In this article we will discuss about the economics of machining process and operations. The important economic aspects of the machining operations are:- 1. Optimizing Cutting Parameters for Minimum Cost 2. Optimizing Cutting Parameters for Maximum Production and 3. Optimum Cutting Speed for Maximum Efficiency.
Since the industry is very closely linked to economics and the machining operations play a predominant role in manufacturing parts, it is not difficult to appreciate the importance of studying the economic implications of the machining operations. The two quantities generally of interest are – (i) cost, and (ii) production time (or rate). Both these depend on the choice of cutting parameters, e.g., cutting speed, feed, and depth of cut. Generally, a component goes through various operations and an exact economic analysis is extremely complicated.
But, at the same time, in mass scale production, often one operation is performed in one special machine; thus, we shall make an attempt to carry out a preliminary analysis, considering single operations. Such an analysis will provide us with some basic information on the important economic aspects of the machining operations. To avoid complications, we shall restrict our analysis to the simple turning operation of cylindrical bars.
Economics of Machining Process and Operations
1. Optimizing Cutting Parameters for Minimum Cost:
The total cost of a part can be written in the form –
R = R1 + R2 + R3 + R4 + R5,… (4.77)
Where R is the total cost / piece, R1 is the material cost / piece, R2 is the set-up and idle time cost / piece, R3 is the machining cost / piece, R4 is the tool changing cost / piece, and R5 is the tool regrinding cost / piece. Let us consider the length and diameter of the cylindrical part to be L and D, respectively, both being in mm. The feed and the cutting speed used are ƒ mm / revolution and v m / min, respectively.
Since we are analyzing only one pass, the depth of cut is fixed either by the final diameter required or by the maximum allowable limit without causing chatter (vibration) depending on which one is less. Thus, the depth of cut does not really come into our analysis. For purposes of cost calculation, let us have the rates –
λ1 = cost / min of labour and overheads,
λ2 = cost of setting a tool for regrinding,
λ3 = cost / mm of tool ground.
The time spent in the various operations is –
ts = set-up time and idle time / piece, min,
tm = machining time / piece, min,
tct = tool changing time, min.
Now, the cost of the component can be expressed in terms of the cutting variables.
Material cost – The material cost does not depend on the cutting conditions and remains as a constant (R1).
Set-up and idle time cost – It is given by the product of the set-up and idle time and the cost / unit time of labour and overheads. Thus,
If the total cost of a new tool is A and the total length that can be ground off (after which the tool cannot be used) is B mm, then the cost per mm of the tool ground is given by –
In a similar manner, it can be shown that the optimum feed for a given cutting speed can be expressed as –
The nature of variation of the total cost / piece with speed and feed is shown in Fig. 4.71. From this figure, we see that the minimum value of R becomes smaller as ƒ increases. However, the feed cannot be indefinitely increased for various reasons, and the most important constraints on feed are the surface finish, cutting force, and power available in the machine tool.
Even if we make the tool strong enough to withstand very large forces, the constraints, because of the surface finish requirement and machine tool power, cannot be avoided. From relation (4.74), we know that, for a given nose radius, the maximum unevenness Hmax varies with ƒ2. In other words, if Hmax lim is the limiting value of the unevenness height, then –
Where k1 is a constant. The power consumption will then be –
The corresponding curve is shown in Fig. 4.72 and any speed-feed combination on the right side of this curve is again not allowed. Since Rmin reduces as the feed is increased, we should follow the line xx and go in the direction of increasing feed as far as possible. So, depending on the relative positions of the constraint boundaries, either P or Q will be the optimum allowable speed-feed combination (whichever is reached earlier). Figure 4.73 shows the nature of variation of the various components of cost with speed for a given feed.
2. Optimizing Cutting Parameters for Maximum Production:
Sometimes, the rate of production is more important than the cost per piece. In fact, from the economic point of view, the best situation is when the profit is maximum. A large rate of production may result in a better return, and therefore it is also useful to investigate the conditions leading to the highest possible rate of production.
The maximum production rate can be achieved if the total time required per piece is reduced to a minimum. For this, we shall, for reasons already explained, keep the feed at the highest possible value and search for the optimum velocity. The total time required per piece is given by –
It is clearly seen that if the value of λ4 in equation (4.80) is put equal to zero, the result we get is identical with that given by equation (4.82). The reader can easily justify this. Thus, it is clear that the optimum speed for minimum time (or maximum production rate) is always more than that for minimum cost.
3. Optimum Cutting Speed for Maximum Efficiency:
The optimum velocity to achieve the maximum efficiency, i.e., the maximum profit rate, can be also found out without much difficulty. Of course, for doing this, a closed form expression for vopt will not be possible, and the numerical or graphical methods have to be employed. If S is the amount received per piece, then the expression for the profit rate is –