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Issue Title: Special Issues: "Manufacturing Intelligence and Innovation for Digital Manufacturing and Operational Excellence" and "Information Control Problems in Manufacturing - INCOM 2012"
We discuss a tool management model for a flexible machine equipped with a tool magazine, variable cutting speed, and sensors to monitor tool wear, when tool life due to flank wear is stochastic. The objective is to adjust the cutting speed as a function of remaining distance, each time a tool change occurs, in order to minimize the expected processing time (sum of cutting and tool setup time). We address the computational aspects of finding optimal decision rules and we present numerical results suggesting that easily computed decision rules of a simple static model are near-optimal for our dynamic programming model. Dynamic adjustment is assessed with simulation experiments.[PUBLICATION ABSTRACT]
J Intell Manuf (2014) 25:11531166 DOI 10.1007/s10845-013-0756-8
Dynamic speed control of a machine tool with stochastic tool life: analysis and simulation
Bernard F. Lamond Manbir S. Sodhi
Martin Nol Ousman A. Assani
Received: 17 October 2012 / Accepted: 9 March 2013 / Published online: 21 March 2013 Springer Science+Business Media New York 2013
Abstract We discuss a tool management model for a exible machine equipped with a tool magazine, variable cutting speed, and sensors to monitor tool wear, when tool life due to ank wear is stochastic. The objective is to adjust the cutting speed as a function of remaining distance, each time a tool change occurs, in order to minimize the expected processing time (sum of cutting and tool setup time). We address the computational aspects of nding optimal decision rules and we present numerical results suggesting that easily computed decision rules of a simple static model are near-optimal for our dynamic programming model. Dynamic adjustment is assessed with simulation experiments.
Keywords Stochastic control Optimal control Dynamic
programming Flexible manufacturing systems Machining
Machine tools Random lifetime Stochastic modelling
Renewal processes
This research was supported in part by the National Science and Engineering Research Council of Canada (0105560).
B. F. Lamond (B)
Operations and Decision Systems Department, Universit Laval, Qubec, QC, Canadae-mail: [email protected]
M. S. SodhiMechanical, Industrial and Systems Engineering Department, University of Rhode Island, Kingston, RI, USAe-mail: [email protected]
M. NolTLUQ, Universit du Qubec, Qubec, QC, Canada e-mail: [email protected]
O. A. AssaniCentre de services partags du Qubec, Gouvernement du Qubec, Qubec, QC, Canadae-mail: [email protected]
Introduction
In metal cutting, a large share of total manufacturing costs is due to tool wear, which is a major source of variability in processing times. Tooling issues are thus key factors in the planning and management of manufacturing operations. The physical phenomenon of tool wear is complex and hardly predictable, but statistical relations between tool life and cutting speed can be tted to experimental data. In this paper, we examine an optimization model, based on a stochastic tool life relation, to adjust the cutting speed dynamically as a function of the amount of work remaining and the number of tools available in the tool magazine.
In Dimla (2000), it is noted that cost savings and productivity improvements have led to minimally manned factories where To change a tool when worn requires on-line wear monitoring. Further, Silva and Wilcox (2008) point out that The traditional ability of the operator to determine the condition of the tool based on his/her experience and senses, i.e. vision and hearing, is now the expected role of the monitoring system. In the absence of tool wear monitoring systems, process planners selected cutting parameters to use 5080% of the mean tool life (see Wiklund 1998). Scheduling tool changes based on predictive degradation can be determined using the work reported in Yang et al. (2008). Subsequent developments relied on the partial observability of the cutting process and tool, and models for optimizing processing metrics under these conditions can be found in Ait-Kadi and Chelbi (2010). Successful implementations of on-line tool wear monitoring systems have been reported in Sharma et al. (2008), Wang and Cui (2012) and Wang et al. (2008) among others. The exploitation of online tool wear detection clearly can lead to signicant savings, as reported from simulation experiments in Nol et al. (2009).
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1154 J Intell Manuf (2014) 25:11531166
Hence, as the extent of automation in manufacturing increases, especially with the widespread use of exible machines, the use of precise tool condition monitoring helps not only to reduce tool costs but also the overall cycle time, by not discarding good inserts and consequently eliminating unnecessary insert changes. Also, if relieved from the duty of monitoring tool condition, machine operators can be given the responsibility for several machines, leading to higher productivity rates and overall lower labor costs. According to Cheng et al. (2010), the capability to remotely monitor tool status and machining conditions could bring further benets by integrating tool management data into the business information systems.
Cutting speed is a key factor in tool management because it affects both the cutting time and the rate of tool wear. Kaspi and Shabtay (2003) point out that cutting speed varies considerably among different part types, from a few inches per minute to hundreds of feet per minute. According to Kayan and Akturk (2005), the manufacturing cost of a turning operation is a nonlinear convex function of its processing time.Also, in many situations an important fraction of the processing time is spent changing or switching tools (see Konak et al. 2008). Therefore, it seems relevant to consider optimization models for cutting speed adjustment whose objective is to minimize the sum of cutting time and tool setup time.
Moreover, numerous scholars have shown that it is more realistic to represent tool life as a random variable rather than a deterministic model, such as Wager and Barash (1971), Ramalingam (1977), Ramalingam and Watson (1977), Wik-lund (1998) and Wang et al. (2001). Nonetheless, few authors have treated optimization of cutting parameters and tool loading through stochastic modeling. In Rosetto and Zompi (1981), a stochastic model for multi-face cutting tools is represented with the use of a normal or lognormal distribution when thermal or mechanical tool fatigue occurs. An extension is proposed by Zhou and Wysk (1992) when tools are used to cut different parts, based on conditional probabilities.Iakovou et al. (1996) discuss a stochastic tool life model with regard to tool inventory management. Kumar et al. (1997) present a tool requirement planning model where tool lives follow a Weibull or gamma distribution and their convolution is approximated. Liu et al. (2001) introduce an optimal preventive tool replenishment model based on reliability function when tool life follows a Weibull distribution. Shabtay and Kaspi (2001) and Shabtay and Kaspi (2003) present models to optimize cutting parameters for a periodic control strategy and an age replacement strategy when tool lives follow a normal distribution. Another relevant study is reported by Berenguer et al. (1997), where a predictive maintenance policy is designed to minimize a function of costs using a semi-Markov decision process.
