INTRODUCTION
Cutting conditions for a cutting tool have been the most critical variables in machining process. Cutting speed, feed rate and depth of cut were considered as three factors of input cutting parameters (Montgomery, 1976). Koren et al. (1991) have described several methods to be used under stepwise constant variation in feed, speed, or depth of cut, but none is practically applicable when two or more cutting conditions are changed. Hence, the method of controlling cutting conditions with fixed material removal rate has been introduced (Balazinski and Ennajimi, 1984; Davim and Antonio, 2001). For most studies with this viewpoint, the material removal rate is fixed because of the expensive observation of control. However, through the computerintegrated interface to program the machining feed rate on modern Computer Numerical Controlled (CNC) machines with fixed cutting speed and depth of cut, the material removal rate is capable of being dynamically controlled (Yeh and Lan, 2003).
In addition, Choudhury and Appa Rao (1999) described that tool life is the critical parameter of the cutting process. Novak and Wilklund (1996) proposed a suitable implementation to predict tool life and Lee et al. (1992) proposed a method of optimal control to ensure maximum tool life. As a matter of fact, maximal tool life will not guarantee the maximum profit of machining. Besides, the various tool checking periods for a tool change from different machine tool operators will decrease the productivity and increase the cost during the machining significantly. In order to well manage the consumption of tools, a fixed tool life is practically considered for the cutting process in this study.
Moreover, the cost to machine each part is a function of the machining time (Jung and Ahluwalia, 1995). From Kamien and Schwartz (1991), the marginal cost of production is a linear function of production rate (Lan et al., 2002, 2008). Therefore, the marginal cost of operation is also proposed to be a linear function of the material removal rate in this study. This explains that more machining rate causes more operational cost such as machine maintenance, loadingunloading and machine depreciation costs.
According to Galante et al. (1998), the dependency of a reliability
model on the cutting conditions is the aim to optimize the manufacturing system.
Although several time series modeling on the control of machining process are
mentioned for decisionmaking (Kim et al., 1996), none is capable to
achieve the maximum profit. They are all emphasizing on the maximum tool usage
or minimum tool cost. Practically, the profit and the productivity of a machining
process are the mostly concerned problems confronting the manufacturing industry.
Besides, the need of operating CNC machines efficiently to obtain the required
payback is even greater in the case of rough machining, since a greater amount
of material is removed thus increasing possible savings (Meng et al.,
2000). With the reasons above, there is an economic need to control the material
removal rate of rough machining operation for a cutting tool. Hence, the need
of controlling the material removal rate with fixed tool life to achieve the
maximal profit for a cutting tool is absolutely arising.
Nevertheless, the traditional machining model (fixed machining rate model) may surpass the DMC model under certain conditions. Thus, the demand to compare two models is also appeared. Therefore, the decision criteria for selecting the optimal machining model are then desired in this field.
ASSUMPTIONS AND NOTATIONS
Assumptions
• 
The cutting process is a continuous rough turning operation
with one type of tool. 
• 
Each tool performs the same fixed length of cutting time (tool life).

• 
The upper speed limit of material removal rate is generated from the maximal
machining conditions (speed, feed rate and depth of cut) suggested in the
machining handbook and the fixed tool life is obtained from the Taylor’
tool life equation (DeGarmo et al., 1997) with these selected conditions.
Thus, no tool will break before this fixed tool life even with the upper
speed limit. 
• 
There is no chattering or scrapping of parts occurs during the machining
process. 
• 
All chip from cutting and finished parts are held in the machine until
a tool change. 
• 
All parts are moved to other department and paid at a given price immediately
after machining operation at the tool change. 
Parameters and Notations
A 
: 
Fixed MRR of the traditional machining model 
a 
: 
Average volume of material removed per unit part machined. 
B 
: 
Upper speed limit of material removal rate. 
bM_{R}^{′}(t) 
: 
Marginal operation cost, including all labor and machine costs, at
the material removal rate M_{R}′(t); where b is a constant.

