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Optimization of the machining economics problem under the failure replacement strategy
Int. J. Production Economics 80 (2002) 213–230
Optimization of the machining economics problem under the failure replacement strategy Dvir Shabtay*, Moshe Kaspi Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel Received 21 September 2000; accepted 4 May 2002
Abstract This paper presents a model for calculating the optimal cutting feed rate and spindle speed in a stand-alone cutting machine. The optimal cutting conditions are determined for three different objective functions—minimum expected cycle time, minimum expected cost per unit, and maximum expected profit-rate—under the failure replacement strategy, taking into account cutting tool constraints and machine limitations. We also examine the relationships between the optimal solutions, and present the efficiency range of feed rate. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Machining economics; Tool life; Failure replacement
1. Introduction In this paper we develop three algorithms to optimize the machining economics problem for three different objective functions—minimum expected cycle time, minimum expected cost per unit, and maximum expected profit-rate—under the failure replacement strategy (FRS). According to this strategy a tool is replaced only when a failure occurs. The relationships between the optimal solutions are exhibited via a definition of the ‘‘efficiency range’’ that extends Hitomi’s (1976) earlier definition. Tool life has long been recognized as a factor that has to be taken into consideration for operating a cutting system. In roughing operations, the various cutting speeds, feed rates, and tool angles are usually chosen according to criteria that will provide an economical tool life. The life of a cutting tool ends in two different ways: the gradual or progressive wearing away of certain regions of the face and flank of the cutting tool or a failure that can bring the life of the tool to a premature end. The fundamental nature of the mechanism of wear can vary significantly. Boothroyd and Knight (1989) described three main forms of wear: (1) Adhesion wear—wear caused by the fracture of welded asperity junctions between two metals. (2) Abrasion wear—wear caused by hard particles on the underside of the chip that pass over the tool face and remove tool material by mechanical action. (3) Diffusion wear—wear caused by atoms in a metallic crystal lattice moving from a region of high atomic concentration to one of low concentration. *Corresponding author. Fax: +972-86472958. E-mail address: [email protected] (D. Shabtay). 0925-5273/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 2 ) 0 0 2 5 5 - 4
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In metal cutting, increases in speed or feed result in increases in temperature on the tool face. At low speeds, increases in tool-face temperatures tend to reduce friction at the chip–tool interface. At high speeds, increases in tool–face temperatures tend to increase the rate of crater wear. Tool-life criterion is defined as a predetermined threshold value of a tool-wear measure or the occurrence of a phenomenon such as a tool break. Tool life is defined as the cutting time required to reach the tool-life criterion. Taylor (1906) conducted the earliest research on this subject, the main achievement of which was the development of the well-known tool-life equation. Taylor’s tool life equation bonds the tool life to the cutting speed; namely, V T% n ¼ C; where V is the cutting speed (ft/min), T% is the tool life (min), n is the slope of the line on log * log paper, and C is a constant. The index n depends mainly on the tool material: for HSS, nE0:125; for carbide, 0:25ono0:3; and for ceramics, 0:5ono0:7: Recommended values of n; C have been published extensively in various handbooks (e.g., Rembold et al., 1985). The machining economics problem has been optimized under three different objective functions— minimum expected cycle time, minimum expected cost per unit, and maximum expected contribution to profit per unit of time (usually defined as the ‘‘profit-rate’’). If a machining operation is a bottleneck operation in the production sequence, it might be necessary to operate at the cutting conditions that provide a maximum production rate. In this case, too high a cutting speed and feed rate can cause a low production rate, since a great deal of time will be wasted in changing tools. On the other hand, too low a cutting speed and feed rate will cause a low production rate. Hence, we have to find the optimal speed and feed that provides the highest production rate. This objective function has been analyzed by Hitomi (1976) and Kaspi (1993), among others. The expected cycle time function is composed of four different terms: tool approach time; load, unload and set-up time; retract time; and tool changing time. In many cases, the cutting operation is not necessarily a bottleneck in the production sequence, so the tool cost must be taken into account. Cutting conditions are usually selected to minimize cost under the assumption that operating at minimum cost conditions will lead to an increase in long-term profits (Taha, 1966; Hitomi, 1976; La Commare et al., 1983; Koulamas et al., 1987; Iakovou et al., 1996, and others). The costs can be divided into two major groups: labor and machining cost (proportional to the cycle time), and tool changing cost. Maximum profit per component (which is equivalent to minimum cost per unit) and maximum production rate are not the only objective functions since these criteria cause, respectively, either high cost or low production rate. Therefore, the maximum profit per unit of time also has to be considered. The profit-rate is calculated by the ratio revenue per unit expected cost per unit : expected cycle time Hitomi and Okushima (1964) were the first to discuss maximum profit criteria. Wu and Ermer (1966) also analyzed the maximum profit cutting speed, and discussed maximum profit conditions, not only in terms of cutting speed, but also in terms of feed rate. This paper differs from most works in the machining economics literature according to the following: 1. In the literature, many papers deal with the machining economics problem, but disregard some of the cutting constraints and the limitations of the machine. We consider those constraints in the analysis of the problem, the modeling, and in the optimal solution procedure. 2. Many works examine the problem under one of the three objective functions. We examine all the three different objective functions. 3. We extend Hitomi’s (1976) definition for the ‘‘efficiency range’’ by representing the ‘‘efficiency range’’ with respect to the cutting constraints.
