DOCUMENT DE TRAVAIL 2003-028 - Université Laval
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DOCUMENT DE TRAVAIL 2003-028
MINIMIZING THE EXPECTED PROCESSING TIME ON A FLEXIBLE MACHINE WITH RANDOM TOOL LIVES Bernard F. Lamond Manbir S. Sodhi
Version originale : Original manuscript : Version original :
ISBN – 2-89524-178-3
Série électronique mise à jour : On-line publication updated : Seria electrónica, puesta al dia
08-2003
Minimizing the Expected Processing Time on a
Flexible Machine With Random Tool Lives
Bernard F. Lamond1 Manbir S. Sodhi2
1 Faculté des Sciences de l’Administration,Université Laval, Québec, Canada G1K 7P4.
Email: [email protected] Industrial and Manufacturing Engineering,
University of Rhode Island, Kingston, RI, USA 02881.Email: [email protected]
29 July 2003
Abstract
We present a stochastic version of economic tool life models for machines with finite
capacity tool magazines and variable processing speed capability, where the tool life
is a random variable. Using renewal theory to express the expected number of tool
setups as a function of cutting speed and magazine capacity, we extend the (determin-
istic) mathematical programming models of Lamond and Sodhi (1997) to the case of
minimizing the expected total processing time. A numerical illustration with typical
cutting tool data shows the deterministic model underestimates the optimal expected
processing time by more than 8% when the coefficient of variation equals 0.3 (typi-
cal for carbide tools), and the difference exceeds 15% for single injury tools having an
exponentially distributed economic life (worst case).
1
1 Introduction
Modern manufacturing facilities now routinely utilize computer controlled machines with
integrated tool magazines and programmable speed capabilities. These capabilities enable
parts to be processed with low setup times, fewer machine changes, and under pre-selected
optimal processing conditions. Such machines, which we refer to as flexible machines, are
a basic component of flexible manufacturing systems (FMSs) [15], and are essential for
automated/lights-out operation. Because of the relatively high cost of these machines, the
tool loading policy can have a significant impact on the overall operational efficiency (OOE)
of these systems.
Methods to minimize the processing time of a batch of parts on a single flexible machine,
using tool life models to account for the tool wear process, were presented in [7]. Under the
same assumptions, [14] examined heuristics for allocating parts to several identical machine
tools. Two mathematical programming models were considered in [7], where the tool wear
process is represented by Taylor’s empirical formula [18] relating tool life and cutting speed
under normal operating conditions. The first one is a nonlinear programming (NLP) model
in which both the cutting speed and the number of tool setups, for all part types, are
continuous variables. The second one is a nonlinear integer programming (NLIP) model in
which the speed is a continuous variable but the number of tool setups is discrete. Both
models minimize the total processing time, which is the sum of the time needed to cut the
parts plus the time required to setup the extra tools in excess of the tool magazine capacity.
The first model being a continuous relaxation of the second one, its optimal solution gives a
lower bound on the optimal processing time of the discrete model.
The life of a cutting tool is an important component in the economics of machining
operations. Under normal cutting conditions, flank wear is the primary mechanism of tool
wear. The rate of flank wear increases with time, and can ultimately lead to rapid breakdown
of the cutting tool [2]. The tool should therefore be changed before it enters the phase
where rapid breakdown is possible. The decision to change a tool is thus defined by some
tool wear criterion that is a function of flank tool wear, such as criteria recommended by
2
the International Standards Organization [5]. Tool life is then defined as the cutting time
required to reach a tool life criterion.
In conventional systems, the machine operator is responsible for tool changes. However,
it is now possible to integrate tool monitoring capabilities with the tool changing devices.
Other than off-line sensing, there are several methods for tool wear monitoring: direct sensing
methods using mechanical and physical sensors such as vision sensors, radioactive sensors
and proximity sensors; indirect methods that monitor cutting force, vibration and acoustic
emission monitoring etc. [6]. The incorporation of such sensors allows a greater degree of
process control and reaction to changes in parameters governing the cutting process. Related
advances in condition monitoring methods have improved the resolution and signal to noise
ratio of the data collected, enhancing the gain potential.
In a deterministic environment, it is assumed that the tool life of each tool is known
exactly and can be determined by some function of the cutting parameters (such as cutting
material, depth of cut and cutting speed). In [7], for instance, the cutting tools are assumed
to behave according to Taylor’s empirical relation so that the tool life is given as a function
of the cutting speed. Under the assumption of determinism, the scheduling becomes strictly
a problem of selecting the cutting speed and of assigning tools to the magazine, because
the instants when tool changes will occur can be programmed in advance and hence, tool
wear detection capability is not required (at least in theory). This assumption was used, for
instance, in [7, 14, 16, 17, 19].
In practice, however, it has been observed that identical tools used under similar operating
conditions usually have unequal tool lives. In fact, tool life experiments used to estimate the
parameters for tool life have provided ample experimental evidence that the tool life itself is a
random variable whose distribution is related to the cutting conditions. For example, if tool
failure is assumed to be the result of a single injury, the use of an exponential distribution
for the tool life was suggested in [8]. The effect of multiple injuries, or shocks, on tool
failures was also studied in [9], with the conclusion that the gamma, normal or lognormal
distributions best represent the life of a tool under these conditions. Based on extensive
cutting experiments, [20] computed a coefficient of variation of 0.3 for tool life when turning
3
steel. Given that the variance of tool lives is significant, an immediate question of interest is
the practical utility of solutions obtained under deterministic assumptions when operating
in stochastic environments.