A formulation was proposed in Lamond et al. (1998) for the expected processing time when tool lives follow an Erlang
distribution. Lamond and Sodhi (2006) extended that study to include an illustration that shows the deterministic model underestimates the optimal expected processing times. Nol et al. (2009) propose a simulation of a deterministic optimization model in a stochastic environment. These results clearly show that scheduling according to the deterministic model can be very different from what can be expected in more realistic settings. Finally, it is shown in Chapter 4 of Nol (2006) that the standard deviation of the processing time can be quite important, especially in states where a tool change is likely.
Here, we address the numerical solution of stochastic optimization models presented in Lamond and Sodhi (2006) for the static case, and in Lamond (2007) for the dynamic case, using the analytical results of Lamond (2010) on the structure of optimal decision rules. The paper is organized as follows: tool life model in section Machine tool operation, stochastic optimization models in section Optimal control of machine tool, parametric analysis of static model in section Parametric analysis of static model, dynamic control in section Dynamic control of cutting speed, numerical analysis in section Numerical analysis, and optimization strategy in section Optimization strategy. Moreover, a simulation experiment to assess dynamic online adjustment strategies for a exible machine equipped for multiple parts is presented in sections Simulating online adjustment strategies, Multiple part types on a exible machine, Monte Carlo simulation experiments, with concluding remarks in section Conclusion.
Machine tool operation
We suppose a exible machine tool has been set up for processing a batch of identical parts for a single cutting operation. Extension to multiple part types as in Lamond and Sodhi (1997) is straightforward and is reviewed in section Multiple part types on a exible machine, with a simulation study in section Monte Carlo simulation experiments. Finished parts are replaced instantly by a material handling system, and tool wear is monitored by an on-line sensor. It is assumed that cutting speed changes do not degrade performance efciency or quality. There is an automatic tool charger, so tool changes occur instantly whenever there are fresh tools in the tool magazine. But if the magazine is empty, the intervention of a human operator is required, so there is a setup time S to make a tool change in this case.
At a cutting speed of v m/s, we suppose the tool life is a random variable with mean t s and coefcient of variation , where t obeys Taylors relation
vvr =
tr t
, (1)
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J Intell Manuf (2014) 25:11531166 1155
where tr is the nominal tool life at a reference speed vr and is Taylors exponent such that 0 < < 1. Typical values are = 0.08 to 0.2 for high speed steel tool, = 0.2 to 0.5 for
carbide tool, and = 0.5 to 0.7 for ceramic tool (see Shin
et al. 2009). For carbide tool, the coefcient of variation is about 0.3 (see Wager and Barash 1971).
While it is natural to control a machine tool by setting its cutting speed v, because of (1) it is mathematically equivalent to specify the nominal tool life t, or the nominal distance y = vt traversed by a tool during its lifetime, or the nominal
amount of tool = x/y used to cut a distance x. Conversion
formulas are given in section Appendix. For a given tool, we dene a (generic) random variable W such that E [W] = 1
and Var(W) = 2. Then at cutting speed v, the (random) tool
life is tW, with t given by (35), so the mean tool life and coefcient of variation are t and , respectively. Also, the amount of cutting performed by a tool during its economic life is yW, with y given by (36).
Now suppose several tools are used consecutively at cutting speeds v1, v2, . . ., until a total cutting distance x > 0 is completed. Let W1, W2, . . ., be a sequence of independent, identically distributed random variables such that E [Wi] = 1
and Var(Wi) = 2, for i = 1, 2, . . .. Then the cutting dis
tance of the ith tool is yi Wi so the number of tools M needed to process a distance x is a positive random variable such that M tools are enough to cut a distance x but M 1 tools are
not sufcient. In the (static) case when vi = v and yi = y
for all i, it was shown in Lamond and Sodhi (2006) that the expected number of tools used to process a distance x is E [M] = (), with = x/y given by (37), where () is
the renewal function of a renewal process with interarrivals Wi, and with (0) = 1.
Optimal control of machine tool
To cut a distance x of a given part type, we want to select the cutting speeds v1, v2, . . . of successive tools so as to minimize the expected total processing time:
V (x) = min E[(total cutting time)+ S (# of manual tool setups)]. (2)
Before addressing the case of dynamic speed adjustment, it is useful at rst to look at the static case with equal speed vi = v for all i. In the classical deterministic approach, and
without a tool magazine, (2) is replaced by
min
v
Fig. 1 In (5), for a given state
, the objective function is the sum of a convex decreasing function (
, ) and an increasing, squigly function
()
xvr S . (4)
As in Lamond (2010), it is useful to take the dimensionless quantities
as state variable and as control variable. Then, with random tool life and without a tool magazine, (2) can be reformulated in standard form as
T ( ) = min
>0 (
, ) + (), (5)
where the function
(
, ) =
= x/ =
1 1
S tr
1
/(1)
(6)
gives the cutting time divided by the manual setup time S, and T (
) is the expected processing time of state
when
all tools have the same cutting speed vi = v given by (37)
with optimal control = . We refer to (5) as the static
model. Typically, in (5) the objective function looks as in Fig. 1. This model is said to be in standard form because (5) has dimensionless state and control variables (
and ) and it depends on dimensionless tool data (tool life exponent , type of tool life distribution, and coefcient of variation ).