bM_{R}^{′} ^{2}(t) 
: 
Operational cost at time t. 
c 
: 
Overall holding cost per unit chip per unit time, where c = h_{1}
+ h_{2}/a. 
h_{1} 
: 
Chip holding cost per unit chip per unit time. 
h_{2} 
: 
Part holding cost per unit finished part per unit time. 
P 
: 
Contribution per unit part machined. 
[0,T] 
: 
Controlling time interval of tool life. 
Decision functions
M_{R}(t) 
: 
Cumulated volume of material removed during time interval
[0,t]. 
M_{R}′(t) 
: 
Material removal rate at time t. 
THEORETICAL BACKGROUND
In the previous research (Yeh and Lan, 2002), the cutting process is regarded as a continuous singletool turning operation without breakdown. That is, the machine is operated within the tool life and it will not break even with the highest machining rate from the handbook. The data for both the fixed tool life time and the upper MRR limit are obtained from the maximal machining conditions suggested in the machining handbook. In general, if the machine is operated within the tool life time, it will not break even with the highest machining rate.
In this study, means the contribution of one tool under machining operation
with a fixed tool life T. Whilst Kamien and Schwartz (1991) described the marginal
operation cost of production is a linear increasing function of production rate
and the operational cost is directly proportional to the square of the production
rate, the operational cost of a machine is also proposed in direct proportion
to the square of the material removal rate in this study. Thus, represents the
operational cost during time interval [o,T].
denote the chip holding cost and finished part holding cost during time interval [0, T], respectively. Therefore, the objective function for maximum profit of each tool is described as:
Set c = h_{1} + h_{2}/a be the overall holding cost per unit chip per unit time. Thus, the DMC model is developed as:
Set M_{R}′*(t) and M_{R}*(t) be the optimal solution
of DMC model and assume that time interval is
the maximal subinterval of [0,T] to satisfy Euler Equation (Kamien and Schwartz,
1991; Chiang, 1992).
There are two feasible situations to be discussed. They are proposed as follows.
Situation 1: M_{R}*′(t) will not touch B before T.
In this case, it is assumed that M_{R}′(t) will never reach the
upper speed limit B before tool life time T. From Euler Equation (Kamien and
Schwartz, 1991; Chiang, 1992), the transversality condition of salvage value
for free M_{R}(t) (Kamien and Schwartz, 1991; Chiang, 1992) and the
boundary conditions, one can obtain the optimal solution (Yeh and Lan, 2002)
for Situation 1 as:
Before finding the optimal solution for Situation 2, one Property is proposed by Yeh and Lan (2002) as follows:
Property: If the line y = M_{R}*′(t) touches the line
y = B, these two lines should overlap to be y = B from the touch point to
the end point T.
Proof: From Eq. 1, the optimal control of M_{R}*′(t) is
a strictly increasing linear function of t. And it holds for the subinterval
of [0,T] subject to 0 ≤ M_{R}*′(t) ≤B. Since the straight
line in the time interval cannot
exist because it contradicts Euler Equation (Kamien and Schwartz, 1991; Chiang,
1992) to be a decreasing linear function of t, the property is verified.
Situation 2: M_{R}*′(t) will touch B before T.
In this case, we assume that M_{R}′(t) will reach the upper limit
B at time .
From the transversality condition of salvage value for free end value M_{R}() (Kamien
and Schwartz, 1991; Chiang, 1992) and the Property, one will have
Therefore, the optimal solution (Yeh and Lan, 2002) for Situation 2 is shown
as follows:
COMPARISON OF TRADITIONAL AND DMC MODEL
After the dynamic solution of machining control is achieved, the comparison to the traditional model with fixed material removal rate A (fixed speed, feed rate and depth of cut) is then necessarily taken. Here, both models are assumed to receive positive profits. Thus, the two feasible cases are then discussed as follows.
Case 1: The optimal control of MRR does not touch the upper speed limit B.
When
the DMC model is the optimal solution for the control of material removal rate.
When
the traditional machining control model may be the optimal solution. For
the traditional model is optimal, where
The detailed process is developed in Appendix A.
Case 2: The optimal control of MRR touches the upper speed limit B.
When
the DMC model is the optimal solution for the control of material removal rate.
When
the traditional machining control model may be the optimal solution. For
the traditional model is optimal,
where
The detail is developed in Appendix B.