The purpose of this paper is to investigate the influence of stochastic tool life in the
scheduling and control of a FMS. Assuming the lives of successive cutting tools are inde-
pendent and identically distributed random variables, we propose a model based on renewal
theory for evaluating the expected processing time of a batch of parts on a single machine.
Our approach further assumes the flexible machine has perfect tool wear detection capabil-
ity. A numerical illustration with typical cutting tool data shows the deterministic model
underestimates the optimal expected processing time by more than 8% when the coefficient
of variation equals 0.3 (typical for carbide tools), and the difference exceeds 15% for single
injury tools having an exponentially distributed economic life (worst case).
The models presented in this paper do not allow for the error in tool condition monitoring.
Nonetheless, these models are useful in at least two ways: first, although the situation
is idealized, our model provides valuable insights into the effect of tool life randomness;
second, our results show that deterministic analysis significantly underestimates the expected
processing time of a flexible machine with perfect tool wear detection, therefore we argue
that the expected processing time will be even longer on a flexible machine with less than
perfect tool wear detection. Our model thus provides a lower bound on a realistic estimate
of the expected processing time, and this lower bound is better than the one previously
obtained using deterministic analysis. This analysis is necessary if flexible machines are to
be operated in a lights-out/unmanned mode to plan minimum run lengths during periods
when the machine is set up to operate autonomously. Even though this form of operation has
long been a quest of manufacturing systems designers, it is only recently that manufacturers
have begun to realize the economic advantage of unmanned operation [1], and these models
add to a foundation of exploratory work in lights-out operation.
The paper is organized as follows. The deterministic tool life model is briefly reviewed
in §2. Then a stochastic tool life model is presented in §3 with the corresponding renewal
process in §4 for the expected number of tool setups. Next, an extension to a machine tool
4
with a tool magazine is discussed in §5. The minimization problem is then discussed in §6
and a numerical example is solved in §7. Some detailed algebraic derivations and numerical
results are given in the appendix.
2 Deterministic analysis
The nominal tool life t` can be expressed as a function of the cutting speed v, using Taylor’s
empirical formula [18]:v
vr
=(trt`
)n
⇒ t` = tr
(v
vr
)−1/n
, (1)
where tr is the nominal tool life at some reference speed vr and n is a given constant. Under
the deterministic tool life assumption, the tool life t` is known with certainty. Following [7],
if S is the setup time for manually changing a tool, then the processing time of a part type
on a machine-tool without a tool magazine is given by
TP (v) = τ(v) + dθ(v)eS, (2)
where
τ(v) = K1/v (3)
is the cutting time,
θ(v) =τ(v)
t`= K2v
α (4)
is the quantity of tool used for cutting the part type, and we define the constants
α = (1− n)/n, K1 = vrτ(vr) and K2 = τ(vr)/(vαr tr).
In this context, selecting the best cutting speed is a discrete minimization problem because
at the optimal speed v∗, only an integer number θ∗ = θ(v∗) of tools is used (see [7, §4]).
Taylor’s relation is usually considered valid for cutting speeds within an admissible range
v` ≤ v ≤ vu, but for simplicity, we will omit this constraint throughout the paper. This is
equivalent to taking v` = 0 and vu = ∞.
5
Table 1: Inputs for numerical example
System parameter Symbol ValueSetup time (sec) S 300Reference speed (m/s) vr 1Cutting time at vr (sec) τ(vr) 900Taylor’s exponent n 0.25Tool life at vr (sec) tr 160
Previous studies, such as [2], neglected the discrete nature of tool setups and minimized
a continuous relaxation
tp(v) = τ(v) + θ(v)S, (5)
giving a lower bound on the optimal integer solution of eq. (2). Figure 1 shows the processing
time plotted against the cutting speed, as given by equations (2) and (5), for the numerical
example described in Table 1. These data were used in [7] and are representative of task
processing times found in the industry.
3 Stochastic tool life
Experimental evidence reported in [20] indicates that the cutting time would be more ad-
equately modeled as a random variable whose expected value is given by Taylor’s formula,
with a coefficient of variation of approximately 0.3 or more, and a normal or lognormal
distribution. Unlike the mean tool life, the coefficient of variation appears to be relatively
insensitive to the cutting speed. With such variance, the deterministic model of [7] is, at
best, a very coarse approximation of reality. The approach we propose here extends the
model of [7] by taking a sequence of random variables to represent the lives of successive
tools and by replacing dθ(v)e in eq. (2) by the expected number of tools used.