If the cutting speed is adjusted dynamically, the speed vi may depend on the distance remaining to cut when the ith tool is engaged. In the stochastic dynamic programming (SDP) model of Lamond (2007), (2) becomes
V (x) = min
v>0 E S +
v + V (x yW) , (7)
for x 0, with V (x) = 0 for all x < 0, and y related to v by
(36). In standard form, dene V (
) = V (x)/S and rewrite
(7) with state variable
and control variable :
V ( ) = min
min{x, yW}
xv + S
>0 E
1 + (
xvt , (3)
so the nominal tool life at optimal speed is t = S(1 )/.
The optimal speed v is found by substituting t in (35), with
the nominal distance = vt and number of tools
, ) min (1, W/)
+ V
(1 W/)
. (8)
As shown in Lamond (2007), when tool life has an exponential distribution the cutting speed v is optimal.
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1156 J Intell Manuf (2014) 25:11531166
Exponential
5 Exponential
150
4
y(x)
100
3
* (x)
2
50
0-1 tool 2 tools 3 tools
1
0-1 tool 2 tools 3 tools
0 0 100 200 300 400 500
0 0 100 200 300 400 500
Erlang (r=11)
Erlang (r=11)
5
150
4
y(x)
* (x)
100
3
0-1 tool 2 tools 3 tools
2
50
0-1 tool 2 tools 3 tools
1
0 0 100 200 300 400 500
0 0 100 200 300 400 500
Erlang (r=100)
Distance to cut (x)
Erlang (r=100)
Distance to cut (x)
5
150
4
3
y(x)
100
* (x)
2
50
0-1 tool 2 tools 3 tools
0-1 tool 2 tools 3 tools
1
0 0 100 200 300 400 500
0 0 100 200 300 400 500
Fig. 2 Optimal decision rules for the SDP model (with Taylors exponent = 0.38)
Proposition 1 W Exp =
and V (
) = 1 +
is the renewal function of a delayed renewal process counting the manual tool setups.
Parametric analysis of static model
A decision rule is a function that prescribes a control for each state
. Let () be an optimal decision rule for the static
model (5). Then = (
/.
)+
For a machine tool equipped with a tool magazine, let
V ( ) denote the optimal expected processing time for a cut
ting job in state
when there are tools available in the tool magazine. Then V0(
) = V (
) satises (8) and, for = 1, 2, . . ., the SDP model of (8) is adapted as
V ( ) = min
>0 E
(
, ) min (1, W/)
+ V 1
(1 W/)
) is optimal for state
. (9)
For = 1, Eqs. (8) and (9) differ only by one setup. Proposition 2 V1(
) = V0(
) 1.
Typical decision rules of the SDP model are shown in Fig. 2 for the three coefcients of variation ( = 1, 0.3 and
0.1) used in Lamond and Sodhi (2006) and up to three tools in the magazine, and for both control variables y (left) and (right). For large states, we see that y (constant) while
(straight line).
In the static case with a tool magazine and constant cutting speed, the static model (5) becomes
T ( ) = min
>0 (
, ) + (), (10)
where () = E (M
. The detailed structure of optimal decision rules for the static model (5) was derived in Lamond (2010). Only a brief summary is reported here.
These results assume certain monotonicity properties of the cutting time function, which hold in the case of (6), and certain regularity properties of the renewal function, such as convexity or what Lamond (2010) dened as () is regularly approaching a straight line. Numerical evidence suggests that this property holds when tool life has a gamma distribution with integer shape parameter r 3. (For r = 1
or 2, () is convex.) In particular, for such a renewal function there is an increasing sequence of critical points
0 = 0, 1, 2, . . ., such that for k 1, k is a local minimum
of the function () . Also, () is convex increasing
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J Intell Manuf (2014) 25:11531166 1157
Exponential with 3 tools in mag.
5 Exponential with 3 tools in mag.
150
4
y(x)
* (x)
100
3
2
50
static dynamic
1
static dynamic
0 0 100 200 300 400 500
0 0 100 200 300 400 500
Erlang (r=11) with 0 tools in mag.
5 Erlang (r=11) with 0 tools in mag.
150
4
y(x)
* (x)
100
3
2
50
static dynamic
1
static dynamic
0
0 100 200 300 400 500
0 0 100 200 300 400 500
5 Erlang (r=100) with 0 tools in mag.
Distance to cut (x)
150
Erlang (r=100) with 0 tools in mag.
Distance to cut (x)
4
y(x)
* (x)
100
3
2
50
static dynamic
1
static dynamic
0 0 100 200 300 400 500
0 0 100 200 300 400 500
Fig. 3 Dynamic (SDP) versus static decision rules ( = 0.38)
on 0
1. The structure of optimal decision rules is summarized in the following propositions.
Proposition 3 (
) and T (
) are increasing functions of
.
Proposition 4 If () is convex and differentiable, then
T ( ) is convex, the reverse decision rule is
= 1() = ()1, (11) where () is the renewal density function, and
T ( ) =
() + (). (12)
Proposition 5 If () is regularly approaching a straight line and
k <
k+1, then
k < (
)
1
The above properties were shown in Lamond (2010). We remark that such decision rules could be tabulated for different values of Taylors exponent , tool life coefcient of variation , and distribution function (gamma, etc.), allowing for fast computation during online tool changes. Also, these results can be extended to the case when there is a tool magazine, with () replaced by () as in (10).
