Economics of machining: optimum speed for cost and production

Why it matters

Running a tool faster cuts the machining time but wears the tool out sooner, so tool changes and tool cost rise. Somewhere between "too slow" and "too fast" there is a speed that gives the lowest cost per part, and a (higher) speed that gives the most parts per hour. Production planners use these two speeds to set cutting data for every job.

Key ideas

Time per piece has three parts:

• Machining (cutting) time tm — falls as speed rises.
• Non-productive (handling) time th — loading, unloading, approach; independent of speed.
• Tool-change time per piece — tc × (tm/T), because one tool change is needed every T minutes of cutting. It rises steeply with speed because T falls.

Cost per piece = machine-and-operator cost on all that time + tool cost per piece:

• Cm = cost rate of machine + labour + overheads (₹/min).
• Ct = cost of one tool cutting edge (insert edge cost, or regrind cost plus a share of the tool price), ₹ per tool life.

Minimum-cost criterion (Gilbert). Writing tm ∝ 1/V and T = (C/V)^(1/n), and setting d(cost)/dV = 0, gives the optimum tool life To and speed Vo. The machining-cost term falls with speed while the tool-cost and tool-change terms rise; Vo balances them.

Maximum-production-rate criterion. Minimising the time per piece gives Tp and Vp. Since only the tool-change time (not tool cost) penalises speed here, Tp < To and so Vp > Vo always.

High-efficiency (Hi-E) range. The speed band between Vo and Vp. Maximum-profit speed lies inside it. Below Vo both cost and time are worse; above Vp both are worse.

Practical points

• Disposable carbide inserts have low Ct and short tc, so To is short and Vo is high — that is why indexable tooling runs fast.
• Expensive tools (form tools, broaches, gear hobs) with long change times call for low speeds and long tool lives.
• If the machine is a bottleneck, run nearer Vp; if it has spare capacity, run nearer Vo.
• Feed and depth are normally fixed first at the highest values the machine, part and finish allow; speed is then optimised, because tool life is least sensitive to feed and depth.

Formulas

• tm = π·D·L / (1000·V·f) (turning; tm in min, D and L in mm, V in m/min, f in mm/rev)
• T = (C / V)^(1/n) (Taylor)
• Cost per piece = Cm·th + Cm·tm + (Cm·tc + Ct)·(tm / T)
• Time per piece = th + tm + tc·(tm / T)
• To = (1/n − 1)·(tc + Ct/Cm), Vo = C / To^n (minimum cost)
• Tp = (1/n − 1)·tc, Vp = C / Tp^n (maximum production rate)
  • Cm = machine + labour + overhead rate (₹/min); Ct = tool cost per cutting edge (₹); tc = tool-change time (min); th = handling time per piece (min); n, C = Taylor constants (C in m/min with T in min).
• At the minimum-cost speed: (tool cost + tool-change cost per piece) / (machining cost per piece) = n / (1 − n).

Worked examples

Example 1 (standard). Taylor constants n = 0.25, C = 300 m/min. Machine + labour + overheads Cm = ₹10/min, tool-change time tc = 2 min, cost per cutting edge Ct = ₹60. Find the minimum-cost and the maximum-production tool lives and speeds.

1. To = (1/n − 1)·(tc + Ct/Cm) = (4 − 1) × (2 + 60/10) = 3 × 8 = 24 min.
2. Vo = C / To^n = 300 / 24^0.25 = 300 / 2.2134 = 135.5 m/min.
3. Tp = (1/n − 1)·tc = 3 × 2 = 6 min.
4. Vp = C / Tp^n = 300 / 6^0.25 = 300 / 1.5651 = 191.7 m/min.
5. Hi-E range: 135.5 to 191.7 m/min.

Example 2 (GATE level). With the data of Example 1, a bar of diameter D = 100 mm is turned over L = 300 mm at f = 0.25 mm/rev; handling time th = 1 min per piece. Find the cost and time per piece at Vo and at Vp. At Vo = 135.5 m/min:

1. tm = π·D·L / (1000·V·f) = π × 100 × 300 / (1000 × 135.5 × 0.25) = 94 248 / 33 885 = 2.781 min.
2. Tool-related cost per piece = (Cm·tc + Ct)·(tm/T) = (20 + 60) × 2.781/24 = ₹9.27.
3. Cost = 10 × 1 + 10 × 2.781 + 9.27 = ₹47.08.
4. Time = 1 + 2.781 + 2 × 2.781/24 = 4.01 min. At Vp = 191.7 m/min:
5. tm = 94 248 / (1000 × 191.7 × 0.25) = 1.967 min; T = 6 min.
6. Cost = 10 + 19.67 + 80 × 1.967/6 = 10 + 19.67 + 26.22 = ₹55.89.
7. Time = 1 + 1.967 + 2 × 1.967/6 = 3.62 min. Running at Vp saves about 10 % of the time per piece but costs about 19 % more per piece. Check of the ratio rule at Vo: tool-related ₹9.27 / machining ₹27.81 = 0.333 = n/(1 − n) = 0.25/0.75. ✓

Common mistakes

• Using Ct/T directly as "tool cost per piece" — it must be multiplied by tm (the fraction of a tool life used per piece).
• Forgetting the operator's cost during the tool change (Cm·tc).
• Putting Ct in the maximum-production formula; tool cost does not affect production rate.
• Concluding Vp < Vo — it is always the other way round.
• Mixing units in Ct/Cm: Ct in ₹ and Cm in ₹/min give minutes, as they must.
• Raising T to n when you need V = C/Tⁿ (and vice versa).

For GATE PI

• NAT: optimum tool life and speed for minimum cost or maximum production, given n, C, Cm, Ct and tc.
• Cost per piece and time per piece at a given speed, combining Taylor's law with machining time.
• Comparing two tools (e.g. HSS vs carbide) on cost per piece.
• MCQs on the Hi-E range and why Vp > Vo.

Quick check

1. n = 0.2, tc = 1 min. Tool life for maximum production?
2. n = 0.25, tc = 2 min, Ct = ₹40, Cm = ₹8/min. Tool life for minimum cost?
3. Which is higher, the minimum-cost speed or the maximum-production speed?
4. Does handling time affect the optimum speed?
5. C = 400 m/min, n = 0.25, To = 16 min. Optimum speed?

Answers: 1. Tp = 4 × 1 = 4 min. 2. To = 3 × (2 + 5) = 21 min. 3. The maximum-production speed. 4. No — it adds the same time and cost at every speed. 5. 400/16^0.25 = 400/2 = 200 m/min.