Other authors have also studied tool life uncertainty. [8] examined the case where a tool
failure is the result of a single injury and then relate to an exponential distribution. [9]
discussed the case of multiple injury tool life which can be treated as a gamma, normal or
lognormal distribution. In [10], a stochastic model for multi-face cutting tools was repre-
6
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.91800
1900
2000
2100
2200
2300
2400
2500
Cutting velocity
Pro
cess
ing
time
__ deterministic tool life
−− continuous relaxation
.... exponential distribution
Figure 1: Processing time versus cutting speed
sented with the use of a normal or lognormal distribution when thermal or mechanical tool
fatigue occurs. An extension was proposed in [21] when tools are used to cut different parts,
based on conditional probabilities. Recently, [4] discussed a stochastic tool life model with
regard to tool inventory management.
In the present paper, we suppose the lives of the successive cutting tools are i.i.d. (in-
dependent and identically distributed) random variables Xi, i = 1, 2, . . ., with a given dis-
tribution. Let N(v) be the discrete random variable giving the number of tools required to
finish a part type at constant cutting speed v. We present a simple model to evaluate the
expected value N(v) = E[N(v)] of the number of tools used to process a single part type.
Decisions based on expected values are often justified in common manufacturing environ-
ments, because of the repetitive aspect of processing successive tasks. Equation (2) can then
be replaced by
TP (v) = τ(v) +N(v)S. (6)
Unlike equation (2), here we do not need to round up to the nearest integer since we are
7
calculating the expected value of a discrete, integer valued random variable. The expected
value itself does not have to be integer.
Let µ = E[Xi] be the mean tool life and σ2 = Var[Xi] = E[(Xi − µ)2] its variance.
We suppose the mean tool life is related to the cutting speed by Taylor’s formula, i.e.,
µ = t` = tr(v/vr)−1/n. Moreover, we suppose the coefficient of variation ξ = σ/µ is given
and is independent of the cutting speed v. This information is sufficient for determining
a tool life distribution with two parameters. For example, when Xi ∼ Gamma(r, λ), then
µ = r/λ and σ2 = r/λ2, so we obtain the two parameters:
r = 1/ξ2 and λ =r(v/vr)
1/n
tr. (7)
For the gamma distribution, there are three special cases of particular interest:
• Deterministic. Xi = t`, ξ = 0, r →∞.
• Exponential. Xi ∼ Exp(λ), ξ = 1, r = 1.
• Carbide tool. From Wager and Barash [20], ξ ≈ 0.3, so r ≈ 11 typically.
The exponential distribution represents tools with a highly variable lifetime, for example
when failure results from single injury. The expected processing time is plotted in Figure 1.
It is consistently larger than for a deterministic tool life, however the difference is always
bounded above by one setup time S. This is shown in the next section, where a method for
computing N(v) is also given.
It is well known (see, e.g., [11, 12, 13] that the exponential and gamma (with r ≥ 1)
distributions have an increasing failure rate (IFR). Also, any distribution such that ln(1 −
F (x)) is concave in x > 0 is IFR. On the other hand, a distribution such that ln(1− F (x))
is convex has a decreasing failure rate (DFR). The above discussion about the tool wear
process (and the discussion in [7]) would suggest that IFR distributions should be used to
model the tool life. Another reason for not using DFR distributions is that our convention
of a fixed coefficient of variation ξ may not be appropriate in this case.
8
In fact, as pointed out in [11], a common class of DFR distributions consists of mixtures of
exponential distributions. For example, one could imagine a certain tool has a probability p
of being of type 1, with Taylor parameters n1 = 0.18 and tr1 = 162.5, and a probability
1− p of being of type 2, with n2 = 0.28 and tr2 = 168.2. (In both cases, assume a reference
speed vr = 1.) This could be the case if the tool has been taken at random from one of two
toolboxes. For a tool of type i, i = 1, 2, Taylor’s relation will assign the parameter values
λi =1
tri
(v
vr
)1/ni
to the corresponding exponential distributions. Then each exponential distribution has a
coefficient of variation equal to 1, independent of the cutting speed, but the coefficient of
variation of the resulting hyperexponential distribution varies significantly with the cutting
speed, in contradiction with our convention and experimental evidence. For example, with
p = 0.7, we get ξ = 0.7609 at speed v = 0.7 and ξ = 0.6550 at speed v = 1.
4 Renewal process
We now find an expression for N(v). First, we denote τ = τ(v) as the cutting time used
to process a part. Let M(τ) be the random variable giving the number of tools needed for
processing during a certain time τ (so that the part is completed), i.e.,
M(τ)−1∑j=1
Xj < τ ≤M(τ)∑j=1
Xj.
Let M(τ) = E[M(τ)]. By construction N(v) = M(τ) so that N(v) = M(τ). Now {M(t)−
1, t ≥ 0} is a renewal process whose interarrival times are the random variables Xi, i =
1, 2, . . .. Then M(t) − 1 represents the corresponding renewal function. It is thus possible
to obtain this function from renewal theory, for example using Laplace transforms or a
numerical method. From the renewal function, we get N(v) = M(K1/v) by setting t = τ(v)
and using eq. (3).
There are two special cases for which the renewal function is trivial and thus M(t) is
easily obtained.
9
• Deterministic tool life: then Xj = t` with certainty and is given by equation (1).
So we get M(t) = 1 + bt/t`c. With t = τ(v), we derive N(v) = 1 + bK2vαc. Then eq.
(6) reduces to eq. (2), except at cutting speeds for which an integral number of tools
are used. (By convention, the renewal function counts all tool changes up to time t
inclusively, while in [7] the last tool setup was not counted if that tool was not used.)
