Iq2515441549

S. S. K. Deepak / International Journal of Engineering Research and Applications (IJERA)
ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.1544-1549

A Geometric Programming Model for Production Rate
Optimization of Turning Process with Experimental Validation
S. S. K. Deepak
(Department of Mechanical Engineering, Rungta Engineering College, Raipur (C.G.), India-492099)

ABSTRACT
𝜋𝑑𝑙
Every manufacturing unit wants to 𝑡 𝑚 = time required to machine a work piece =
1000 𝑣𝑓
maximize its production rate because production
(min.)
rate directly affects the profit and the growth of
T = tool life (min.)
the manufacturing unit. There are many
𝑣 = the cutting speed (m/min.)
approaches for achieving the same, some of the
Z = constant
methods are experimental and some are based on
η = efficiency of cutting
very lengthy and time consuming statistical
𝜆01 , 𝜆02 𝑎𝑛𝑑 𝜆11 are lagrange multipliers.
techniques. The manufacturing firms want a
quick approximate solution to the optimization
problem, so as to gain a competitive advantage in 1. Introduction
the market. In this research paper, a geometric Traditionally, the selection of cutting
programming based approach to maximize the conditions for machining was left to the machine
production rate of the turning process within in operator. In such cases, the experience of the
some operating constraints is proposed. It is operator plays a major role, but even for a skilled
achieved by taking production time as the operator it is very difficult to attain the optimum
objective function of the optimization problem values each time. Machining parameters in metal
and then minimizing the same. It involves cutting are cutting speed, feed rate and depth of cut.
mathematical modeling for production time of The setting of these parameters determines the
turning process, which is expressed as a function quality characteristics of machined parts. The first
of the cutting parameters which include the necessary step for process parameter optimization in
cutting speed and feed rate. Then, the developed any metal cutting process is to understand the
mathematical model was optimized with in some principles governing the cutting processes by
operating constraints. The results of the developing an explicit mathematical model, which
experimental validation of the model reveal that may be of two types: mechanistic and empirical [1].
the proposed method provides a systematic and To determine the optimal cutting parameters,
efficient technique to obtain the optimal cutting reliable mathematical models have to be formulated
parameters that will maximize the production to associate the cutting parameters with the cutting
rate of turning process. performance. However, it is also well known that
reliable mathematical models are not easy to obtain
Keywords - geometric programming, [2-3]. The technology of metal cutting has grown
optimization, production rate, model. substantially over time owing to the contribution
Nomenclature: from many branches of engineering with a common
𝜋𝑑𝑙 goal of achieving higher machining process
𝐶01 = constant = 1000 efficiency. Selection of optimal machining condition
𝜋𝑑𝑙 𝑡 𝑐 is a key factor in achieving this condition [4]. In any
𝐶02 = constant = 1 multi-stage metal cutting operation, the
1000 𝑍 𝑛
𝐶11 = constant manufacturer seeks to set the process-related
𝐶0 = machine cost per unit time ($/min. ) controllable variables at their optimal operating
𝐶 𝑚 = machining cost per piece ($/piece) conditions with minimum effect of uncontrollable or
𝐶 𝑡 = tool cost ($/cutting edge) noise variables on the levels and variability in the
d = diameter of the work piece (mm.) output. To design and implement an effective
𝑓 = feed rate (mm/revolution) process control for metal cutting operation by
F = cutting Force (N) parameter optimization, a manufacturer seeks to
l = length of the work piece (mm.) balance between quality and cost at each stage of
n, a, b and p are constants. operation resulting in improved delivery and
𝑃𝑡 = production time per piece (min./piece) reduced warranty or field failure of a product under
R= nose radius of the tool (mm) consideration. Gilbert [5] presented a theoretical
𝑅 𝑎 =average surface roughness (μm) analysis of optimization of machining process and
proposed an analytical procedure to determine the
𝑡 𝑐 = tool changing time (min.)
𝑡ℎ = tool handling time (min.) cutting speed for a single pass turning operation
with fixed feed rate and depth of cut by using two

