Turning EN8 shafts: minimum-cost speed versus a 60-hour delivery window
A job shop turns a batch of 500 EN8 steel shafts (Ø60 × 250 mm) on one CNC lathe. For the TiN-coated carbide insert the planner has fixed Taylor constants n = 0.25 and C = 300 with V in m/min and T in min. The machine hour rate is Rs 360, one throwaway insert edge costs Rs 180, and a tool change costs 3 min of machine downtime. Because the same 900 m of surface is removed at any speed, cutting time per shaft is 900/V minutes. The lathe currently runs at V = 200 m/min, which gives a tool life of only about 5 min; the operator changes an edge roughly once per shaft and has complained to the supervisor. The customer needs all 500 shafts within 60 hours of available machine time. Overtime is available from another shift at Rs 480 per hour. The owner wants the machine slowed to the minimum-cost speed and is willing to let the batch run into overtime rather than keep burning inserts.
1. Compute tool life and cutting speed for minimum cost and for maximum production rate. Then compare cost per shaft and floor-to-floor time per shaft at the present 200 m/min against the minimum-cost setting.
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Minimum cost: T_opt = (1/n − 1)(t_c + C_t/M) = (4 − 1)(3 + 180/6) = 3 × 33 = 99 min, giving V = 300/99^0.25 = 300/3.155 = 95.1 m/min.
Maximum production rate: T = (1/n − 1)t_c = 3 × 3 = 9 min, giving V = 300/9^0.25 = 300/1.732 = 173.2 m/min.
At 200 m/min: T = (300/200)^4 = 5.06 min; t_m = 900/200 = 4.50 min; edges per shaft = 4.50/5.06 = 0.889. Cost = 6 × 4.50 + (3 × 6 + 180) × 0.889 = 27.0 + 176.0 = Rs 203.0 per shaft. Time = 4.50 + 0.889 × 3 = 7.17 min per shaft.
At 95.1 m/min: t_m = 900/95.1 = 9.46 min; edges per shaft = 9.46/99 = 0.0956. Cost = 6 × 9.46 + 198 × 0.0956 = 56.8 + 18.9 = Rs 75.7 per shaft. Time = 9.46 + 0.29 = 9.75 min per shaft.
The answer is not a single number: cost per shaft drops by about 63% while floor-to-floor time rises by only 36%. Note also the tool consumption, 445 edges against 48 edges for the batch, which is what the operator is actually reacting to.
2. Can the 500 shafts be delivered within 60 hours if the lathe runs at the minimum-cost speed? Recommend a concrete operating point and quantify the economics of your choice.
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At 95.1 m/min the batch needs 500 × 9.75 = 4 875 min = 81.2 h, i.e. about 21.2 h beyond the 60-hour window. At the present 200 m/min it needs 500 × 7.17 = 3 583 min = 59.7 h, which fits only by a 0.3 h margin, with no allowance for a breakdown, a gauging stop or a tooling shortage. That is not a plan, it is a coincidence.
The decision is therefore economic, not technical:
- Extra cost of running at the minimum-cost speed = 21.2 h × Rs 480 = Rs 10 176 of overtime.
- Saving on the batch = (203.0 − 75.7) × 500 = Rs 63 650.
- Net gain ≈ Rs 53 500, plus roughly 400 fewer tool changes over the batch.
Recommendation: run at V ≈ 95 m/min and authorise the overtime, provided the second shift can actually absorb 21 h. This holds because the machine-hour cost of the extra time (Rs 480/h overtime) is far below the marginal tooling cost saved by running slowly.
If overtime is capped, do not fall back to 200 m/min. Run at the maximum-rate speed of 173 m/min: 6.93 min/shaft → 57.7 h, cost Rs 145.5/shaft → Rs 72 750 per batch. That still undercuts the 200 m/min plan by about Rs 29 000 and leaves no reliance on overtime.
3. The insert supplier offers an alternative grade at Rs 90 per edge with the same n = 0.25 but C = 260. The buyer says it halves tool cost. Should the shop switch? Support the answer with numbers.
Read the worked answer
Re-optimise with the new constants rather than comparing edge prices.
T_opt = 3 × (3 + 90/6) = 3 × 18 = 54 min; V_opt = 260/54^0.25 = 260/2.711 = 95.9 m/min. t_m = 900/95.9 = 9.38 min; edges per shaft = 9.38/54 = 0.174. Cost = 6 × 9.38 + (3 × 6 + 90) × 0.174 = 56.3 + 18.8 = Rs 75.1 per shaft.
Against Rs 75.7 for the present grade, the saving is Rs 0.6 per shaft, about Rs 300 on the whole batch - inside the uncertainty of the cost model, and far less than the cost of requalifying a grade, re-proving the program and scrapping trial parts.
The bigger problem is the lower C. At 200 m/min the new grade gives T = (260/200)^4 = 2.86 min against 5.06 min, so it is strictly worse whenever the shop needs to run fast to recover a schedule, and at 173 m/min it gives only 5.1 min of life. Verdict: do not switch at Rs 90. Accept only if the price falls to roughly Rs 80 or below, or if the supplier can offer C ≥ 300 at the same price.