• Exponential tool life: then Xj ∼ Exp(λ), which gives
M(t) = 1 + λt, for t ≥ 0. (8)
Replacing λ and t, we get
N(v) = 1 +K2vα. (9)
We remark that (6, 9) give exactly one extra setup time S when compared to (4, 5). This
explains the observation, in Figure 1, that when the tool life has an exponential distribution,
the expected processing time exceeds the lower bound of the continuous relaxation by exactly
one setup time.
Now let F (t) be the cumulative distribution function of the tool life, i.e., F (t) = P (Xi ≤
t). Let also the random variable Sk be the time at which the life of the kth tool ends, i.e.,
Sk = X1 + . . . + Xk, for k = 1, 2, . . .. By convention, let S0 = 0. For k = 0, 1, 2, . . ., define
the cumulative distribution functions Fk(t) = P (Sk ≤ t). Then F0(t) ≡ 1, F1(t) = F (t) and
Fk(t) is the k-fold convolution of F (t) with itself. Then from [11, Prop. 3.2.1], we have
M(t) =∞∑
k=0
Fk(t). (10)
If the tool life follows a gamma distribution with shape parameter r and scale parameter λ,
then Sk also has a gamma distribution but with shape parameter kr and scale parameter λ.
Then the functions Fk(t) can be computed using the incomplete gamma function, which is
available in standard scientific subroutine libraries. In this case, M(t) can be computed by
truncating the infinite sum in eq. (10). Similarly, if the tool life follows a discrete distribution
with a finite number of mass points, then the functions Fk(t) can be obtained using a standard
subroutine for discrete convolutions.
10
Another expression for M(t) can be obtained using Laplace transform analysis. We now
suppose the tool life Xi has an arbitrary continuous distribution with a Laplace transform.
Lemma 1 Let f(x) be the density function of Xi, and Lf (s) its Laplace transform. Then,
the Laplace transform of M(t) is given by
LM(s) =1/s
1− Lf (s). (11)
Proof. Let M̂(t) = M(t)− 1 be the renewal function of Xi, i = 1, 2, . . ., and m̂(t) = M̂ ′(t)
its renewal density. Then, their Laplace transforms are given by:
Lm̂(s) =Lf (s)
1− Lf (s)
LM̂(s) =Lf (s)/s
1− Lf (s).
Knowing that M(t) = M̂(t) + 1, ∀t ≥ 0, we obtain LM(s) = LM̂(s) + 1/s and the result
follows. 2
In some cases, it is possible to invert equation (11) using an expansion in partial frac-
tions. We now obtain the function M(t) in closed form when the tool life follows an Erlang
distribution (the gamma distribution with integer shape parameter r is called an Erlang
distribution). Recall the Laplace transform of an Erlang density is
Lf (s) =
(λ
s+ λ
)r
. (12)
The proofs of the following lemma and theorem are given in the appendix.
Lemma 2 Suppose Xi ∼ Erlang(r, λ), and let φk = ei2πk/r (the r-th unit root) and γk =
2(1− cos(2πk/r)) for k = 1, . . . , r − 1. Then
LM(s) =r + 1
2rs+
λ
rs2+
r−1∑k=1
(1− φk)/(rγk)
s+ (1− φk)λ. (13)
11
Theorem 3 Suppose Xi ∼ Erlang(r, λ) and let Ck = cos(2πk/r) and Sk = sin(2πk/r).
Then
M(t) =r + 1
2r+λt
r+
[1− (r mod 2)]
2re−2λt
+1
r
b(r−1)/2c∑k=1
e−(1−Ck)λt
{cos(Skλt) +
Sk sin(Skλt)
1− Ck
}. (14)
The function N(v) is obtained from (14) by taking r = 1/ξ2 (rounded to the nearest
integer) and substituting t = K1/v and λ = r(v/vr)1/n/tr as in (7). We can see that if r = 1
(exponential case) then equation (14) becomes M(t) = 1 + λt, as in (8). When r → ∞,
numerical computations show that TP (v) tends to the deterministic case. For example,
compare figures 1 and 2 (r = 10000).
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.91800
1900
2000
2100
2200
2300
2400
2500
Cutting velocity
Pro
cess
ing
time
__ r=10000
−. r=50
.... r=11
−− r=6
−. r=1
Figure 2: Processing time for gamma distribution
Figure 2 shows the processing time plotted against the cutting velocity when the tool life
follows an Erlang distribution. For all values of the shape parameter r, the processing time
is bounded above by the exponential (r = 1) and below by the deterministic, continuous
relaxation (see Figure 1). For small values of r (say, r ≤ 9), the coefficient of variation is
12
large (ξ ≥ 1/3), and the curves are in the upper half of the graph, with almost no oscillations.
On the other hand, for larger r there is less variability in the tool life and oscillations appear,
of greatest amplitude in the deterministic case, when r →∞.
5 Machine with tool magazine
In many manufacturing environments, machines are often equipped with a tool magazine.
