 = 1.204 h.

Thus natural tolerance limits = 27 ± 1.204 h = 25.796 h to 28.204 h.

(a)(ii) Proportion failing the 27.5 h goal

Let T be completion time. Fail the goal if T > 27.5 h.

z = (27.5 − 27)/0.4014 = 0.5/0.4014 = 1.246.

From standard normal table, Φ(1.246) ≈ 0.8936.

P(T > 27.5) = 1 − 0.8936 = 0.1064.

Proportion failing = 10.64% ≈ 10.6%.

(a)(iii) Capability index

Since only an upper goal is specified, the appropriate index is the upper capability index:

C_pu = (USL − μ)/(3σ) = (27.5 − 27)/1.204 = 0.415.

C_pu ≈ 0.415. Since C_pu < 1, the process is not capable of meeting the 27.5 h goal; about 10.6% orders fail.

(a)(iv) After improving operation 3

Operation 3 mean changes from 9 h to 8 h, so new total mean:

μ_T′ = 7 + 5 + 8 + 6 = 26 h.

σ_T remains 0.4014 h. Meet goal if T ≤ 27.5 h.

z = (27.5 − 26)/0.4014 = 1.5/0.4014 = 3.737.

P(T ≤ 27.5) = Φ(3.737) ≈ 0.999907.

Proportion meeting goal ≈ 99.99%.

(b)(i) Factors in facility location

Facility location selection depends on both tangible and intangible factors. Raw-material and supplier proximity reduce inbound transport cost and improve reliability. Market proximity lowers outbound distribution cost and improves customer service. Labour availability, skill level, wage rates, productivity and union conditions are critical. Infrastructure such as power, water, transport, roads, rail, ports and communication must be adequate. Land cost, size, topography, drainage and scope for expansion matter. Government policies, taxes, subsidies, environmental regulations and political stability influence long-term viability. Community attitude, housing, schools, hospitals and quality of life affect employee retention. Climate, seismic risk and environmental conditions also matter. Finally, competitors, supporting industries and agglomeration economies may attract or repel a plant.

(b)(ii) Factor rating of locations

Use the weighted scoring method.

X = 70×0.40 + 60×0.35 + 45×0.25 = 28 + 21 + 11.25 = 60.25.

Y = 60×0.40 + 45×0.35 + 90×0.25 = 24 + 15.75 + 22.5 = 62.25.

Z = 55×0.40 + 95×0.35 + 50×0.25 = 22 + 33.25 + 12.5 = 67.75.

Best location: Z (67.75). Worst location: X (60.25).

(c)(I) Heat generated

Joule heating law: H = I² R t.

I = 20,000 A, R = 200 μΩ = 200×10⁻⁶ Ω, t = 0.15 s.

H = (20,000)² × (200×10⁻⁶) × 0.15 = 4×10⁸ × 2×10⁻⁴ × 0.15 = 12,000 J.

Heat generated = 12,000 J = 12 kJ.

(c)(II) Volume of the weld nugget

Heat used for melting = 60% of generated heat:

Q_m = 0.60 × 12,000 = 7,200 J.

Heat required per unit volume = 10 J/mm³.

Volume of weld nugget, V = Q_m / 10 = 7,200/10 = 720 mm³.

Volume of weld nugget = 720 mm³.