An example of the above structure with K = 1 is the dot
ted line in the middle of Fig. 3, where r = 11, = 0.38, and y = 117.7. At state 1 = 1.228(x = 144.2) the optimal con
trol jumps from 1 = 1.076 to 1 = 1.31. These are inside
the interval bounded by the relative minima
1 = 0.698 and 2 = 1.753 of the function () .
We remark that the number of jumps K as 0, in
the case of deterministic tool life. Then, as shown in Lamond (2007), the optimal decision rule is = (
) = k when
k1 <
k with 0 = 0 and, for k = 1, 2, . . .,
k =
k+1.
Proposition 6 If () is regularly approaching a straight line, then (
) is continuous except at a nite number of critical states
1, . . . ,
K such that
k <
k <
k+1, where
it jumps from k to k, with
k < k <
k < k <
k+1.
Between jumps, the reverse decision rule is given by (11).
Proposition 7 Let = (
). Then =
1
1
, (13)
1
1
1
1
k
(k + 1)
if (5) has a
, else is one of the local minima immediately to the left or to the right of
local minimum at =
and these critical points satisfy the inequality
k < k < k +
1
2. (14)
.
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1158 J Intell Manuf (2014) 25:11531166
Dynamic optimization of processing time (r=100 =0.38, x=150)
440
J(x,y) y^y*
430
420
410
400
390
380
50 100 150 200 250
Nominal cutting distance of 1st tool (y)
440
430
420
J ^( ^, )
^ *
410
400
390
380
1 1.5 2 2.5 3
Nominal amount of tool used ()
Fig. 4 SDP objective function versus y and for given state
We observed in many numerical examples that the structure of optimal decision rules for the SDP model resembles the above. In particular, we exploited properties akin to Propositions 5 (search limits) and 7 (adjacent local minima). The latter property is illustrated in Fig. 4 where the state is shown as a triangle and the optimal control as a star. Also, the deterministic bounds of (14) suggest that the search interval for can be restricted to
1.5.
Dynamic control of cutting speed
In section Parametric analysis of static model, a decision rule (
) was found for selecting the cutting speed of all tools given the initial state
, to optimize the static model (5). Now let (
) be an optimal decision rule for the SDP model (8). Figure 3 shows a numerical comparison of (
Clearly, their expected processing times satisfy
(
,
) + (
) T (
) V (
) V (
), (15)
where the expression for the classical approach is obtained from (5) with =
instead of . Expressions for the dynamic SDP and mixed approaches are obtained next.
Suppose the random variable W of tool life has density f (x) and cumulative distribution function (cdf) F(x). Let
H() =
1
0
u f (u) du + 1 F(), (16)
Q(
, ) =
V
(1 u/)
f (u) du, (17)
)
and (
) for three coefcients of variation ( = 1, 0.3 and
0.1). The decision rules for SDP look similar to the static model, although a bit smoother.
One can consider four methods for selecting the cutting speed of successive tools:
Classical: always use the cutting speed v optimal for (3);
Static: always use the cutting speed v from = (
Q(
) =
(
)
0
V
(1 u/(
))
f (u) du. (18)
Then the SDP (8) can be expressed as
V ( ) = min
>0
1 + ( , )H() + Q( , ) , (19)
) and, using = (
) in (19) instead of minimizing, we get
V ( ) = 1 + ( , ( ))H(( )) + Q( ) (20)
for the mixed approach.
In the case when > 0 tools are available in the tool magazine, we develop (9) as
where
is the initial state (when 1st tool is engaged); Mixed: use cutting speed vi from i = (
) where
is
the current state when the ith tool is engaged; Dynamic: use cutting speed vi from i = (
) where
is the current state when the ith tool is engaged.
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0.4 Exponential with 3 tools in mag.
0.3
mixed static constant
0.2
0.1
0 0 100 200 300 400 500 600
0.15 Erlang (r=11) with 0 tools in mag.
0.1
mixed static constant
0.05
0
-0.05 0 100 200 300 400 500 600
0.6 Erlang (r=100) with 0 tools in mag.
0.4
mixed static constant
0.2
0
-0.2 0 100 200 300 400 500 600
Distance to cut (x)
Fig. 5 Penalty of other methods from SDP optimal value
V ( ) = min
>0
( , )H() + Q 1( , )
(21)
where
Q 1(
, ) =
0
V 1 (1 u/) f (u) du, (22)
and similarly for V (
) and Q 1(
) in the case of the mixed
method.
Figure 5 shows the difference between the expected processing times of the classical (constant), static and mixed methods, respectively, and SDP. It is the penalty incurred for not using an optimal decision rule. The solid line shows
V ( ) V ( ) which is the difference between the expected
processing time of the static decision rules (used dynamically) and the optimal control of the dynamic model. For the exponential distribution with a tool magazine, it is
V 3( ) V3( ). In all cases, the difference is very small,
less than 0.02 times the manual setup time S.The telegraphic line indicates T (
) V (
of (4). We suppose that there is a given amount n of cutting to be executed (a distance of nh meters). Then
there are n + 1 discrete states
= i for i = 0, 1, 2, . . . , n.
Initializing rst V (0) = 1, the main loop of our DP algo
rithm successively approximates V (i) for i = 1, . . . , n,
using the previously computed values V ( j), j = 0, . . . ,
i 1. The ith iteration thus searches an optimal action (
)
for
) which shows that for the Erlang distribution, there is an increasing bias for not dynamically adjusting the cutting speed. The bias approaches 0.2S for the case withr = 100. Finally, the dotted
line gives the penalty for always using the classical cutting speed v. This penalty reaches over 0.25S for the exponential
with = 3 tools, about 0.12S for the Erlang with r = 11
and about as high as 0.5S for r = 100. More detailed results
are given in Assani (2008).