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In our analysis of the single-machine economics problem, we make the following assumptions: 1. The machine tool permits a stepless variation of speed and feed rate. Thus, all functions described in this paper are continuous. 2. For HSS, m > 2n; mo1 (n and m are the Taylor exponents for the tool life and the feed, respectively). 3. The revenue per component is not dependent on the production rate. 4. After a tool replacement, the process continues from the point at which it stopped. The paper is divided into five sections. In Section 2 we describe the basic notations, the constraints, and suggest an algorithm to optimize the expected production rate. Section 3 presents an algorithm to minimize the expected cost per unit, and in Section 4 we describe an algorithm to maximize the expected profit-rate. A numerical example is given in Section 5. Without loss of generality, we consider a drilling machine; however, adjustments to other cutting machines require only a few modifications. 2. Production rate criterion Using assumption (4), we can define the expected cycle time of the machining operation using the following equation: Tc EðTpÞ ¼ Ta þ Te þ Tt þ Td ; ð1Þ EðTÞ where: EðTpÞ Ta Te Tt Td Tc
expected production time per component (min), tool approach and machining time (min), sum of load, unload and set-up time per component (min), retract time for the drill head (min), tool-changing time (min), drilling time per component (min).
The production time may be expressed in terms of the cutting parameters by the following relationships: L Tc ¼ ; ð2Þ Hf V¼
PDN ; 12
ð3Þ
H ; Hf
ð4Þ
Ta ¼ f ¼
Hf : N
ð5Þ
An extension of the Taylor tool-life equation bonds the tool life to the cutting speed, feed rate, and depth of cut via the following equation: V T% n f m d w ¼ C: We consider the depth of cut as a predetermined parameter. Thus, the Taylor tool-life equation can be represented as follows: PDN 1=n m=n 1=n EðTÞ ¼ T% ¼ C f ; ð6Þ 12
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where: L H N f D Hf
the length of the machining operation (in), the tool distance traveled by the drill head (in), spindle speed (rev/min), feed rate (in/rev), drill diameter (in), feed rate (in/min).
Inserting Eqs. (2)–(6) into Eq. (1), one obtains H Td L=Hf þ Te þ Tt þ EðTpÞ ¼ : 1=n Hf C ðPDN=12Þ1=n ðHf =NÞm=n
ð7Þ
Eq. (7) is a two-variable function (N; Hf ), so in order to reduce the expected cycle time we have to determine the optimal feed rate and spindle speed, under the following constraints. 2.1. The cutting constraints 1. Feed rate constraint: The maximum feed rate for drilling is restricted by its maximum permissible shear stress. Bhattacherya and Hum (1969) suggested that for standard geometry drills the maximum permissible feed rate, fmax is given by the relationship fmax ¼ Cs KD0:6 ;
ð8Þ
where Cs is a constant dependent upon a tool-workpiece material combination, and K is the ratio of the drill length to the drill diameter. Therefore, Hf NX : ð9Þ fmax Since the selected feed rate of the drill head, Hf ; should be within the drill head feed range available for the machine, Hf has to satisfy the following constraints: Hf pHfmax
ð10Þ
Hf XHfmin :
ð11Þ
and
2. Cutting speed constraints: The Taylor tool-life equation is valid for a wide range of cutting speeds (Taylorized speed range). Let Vtmin and Vtmax represent the limits of this range. Then, Ntmin ¼
12Vtmin ; Pd
ð12Þ
Ntmax ¼
12Vtmax : Pd
ð13Þ
Therefore, NXmaxðNmin ; Ntmin ; Hf =fmax Þ;
ð14Þ
NpminðNmax ; Ntmax Þ;
ð15Þ
where: Nmin
– minimum value of available spindle speed (rev/min),
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Nmax – maximum value of available spindle speed (rev/min), Ntmin – minimum spindle speed within the Taylorized speed range (rev/min), Ntmax – maximum spindle speed within the Taylorized speed range (rev/min). 2.2. Minimum expected cycle time Let K denote the following: L : K¼ 1=n C ðPD=12Þ1=n By inserting Eq. (16) into Eq. (7), we obtain H þ Te þ Tt þ K Td Hf ðmnÞ=n N ð1mÞ=n : EðTpÞ ¼ Hf
ð16Þ
ð17Þ
Proposition 1. For any given feed rate, we have to choose the minimum available spindle speed in order to achieve the minimum expected cycle time. Proof. Differentiating Eq. (17) with respect to N; we obtain qEðTpÞ 1m ¼ N ð1mnÞ=n K Td Hf ðmnÞ=n > 0 8Hf : qN n
ð18Þ
Thus, we are operating at the minimum available spindle speed. We now divide the expected cycle time function into two different functions, each of which is feasible in a different range. Definition. Range 1: maxðNtmin ; Nmin ÞXHf =fmax ; therefore, N ¼ maxðNtmin ; Nmin ; Hf =fmax Þ ¼ max ðNtmin ; Nmin Þ: Increasing the feed rate causes no increase in the spindle speed. Range 2: maxðNtmin ; Nmin ÞpHf =fmax ; therefore, N¼ maxðNtmin ; Nmin ; Hf =fmax Þ ¼ Hf =fmax : Increasing the feed rate directly increases the spindle speed. In order to obtain the optimal feed rate, we now derive the expected cycle time with respect to the feed rate separately for each range. For the first range: H þ Te þ Tt þ K Td Hf ðmnÞ=n N ð1mÞ=n : EðTpÞ ¼ ð19Þ Hf m n qEðTpÞ H ¼ þ N ð1mÞ=n K Td Hf ðm2nÞ=n : qHf n ðHf Þ2 For the second range: H ðm1Þ=n þ Te þ Tt þ K Td Hf ð1nÞ=n fmax ; EðTpÞ ¼ Hf qEðTpÞ H 1 n ðm1Þ=n ð12nÞ=n ¼ þ K Td Hf : fmax qHf n ðHf Þ2
ð20Þ
ð21Þ ð22Þ
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Taking into consideration assumption (2) (1 > m > 2n), it is easy to show that the second derivative of the two functions is positive, meaning that they are both convex, each having one minimum point (not necessarily in the feasible range). Figs. 1a–d describe the four possible options for the expected cycle time function. In each figure, there are two convex functions, each belonging to a different range and appearing in different shades of gray. The feasible function is the one appearing in bold print. Fig. 1a exhibits the situation in which no minimum ðHfp ; Hfp Þ is in its feasible range; therefore the optimal feed rate is Hfp ¼ maxðNmin ; Ntmin Þfmax : Fig. 1b displays the situation in which each minimum ðHfp ; Hfp Þ is in its feasible range; therefore, the optimal feed rate is the one with the smaller expected cycle time. Fig. 1c exhibits the situation in which Hfp is in its feasible range, but Hfp is not; therefore, the optimal feed rate is Hfp : Fig. 1d displays the situation in which Hfp is in its feasible range, but Hfp is not; therefore, the optimal feed rate is Hfp ; where Hfp Hfp Hfp
optimal feed rate for the minimum expected cycle time criteria, optimal feed rate for the first range of the minimum expected cycle time criteria, optimal feed rate for the second range of the minimum expected cycle time criteria.
