Calculation of different Cost elements:

Let ; D= Diameter of job in mm

L= length of the job in mm

f=feed in mm/rev

N= rotational speed.=$\frac{1000 V}{π ×D}$

$t_m$ =machining speed of work piece. =$\frac{L}{f ×N}$

$t_h$= handling or loading time.

$K_1$= Operating cost.

$t_c$= tool changing time for one edge

We know that;

Cost of manufacturing one piece= handling cost per piece+ Machining cost per pc.+ Tool change cost per pc.+ Tool cost per pc. ………..1

Handling Cost per pc.

Handling Cost per pc (Rs/pc)=$t_h \times K_1$

$= t_h . K_1$………2

Machining cost per pc.

Machining cost per pc. = $t_m \times K_1$

=$K_1 .t_m$…………………3

$=K1.\frac{L}{f ×N}$

$= K1. \frac{L π D}{f ×1000 V}$

$= K_1 .\frac{L π D}{f ×1000} .\frac{1}{V}$

$ = K_1. \frac{K}{V}$………………………4

Tool change cost per pc.

$t_m$=machining time (min)

T=tool life (min)

Number of edges required to manufacture one piece = t_m/T

tool changing cost per pc. = $t_c \times \frac{t_{m}}{T} x K_1$

= $K_1 \times t_c \times \frac{t_{m}}{T}$ ……………..5

= $K_1 \times t_c \times \frac{K}{V}\times \frac{1}{T}$ …………6 (from eqn. 3 & 4)

Taylors tool life eqn.

$VT^{n}$=C

∴ $T=\frac{C^{1⁄n}}{V^{1⁄n}}$ ……………7

put in eqn. 6

tool changing cost per pc. =$ K_1 \times t_c \times \frac{K}{V}×\frac{V^{1⁄n}}{C^{1⁄n}}$

= $K_1x t_c \times \frac{K}{C^{1⁄n}} x V^{1⁄n} \times V-1$

= $K_1 \times t_c \times \frac{K}{C^{1⁄n}} × V^{\frac{1-n}{n}}$……….8

Tool cost per pc. Let; K2=tool cost per edge.

tool cost per piece = (no. of tool edges required to produce one pc.) x (tool cost per edge.)

=$ \frac{t_{m}}{T} \times K_2$

= $K_2 \times \frac{t_{m}}{T}$

=$ K_2 \times \frac{K}{C^{1⁄n}} × V^{\frac{1-n}{n}}$……….9

Expression of optimum cutting speed for minimum cost of production: from eqn. 1, 2, 4, 8, 9

Cost of manufacturing one piece= $t_h .K_1 + K_1.\frac{K}{V} + K_1 \times t_c \times \frac{K}{C^{1⁄n}} \times V^{\frac{1-n}{n}}+ K_2 \times \frac{K}{C^{1⁄n}} \times V^{\frac{1-n}{n}}$

=$t_h .K_1 + K_1.\frac{K}{V} + \frac{K}{C^{1⁄n}} (K_1. t_c+ K_2) .V^{\frac{1-n}{n}}$

for min cost of production

.d/dV (RHS)=0

$0+(-K_1. \frac{K}{V^{2}} ) + \frac{K}{C^{1⁄n}} (K_1. t_c+ K_2).V^{\frac{1-2n}{n}}$ = 0

⇛$V^{\frac{1-2n}{n}} .V^2= K_1 .C^{1⁄n} \frac{1}{(K_1 t_c+K_2 )} .\frac{n}{1-n}$

$⇛V_{min}=C.[\frac{n}{1-n}\times \frac{K_{1}}{K_1 t_c+K_2}]^n$

this is an eqn. of cutting speed for min cost of production. To find corresponding tool life

$VT^{n}$=C

$T_{min}=[\frac{C}{V_min}]^{\frac{1}{n}}$

$T_{min}=\frac{n}{1-n} × \frac{K_1 t_c+K_2}{K_{1}}$

this is the eqn. for tool life for min cost of production

⇛$C^{\frac{1}{n}}{t_{c}} . \frac{n}{1-n}= V^{\frac{1-2n}{n}} . V^{2} =V^{1⁄n}$

therefore, $V_{max}= C[\frac{1}{t_{c}} .\frac{n}{1-n}]^{n}$

this is an eqn. of cutting speed for max. rate of production. To find corresponding tool life

$VT^{n}=C$

$T_{max}=[\frac{C}{V_{max}} ]^{\frac{1}{n}}$

$T_{max}=[\frac{C}{C[\frac{1}{t_c}} .\frac{n}{1-n}]^{n})]^\frac{1}{n}$

$T_{max}=\frac{t_{c}(1-n)}{n}$

this is the eqn. for tool life for max. Rate of production.