Numerical analysis
We now discuss numerical algorithms for solving (19) when tool life is not exponential. We discretize the state variable using a xed step size h for the cutting distance x (in meters) with corresponding step size = h/ for the standardized
state variable
= i.
We write (17) as the sum of i integrals, giving
Q(i, ) =
i
j=1
j/i
( j1)/i
V
i(1 u/) f (u) du. (23)
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V
i 1, =
0 if i = 1,
V
i 2,
if i 2.
Fig. 6 Two methods for interpolating the value function
The value function V () is unknown except at discrete states
j, so it is interpolated as in Fig. 6. For a state u such that j 1
i < u
(30)
In the special case i = 1, a rst optimization is performed
using the above value, then a second optimization is performed using the result of the rst optimization.
When there are > 0 tools in the tool magazine, the integral in (22) can be approximated by Q0 1() or Q1 1()
with V () and V () replaced by V 1() and V 1() in
Eqs. (27) and (29). The value function is initialized with
V (0) = 0. The above remark about an unknown value does
not apply here since the quantity V 1(i) has been computed
at step 1, so (30) is not needed for = 1, 2, . . .. We further
remark that, because of Proposition 2, the iteration with = 1
is not needed, although it may be useful to execute it to verify numerical accuracy.
Optimization strategy
At each discrete state
= i, a line search is performed to
nd an optimal control that minimizes one of the functions (
, )H() + Q0(
ji , we write
V
i(1 u/) V (i j)
(24)
+i V (i j,)
1
ji u , (25)
where
V (i j,) = V
(i j + 1) V
(i j)
. (26)
Equation (24) is a piecewise constant approximation, and adding (25) gives a piecewise linear approximation.
Now let G,(u) be the cdf of an Erlang distribution with shape parameter and rate . Then W has cdf F(u) =
Gr,r(u) and partial moment
, ) or (
, )H() + Q1(
, ). Based on the remarks at the end of section Parametric analysis of static model, the strategy we implemented is to search for local minima in the two intervals
1.5 <
and
< + 1.5 and keep the best one. This is because in all
our numerical tests, we observed that either
was optimal, or else there was one or two nearby local minima (separated by
), one of them being globally optimal. See Assani (2008) for details.
We now discuss some issues about the choice of the decision variable to use in practice. The machine tool operator (or the computer program that controls the machine tool) needs to know the cutting speed v which, in our models, goes to
as the distance goes to 0. Mathematically, the variable has nice monotonicity properties, starting at 0 and approaching a straight line for large states. On the other hand, the variable y also starts at 0 but it approaches a limiting value for large
states. This way, the variable y would seem better suited for graphical display of decision rules since it avoids divergence for small and large states.
Another issue concerns numerical computations. Although the monotonicity of is useful for bounding the search of an optimal action, it is not clear that the objective function is smoother with rather than y. For example, a typical objective function is plotted in Fig. 4 for a given state, rst as a function of y then as a function of . The function of y appears somewhat smoother than the function of . Therefore, it is possible that a more stable approach for optimization would be to use the bounds on variable for restricting the search, but to perform the search using variable y instead of .
u0 f (t) dt = Gr+1,r(u). If we
use piecewise constant approximation, (23) becomes
Q0(i,, ) =
i
j=1
V
(i j) Gr,r( j 1, /i), (27)
where
G,( j 1, t) = G,( jt) G, (( j 1)t) . (28)
But using piecewise linear approximation (23), becomes
Q1(i,, ) = Q0(i,, )
+
i
j=1
j V (i j,) Gr,r( j 1, /i)
i V (i j,)
1 Gr+1,r( j 1, /i) .
(29)
We remark here that, during the ith iteration, the quantity
V (i) has not yet been computed. This is not a problem
when using the piecewise constant approximation because that quantity is not used in the computation of Q0(i,, )
with (27). But some estimate of V (i 1,) is needed for
computing Q1(i,, ) with (29). In our implementation of
the algorithm, we used the following initial values:
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We have also encountered numerical instability with variable for small states. To illustrate the situation, consider (21) for some > 1. For i = 1, we have the small state =. We also have V 1(0) = 0, so if we use piecewise
constant approximation, we get Q0(1,, ) = 0. This means
that we have to nd to minimize (
, )H(). For example, with exponential tool life we have that
(
, )H() = constant 1
e
/ (31)
is a decreasing function of , where = 1/(1 ) > 1,
so it would seem optimal to take = . But we know the
correct value should be close to = 0. That is one reason
why we have decided to use piecewise linear approximation instead of a piecewise constant one. However, even then we have found numerical instability for larger values of as well. These difculties are avoided if we use variable y = x/ for
the search, instead of , because (31) with replaced by x/y increases with y. (Alternatively, one could approximate V (
)
by T (
) for very small states.)
According to our models, the cutting speed v diverges for small cutting distances (v as x 0) and settles
around v for large distances. In practice, however, there is an
upper bound on the allowable cutting speed (imposed by the limited capacity of the machine itself but also by the specications of the tool and cutting material combination). This means a more realistic model would have an upper constraint on the cutting speed, so that for small states, all optimal decision rules would have the cutting speed equal to the upper limit, hence the optimal processing time would increase linearly for small states. Let vu be an upper bound on v. There corresponds a lower bound yu on y and a constraint of the form (/yu)
. This constraint would be active for small enough states.
A practical advantage of using the static model for optimization is that it is very fast to compute. By contrast, it is much longer to nd an optimal decision rule with the dynamic model. Using a state discretization of n = 550 equally spaced
nodes with piecewise linear approximation of the value function, our prototype implementation of the SDP model took about 3min to optimize on a standard laptop computer for the case with no tool magazine. By comparison, the optimal decision rule of the static stochastic model took only a fraction of a second to compute and its evaluation with dynamic adjustment took about 5s. Moreover, an optimal decision rule can be computed off-line and tabulated for easy retrieval during online machine tool control.