E (Tp)
E (Tp )
Proposition 2. The case described in Fig. 1b is infeasible.
Hf***
Hflim
Hf (inch / min)
Hf*** (c)
Hflim
Hf***
Hf (inch /min )
(b)
E (Tp)
E (Tp)
(a)
Hf**
Hf**
Hf**
Hflim
Hflim
Hf (inch / min)
(d)
Hf *** Hf** Hf (inch / min)
Fig. 1. The expected cycle time as a function of the feed rate: (a) case 1; (b) case 2; (c) case 3; and (d) case 4.
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Proof. Let N ¼ maxðNmin ; Ntmin Þ and Hflim ¼ Nfmax where Hflim is the breakpoint between Ranges 1 and 2. The limit of the differential of the expected cycle time for both sides of Hflim is m n qEðTpÞ H ðm2nÞ=n ð12nÞ=n ¼ lim þ K Td N ; fmax Hf -Hflim qHf Hf 2 n qEðTpÞ H 1 n ðm2nÞ=n ð12nÞ=n ¼ lim þ þ K Td N : fmax Hf 2 n Hf -Hflim qHf It is easy to see that since nomo1; qEðTpÞ qEðTpÞ lim þ > lim : Hf -Hflim qHf Hf -Hflim qHf In Fig. 1b we can see that qEðTpÞ qEðTpÞ lim þ o lim : Hf -Hflim qHf qHf Hf -Hflim Therefore this case is infeasible. In the following we present the optimization algorithm for minimizing the expected cycle time. Algorithm 1. Step 1: Assume that range 1 is feasible. By setting Eq. (20) to zero, we obtain !n=m H Hfp ¼ : N ð1mÞ=n ðm nÞ=n K Td
ð23Þ
If Hfp pHflim ; then Hfp ¼ Hfp (see Fig. 1c), and we go to step 4; otherwise Hfp ¼ Hflim : Step 2: Assume that range 2 is feasible. By setting Eq. (21) to zero, we obtain Hfp ¼
ðm1Þ=n
fmax
H
ð1 nÞ=n K Td
!n :
ð24Þ
If Hfp XHflim ; then Hfp ¼ Hfp (see Fig. 1d), and we go to step 4; otherwise Hfp ¼ Hflim : Step 3: Hfp ¼ Hflim (see Fig. 1a). Step 4: If Hfp violates constraint (10), then Hfp ¼ Hfmax : If Hfp violates constraint (11), then Hfp ¼ Hfmin : Step 5: Np ¼ max Nmin ; Ntmin ; Hfp =fmax ;where Np is the optimal spindle speed for the minimum expected cycle time criteria.
3. Minimum expected cost per unit If the machine is not a bottleneck in the sequence, it is necessary to take into account not only the production rate as a criterion of optimization but also the cutting tool cost. Define: Ct cutting tool cost ($/edge), Co labor and machine cost ($/min).