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different objectives (i) maximum production rate model and program can be used to determine the
and (ii) minimum machining cost. Hinduja et. al [6] optimal cutting parameters to satisfy the objective of
described a procedure to calculate the optimum obtaining maximum production rate of turning
cutting conditions for machining operations with process under different operating constraints. The
minimum cost or maximum production rate as the results of the mathematical model were obtained by
objective function. For a given combination of tool using MS-Excel. This research paper proposes a
and work material, the search for the optimum was very simple, effective and efficient way of
confined to a feed rate versus depth-of-cut plane optimizing the production rate of the turning process
defined by the chip-breaking constraint. Some of the with some operating constraints such as the
other constraints considered include power maximum cutting speed, maximum feed rate, power
available, surface finish and dimensional accuracy. requirement, surface roughness. This paper also
In any optimization problem, it is very crucial to highlights the advantages of using geometric
identify the prime objective called as the objective programming optimization technique over other
function or optimization criterion. In manufacturing optimization techniques
processes, the most commonly used objective
function is the specific cost [7]. 2. Mathematical Modeling for Optimization:
Walvekar and Lambert used geometric The maximum production rate for turning
programming for the selection of machining process is obtained when the total production time is
variables. The optimum values of both cutting speed minimal. So, the objective function is to minimize
and feed rate were found out as a function of depth the total production time of turning operation. The
of cut in multi pass turning operations [8]. Wu et. al. production time to produce a part by turning
analyzed the problem of optimum cutting operation can be expressed as follows:
parameters selection by finding out the optimal 𝑃𝑡 = Machining Time + Tool Changing Time +
cutting speed which satisfies the basic Set-up Time
manufacturing criterion [9]. Basically, this (1)
optimization procedure, whenever carried out, 𝑡
𝑃𝑡 = 𝑡 𝑚 + 𝑡 𝑐 𝑇𝑚 + 𝑡ℎ (2)
involves partial differentiation for the minimization
of unit cost, maximization of production rate or The Taylor's tool life (T) used in Eq. (2) is given by:
1
maximization of profit rate. These manufacturing Z n
T = fp v (3)
conditions are expressed as a function of cutting
speed. Then, the optimum cutting speed is Where, n, p and Z depend on the many factors like
determined by equating the partial differentiation of tool geometry, tool material, work piece material,
the expressed function to zero. This is not an ideal etc.
approach to the problem of obtaining an economical On substituting Eq. (3) in Eq. (2), we get
1 𝑝
metal cutting. The other cutting variables, 𝑃𝑡 𝑣, 𝑓 = 𝐶01 𝑣 −1 𝑓 −1 + 𝐶02 𝑣 𝑛 −1 𝑓 𝑛 −1 + 𝑡ℎ (4)
particularly the feed rate also have an important 𝑡ℎ does not depend on cutting speed or feed rate. So,
effect on cutting economics. Therefore, it is the modified objective function from Eq. (4) can be
necessary to optimize the cutting speed and feed rate written as:
1 𝑝
simultaneously in order to obtain an economical −1 −1
𝑃𝑡 𝑣, 𝑓 = 𝐶01 𝑣 −1 𝑓 −1 + 𝐶02 𝑣 𝑛 𝑓 𝑛 (5)
metal cutting operation [10]. Once the reliable
2.1 Machining constraints:
model for turning operations has been constructed,
There are many constraints which affect the
an optimization algorithm is then applied to the
selection of the cutting parameters. These
model for determining optimal cutting parameters.
constraints arise due to various considerations like
The geometric model for machined parts and
the maximum cutting speed, maximum feed rate,
various time and cost components of the multistage
power limitations, surface finish, surface roughness
turning operation are also given in the optimization
and the temperature at the tool-chip interface.
process.
2.1.1 Maximum cutting speed:
In the optimization of cutting parameters,
The increasing of cutting speed also increases
several methods are used. Some are based on
the tool wear, therefore, the cutting speed has to be
extensive experimentation which is quite laborious
kept below a certain limit called the maximum
and lengthy process and its result may also vary in
cutting speed.
different conditions. Testing of materials like tool
𝑣 ≤ 𝑣 𝑚𝑎𝑥 . (6)
life test may require large amount of metals and
𝐶11 𝑣 ≤ 1 (7)
considerable tool wear, so, it cannot be used for
Where
precious metals. The aim of this research paper is 1
the construction of a mathematical model describing 𝐶11 = (8)
𝑣 𝑚𝑎𝑥 .
the objective function in terms of the cutting By the method of primal and dual
parameters with some operating constraints, then; programming of geometric programming, the
the mathematical model is optimized by using maximum value of dual function or the minimum
geometric programming approach. The developed value of primal function is given by:

1545 | P a g e

𝐶01 𝜆 01 𝐶02 𝐹×𝑣
𝜈 𝜆 = 𝜆01 + 𝜆02 𝜆01 + 𝑃= 6120 𝜂
≤ 𝑃 𝑚𝑎𝑥 . (30)
𝜆 01 𝜆 02
𝜆 02 𝐶11 𝜆 11
𝐶11 𝑣 ≤ 1 (31)
𝜆02 (9) Where
𝜆 11
𝐹
Subject to the following constraints: 𝐶11 = 6120 𝜂𝑃 (32)
𝑚𝑎𝑥 .
𝜆01 + 𝜆02 = 1 (10)
1
−𝜆01 + − 1 𝜆02 + 𝜆11 = 0 (11) Following the same procedure as described for the
𝑛
𝑝 first constraint, we get the following values:
−𝜆01 + − 1 𝜆02 = 0 (12)
𝑛 𝜆01 = 1 − 𝑛 (33)
And the non-negativity constraints are: 𝜆02 = 𝑛 (34)
𝜆01 ≥ 0, 𝜆02 ≥ 0 and 𝜆11 ≥ 0 (13) 𝜆11 = 1 − 𝑝 (35)
On adding Eq. (10) and Eq. (12), we get 𝐶 01 1−𝑛 𝐶 02 𝑛
𝜆 11 𝐶11 1−𝑝
𝜆02 = 𝑛 (14) 𝑣= 1−𝑛 𝑛
(36)
𝐶11
From Eq. (10) and Eq. (14), we get 𝐶01 ×𝐶11
𝜆01 = 1 − 𝑛 (15) 𝑓= 𝐶 1−𝑛 𝐶 02 𝑛
(37)
1−𝑛 ×𝜆 1 × 01 𝐶11 1−𝑝
From Eq. (11), (14) and (15), we get 1−𝑛 𝑛