Simulating online adjustment strategies
It was argued in the previous sections, with a numerical example, that the expected processing time can be decreased considerably by adjusting the cutting speed dynamically rather
than using the same speed for all successive cutting tools. Also, we saw that the easily computed decision rules from the static stochastic model, when used for dynamic adjustment of the cutting speed, lead to an expected processing time that is nearly as good as the optimal decision rules of the dynamic programming model. In the sequel, we use a Monte Carlo simulation experiment to investigate the savings that could be realized in practice, through a wide array of cutting conditions, by using a stochastic optimization model with dynamic adjustment of the cutting speed. Our simulation approach extends the results of Nol et al. (2009).
In a production environment, it may be useful to look at ways to reduce total processing time by adjusting to new information acquired. Altering the schedule and adapting it to new status and production requirements, often called dynamic scheduling, can help increase output without investing in equipment. Dynamic scheduling can be done by either regeneration or net change. According to Song et al. (2003), Regeneration aims to produce a new schedule covering all unstated operations while net change only produces a schedule for part of operations. In this section, we consider improving production planning for a given part type by reacting to early or late tool breakage. The interest turns to possibly reducing cutting time but mostly avoiding unnecessary and costly extra tool setups. Nol et al. (2009) present a dynamic strategy based on the deterministic model: use = k when k1 <
k, with k given by (13). Adjust
ments were made by reducing the computed cutting speed for the last planned tool to take into account the effect of randomness; otherwise costly extra tool setups would most probably be necessary to complete the tasks.
As discussed earlier, one can proceed with the same speed throughout processing, a static strategy, or update velocity in response to observed conditions, a dynamic strategy. A variety of options are possible in the dynamic case, based on revision schedules, such as: continuous adjustment, xed interval adjustments, after tool change adjustments or others that can be synchronized with the tool wear inspection schedule. The strategy considered here revises the processing speed each time a tool change takes place. For example, if previous tools used for processing a part type have not been satisfactoryi.e. lasted less than expected, cutting speed could be slowed down to try to avoid setting up extra tools, or sped up in an attempt to save on cutting time. As seen before, those setups occur when work remains after using all the preprogrammed tools.
Consider a metal cutting operation for an amount x of material to be removed, with an exponent in Taylors equation (1) and a nominal tool life tr at reference speed vr = 1
meter per second, and let S be the manual setup time. Illustrative values are given in Table 1. Suppose now a fresh tool is engaged at cutting speed v, and let to represent its observed economical life and T be the number of tools preloaded into
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1162 J Intell Manuf (2014) 25:11531166
Table 1 Data for numerical example
Parameter Value
x 2,000m
S 115s
0.25
tr 105s
Table 2 Algorithm for solving the dynamic problem
Step 0. Get values for x, , S, tr and determine TStep 1. Compute optimal stochastic cutting speed; observe tool life to Step 2. x = max(0, x vto)
Step 3. If x = 0, stop; else T = max(0, T 1)
Step 4. Set x = x and T = T ; go to Step 1
the magazine to process that part type. Then the following updates are made each time a tool is consumed and the part type is not yet nished:
x = max(0, x vto);
T = max(0, T 1),
where x is the remaining distance to cut and T is the number of tools left in the magazine. With this updated information, stochastic optimization can be used to re-optimize the cutting process: the new state
is obtained from (4) and the optimal control for the next tool is found by solving (5) with
.
An example of an online adjustment policy is constructed using the parameters presented in Table 1. Note that deterministic optimization requires using eight tools at a cutting speed of 0.7489m/s, yielding a total processing time of 3,590.6s. Also, static stochastic optimization gives an expected total processing time of 3,653.0s at 0.7427m/s and will require 8.35 tools on average.
Figure 7 illustrates a simulation of an online adjustment as was considered in this experiment. For the rst tool, the cutting speed is set at the optimal stochastic cutting speed of 0.7427m/s. Subsequently, cutting velocity is computed according to the work remaining on the part type, again using the stochastic model (5). We can notice that cutting speed remains quite stable when there are many more tools expected to be used. For example, the rst tool is changed much earlier than expected, yet cutting speed barely changes. However, as there are fewer tools expected to be used, cutting speed is reduced, which will ensure avoiding setting up an extra tool.
Multiple part types on a exible machine
We now examine the case when several part types are to be processed on a exible machine with a tool magazine of limited capacity. Let {1, 2, . . . , I} be the set of part types to
be produced, and suppose the machine tool has a magazine capacity of tool slots. No tool sharing is assumed, i.e. a tool is only used for one type of operation on a given part type. In addition, the initial setup time required for preloading the magazine is assumed constant and therefore does not play a role in time minimization. Then, in a deterministic
Table 2 presents the procedure.
Fig. 7 Illustration of online adjustments
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Table 3 Algorithm for allocating tool magazine capacity to multiple part types by marginal analysis
Let T = (T0, T1, . . . , TI )
Step 0. Initialize: T0 = ; Ti = 0, for i = 1, . . . , IStep 1. Let I (T ) = mini{hi(Ti)}, and D(T ) = max j|Tj >0 h
j (Tj 1)
Step 2. If I (T ) > D(T ), stop; else go to Step 3
Step 3. Ti = Ti + 1 where hi(T i) = I (T ), Tj = Tj 1 where h j (Tj 1) = D(T )
Step 4. Go to Step 1
environment, it was shown in Lamond and Sodhi (1997) that the tool magazine can be allocated to the different part types by the algorithm of Table 3, which is based on the marginal analysis approach of Fox (1966) and Dernardo (1982). To do this, one can compute the savings in processing time obtained by allocating an extra tool slot to each part type into the tool magazine. Note that if we let hi(Ti) be the minimum processing time for part type i when Ti tools are preloaded into the magazine, then the difference in processing time between using Ti and Ti + 1 tools is hi(Ti) = hi(Ti + 1)
hi(Ti). Starting from an arbitrary allocation, the algorithm successively reallocates a tool slot from the part type with worst marginal savings to the part type with best marginal savings, until no further gain is possible.