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The expected cost value is Tc : T% Inserting Eqs. (2)–(6) and (16)–(17) into Eq. (25), yields Co H þ CoðTe þ TtÞ þ K Hf ðmnÞ=n N ð1mÞ=n ðCo Td þ CtÞ: EðcostÞ ¼ Hf EðcostÞ ¼ Co EðTpÞ þ Ct
ð25Þ
ð26Þ
Eq. (26) is a two-variable function (N; Hf ), so we have to determine the optimal feed rate and spindle speed that minimize the expected cost per unit. Proposition 3. For any given feed rate, we have to choose the minimum available spindle speed in order to achieve the minimum expected cost per unit. Proof. Differentiating Eq. (26) with respect to N; we obtain qEðcostÞ 1m ¼ N ð1mnÞ=n K Hf ðmnÞ=n ðCo Td þ CtÞ > 0 qN n
8Hf :
ð27Þ
Thus, we are operating at the minimal feasible spindle speed. In order to obtain the optimal feed rate, we now derive the expected cost per unit with respect to the feed rate separately for each range. For the first range: Co H þ CoðTe þ TtÞ þ K Hf ðmnÞ=n N ð1mÞ=n ðCo Td þ CtÞ; EðcostÞ ¼ ð28Þ Hf qEðcostÞ Co H m n ð1mÞ=n ¼ þ K Hf ðm2nÞ=n ðCo Td þ CtÞ: ð29Þ N qHf n ðHf Þ2 For the second range:
ð1mÞ=n Co H ðmnÞ=n Hf þ CoðTe þ TtÞ þ K Hf ðCo Td þ CtÞ; EðcostÞ ¼ Hf fmax qEðcostÞ Co H 1 n ðm1Þ=n ð12nÞ=n ¼ þK Hf ðCo Td þ CtÞ: fmax qHf n ðHf Þ2
ð30Þ ð31Þ
Taking into consideration assumption (2) (1 > m > 2n), it is easy to show that the second derivative of the two functions is positive, meaning that they are both convex and that each has one minimum point (not necessarily within its feasible range). Figs. 2a–d exhibit the four possible options for the expected cost per unit. In each figure there are two convex functions, each belonging to a different range and appearing in different shades of gray. The feasible function is the one appearing in bold print. Fig 2a displays the case in which, neither Hfc nor Hfc is in its feasible range; therefore the optimal feed rate is Hfc ¼ maxðNmin ; Ntmin Þfmax : Fig. 2b exhibits the situation in which each minimum ðHfc ; Hfc Þ is in its feasible range; therefore, the optimal feed rate is the one with the smaller expected cost per unit. Fig. 2c displays the situation in which Hfc is in its feasible range, but Hfc is not; therefore, the optimal feed rate is Hfc : Fig. 2d exhibits the situation in which Hfc is in its feasible range, but Hfc is not; therefore, the optimal feed rate is Hfc ; where: Hfc optimal feed rate for the minimum expected cost per unit criteria; Hfc optimal feed rate for the first range of the minimum expected cost per unit criteria; Hfc optimal feed rate for the second range of the minimum expected cost per unit criteria.
Hf** (a)
Hflim
Hf**
Hf** (b)
Hflim
Hf***
Hf (inch / min)
E (cost)
E (cost)
Hf (inch / min)
Hf *** (c)
221
E (cost)
E (cost)
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Hf **
Hflim
Hflim
Hf (inch / min)
(d)
Hf***
Hf** Hf (inch / min)
Fig. 2. The expected cost per unit as a function of the feed rate: (a) case 1; (b) case 2; (c) case 3; and (d) case 4.
Proposition 4. The case described in Fig. 2b is infeasible. Proof. The limit of the differential of the expected cost per unit for both sides of Hflim is m n qEðcostÞ Co H ðm2nÞ=n ð12nÞ=n ¼ lim þK N ; ðCo Td þ CtÞ fmax 2 Hf -Hflim qHf Hf n qEðcostÞ Co H 1n ðm2nÞ=n ð12nÞ=n ¼ lim þ þK N : ðCo Td þ CtÞ fmax qHf Hf 2 n Hf -Hflim It is easy to see that since nomo1; qEðcostÞ qEðcostÞ lim þ > lim : Hf -Hflim qHf qHf Hf -Hflim In Fig. 2b we can see that qEðcostÞ qEðcostÞ lim þ o lim : Hf -Hf qHf qHf Hf -Hflim lim Therefore, this case is infeasible.
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In the following we present the optimization algorithm for minimizing the expected cost per unit. Algorithm 2. Step 1. Assume that range 1 is feasible. By setting Eq. (29) to zero, we obtain !n=m Co H Hfc ¼ : N ð1mÞ=n ðm nÞ=n K ½Co Td þ Ct If Hfc pHflim ; then Hfc ¼ Hfc (see Fig. 2c) and we go to step 4; otherwise Hfc ¼ Hflim : Step 2: Assume that range 2 is feasible. By setting Eq. (31) to zero, we obtain !n Co H : Hfc ¼ ðm1Þ=n fmax ð1 nÞ=n K½Co Td þ Ct
ð32Þ
ð33Þ
If Hfc XHflim ; then Hfc ¼ Hfc (see Fig. 2d) and we go to step 4; otherwise Hfc ¼ Hflim : Step 3: Hfc ¼ Hflim (see Fig. 2a). Step 4: If Hfc violates constraint (10), then Hfc ¼ Hfmax : If Hfc violates constraint (11), then Hfc ¼ Hfmin : Step 5: Nc ¼ maxðNmin ; Ntmin ; Hfc =fmax Þ; where Nc is the optimal spindle speed for the minimum expected cost per unit criteria.
