(a) A strip of width 250 mm is reduced from 30 mm to 20 mm thickness in a rolling mill of roll radius 400 mm. Considering…

(a) Flat rolling of a strip (10 marks)

Data: width w = 250 mm, h₀ = 30 mm, h_f = 20 mm, roll radius R = 400 mm, μ = 0.25 and average flow stress Y = 300 MPa. The draft is Δh = 30 − 20 = 10 mm.

(i) Contact length. L = √(R Δh) = √(400 × 10) = 63.2 mm.

(ii) Rolling force. F = Y L w = 300 × 63.25 × 250 = 4.74 × 10⁶ N ≈ 4.74 MN. This uses the average flow stress over the contact area and ignores the extra force needed for friction.

(iii) Biting condition. The strip is drawn in by the rolls if μ ≥ tan α, where cos α = 1 − Δh/(2R) = 1 − 10/800 = 0.9875. So α = 9.07° and tan α = 0.160 (approximately √(Δh/R) = 0.158). Since μ = 0.25 > 0.160, biting is satisfied. The maximum draft possible for this friction is Δh_max = μ²R = 0.0625 × 400 = 25 mm, which is larger than the 10 mm taken.

(b) Non-conventional machining of a biomedical implant (20 marks)

(i) Suitability. An implant needs a sound, contamination-free surface with no thermal damage, since heat affected material can crack, corrode or shed particles in the body.

• USM (ultrasonic machining): material is removed mechanically by abrasive grains driven by a vibrating tool, with no heat or electrical effect. It suits hard, brittle, biocompatible materials such as zirconia, alumina, bioglass and hydroxyapatite, and it cuts non-conductors. Its material removal rate is low, and it is slow on ductile titanium and cobalt–chromium alloys.
• LBM (laser beam machining): removal is by melting and vaporisation. It is fast and flexible, but it is thermal, so it leaves damage on the surface of an implant.
• EDM (electrical discharge machining): removal is by spark erosion at 8000–12 000 °C. It cuts hard conductive metals (titanium alloys) accurately, but it is thermal and needs conductive material.

(ii) Surface defects.

• USM: embedded abrasive particles, small edge chipping at the exit, taper and overcut, a matt surface (Ra of about 0.4–1.6 μm), and possible micro-cracks in very brittle ceramics. There is no heat-affected zone and no recast layer, and the residual stress is compressive.
• LBM: a recast layer, heat-affected zone, micro-cracks, tensile residual stress, dross and spatter, taper, and oxidation (an oxygen-rich alpha case on titanium).
• EDM: a white (recast) layer with micro-cracks, a heat-affected zone, craters, tensile residual stress, and contamination from electrode material and carbon from the dielectric, which is a biocompatibility concern.

(iii) Best process. USM is the best of the three for this requirement, because it is a cold, purely mechanical process that leaves no recast layer, heat-affected zone or tensile residual stress, and it leaves no metallurgical change to the implant material. Its slow removal rate is acceptable for a small, high-value part. For tough metallic implants, rotary or ultrasonic-assisted machining can raise the rate. (Electrochemical machining, which is outside the three, is also thermal-damage free.)

(iv) Post-processing of the USM part.

• Thorough ultrasonic cleaning in solvent and water, to remove the abrasive slurry and any embedded grains.
• Light lapping, polishing or electropolishing to lower roughness and remove edge chips and micro-cracks, since roughness affects fatigue life and cell attachment.
• Deburring and edge finishing, and passivation of titanium or stainless steel (nitric or citric acid) to restore the protective oxide film.
• Inspection of dimensions, surface roughness and cracks (SEM or dye penetrant), followed by sterilisation (autoclave, gamma or ethylene oxide) and packaging.

(c) Job shop scheduling (20 marks)

(i) Critical ratio. The critical ratio is the time remaining to the due date divided by the remaining processing time: CR = (d − t)/(remaining work). CR < 1 means the job will be late, CR = 1 means it is just on time, and CR > 1 means it has slack.

At t = 0:

• J₁: 28/21 = 1.333
• J₂: 25/20 = 1.250
• J₃: 30/20 = 1.500
• J₄: 35/21 = 1.667
• J₅: 32/20 = 1.600

The priority order (lowest CR first) is J₂ (1.25) > J₁ (1.33) > J₃ (1.50) > J₅ (1.60) > J₄ (1.67). J₂ is the most critical job, although all CR values exceed 1, so none is yet late.

CR is more informative than the due date alone because it says how much work still has to fit in the time left, and than the processing time alone because it says how urgent the job is; it combines both in one dimensionless number that can be recomputed as time advances.

(ii) SPT dispatching. Rule: whenever a machine is free, it takes the waiting job with the shortest operation on it. The operations, as (start–end), are:

• M₁: J₁ (0–6), J₂ (6–13), J₄ (13–22), J₅ (23–25), J₃ (25–28)
• M₂: J₂ (0–5), J₅ (5–11), J₁ (11–15), J₃ (15–23), J₄ (25–30)
• M₃: J₃ (0–4), J₅ (11–16), J₂ (16–22), J₄ (22–25), J₁ (25–33)
• M₄: J₂ (13–15), J₅ (16–23), J₃ (28–33), J₁ (33–36), J₄ (36–40)

At t = 23, J₃ and J₅ both wait for M₁, and SPT selects J₅ (2 h) before J₃ (3 h). At t = 33, J₄ (4 h) and J₁ (3 h) both wait for M₄, and SPT selects J₁.

Job results. All jobs are released at t = 0, so F_j = C_j, L_j = C_j − d_j and T_j = max(0, L_j).

• J₁: C = 36, F = 36, L = +8, T = 8
• J₂: C = 22, F = 22, L = −3, T = 0
• J₃: C = 33, F = 33, L = +3, T = 3
• J₄: C = 40, F = 40, L = +5, T = 5
• J₅: C = 25, F = 25, L = −7, T = 0

Summary. Makespan = 40 h. Mean flow time = (36 + 22 + 33 + 40 + 25)/5 = 31.2 h. Total tardiness = 8 + 0 + 3 + 5 + 0 = 16 h. Maximum tardiness = 8 h (J₁). Number of tardy jobs = 3 (J₁, J₃, J₄).