A manager usually has many machine tools that can inuence the way part types are allocated into the system. Sodhi et al. (2001) present an extended version of the deterministic model in the case of multiple machine tools. However, in this paper we do not look at the part type allocation problem on multiple machine tools. Yet, one could use the allocation heuristics developed in that paper and then individually simulate processing on each machine in order to determine the systems total processing time.
In a stochastic environment, the problem of selecting processing speeds for a set of part types and the determination of tool loading can be represented by the following formulation:
min
I
!i=1Pi(vi, Ti) = i + MTi (i)Si
s.t.
I
!i=1Ti
vi > 0, Ti 0 integer, i = 1, . . . , I,where Pi(vi, Ti) is the expected processing time if Ti tools are preloaded into the magazine for processing part type i at cutting speed vi, i is the cutting time for the part type at cutting speed vi and MTi (i) is the expected number of manual tool setups. With respect to our static stochastic model (10), we have Pi(vi, Ti) = T (
)Si, i = (
, )Si and MTi (i) = () with = Ti. Except for the randomness
in the number of tools required to process the part types, this model is similar to the deterministic model dened in Lamond and Sodhi (1997).
Now, let h(Ti) be the expected minimum processing time
for part type i when Ti tools are preloaded into the magazine. Then the difference in expected processing time between using Ti and Ti + 1 tools is h(Ti) = h(Ti + 1) h(Ti).
Lamond and Sodhi (2005) argue that the optimal values for a given part type i have increasing increments, i.e.
h(Ti) h(Ti + 1), for Ti = 0, 1, . . . , 1. Assuming forward non-increasing values of h(Ti), an
optimal tool loading can easily be determined. Indeed, using a stack sort, as was done for the deterministic case in Lamond and Sodhi (1997), we can determine the optimal stochastic tool loading with the algorithm of Table 3 if we replace hi(Ti) by h(Ti).
We now turn our attention to the simulation of a set of typical part types to be processed on a machine tool. The intent is to see the difference between deterministic, static stochastic and dynamic stochastic models.
Monte Carlo simulation experiments
The parameters needed for the simulation are generated from uniform distributions where the limits are dened to include typical values for the metal cutting industry, as in Sodhi et al. (2001). These parameters are randomly generated from distributions presented in Table 4. On the one hand, this information sufces to easily compute the optimal deterministic cutting speed and processing time. On the other hand, we also need to determine a coefcient of variation and a tool life distribution in order to evaluate the optimal stochastic cutting speed and processing time.
Table 4 Distributions used for generating test problems
Parameters Probability distributions
Setup time Si U[100 s, 500 s]
Reference speed vr = 1 m/s
Amount of work di U[400 m, 600 m]
Taylors parameter i U[0.1, 0.7]
Tool life constant Ci U[2, 7], where Ci = vr tiri
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Table 5 Tool life distributions with two parameters
Dist First parameter Second parameter
G = r(v/vr )1/n/tr r = 1/2N = tr (v/vr )1/n = tr (v/vr )1/n
LN = ln
tr (v/vr )1/n
(2+1)
= "ln(2 +
1)
W a = tr(v/vr)
1/n
(1+1/b) b = #
3.7138 if = 0.3
12.1534 if = 0.1 G gamma, N normal, LN lognormal, W Weibull
The distribution can be the gamma, normal, lognormal or Weibull. These are the typical laws used for the fatigue failure of material and the life length of components (see Barlow and Proschan 1965, p. 13). The two parameters needed for each distribution can be derived from the coefcient of variation and the expected tool life t [see Taylors equation (1)]. Table 5 summarizes those parameters. We can note that tool life is set fairly low because it is assumed that part material is made of very hard metal like stainless steel or titanium. In such cases, having extra manual tool setups will have an important impact on total processing times.
For the static approach, one thousand randomly generated part types were simulated for each case of interest, i.e. two different coefcients of variation (0.3 and 0.1). For each generated part type, the simulation would run through deterministic, gamma, normal, lognormal and Weibull distributions, allowing a comparison between deterministic and stochastic values. Many metrics can be used in order to analyze the stochastic environment and make comparisons. More precisely, we are interested in three specic metrics:
e = E $
Pst,i(vd,i) Pd,i(vd,i) Pd,i(vd,i)
% , (32)
r = E $
Pst,i(vd,i) Pst,i(vst,i) Pst,i(vd,i)
% , (33)
o = E $
Pst,i(vst,i) Pd,i(vd,i) Pd,i(vd,i)
% , (34)
where E [ ] denotes the expected value, Pst,i and vst,i rep
resent processing time and optimal cutting speed for a part type i, obtained from the static stochastic model with distribution st, and Pd,i and vd,i are processing time and optimal cutting speed for the same part type i, but computed from the deterministic model. In short, e compares processing times for the same optimal deterministic cutting speed along two curvesdeterministic and stochastic; r evaluates processing times for two different optimal cutting speedsdeterministic and stochasticalong the same stochastic curve and o compares the two optimal values. The results are shown in Table 6 when processing part types on a traditional machine, i.e. without a tool magazine.