3.1. Efficiency range Hitomi (1976) showed that the efficiency range is the range of the economic cutting feed rate (or cutting speed). Within this range, there exists an optimal feed rate that satisfies the economic needs of the factory. In this paper we consider the efficiency range regarding the machining constraints (which was disregarded in Hitomi’s paper). Definition 1. The efficiency range is defined as 8 > ½Hfp ; Hfc > < E ¼ Hfp > > : ½Hf ; Hf c
p
if Hfp oHfc ; if Hfp ¼ Hfc ; if
Hfp
>
ð34Þ
Hfc :
Proposition 5. 0oHfc pHfp oN; so that E ¼ ½Hfc ; Hfp : Proof. Let us note that n=m Co Td Hfc ¼ Hfp ; Co Td þ Ct and Hfc
¼
Hfp
Co Td Co Td þ Ct
for Range 1;
n ;
for Range 2:
Since, Co Td=ðCo Td þ CtÞo1; and n is positive, then Hfc oHfp :
ð35Þ
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If Hfc > Hflim ; then from Eq. (35) we know that Hfp > Hflim ; therefore, Hfc ¼ Hfp ¼ maxðNmin ; Ntmin Þfmax ; otherwise Hfc oHfp : Thus, we can conclude that Hfc pHfp : Using the same analysis, we can see that Hfc pHfp : The conclusion is that Hfc pHfp ; i.e., E ¼ ½Hfc ; Hfp : Let us assume that the cutting machine is a machine in a multi-machine production line. If this machine is not a bottleneck in the sequence when operating at the optimal feed rate for the minimum expected cycle time criteria (Hfc ), we can find the bordered feed rate for which the machine becomes a bottleneck machine in the sequence—Hfb : To operate the system economically, we choose the feed rate as Hf ¼ maxðHfc ; Hfb Þ: 4. Maximum expected profit-rate A natural criterion for the selection of the optimal cutting conditions is the maximum profit-rate. We define r as the revenue per component ($/unit). Due to assumption (3), the revenue per component is not dependent on the production rate. Thus, the expected profit-rate is EðOÞ ¼
r EðcostÞ r Ct KN ð1mÞ=n Hf ðmnÞ=n ¼ Co: EðTpÞ H=Hf þ Te þ Tt þ Td KN ð1mÞ=n Hf ðmnÞ=n
ð36Þ
Eq. (36) is a function of two decision variables ðN; Hf Þ; so we have to determine the optimal feed rate and spindle speed that maximizes the expected profit-rate. Proposition 6. For any given feed rate, we choose the minimum feasible spindle speed available in order to maximize the expected profit-rate. Proof. Differentiating Eq. (36) with respect to N; we obtain
qEðOÞ 1m H 1 ð1mnÞ=n ðmnÞ=n ¼ K þ Te þ Tt þ Td r Hf Ct o0 N qN n Hf ½EðTpÞ 2
8Hf > 0:
ð37Þ
Thus, we are operating at the minimal available spindle speed. In order to obtain the optimal feed rate, let us derive the expected profit-rate with respect to the feed rate separately for each range. For the first range, the numerator from the derivative of Eq. (36) with respect to Hf is m m n Hr ð1mÞ=n ðm3nÞ=n K Ct H Hf ð38Þ N KN ð1mÞ=n Hf ðm2nÞ=n ½ðTe þ TtÞCt þ rTd : n n ðHf Þ2 For the second range, it is ðm1Þ=n
EðOÞ ¼
r Ct Kfmax
Hf ð1nÞ=n ðm1Þ=n
H=Hf þ Te þ Tt þ Td Kfmax
Hf ð1nÞ=n
:
The numerator from the derivative of Eq. (39) with respect to Hf is Hr 1 ðm1Þ=n 1n ð13nÞ=n ðm1Þ=n K Ct H Hf Hf ð12nÞ=n ½ðTe þ TtÞCt þ rTd ; f Kfmax n max n ðHf Þ2
ð39Þ
ð40Þ
since: (a) The expected profit-rate function and Eqs. (38) and (40) are continuous; (b) Eqs. (38) and (40) become smaller as the feed rate grows; and (c) if Hf ) 0; Eqs. (38) and (40) tend to infinity, and if Hf ) N; Eqs. (38) and (40) tend to negative infinity. Therefore, we conclude that if we set Eqs. (38) and (40) to zero to find the local maximum, there is only one solution to each equation.
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Figs. 3a–d display the four possible options for the optimization process. In each figure there are two functions, each belonging to a different range and appearing in different shades of gray. The feasible function is the one appearing in bold print. Fig. 3a exhibits the situation in which no maximum ðHfO ; HfO Þ is in its feasible range; therefore the optimal feed rate is HfO ¼ maxðNmin ; Ntmin Þfmax : Fig. 3b displays the situation in which each maximum ðHfO ; HfO Þ is in its feasible range; therefore the optimal feed rate is the one for which the expected profitrate is bigger. Fig. 3c displays the situation in which HfO is in its feasible range, but HfO is not; therefore the optimal feed rate is HfO : Fig. 3d exhibits the situation in which HfO is in its feasible range, but Hfp is not; therefore the optimal feed rate is HfO ; where: HfO the optimal feed rate for the maximum expected profit-rate criteria; HfO the optimal feed rate for the first range of the maximum expected profit-rate criteria; and HfO the optimal feed rate for the second range of the maximum expected profit-rate criteria. Proposition 7. The case described in Fig. 3b is infeasible. Proof. The limit of the differential of the expected profit-rate for both sides of Hflim is lim
Hf -Hflim
¼
qEðOÞ qHf
m n Hr ðm3nÞ=n ð13nÞ=n ðm2nÞ=n ð12nÞ=n Kðm=nÞCt Hfmax N K N ½ðTe þ TtÞCt þ rTd fmax 2 n ðNfmax Þ ðmnÞ=n
ðH=Nfmax þ Te þ Tt þ K Td fmax
lim
þ Hf -Hflim
qEðOÞ qHf 2
Hr=ðNfmax Þ
ðm3nÞ=n ð13nÞ=n Kð1=nÞCt Hfmax N
¼
N ð1nÞ=n Þ2
;
1n ðm2nÞ=n ð12nÞ=n K N ½ðTe þ TtÞCt þ r Td fmax n ðmnÞ=n
ðH=Nfmax þ Te þ Tt þ K Td fmax
N ð1nÞ=n Þ2
:
It is easy to see that since nomo1; lim
þ Hf -Hflim
qEðOÞ qEðOÞ o lim : Hf -Hflim qHf qHf
In Fig. 3b, we can see that lim
þ Hf -Hflim
qEðOÞ qEðOÞ > lim : Hf -Hflim qHf qHf
Hence, this case is infeasible. In the following we present the optimization algorithm that maximizes the expected profit-rate. Algorithm 3. Step 1: Assume that range 1 is feasible. Set Eq. (38) to zero and find HfO using any enumeration method. If HfO pHflim ; then HfO ¼ HfO (see Fig. 3c), and we go to step 4; otherwise HfO ¼ Hflim :
225
E ()
E ()
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Hf***
(a)
Hf ** Hflim
Hflim Hf**
(b)
Hf ***
Hf (inch / min)
E ()
E ()
Hf (inch / min)
Hf ** Hf ***
(c)
Hflim
Hf (inch / min)
Hflim Hf ***
(d)
Hf** Hf (inch / min)
Fig. 3. The expected profit-rate as a function of the feed rate: (a) case 1; (b) case 2; (c) case 3; and (d) case 4.
