Introduction

• The basic objective of any manufacturing activity is to produce components of acceptable quality at the most economical cost.
• The cost of a manufactured component is comprised of the cost of raw material and the cost of conversion of the raw material into the finished product.
• For a machined component, the cost of conversion is decided, to a large extent, by the parameters of machining namely speed, feed and depth of cut selected for the machining operation.
• In order to keep the cost of the component low, it is necessary to select these parameters such that the total cost of the components is minimized.
• On the other hand, sometimes it becomes necessary to produce a component in the shortest time to meet certain planned targets.
• In such situations, it is necessary to minimize production time even though the cost may not be optimal.
• Two modes for selection of optimal cutting parameters, therefore, become evident.
  These are
  
  • Optimal cutting parameters for minimum cost per component.
  • Optimal cutting parameters for maximum production rate.
• In the analysis below these models have been presented for the simple case of single-pass external cylindrical turning on an engine lathe using single-point tool.
• More complicated models on similar lines can be developed for multi-pass, multi-tool turning operations on other lathes or on other machines like shapers, drilling machines and milling machines.
• It may be pointed out here that the optimization presented here can at best be called sub optimization because a single machining operation is seldom the only operation carried out for the manufacture of a component.
• Most components need other operations on the same machine, or operations on more than one machine with in-process storage, handling inspection etc.
• Further, a given machine tool may be used for more than one type of components.
• The company may also be involved in the manufacture of different types of components.
• Any attempt to optimize cutting conditions for one component may affect the production rate of other components.
• An attempt at total optimization for a component should, therefore, involve many more coat and time factors than those considered here.
• Such an optimization is very complex and time consuming and is rarely attempted.

Optimization of cutting Parameters for minimum cost

The total cost, C, for a machined component is composed for the following coast.