This permits having a number of different tools immediately available for processing. We
suppose the setup times are negligible for tools that are preloaded into the magazine. More-
over, extra tools are manually inserted one by one in the machine when needed, and thus
each incurs a setup time S. Let TC be the number of tool slots in the magazine. The special
case TC = 0 corresponds to a machine with no tool magazine, as in the previous sections,
and for which all tools require a setup time.
Now let NTC(v) be the random variable of the number of tool setups needed to complete a
part at cutting speed v. With τ = τ(v) = K1/v the cutting time, we have NTC(v) = MTC(τ)
where the random variable MTC(τ) gives the number of tool setups that occur during a time
interval of duration τ . We now argue that {MTC(t), t ≥ 0} is a delayed renewal process. Let
Sk be the epoch of the kth tool loading. Then
S1 =TC∑j=1
Xj
is the instant when the last preloaded tool must be replaced. For k = 2, 3, . . ., we have
Sk = S1 +k−1∑j=1
XTC+j.
The interarrival times between successive tool loadings are therefore the i.i.d. random vari-
ables XTC+1, XTC+2, . . .. The time S1 of the first tool loading has a different distribution,
however, and the renewal process is thus delayed because of the tool magazine. From [11,
eq. (3.5.1)], the renewal function MTC(t) = E[MTC(t)] is given by
MTC(t) =∞∑
k=TC
Fk(t) = M(t)−TC−1∑k=0
Fk(t). (15)
Particular expressions are straightforward for three special cases.
13
• Deterministic tool life. As before, Xj = t` with certainty. So we get
MTC(t) = 1 +
⌊(t
t`− TC
)+⌋.
• Exponential tool life. Then Xj ∼ Exp(λ) and S1 ∼ Erlang(TC, λ). Let G0(t) ≡ 1
for t ≥ 0 and, for j = 1, 2, . . ., let Gj(t) be the cumulative distribution function of a
Erlang(j, λ). Then
MTC(t) = 1 + λt−TC−1∑k=0
Gk(t).
• Erlang tool life. If Xi ∼ Erlang(r, λ) then
MTC(t) = M(t)−TC−1∑`=0
Gr`(t), (16)
with M(t) given by eq. (14).
0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
Cutting velocity
Too
l set
ups
__ exponential distribution
.... gamma distribution (r=50)
−− deterministic tool life
Figure 3: Tool setups versus cutting speed
The expected number of tool setups NTC(v) is obtained by substituting the correct values
for r, λ and t in (14, 16). Figure 3 shows N(v) and NTC(v) plotted against the cutting
14
speed, for a tool magazine of capacity TC = 2, compared to TC = 0. For the exponential
distribution, the expected number of tool setups when a tool magazine is present is near zero
at small speed but it remains positive. On the other hand, for tool life distributions with
small variance, the expected number of tool setups is much nearer zero in a significant range
of the cutting speed.
6 Minimizing the expected processing time
We now look at the problem of finding the cutting speed giving the minimum expected
processing time. For a flexible machine without a tool magazine, we simply minimize TP (v)
given by eq. (6). For a flexible machine with a tool magazine of capacity TC, we minimize
TP (v, TC) = τ(v) +NTC(v). (17)
Clearly, the special case with TC = 0 corresponds to a machine without a tool magazine
since TP (v, 0) = TP (v).
From this point on, we find it convenient to make a change of variable. Rather than
using the cutting speed v, we will use θ = θ(v), the corresponding quantity of tool used for
cutting the part type at speed v under Taylor’s deterministic tool life model, as given by eq.
(4). The cutting speed can easily be recovered from
v = (θ/K2)1/α.
Redefining the cutting time as ϕ(θ) = τ(v)/S, where S is the tool setup time, we get
ϕ(θ) = K3θ−1/α, (18)
with K3 = τ(vr)θ1/αr and θr = (τ(vr)/tr)
1/α is the amount of tool used at Taylor’s reference
cutting speed. We remark that the function ϕ(θ) is decreasing and convex provided α > 0.
This is true whenever the Taylor exponent n satisfies 0 < n < 1. Hence our analysis
is applicable to a wider range than was claimed in [7, Table 2], where only the interval
0 < n < 1/2 was considered.
15
We further remark that by setting t = τ(v) as in §4, then eq. (4) gives θ = t/t`. This
means θ expresses time in units of the mean tool life. Hence in the renewal process {M(t)−
1, t ≥ 0}, we can replace t by θ provided we choose the scale parameter of the interarrival
time distribution so that the expected tool life E[Xi] = 1. For the exponential distribution,
we would then have λ = 1, while for an Erlang distribution with shape parameter r, we
would have λ = r.
Under this convention, we then define the function
ωk(θ) = Nk(v) = Mk(θ) (19)
giving the expected number of tool setups under the cutting speed v, for a flexible machine
with a tool magazine of capacity TC = k tools. By construction, ωk(θ) gives the expected
number of renewals in excess of k−1 at time θ of a renewal process with unit mean interarrival
time. It increases with θ and it decreases with k. Moreover, it is easily shown that
ωk+1(θ)− ωk(θ) ≥ ωk(θ)− ωk−1(θ)
for any fixed value of θ, and for general tool life distribution. Hence the expected number
of tool setups is a discretely convex function of the magazine capacity. On the other hand,
for fixed k, the function ωk(θ) is not convex in θ, in general. There is one notable exception,
however.