Step 2: Assume that range 2 is feasible. Set Eq. (40) to zero and find HfO using any enumeration method. If HfO XHflim ; then HfO ¼ HfO (see Fig. 3d), and we go to step 4; otherwise HfO ¼ Hflim : Step 3: HfO ¼ Hflim (see Fig. 3a). Step 4: If HfO violates constraint (10), then HfO ¼ Hfmax : If HfO violates constraint (11), then HfO ¼ Hfmin : Step 5:
Hf NO ¼ max Nmin ; Ntmin ; O ; fmax
where NO is the optimal spindle speed for the maximum expected profit-rate criteria. Proposition 8. If r ) N; then the optimal solution for the maximum profit-rate criteria is the same as the optimal solution for the maximum production rate criteria. Proof. For the first range, if we assign r ) N to Eq. (38), we obtain m n qEðOÞ Hr 1 ð1mÞ=n ðm2nÞ=n ¼ Hf r Td : KN 2 qHf limr)N ðHf Þ n ½E-ðTpÞ 2
ð41Þ
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226
By setting Eq. (41) to zero, we obtain HfO ¼
H N ð1mÞ=n ðm nÞ=n K Td
!n=m ¼ Hfp ;
ð42Þ
which is exactly what we obtained for the first range of the minimum expected cycle time criteria (Eq. (23)). Using the same analysis method, it is easy to show that for the second range, if we assign r ) N to Eq. (40) we obtain !n H HfO ¼ ¼ Hfp ; ð43Þ ðm1Þ=n fmax ðð1 nÞ=nÞK which is exactly what we obtained for the second range of the minimum expected cycle time criteria (Eq. (24)). This proposition is quite intuitive since if the item-selling price is much bigger than any item cost, the item should be produced at the maximum production rate. 4.1. Efficiency range and profit-rate function It has long been recognized that the optimal feed rate to operate a machining system is selected from the efficiency range, fEg: Later on Hitomi and Okushima (1964) suggested a procedure to maximize the expected profit-rate. Hitomi (1976) proved that the optimal cutting speed for the maximum expected profitrate exists within the efficiency range. The following proposition extends Hitomi (1976) definition to the case in which the cutting constraints are considered. Proposition 9. EðOÞ is maximized by HfO which is taken from the efficiency range fEg: Proof. In fEg; qEðcostÞ=qHf X0 and qEðTpÞ=qHf p0: Also, we know that qEðcostÞ ¼0 qHf
and
qEðTpÞ o0; qHf Hfc ;Hfc
ð44Þ
qEðcostÞ > 0 and qHf Hfp ;Hfp
qEðTpÞ ¼ 0: qHf Hfp ;Hfp
ð45Þ
Hfc ;Hfc
and
From Eqs. (36) and (44), we conclude that qEðOÞ=qHf jHfc ;Hfc > 0; and from Eqs. (36) and (45), we conclude that qEðOÞ=qHf jHfp ;Hfp o0: Hence, since EðOÞ is a continuous function, the optimal point is within fEg:
5. Numerical example In this section we examine a numerical example that illustrates the optimal algorithms demonstrated in the previous sections. Numerical data: Te ¼ 0:5 min; Tt ¼ 0:1 min; Td ¼ 6 min; D ¼ 1 in; C ¼ 4:77; L ¼ 4 in; H ¼ 5 in; n ¼ 0:125; m ¼ 0:75; Hfmin¼ 1 in/min; Hfmax ¼ 100 in/min; fmax ¼ 0:055 in/rev; maxðNtmin ; Nmin Þ ¼ 100 rev/min; Ct ¼ 10 $; Co ¼ 0:2 $/min; and r ¼ 4 $/unit. The tool parameters were taken from Hoffman and Haavisto (1962). First we calculate K; which is used as a parameter over the various optimization
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algorithms: K¼
L C 1=n ðPD=12Þ1=n
¼
4 4:771=0:125 ðP1=12Þ1=0:125
¼ 3:293 1010 :
The following presents three optimization algorithms for the three objective functions. 5.1. Minimum expected cycle time Hflim ¼ fmax maxðNtmin ; Nmin Þ ¼ 5:5 in=min: Step 1: n=m 0:125=0:75 H 5 Hf ¼ ¼ N ð1mÞ=n ððmnÞ=nÞTd K 100ð10:75Þ=0:125 ðð0:750:125Þ=0:125Þ6 3:293 1010 ¼ 6:082 in=min; Hfp ¼ 6:082 > 5:5; then Hfp ¼ Hflim ¼ 100 0:055 ¼ 5:5 in=min: Step 2: ! 0:125 H 5 Hf ¼ ðm1Þ=n ¼ 0:05ð0:751Þ=0:125 ðð1 0:125Þ=0:125Þ6 3:293 1010 fmax ðð1 nÞ=nÞTd K ¼ 5:687 in=min; Hfp
¼ 5:687 > 5:5; then Hfp ¼ Hfp ¼ 5:687 in=rev; and go to step 4.