𝜆11 = 1 − 𝑝 (16)
Therefore, the maximum value of dual function and 2.1.4 Surface roughness:
the minimum value of primal function is given by: Surface roughness can be used as a
𝐶 1−𝑛 𝐶02 𝑛 constraint in finishing operations. Therefore, it
01 1−𝑝
𝜈 𝜆 = 𝐶11 1 − 𝑝 (17) becomes a very important factor in determining
1−𝑛 𝑛
Now, finish cutting conditions. Surface roughness can be
𝐶 𝑣 expressed in terms of feed as follows:
𝜆11 = 𝜈11𝜆 (18)
𝑓2
From Eq. (17) and Eq. (18), we get the optimum 𝑅𝑎 = 32𝑅
(38)
values of cutting speed as: 𝑓2
𝑅 𝑎 𝐶11 ≤ 32𝑅 (39)
𝐶 01 1−𝑛 𝐶 02 𝑛
𝜆 11 𝐶11 1−𝑝
𝑣= 1−𝑛 𝑛
(19) Following the same procedure as described for the
𝐶11 first constraint, we get the following values:
And, 𝜆01 = 1 − 𝑛 (40)
𝐶01 𝑣 −1 𝑓 −1
𝜆01 = (20) 𝜆02 = 𝑛 (41)
𝜈 𝜆
1−𝑝
Therefore, from (19) and (20), we get optimum feed 𝜆11 = 2
(42)
rate as: 1−𝑝 𝐶 01 1−𝑛 𝐶 02 𝑛
𝐶11 1−𝑝
𝐶01 ×𝐶11 2 1−𝑛 𝑛
𝑓= 1−𝑛 𝐶 𝑛 (21) 𝑣= 𝐶11
(43)
𝐶 01 02
1−𝑛 ×𝜆 1 × 𝐶11 1−𝑝 𝐶01 ×𝐶11
1−𝑛 𝑛
𝑓= 2 (44)
1−𝑝 𝐶 01 1−𝑛 𝐶 02 𝑛
1−𝑛 𝐶11 1−𝑝
2 1−𝑛 𝑛
2.1.2 Maximum feed rate:
In rough machining operations, feed rate is
taken as a constraint to achieve the maximum 2.1.5 Chip-tool interface temperature constraint:
production rate. The temperature at the chip tool interface
f≤ 𝑓 𝑚𝑎𝑥 . (22) should be with in a permissible limit otherwise
overheating may lead to excessive tool wear damage
𝐶11 𝑓 ≤ 1 (23)
Where to metal. The temperature at the chip tool interface
1 is represented by
𝐶11 = 𝑓 (24) 𝑄𝑖 = 𝑘 𝑣 𝑎 𝑓 𝑏 ≤ 𝑄 𝑢 (45)
𝑚𝑎𝑥 .
Following the same procedure as described Or
for the first constraint, we get the following values: 𝐶11 𝑘 𝑣 𝑎 𝑓 𝑏 ≤ 1 (46)
𝜆01 = 1 − 𝑛 (25)
𝜆02 = 𝑛 (26) 1
Where 𝐶11 =
𝑄𝑢
𝜆11 = 1 − 𝑝 (27)
𝐶 01 1−𝑛 𝐶 02 𝑛 By using the method of primal and dual
𝜆 11 𝐶11 1−𝑝
𝑓= 1−𝑛 𝑛
(28) programming of geometric programming, the
𝐶11 maximum value of dual function or the minimum
𝐶01 ×𝐶11
𝑣= 𝐶 1−𝑛 𝐶 02 𝑛
(29) value of primal function is given by:
1−𝑛 ×𝜆 1 × 01 𝐶11 1−𝑝 𝜆 01
1−𝑛 𝑛 𝐶05 𝐶06
𝜈 𝜆 = 𝜆 01
𝜆01 + 𝜆02 𝜆 02
𝜆01 +
2.1.3 Power constraint: 𝜆 02 𝐶11 𝜆 11
The maximum power available for the 𝜆02 𝜆 11
(47)
turning operation will be a constraint in the turning Subject to the following constraints:
operation, which has to be taken in to consideration. 𝜆01 + 𝜆02 = 1 (48)
The power available for the turning operation is 1
−𝜆01 + 𝑛
− 1 𝜆02 + 𝑎 𝜆11 = 0 (49)
given by:

1546 | P a g e

𝑝 Table 1: Experimental values of cutting speed (v)
−𝜆01 + 𝑛
− 1 𝜆02 + 𝑏 𝜆11 = 0 (50)
and feed rate (f):
On solving we get the following values of lagrange
S. No. Cutting speed Feed rate (f)
multipliers:
𝑎𝑛 (v ) mm./rev.
𝜆01 = 1 − 𝑎𝑝 +𝑏𝑛 −𝑏 (51) m/min.
𝑎𝑛
𝜆02 = (52) 1 105 0.02
𝑎𝑝 +𝑏𝑛 −𝑏
𝑛 −1 2 115 0.04
𝜆11 = (53)
𝑎𝑝 +𝑏𝑛 −𝑏 3 125 0.06
So, 4 135 0.08
𝜈 𝜆 = 5 145 0.10
𝑎𝑛 𝑎𝑛 𝑛 −1
1−
𝐶05
𝐶06
𝐶11
𝑎𝑝 +𝑏𝑛 −𝑏 6 155 0.12
𝑎𝑛 𝑎𝑛 𝑛 −1
1−
𝑎𝑝 +𝑏𝑛 −𝑏 𝑎𝑝 +𝑏𝑛 −𝑏 𝑎𝑝 +𝑏𝑛 −𝑏
7 165 0.14
(54) 8 175 0.16
9 185 0.18
On solving we get the optimum values of cutting 10 195 0.20
speed and feed rate as:
𝑎−𝑏
𝜆 11 𝜈(𝜆)1−𝑏 𝜆 01 𝑏 Table 2: Values of constants:
𝑣= (55)
𝐶11 𝐶03 𝑏
𝑎−𝑏
𝐶11 𝐶03 𝑏 S. Parameter Formulae Value
𝑓 = 𝐶05 𝜈 𝜆 𝜆 11 𝜈 (𝜆)1−𝑏
(56)
No.
1 𝐶01 𝐶01 = constant = 47
2.2 Experimental validation of the mathematical 𝜋𝑑𝑙
model: 1000
For validation of the mathematical model,
experimental values were used from [11] and [12]. 2 𝐶02 𝐶02 = constant = 18
The values taken from the Chen-Tsai cutting model 𝜋𝑑𝑙
1
and experimental data were incorporated in the 1000 𝑍 𝑛
mathematical model developed to analyze the
variations in the production time against the cutting
speed and feed rate. The missing values required for Table 3: Optimum cutting parameters for minimum
analysis were assumed suitably. The values of the production time:
various parameters used in the experimental S.No. Optimum Optimum Optimum
validation are as follows: cutting feed rate Production
a=b=1 speed (v) (f) time
d = 50 mm. (m/min.) (mm. /rev.) (𝑷 𝒕 )
n = 0.9
l = 300 mm (min./piece)
p=1
R= 1.2 mm. 1 155 0.12 28
𝑅 𝑎 = 10 μm.
𝑡 𝑐 = 0.5 min.
𝑡ℎ = 0.5 min.
t1 = 45 min.
t2 = 0.75 min. /piece.
t3 = 0.5min.
η = 0.85.
Z = 10

3. Results and Discussion:
3.1 Figures:
The figures obtained from the
implementation of the mathematical model are as
follows:

1547 | P a g e

43
the production rate will be maximum at the same
42
cutting speed.
41

40
The curves between the production time and
39
the feed rate indicate that a small feed rate will
38 result in high production time. A smaller feed rate
means the number of revolutions should be
Production Time (Pt) (min./piece)

37

36 increased. The more the number of revolutions, the
35
more will be the production time. Even a very high
34
feed rate is not advisable as it will increase the tool
33

32
wear and surface roughness resulting in increased
31
machining time and machine downtime resulting in
30
high production time. So, the optimum feed rate is
29 somewhere between “too small” and “too high”
28 which will result in the minimum production time
27
and the production rate will be maximum at the
26
same feed rate.
25

24

23
4. Conclusion:
100 120 140 160 180 200
In this research paper, the cutting speed and
Cutting speed (v) (m/min.)
feed rate were modeled for the maximum production
Figure 1: Variation of production time versus rate of a turning operation. The maximum cutting
cutting speed speed, the maximum feed rate, maximum power
43 available and the surface roughness was taken as
42 constraints. The results of the model reveal that the
41
Production Time (Pt) (min./piece)

proposed method provides a systematic and efficient
40
39
methodology to obtain the maximum production
38 rate for turning. It can be concluded from this study
37 that the obtained model can be used effectively to
36 determine the optimum values of cutting speed and
35
feed rate that will result in maximum production
34
33
rate. The developed model saves a considerable time
32 in finding the optimum values of the cutting
31 parameters. It has been shown that the method of
30 geometric programming can be applied successfully
29 to optimize the production rate of turning process.
28
The coefficients n, p and Z of the extended Taylor's
27
26
tool life equation are not described in depth for all
25 cutting tool and work piece combinations. Obtaining
24 these coefficients experimentally requires lot of
23 time, resources and then, the analysis of the
0 0.1 0.2
obtained values increases the complexity of the
Feed rate (f) (mm./rev.) process.

Figure 2: Variation of production time versus feed
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