Editorial & Advisory Board - Acta Technica Corviniensis

MATHEMATICAL MODEL MINIMALIZATION OF COSTS
In present, the cutting conditions are mainly chosen
from norms that are not optimal.
The nature of cutting conditions optimization is to
determine optimal values of given conditions (cutting
depth – a
p
, underthrust depth – f, cutting speed – v)
c
and the optimization of a machine durability.
For a machine with replaceable cutting plates that are
re‐sharpened is valid:
C
p
.z
p
.z
0
C
tn
NnT
=
+ (1+
kut
).
zh
.s
p
(z0
+ 1) zu
M RNO
(1)
as
pl z0
+ τ
as.
.k
c
(1+
).
60 100 z0
+ 1
where:
z0
is the number of possible re‐sharpening of a plate.
Costs to exchange of a device can be given by:
M RNS
n
pl Os
N
vn
= τvn
kc
.(1+
) + = τvnNvmnzv
(2)
60 100 60
where:
Nvmn
are costs to exchange of a device per, [min],
M
n
– wages of a setup man including social and health
1
insurance, [ € ],
τ
vn
– time to exchange a device,[ min],
The criterion for minimal production costs can be
given (production costs to operational department
shall be minimum) by the relation:
N
c
= Ns
+ Nn
+ Nm
(3)
where:
N – production costs to calculate an operational
section,[€],
N
s
‐ costs to machine labor per an operational
section, [€],
N
n
– costs to machines related to the operational
section, [€],
N
vn
– costs to exchange or offset of a worn‐out device
related to the operational section, [€].
Having substituted the above‐mentioned relations
into this criterion the optimization criterion to be
reached from the point of view of production costs as
follows:
or :
τ
N +
T
As
As
= τAsNsm
+ krNnT
τvnNvmn
. kr
→min (4)
τAs
N= τAs Nsm
+ kr
(NnT
+ τvnNvmn)
→ min (5)
T
The machine time can be given by:
Lch
τ
As
= (6)
n.f
τ
T
ACTA TECHNICA CORVINIENSIS – Bulletin of Engineering
where:
Lch
is the length of machine automatic operation run,
[mm],
n‐ rotational frequency,
f – displacement,[ mm]
Substitution (4) in (5) results in the criterion equation:
Lch
Lchk
r
N = Nns
+ (NnT
+ τvnNvmn
) →min (7)
n.f n.f.T
Having modified:
where:
K
K1
K
2
+ →min (8)
n.f n.f.T
1
Lch
. Nsm
2
Lchk
T
(NnT
+ τvn
K = (9)
= N ) (10)
When milling operation, proportional parameter of
displacement per rotation f and displacement per
tooth fz are considered. Total production costs per
a work‐piece can be given by the relation:
N N
vmn
∑ nu d šz
c
= Ni
+ Nv
+
(11)
1 n n
N +
where:
Nare total production costs per a work‐piece, [€],
c
N – production costs to i‐ operational section, [€],
i
Nšz
‐ costs to a special device necessary for production
of a given work‐piece, [€],
n – number of produced pieces,
n
u
– number of operational sections within one workpiece,
Costs to secondary work:
M RNS
0
pl Os
N
v
= τ
Av
k
c
(1+
) + = τ
AsNvn
(12)
60 100 60
Where:
N – costs to secondary work, [€],
vm
τ
Av
– unit secondary time,[ min.],
Rate costs:
M RNS
s
pl Os
N
B
= k
c
(1+
) + = τBCNBm
(13)
60 100 60
where:
1
N are rate costs, [ € ],
Bm
τ
BC
– rate time with shift time over plus,[ min]
Criterion of minimum production costs can be also
given by the method of hourly operational costs.
Fixed costs whose share in total costs continually
raises are just those unwelcome costs that burden
production. This is one reason why it is success to
produce with optimal capacity employment.
For practical utilization it is appropriate to express the
capacity utilization in time units (hours, norm hours).
104
2012. Fascicule 2 [April–June]