Step 4: Hfmin oHfp oHfmax : Step 5: Np
5:687 ¼ max 100; ¼ 103:39 rev=min; 0:055
E ðTpÞ ¼
5 þ 0:5 þ 0:1 þ 3:293 1010 6ð5:687Þð0:750:125Þ=0:125 ð103:39Þð10:75Þ=0:125 ¼ 1:605 min : 5:687
5.2. Minimum expected cost per unit Step 1: Hfc
n=m Co H ¼ N ð1mÞ=n ððm nÞ=nÞK½Co Td þ Ct 0:125=0:75 0:2 5 ¼ 100ð10:75Þ=0:125 ðð0:75 0:125Þ=0:125Þ3:293 1010 ½0:2 6 þ 10 ¼ 4:191 in=min:
Hfc ¼ 4:191o5:5; then Hfc ¼ Hfc ¼ 4:191 in=min; and go to step 4.
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Step 4: Hfmin oHfc oHfmax : Step 5:
4:191 Nc ¼ max 100; ¼ 100 rev=min; 0:055 E ðcostÞ ¼
0:2 5 þ 0:2ð0:5 þ 0:1Þ 4:191 þ 3:293 1010 ð4:191Þð0:750:125Þ=0:125 ð100Þð10:75Þ=0:125 ½0:2 6 þ 10
¼ 0:406 $=unit: As a result of the optimization of the expected cycle time and the expected cost per unit, the efficiency range of economic cutting feed rates is E ¼ ½4:191; 5:687 : 5.3. Maximum expected profit-rate Step 1: Using an enumeration method we solve Eq. (38) for our numerical example, and obtain: HfO ¼ 5:567 in=min; HfO ¼ 5:567 > 5:5: Thus, HfO ¼ Hflim ¼ 100 0:055 ¼ 5:5 in=min : Step 2: Using an enumeration method we solve Eq. (40) for our data, and obtain HfO ¼ 5:323 in=min; HfO ¼ 5:523o5:5; then HfO ¼ Hflim ¼ 100 0:055 ¼ 5:5 in=min : Step 3: HfO ¼ Hflim ¼ 100 0:055 ¼ 5:5 in=rev : Step 4: Hfmin oHfO oHfmax : Step 5: NO
5:5 ¼ max 100; ¼ 100 rev=min; 0:055
EðOÞ ¼ ¼
r Ct KN ð1mÞ=n Hf ðmnÞ=n Co H=Hf þ Te þ Tt þ Td KN ð1mÞ=n Hf ðmnÞ=n 4 10 3:293 1010 100ð10:75Þ=0:125 5:5ð0:750:125Þ=0:125 0:2 ¼ 2:184 $=min : 5=5:5 þ 0:5 þ 0:1 þ 6 3:293 1010 100ð10:75Þ=0:125 5:5ð0:750:125Þ=0:125
A summary of the numerical results is presented in Table 1. Table 1 shows that Hfc is 4.191 in/min while Hfp is 5.687 in/min. These values define the efficiency range E ¼ ½4:191; 5:687 : Although the expected cycle time and the expected cost per unit functions are insensitive to small changes in the feed rate within the close neighborhood of the optimal feed rate, in cases where the efficiency range is wide, operating in high speeds can cause a very significant increase in the production costs even within the efficiency range. In our example, the expected cost is 0.53 $/unit if operating in the optimal cutting conditions for the expected cycle time criterion while the expected cost is 0.406 $/unit if operating in the optimal cutting conditions for the expected cost per unit criterion. The main cause for this difference is the tool break costs since when operating under the optimal cutting conditions for the expected
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Table 1 The optimal cutting conditions for the three objective functions under the FRS Objective parameter
Ex. cycle time
Ex. cost per unit
Ex. profit-rate
Hf (in/min) N (rev/min) Cycle time (min) Cost ($/unit) Profit ($/min)
5.687 103.39 1.605 0.53 2.162
4.191 100 1.819 0.406 1.976
5.5 100 1.609 0.488 2.184
cycle time criterion a tool gets broken after 47.8 cycles on average, while when operating under the optimal cutting conditions for the expected cost criterion a tool gets broken only after 234.68 cycles on average. In our numerical example the maximal expected profit-rate is positive, and for such a case it has been found that HfO is within the efficiency range (HfO ¼ 5:5 in=min). Because the revenue per unit (4 $/unit) is much larger than the expected cost within the efficiency range (about 0.5 $/unit) the optimal cutting conditions for the expected profit-rate criterion are quite similar to those of the minimum expected cycle time criterion. The expected profit-rate function is insensitive to changes in the feed rate within the close neighborhood of the optimal solution. Thus, selecting the mid-efficiency range usually provides a reasonable solution for the expected profit-rate criterion. In our numerical example the expected profit-rate for operating in the mid-efficiency range (at 4.94 in/min) is 2.137 $/min, which is only 2.1% less than the optimal expected profit-rate even though HfO is close to the right side of the efficiency range rather than to the middle of the efficiency range.
6. Conclusions The optimal cutting conditions are usually determined by an economic model that optimizes one of the following criteria: minimum expected cycle time, minimum expected cost per unit, or maximum expected profit-rate. In this paper, we optimized the machining economics problem under all three criteria taking into consideration the cutting constraints. In addition, we evaluated the relationships between the optimal solutions by defining the efficiency range of the feed rate. The optimization algorithms are demonstrated using a numerical example, in which the optimal cutting conditions for the expected profit function are selected from the efficiency range. This function is not very sensitive to small changes in feed rate within the efficiency range. Therefore, defining the efficiency range is very important not only from a theoretical perspective but also from a practical point of view since any feed-rate selection from this range will provide a practical solution for the machining economics problem.