DECISION CRITERIA
With Eq. 3, there are two situations to be considered. They
are described as follows.
Situation 3: If , will not touch B before
From the discussion of comparing two machining models, the decision criteria to select the optimal machining model for Situation 3 is then proposed and described as follows.
When
the DMC, Dynamic Machining Control, Model is optimal.
When
there are three possible conditions.
For
the traditional machining control model is optimal.
For
the DMC model is optimal.
For
within [0,B], both the DMC and traditional models are optimal.
Situation 4: If , will touch B before T. .
From the discussion of comparing two machining models, the decision criteria to select the optimal machining model for Situation 4 is also proposed and described as follows.
When
the DMC model is optimal.
When
there are three possible conditions.
For
the traditional machining control model is optimal.
For
the DMC model is optimal.
For
within [0,B], both the DMC and traditional models are optimal.
CONCLUSIONS
The tool life, operational cost, holding costs, contribution per unit part
machined, average volume of material removed per unit part machined and upper
speed limit are considered simultaneously to dynamically optimize the control
of material removal rate. It is an extremely hardsolving and complicated issue.
However, with the DMC model, the problem becomes concrete and solvable.
In addition, four characteristics of the optimal solution for DMC model are
illustrated as follows: First, the optimal material removal rate M_{R}*′(t)
is a strictly increasing linear function of t before reaching the upper
limit. Second, by Property, if the optimal material removal rate M_{R}*′(t)
touches the upper speed limit B, it will stay to be B for the rest of the tool
life. Third, by Eq. 3, the overall holding cost per unit
chip must be smaller than the contribution per unit material removed. Otherwise,
the optimal material removal rate may reach zero. Fourth, with Eq.
4, two times of the marginal cost at upper speed limit must be larger than
the contribution per unit material removed. Otherwise, the optimal material
removal rate may reach the upper speed limit.
Moreover, the comparison of DMC model and traditional model to approach the
decision criteria is fully discussed. And the typical conditions for traditional
model to surpass DMC model are also presented. In Situation 3, when and
the traditional model is optimal. In Situation 4, if
and
the traditional model is optimal. Furthermore, with the comparison of two models, the decision criteria for selecting the optimal control of material removal rate are then suggested.
This study not only provides the idea of automatic control on material removal rate to machine tool manufacturer, but also leads the cutting process toward to reach maximum profit. Future researches on the modeling of dynamic optimization on multitool machining processes and the production project control with deadline constraint are surely encouraged. In sum, this study definitely generates an adaptable concept of machining control to the technology and contributes a better and practical tool for decisionmaking to this field.
Appendix A
Substituting Eq. 1 and 2 into the objective
of DMC model, it is derived that
In addition, the objective of the traditional machining control is described and rearranged as below.
With Eq. A1 subtracting Eq. A2, the
difference between the objectives of two models is then found as:
Let
Also, set the difference of two models as a function y(A). Hence, the function
is written as:
It is observable that X, Y >0. And, from Eq. (A1), it
is also known that Z>0
From Eq. (A3), we then have
By Eq. A5, it is noticed that y′′(A)>0.
This denotes that the curve function of A is concave.
The minimum value of curve function y(A) occurs when y′(A) = 0. From
Eq. A4, we have
Therefore, the minimal value of the curve function y(A) is shown as:
From Eq. A6 and A3, the criteria are
then derived.
Appendix B
Substituting Eq. 4 and 5 into the objective
of DMC model, it is then modified as:
With Eq. B1 subtracting Eq. A2, the
difference between the objectives of two models is then obtained as:
Let
Also, set the difference of two models as a function ȳ(A).
Hence, the function is written as:
It is observable that x̄ ȳ >0.
And, from Eq. B1, it is known that .
From Eq. B2, we then have
Again, by Eq. B4, we know that .
This denotes that the curve function is
also concave.
The minimum value of curve function occurs
when .
From Eq. B3, it is obtained that
Therefore, the minimal value of the curve function is
found as:
Substituting Eq. 5 into Eq. B5, it is
derived that
With Eq. B6 and B2, the criteria are
then obtained.