1. Material cost $C_1$. This cost depends upon the type and size of the material used and the cost rate. Material cost does not depend on the cutting conditions and may be taken as constant.
2. Set up and idle time cost $C_2$: This is the cost of that portion of the machining cycle which is not involved in actual cutting i.e. loading, unloading, approach etc.
   The set up cost may be written as
   $C_2 = xt_e$
   Where x is the average operating cost per minute of the machine and operator in including overhead in Rs/minute and $t_c$ is the total non-productive time per piece in minutes.
3. Machining cost $C_3$: This is given by the product of the machining time $t_c$(minutes/piece) and cost per unit time of labour, machine and overheads.
   Thus, $C_3 = x t_c$
4. Tool changing cost $C_4$ : If $t_d$ is the time(min.) required to change and adjust the tool after an edge has become worn out and T is the life of the tool in minutes, then $\frac{t_c}{T}$ is the number of times the tool has to be changed during machining of one component and
   $C_4 = x t_d\frac{t_c}{T}$
5. Tool cost per component $C_5$ : This is equal to tool cost per cutting edge time the number of cutting edges used per component i.e.
   $C_5 = y\frac{t_c}{T}$
   The tool cost y per cutting edge depends on the type of tool used and the cost of grinding. Thus for H.S.S. tool or a brazed tipped tool :
   y = $\frac{initial cost of tool + cost of regrinding}{No. of regrinds + 1}$
   = $\frac{C_t + rC_g}{r+1}$
   Where $C_t$ = the initial cost of the tool Rs.
   r = number of regrinds possible for the tool
   $C_g$ = cost of each regrinding
   And for throw away type of tipped tool
   y = $\frac{Cost of insert}{No. of cutting edges} + \frac{Cost pf tool holder}{No. of cutting edges used in the life of the tool holder}$
   Combining all the above costs,
   $C =C_1 +x[t_e + t_c + \frac{t_ct_d}{T}] + y\frac{t_c}{T}$
   $=C_1 +x[t_e + t_c(1+ \frac{R}{T})]$
   $=C_1 + x[t_e + t_c(1+\frac{R}{T})]$
   Where $R=t_d + \frac{y}{x}$
   The quantities in Equation which vary with velocity v and feed rate for a constant depth of cut are $t_c and T$.
   Now, $t_c = \frac{L}{fN}$
   $= \frac{L}{f*\frac{v * 60000}{\pi D}} = \frac{\pi DL}{60000 V f}$
   Where L= Length of the job to be machined including approach and over travel, mm
   f = feed rate, mm/rev
   V = cutting speed, m/sec.
   N = RPM of the machine
   D = Workpiece diameter, mm
   Tool life T is a strong function of V and f. If depth of cut d is assumed to be constant, the tool life equation can be written as
   $VT^nf^m = Constant C$
   or $T = (\frac{C}{Vf^m})^\frac{1}{n}$
   $= C^\frac{1}{n}V^{-\frac{1}{n}}f^{-\frac{m}{n}}$
   Substituting these values in Equation, the equation for cost per component becomes
   $C = C_1 + x[ t_e+\frac{\pi DL}{60000Vf}(1+\frac{R}{C^\frac{1}{n}V^{-\frac{1}{n}}f^{-\frac{m}{n}}})$
   $= C_1 + x[t_e + \frac{\pi DL}{60000 Vf}(1+RC^{-\frac{1}{n}}V^\frac{1}{n}f^\frac{m}{n})]$
   The optimum cutting conditions $V_o$ and $f_o$ can be found by partially differentiating Equation with respect to v and f and equating to zero.
   $\frac{\delta C}{\delta V} = 0 $ gives
   $\frac{1}{V^2_o} = RC^{-\frac{1}{n}}f^\frac{m}{n}(\frac{1}{n}-1)V_o^{\frac{1}{n}-2}$
   or $1 = RC^{-\frac{1}{n}}f^\frac{m}{n}V_o^\frac{1}{n}(\frac{1}{n}-1) $
   $ = \frac{R}{T_o}(\frac{1}{n}-1)$
   i.e. $T_o = R[\frac{1}{n} - 1]$
   Now since $ R = T_d +\frac{y}{x} = \frac{{xT_d}+y}{x}$
   The above equation reduces to
   $\frac{T_o}{R} = \frac{1-n}{n}$ or $\frac{xT_o}{xT_d+y} = \frac{1-n}{n}$
   $ or \frac{cost of operating between tool changes}{total cost of a tool change} = \frac{1-n}{n}$
   This ratio is sometimes known as cost ratio for economic cutting. $T_o$ is the tool life which gives minimum cost when V is varied. It may be noted that $T_o$ is not a function of f inspite of the fact that T is a function of f.
   Partially differentiating the cost equation with respect of f similarly gives
   $T'_o = R(\frac{m}{n}-1) $
   It can be observed that Equations give the same value of $T_o$ only when m = 1 but referring back to tool life Equation, m= 1 means that tool life T varies as much with f as with V which is not true. The reduction in tool life with increase in V is generally twice as much as that with an equivalent increase in f i.e. m<1.
   Since rate of metal removal is proportional to the product of V and f and an increase in V is more harmful to tool life than an increase in f, it is desireble to operate with condition of $T_o = R(\frac{1}{n}-1)$ and select as large a value of feed as is possible without unduly spoiling the surface finish. Thus, first the feed rate is selected on the basis of consideration other than tool life and then V is raised to $V_o$ to satisfy the condition $T_o = R(\frac{1}{n}-1)$.
   In deriving the above expression it has been assumed that the operation is being carried out at a constant feed and depth of cut. Since n depends mainly on the tool work pair and any variation in feed and depth of cut are known to have not much effect on n, the cost ratio relationship can be treated as constant regardless of the value of d and f used. The same thing is, however, not true for value of C which varies considerably with variation of feed and depth of cut. Thus the economic value of tool life will depend significantly on the particular values of f and d for which the concerned Taylors's tool life equation has been derived.