Lemma 4 The function ωk(θ) is convex in θ for any given k when the tool life follows an
exponential distribution.
Proof.
ω′′k(θ) =∞∑
`=k
G′′` (θ) =
∞∑`=k
[g`−1(θ)− g`(θ)] = gk−1(θ) ≥ 0,
where g`(θ) = θ`−1/(`− 1)! is the `th Erlang density. 2
Nonetheless, even under exponential tool life, the function ωk(θ) is not jointly convex in k
and θ. For example, it is easily verified that
ω2(2) = 1.1353 > 1.1245 = 0.5[ω1(1) + ω3(3)].
16
Finally, for k = 0, 1, 2, . . ., we define the functions
ψk(θ) = ϕ(θ) + ωk(θ). (20)
Then the problem of minimizing eq. (20) is equivalent to minimizing eq. (17) with TC = k.
In particular, the optimal solutions θ∗k and v∗k satisty eq. (4) and the optimal values satisfy
TP (v∗k, k) = Sψk(θ∗k). Now let ψ∗k = ψk(θ
∗k), for k = 0, 1, 2, . . ..
Lemma 5 For general tool life distribution, the optimal solutions θ∗k are increasing and the
optimal values are decreasing: for k = 0, 1, 2, . . ., θ∗k+1 ≥ θ∗k and ψ∗k+1 ≤ ψ∗k.
Proof. This follows trivially from the fact that ψk(θ) = Fk(θ) + ψk+1(θ), where Fk(θ), the
probability distribution function of Sk = X1 + . . .+Xk, is increasing in θ. 2
The above argument also shows that ψ∗1 − ψ∗0 = −1 and θ∗0 = θ∗1. Also, in the special case of
the exponential distribution, the first-order optimality conditions give
θ∗0 = θ∗1 = ϕ−1(1) =(nK3
1− n
)1−n
.
Moreover, our numerical experience with the exponential and Erlang distributions of the
tool life suggest that these sequences have much stronger monotonicity properties, in fact.
Conjecture 6 Both the optimal solutions and the optimal values have increasing incre-
ments, with the increments of the optimal solutions being bounded above by 1: for k = 1, 2, . . .,
0 ≤ θ∗k − θ∗k−1 ≤ θ∗k+1 − θ∗k ≤ 1
−1 ≤ ψ∗k − ψ∗k−1 ≤ ψ∗k+1 − ψ∗k ≤ 0.
These monotonicity properties, when present, have two important consequences for com-
putations. One important consequence is that the discrete convexity of the optimal values ψ∗kallows the use of the greedy algorithm of [3] for allocating magazine capacity among multiple
part types as in [7]. Another consequence is to restrict the range for searching the optimal
solution for k+1 when θ∗k is given. Lower and upper bounds on the expected number of tool
17
setups can also be used to restrict the range of search for optimal solutions. For this, we
recall that a nonnegative random variable X is said to be new better than used in expectation
(NBUE) if
E[X − a | X > a] ≤ E[X] for all a ≥ 0.
It is well known that IFR distributions are also NBUE [11, 12, 13].
Lemma 7 For k = 0, 1, 2, . . ., and θ ≥ 0:
(θ − k)+ ≤ ωk(θ) ≤ 1 + θ −k−1∑`=0
G`(θ). (21)
where the lower bound is valid for general tool life distributions and the upper bound is valid
whenever the tool life distribution is NBUE.
Proof. The result is well known in the special case k = 0 (see, e.g., [11, 12, 13]): for a renewal
process with mean interarrival time 1/λ, the renewal function m(θ) is always bounded below
by λθ − 1, and if the distribution of the interarrival times is NBUE, it is bounded above by
λθ. (Here, by convention, we have λ = 1.) When k > 0, the lower bound follows from the
rightmost part of eq. (15) because M(θ) ≥ θ and F`(θ) ≤ 1. The upper bound, on the other
hand, follows from the middle part of eq. (15) by applying [11, Lemma 8.6.5]:
Mk(θ) =∞∑
`=k
F`(θ) ≤∞∑
`=k
G`(θ).
The result follows from the fact that
∞∑`=0
G`(θ) = 1 + θ.
2
Now suppose the tool life follows a NBUE distribution F (θ) and we want to find the
optimal solution θ∗k minimizing ψk(θ). Then Lemma 7 implies that
ϕ(θ) + (θ − k)+ ≤ ψk(θ) ≤ ψ̂k(θ),
where ψ̂k(θ) is the expected processing time when the tool life follows an exponential distri-
bution. But then both the lower and upper bounds are convex functions of θ. Now suppose
18
ψ̂∗k is the minimal value of ψ̂k(θ), then θk ≤ θ∗k ≤ θk where θk and θk are the two roots of the
equation
ϕ(θ) + (θ − k)+ − ψ̂∗k = 0.
7 Numerical example
We now illustrate our stochastic tool life model with a numerical example, taken from [7], in
which the magazine capacity of TC = 15 tools has to be allocated optimally between four
different part types, and whose tool characteristics are given in Table 2. For a given part
type, and for TC = 0, 1, 2, . . ., we define the function
h(TC) = minv≥0
TP (v, TC)
where TP (v, TC) is the expected processing time given by eq. (17). Then h(TC) = Sψ∗TC .