Table 6 Estimated metrics from simulation of various distributions with respect to the deterministic case
Distribution Coefcient of variation (%)
0.1 0.3
e
Normal 15.58 15.82
Gamma 16.15 16.96
Lognormal 15.99 16.98
Weibull 14.09 15.41
r
Normal 8.33 4.89
Gamma 8.68 5.72
Lognormal 8.81 6.20
Weibull 7.05 4.64
o
Normal 4.12 9.04
Gamma 4.16 9.06
Lognormal 4.56 9.72
Weibull 4.07 8.85
The Weibull always yields a lower percentage increase and the lognormal the highest. The difference between any distributions is however, pretty small, at a maximum of 2%. One may expect that a higher coefcient of variation would yield a higher percentage increase. However, we observe the opposite for r. On the one hand, notice that the lower the coefcient of variation, the closer the curve of the total processing time versus cutting speed will resemble the saw-shape curve of the deterministic case. (For deterministic tool life, () is replaced by the staircase function in Fig. 1 so the total
is a sawtooth function.) On the other hand, the curve will be close to the exponential case when we use a higher coefcient of variation. Hence, one plausible explanation could be that a lower coefcient of variation brings greater oscillations along the stochastic curve for two cutting speeds that are close to each other.
Next, we want to simulate the processing time of a batch of ve part types when tool lives are randomly generated from known distributions. A simulation was performed using six different tool magazine capacities, going from a very tight magazine to a very loose one, in other words, ranging from 0 to 25 tool slots.
Table 7 shows how different the stochastic model is from a deterministic model if we arbitrarily assume tool lives follow an Erlang distribution. This conrms the conclusion reached in Nol et al. (2009). In fact, the results in their Table 4 are similar to the third column of our Table 7. It shows the percentage increase to expect if we use the optimal deterministic cutting speed. From the rst column, we notice that the optimal expected processing times are never more than 15% above the deterministic optimal. In fact, the percentage
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J Intell Manuf (2014) 25:11531166 1165
Table 7 Percentage increase over the deterministic case and standard deviation in the number of tools needed for the gamma distribution with parameter r = 11 o e Standard deviation
Static Dynamic Static Static Dynamic
0 8.94 7.81 16.96 0.4294 0.4031
5 13.44 13.05 30.19 0.4930 0.4167
10 14.90 13.41 38.05 0.5210 0.4345
15 14.43 12.22 39.85 0.5728 0.4512
20 13.68 11.16 45.68 0.6205 0.4612
25 13.40 9.76 49.90 0.6716 0.4707
increase between both optimal values declines when magazine capacity is above 10 (twice the batch size). This is quite an improvementover 36% for a large magazinewhen compared with e. Also apparent in the second column of
Table 7 (dynamic approach) is that online adjustments yield better results than constant speed with respect to expected processing times. Although this reduction is rather small(0.393.64%) this can be an interesting approach, as it is simple to implement.
Table 7 also presents the standard deviation of the number of tools needed to process a given part type. The static approach uses the formulation presented in Nol (2006) when tool lives follow an Erlang distribution. However, computing the standard deviation for the dynamic approach was done numerically, i.e. for each set of parameters tested, the dynamic approach was simulated a hundred times and the number of tools needed for each run was recorded in order to compute the standard deviation. We notice that the dynamic approach yields superior results compared to the static approach, and further that variability is also reduced.
Conclusion
This paper presents optimal static and stochastic approaches to the problems of tool magazine loading and cutting speed selection in order to minimize total expected processing time of a set of part types to be manufactured on a machine tool.Based on renewal theory, optimal cutting speed selection was computed when tool lives are random variables following a known distribution with a constant coefcient of variation.In the case where cutting speed is kept constant throughout the processing of a given part type, i.e. the static approach, the algorithm from Lamond and Sodhi (1997) was adapted for the stochastic environment and used for tool loading.
We also present a near-optimal procedure when the cutting speed is dynamically updated to take into account new information obtained from previously used tools. These adjustments can occur whenever there is a tool change and when the observed tool life differs from expectations. The method is illustrated with a numerical example. It can be easily
implemented and can help improve planning operations in a computer-aided manufacturing environment. In fact, results from the simulation show that the dynamic approach always yields better results than the static model with regard to expected processing time and its variance. The optimal value is close to the ideal deterministic model (i.e. less than 10%) when using a large tool magazine as opposed to 13.40% with the static approach. This advantage is due to the dynamic approach that uses information during the cutting process to re-optimize the problem. In a continuously changing environment, this approach will indeed improve productivity.
Using the stochastic model not only can improve productivity (by minimizing processing time) but also help improve resource management by better utilizing tools and machines (by optimizing cutting speed and tool loading). Since the model takes into account randomness in tool lives, the optimization allows for better control of tool management and a more accurate prediction of production times than a deterministic model. With the help of advances in tool-condition monitoring and online control, this model becomes viable to support exible manufacturing system and respond to the increase in competitiveness.
Future work could extend the stochastic model to include many machines, or the entire manufacturing system, as was done in the deterministic case by Sodhi et al. (2001). Stochastic optimization of the system could also be addressed by assigning batches of part types instead of individual part types. This is an even more realistic approach to real life problems, since many operations are often required on the same part before its completion. Processing all those operations on the same machine is therefore more efcient.
Appendix: Control variables
Here we give conversion equations between cutting speed v and other control variables, based on Taylors relation (1).
t = nominal tool life:
v = vr
tr t
and t = tr vr v
1/. (35)
y = nominal cutting distance per tool:
v = vr
vrtr
y
1 and y = vrtr
vr v
1 . (36)
= x/y = nominal quantity of tools used to cut a dis
tance x:
v = vr
vrtr
x
1 and =
x vrtr
v vr
1 . (37)
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