2. Selection of Cutting Conditions
• Selection of cutting speed, or cutting speed
and feed can be made using unconstrained
or constrained mathematical optimization
methods.
2
• Optimization is based on time, cost, or profit
rate criteria.
3. Optimization Criteria
• Commonly employed optimization criteria (objectives)
are:
– Max. production rate or min. production time: aims to
maximize number of parts produced in a unit time interval or
minimizes the time per unit part. Neglects cost and/or profit.
3
– Min. production cost: aims to determine the least
production cost. Coincides with max. profit rate criterion for
constant unit revenue. Ignores time constraints.
• Usually one of the two criteria is used. Sometimes
both criteria are used simultaneously.
4. Decision Variables
• Depth of cut (d)
• Feed (f )
• Cutting speed (V)
4
• Cutting speed (V)
Depth of cut usually decided based on part
geometry and material to be removed and
can be assumed constant.
5. Definitions (1)
• Unit production time, t (min/pt): time to manufacture
a unit of the product.
t = tp + tm + tr
Where,
tp = setup time (min/pt); includes time needed to load/unload parts
into machine, tool setup time, etc.
ME 4210: Manufacturing Processes & Engineering
Instructor: Ramesh Singh; Notes by: Prof. S.N. Melkote
5
into machine, tool setup time, etc.
tm = machining time (min/pt)
tr = total tool replacement time (min/pt) = tc(tm/T)
tc = time needed to replace a worn cutting edge with a new one
(min/edge)
T = tool life (min)
• Production rate = (1/t)
6. Definitions (2)
• Unit production cost, u ($/pt): cost to manufacture a unit of the
product.
u = uo + um + ut = kotp + (ko + km) tm + [kt + kotc] (tm/T)
Where,
uo = capacity utilization cost ($/pt); includes machine cost, labor cost,
overhead etc. = kot = ko[tp + tm + t (t /T)]
ME 4210: Manufacturing Processes & Engineering
Instructor: Ramesh Singh; Notes by: Prof. S.N. Melkote
6
overhead etc. = kot = ko[tp + tm + tc(tm/T)]
ko = machine utilization rate ($/min)
um = machining cost (associated with actual machining time); includes cost
of electricity, cutting fluids etc. = km tm
km = machining overhead ($/min)
ut = tool utilization cost; includes cost of cutting tool, tool resharpening, etc.
= kt (tm/T)
kt = cost per cutting edge ($/edge)
7. Unconstrained Optimization (1)
• Basic problem formulation: Find cutting speed
V that optimizes (Minimizes or Maximizes) Z,
where Z = appropriate optimization criterion.
• In order to solve this problem, the objective
7
• In order to solve this problem, the objective
function Z should be expressed in terms of V.
Feed and depth of cut are assumed to be
fixed.
• Note that only tool life T and the machining
time tm are functions of V.
8. Unconstrained Optimization (2)
• In general,
tm = λ/V
Where λ is a constant based on the machining
process and V is in m/min units.
8
• For turning, boring, drilling, and reaming operations:
tm = L/(Nf) and V = πDN/1000
λ = πDL/(1000f)
where D = diameter of part/tool (mm), L = length of
cut (mm), f = feed (mm/rev), N = spindle speed (rpm)
9. Unconstrained Optimization (3)
• Taylor’s tool life equation:
n
VT C
=
9
Where T = tool life (min), C = 1 min. tool life
cutting speed, n = function of tool-workpiece
material.
10. Unconstrained Optimization (4)
• Expressing t, u in terms of V, we get:
( )
1
1/
n
n
n
p c
t t V t V C
λ λ
−
= + +
10
( )( ) ( )
1
1/
n
n
n
o p o m t o c
u k t k k V k k t V C
λ λ
−
= + + + +
11. Unconstrained Optimization (5)
• Optimal cutting speed (and corresponding
tool life) can be obtained by standard calculus
as:
• Min. time or max. production rate criterion:
1
n
n
−
11
1
1
n
opt c
opt c
n
V C t
n
n
T t
n
−
=
−
=
12. Unconstrained Optimization (6)
• Min. cost criterion:
( )
( )
1
n
o m
opt
k k
V C
n
k k t
+
=
−
+
12
( )
( )
( )
1
1
t o c
t o c
opt
o m
n
k k t
n
k k t
n
T
n k k
−
+
+
−
=
+
13. Unconstrained Optimization Example
(1)
• Given:
– Taylor tool life equation VT0.23 = 430, V (m/min)
– Machining (turning) parameters
• Depth of cut, d = 1.00 mm
• Feed, f = 0.2 mm/rev
– Workpiece geometry
13
– Workpiece geometry
• Workpiece diameter, D = 50 mm
• Workpiece length, L = 200 mm
– Time parameters
• Setup time, tp = 0.75 min/pt
• Tool replacement time, tc = 1.5 min/edge
15. Unconstrained Optimization Example
(3)
• Min. time or max. production rate cutting
speed and tool life:
0.23
1 1 0.23
430 1.5 296 m/min
0.23
n
opt c
n
V C t
n
− −
= = =
15
0.23
1 1 0.23
1.5 5 min
0.23
opt c
n
n
T t
n
− −
= = =
16. Unconstrained Optimization Example
(4)
• Min. cost cutting speed and tool life:
( )
( )
( )
( )
0.23
0.5 0.05
430 216 m/min
1 1 0.23
2.5 0.5 1.5
0.23
n
o m
opt
t o c
k k
V C
n
k k t
n
+ +
= = =
− −
+ + ×
16
( )
( )
( )
( )
0.23
2.5 0.5 1.5
1 1 0.23
19.8 min
0.23 0.5 0.05
t o c
opt
o m
n
k k t
n
T
n k k
+ + ×
− −
= = =
+ +
17. Summary
• Manufacturing process optimization
– Application to machining
• Unconstrained optimization
17
– Optimum cutting speeds
– Production time and cost criteria