Now as in [7], the problem of allocating the magazine capacity among the different part
types can be formulated as follows:
Min∑
i hi(Ti)s.t.
∑i Ti ≤ TC
Ti ≥ 0, integer
Table 2: Inputs for numerical example with four part types
Tool Part typescharacteristics 1 2 3 4S (seconds) 115 300 200 100vr (m/s) 1 1 1 1τ(vr) (seconds) 500 900 400 550n 0.25 0.18 0.28 0.38vrt
nr 3.2 2.5 4.2 5.0
Under the assumption of deterministic tool lives related to cutting speed by Taylor’s
equation, it was shown in [7] that the functions hi(Ti) have nondecreasing marginals, i.e.,
∆hi(Ti + 1) ≥ ∆hi(Ti), with ∆hi(Ti) = hi(Ti + 1) − h(Ti). The magazine capacity, in
this case, can then be allocated using the greedy method of [3], which increments Ti by
19
one whenever ∆hi(Ti) is the smallest marginal among all tool parts. Our computational
experience with stochastic tool life under exponential and Erlang distributions has been that
the same property seems to hold as well.
To investigate the effect of random tool life, we have computed the optimal tool allocation
for three different cases.
1. Worst case: exponential distribution (Erlang with r = 1) which has a coefficient of
variation ξ = 1.
2. Carbide tool: Erlang distribution with r = 11, with ξ ≈ 0.3.
3. Low variance: Erlang distribution with r = 100, with ξ = 0.1.
The detailed allocation results are given in the appendix (tables 4–6) and in [7, Table 8].
These results are summarized in Table 3 where
TP (v∗, TC) = expected processing time at deterministic optimal speed
TP ∗(TC) = optimal expected processing time,
and the numbers in parentheses give the increase in processing time relative to the deter-
ministic optimal value.
Table 3: Effect of random tool lives on optimal processing time
TC = 0 TC = 15Coef. var. TP (v∗) TP ∗ TP (v∗, TC) TP ∗(TC)
0 4500.1 4500.1 2632.1 2632.10.10 4865.9 4683.2 2994.9 2745.3
(8.1%) (4.1%) (13.8%) (4.3%)0.30 4888.1 4865.4 3042.6 2890.7
(8.6%) (8.1%) (15.6%) (9.8%)1 5215.1 5212.6 3559.0 3311.5
(15.9%) (15.8%) (35.2%) (25.8%)
These results show that the deterministic optimal processing times significantly under-
estimate the expected processing times by up to 16% for the exponential distribution (the
20
worst case) in the absence of a tool magazine, and up to 35% with a tool magazine of capac-
ity TC = 15. For a carbide tool, the underestimation is of nearly 9% and 16%, respectively.
Interestingly, the underestimation is only slightly smaller in the low variance case, with 8%
and 14%, respectively. These results thus make a strong case in favor of using a stochastic
model rather than a deterministic model for estimating the processing time when the tool
life is random and the tool setup times are not negligible. Further, Table 3 also shows that
the expected processing times can be reduced significantly by finding the optimal speed with
a stochastic model rather than using the deterministic optimal speed.
8 Conclusion
Using existing empirical relations of Taylor [18] and Wager and Barash [20] to express the
random tool life of a metal cutting tool as a function of cutting speed, we have developed
a model based on renewal theory to obtain the expected processing time of a metal cutting
task on a flexible machine tool. This model assumes that the tool life is a function of
cutting speed, and takes the cutting time as well as the tool setup time into account. A
correction term for machines equipped with a tool magazine has also been developed. We
presented a computational procedure valid when the tool life distribution is exponential
or gamma. Our method is illustrated with a numerical example showing the significant
impact of random tool life on the expected processing time. The model developed above
can be used to evaluate the consequences of machine-tool magazine configuration when
planning for lights-out/unmanned operations. The model supports the current focus in
manufacturing environments on single piece flow, rapid manufacturing response to demand
changes and waste reduction, increasing the competitiveness of manufacturers seeking to
exploit the opportunities that advances in tool-condition monitoring and on-line control now
offer.
21
9 Acknowledgements
This research was supported in part by the National Science and Engineering Research
Council of Canada, under Grant 0105560, and the Fonds pour la Formation de Chercheurs
et l’Aide à la Recherche du Québec, under grant 1570. The authors are also thankful to
the National Science Foundation for supporting this research through grant DMII: 9813177.
Martin Noël’s help with some of the computations is also acknowledged.
Appendix
Proof of Lemma 2
Substituting (12) in (11), we get
LM(s) =(s+ λ)r/s
(s+ λ)r − λr=
1
s+
λr
p(s), (22)
where
p(s) = s[(s+ λ)r − λr] = s2r−1∏k=1
(s+ (1− φk)λ).
Indeed, (φk − 1)λ is a root of p(s) because
[(φk − 1)λ+ λ]r − λr = φrkλ
r − λr = 0.
Writing LM(s) in partial fractions, we get
LM(s) =1
s+A0
s+B
s2+
r−1∑k=1
Ak
s+ (1− φk)λ. (23)
Equating (22) and (23), we get
λr = A0p(s)
s+B
p(s)
s2+
r−1∑k=1
Akp(s)
s+ (1− φk)λ. (24)
To find B, evaluate (24) at s = 0, applying L’Hospital’s rule twice:
λr = B lims→0
p(s)
s2= B
p′′(0)
2.
22
This gives
B =λ
r. (25)
Next, for ` = 1, . . . , r− 1, find A` by evaluating (24) at s = (φ` − 1)λ, applying L’Hospital’s
rule once:
λr = A` lims→(φ`−1)λ
{p(s)
s+ (1− φ`)λ
}= A` p
′((φ` − 1)λ).
Straightforward algebra then gives
A` =1− φ`
rγ`
. (26)
Finally, identify all terms in sr, giving
A0 = −r−1∑k=1
Ak.
If φk and φk′ are complex conjugate, then Ak+Ak′ = 1/r. Otherwise φk = −1 and Ak = 1/2r.
Hence
A0 = −r − 1
2r. (27)
Substitution of (25–27) in (23) yields (13). 2
Proof of Theorem 3
Direct inversion of Laplace transforms gives the first two terms of (14) from the first two
terms of (13). Direct inversion of the third term gives
r−1∑k=1
(1− φk
rγk
)e−(1−φk)λt.
The third term of (14) vanishes except for even r, in which case it corresponds to k = r/2,
so that φk = −1 and γk = 4.
The other terms can be grouped in complex conjugate pairs. Suppose φk and φk′ are a
conjugate pair, then (1− φk
rγk
)e−(1−φk)λt +
(1− φk′
rγk′
)e−(1−φk′ )λt
23
= 2<{(
1− φk
rγk
)e−(1−φk)λt
}
=2e−(1−Ck)λt
rγk
{[(1− Ck)− iSk]
× [cos(Skλt) + i sin(Skλt)]}
=2e−(1−Ck)λt
rγk
{(1− Ck) cos(Skλt)
+ Sk sin(Skλt)}.
Simplification yields the fourth term of (14). 2
Marginal allocation tables for stochastic tool life
We include here the detailed results of the numerical example for allocating a magazine
capacity of TC = 15 among four different part types, for three cases of stochastic tool life:
r = 1 (exponential), r = 11 (carbide tool) and r = 100 (low variance). The corresponding
table for deterministic tool life (r = ∞) is given in [7, Table 8].
24
Table 4: Optimal magazine allocation with r = 1 (exponential tool life)
Magazine Tools preloaded Total time Marginalcap. (TC) T1 T2 T3 T4
∑4i=1 hi(Ti) time
0 0 0 0 0 5212.5 -300.01 0 1 0 0 4912.5 -201.52 0 2 0 0 4711.0 -200.03 0 2 1 0 4511.0 -139.44 0 2 2 0 4371.6 -129.65 0 3 2 0 4242.0 -115.06 1 3 2 0 4127.0 -100.27 2 3 2 0 4026.9 -100.08 2 3 2 1 3926.9 -99.19 2 3 2 2 3827.8 -95.110 2 3 2 3 3732.7 -90.611 2 4 2 3 3642.2 -89.412 2 4 3 3 3552.8 -87.013 2 4 3 4 3465.8 -77.614 3 4 3 4 3388.1 -76.715 3 4 3 5 3311.5 -67.8
Table 5: Optimal magazine allocation with r = 11 (carbide tool)
Magazine Tools preloaded Total time Marginalcap. (TC) T1 T2 T3 T4
∑4i=1 hi(Ti) time
0 0 0 0 0 4865.4 -300.01 0 1 0 0 4565.4 -225.02 0 2 0 0 4340.4 -200.03 0 2 1 0 4140.4 -163.34 0 2 2 0 3977.0 -116.75 0 3 2 0 3860.3 -115.06 1 3 2 0 3745.3 -114.27 2 3 2 0 3631.1 -100.08 2 3 2 1 3531.1 -100.09 2 3 2 2 3431.1 -100.010 2 3 2 3 3331.1 -99.811 2 3 2 4 3231.3 -93.112 2 3 2 5 3138.2 -91.513 3 3 2 5 3046.7 -79.314 3 3 3 5 2967.3 -76.615 3 3 3 6 2890.7 -75.9
25
Table 6: Optimal magazine allocation with r = 100 (low variance)
Magazine Tools preloaded Total time Marginalcap. (TC) T1 T2 T3 T4
∑4i=1 hi(Ti) time
0 0 0 0 0 4683.3 -300.01 0 1 0 0 4383.3 -207.12 0 2 0 0 4176.3 -200.03 0 2 1 0 3976.3 -151.54 0 2 2 0 3824.7 -115.05 1 2 2 0 3709.7 -115.06 2 2 2 0 3594.7 -105.47 2 3 2 0 3489.3 -100.08 2 3 2 1 3389.3 -100.09 2 3 2 2 3289.3 -100.010 2 3 2 3 3189.3 -100.011 2 3 2 4 3089.3 -100.012 2 3 2 5 2989.4 -91.913 3 3 2 5 2897.5 -82.014 3 3 2 6 2815.4 -70.215 3 3 3 6 2745.3 -68.6
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