Acknowledgements This research has been partially supported by the Paul Ivanier Center for Robotics and Production Management, Ben-Gurion University of the Negev.
References Bhattacherya, A., Hum, I., 1969. Design of Cutting Tools—Use of Metal Cutting Theory. American Society of Tool and Manufacturing Engineers, Dearbon, MI. Boothroyd, G., Knight, W.A., 1989. Fundamentals of Machining and Machine Tools. Marcel Dekker, New York.
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Iakovou, E., Ip, C.M., Koulamas, C., 1996. Optimal solutions for the machining economics problem with stochastically distributed tool life. European Journal of Operational Research 92, 63–68. Hitomi, K., 1976. Analyses of production models, Part 1: The optimal decision of production speeds. AIIE Transactions 8 (1), 96–105. Hitomi, K., Okushima, K., 1964. A study of economic machining: An analysis of the maximum-profit cutting speed. International Journal of Production Research 3 (1), 73–78. Hoffman, E., Haavisto, M., 1962. Fundamentals of Tool Design. SME Publication, Dearborn, MI. Kaspi, M., 1993. Optimal machining operation for multiple-tool setups with stochastic tool life. Proceedings of the 12th International Conference on Production Research (ICPR), Lappeenranta, Finland. Koulamas, C.P., Lambert, B.K., Smith, M.L., 1987. Optimal machining conditions and buffer space size for the two stage case. International Journal of Production Research 25 (3), 327–336. La Commare, U., La Diega, S.N., Passannanti, A., 1983. Optimum replacement policies with penalty cost for unforeseen tool failure. International Journal of Machine Tool Design Research 23, 237–243. Rembold, U., Blume, C., Dillmann, R., 1985. Computer-Integrated Manufacturing Technology and Systems. Dekker, New York. Taha, H., 1966. A policy for determining the optimal cycle length for a cutting tool. Journal of Industrial Engineering 17 (3), 157–162. Taylor, F.W., 1906. On the art of cutting metals. ASME Journal of Engineering for Industry 28, 310–350. Wu, S.M., Ermer, D.S., 1966. Maximum profit as the criterion in the determination of optimum cutting conditions. ASME Journal of Engineering for Industry 88, 435–442.
Optimization of the machining economics problem under the failure replacement strategy Dvir Shabtay*, Moshe Kaspi Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel Received 21 September 2000; accepted 4 May 2002
Abstract This paper presents a model for calculating the optimal cutting feed rate and spindle speed in a stand-alone cutting machine. The optimal cutting conditions are determined for three different objective functions—minimum expected cycle time, minimum expected cost per unit, and maximum expected profit-rate—under the failure replacement strategy, taking into account cutting tool constraints and machine limitations. We also examine the relationships between the optimal solutions, and present the efficiency range of feed rate. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Machining economics; Tool life; Failure replacement
1. Introduction In this paper we develop three algorithms to optimize the machining economics problem for three different objective functions—minimum expected cycle time, minimum expected cost per unit, and maximum expected profit-rate—under the failure replacement strategy (FRS). According to this strategy a tool is replaced only when a failure occurs. The relationships between the optimal solutions are exhibited via a definition of the ‘‘efficiency range’’ that extends Hitomi’s (1976) earlier definition. Tool life has long been recognized as a factor that has to be taken into consideration for operating a cutting system. In roughing operations, the various cutting speeds, feed rates, and tool angles are usually chosen according to criteria that will provide an economical tool life. The life of a cutting tool ends in two different ways: the gradual or progressive wearing away of certain regions of the face and flank of the cutting tool or a failure that can bring the life of the tool to a premature end. The fundamental nature of the mechanism of wear can vary significantly. Boothroyd and Knight (1989) described three main forms of wear: (1) Adhesion wear—wear caused by the fracture of welded asperity junctions between two metals. (2) Abrasion wear—wear caused by hard particles on the underside of the chip that pass over the tool face and remove tool material by mechanical action. (3) Diffusion wear—wear caused by atoms in a metallic crystal lattice moving from a region of high atomic concentration to one of low concentration. *Corresponding author. Fax: +972-86472958. E-mail address: [email protected] (D. Shabtay). 0925-5273/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 2 ) 0 0 2 5 5 - 4
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In metal cutting, increases in speed or feed result in increases in temperature on the tool face. At low speeds, increases in tool-face temperatures tend to reduce friction at the chip–tool interface. At high speeds, increases in tool–face temperatures tend to increase the rate of crater wear. Tool-life criterion is defined as a predetermined threshold value of a tool-wear measure or the occurrence of a phenomenon such as a tool break. Tool life is defined as the cutting time required to reach the tool-life criterion. Taylor (1906) conducted the earliest research on this subject, the main achievement of which was the development of the well-known tool-life equation. Taylor’s tool life equation bonds the tool life to the cutting speed; namely, V T% n ¼ C; where V is the cutting speed (ft/min), T% is the tool life (min), n is the slope of the line on log * log paper, and C is a constant. The index n depends mainly on the tool material: for HSS, nE0:125; for carbide, 0:25ono0:3; and for ceramics, 0:5ono0:7: Recommended values of n; C have been published extensively in various handbooks (e.g., Rembold et al., 1985). The machining economics problem has been optimized under three different objective functions— minimum expected cycle time, minimum expected cost per unit, and maximum expected contribution to profit per unit of time (usually defined as the ‘‘profit-rate’’). If a machining operation is a bottleneck operation in the production sequence, it might be necessary to operate at the cutting conditions that provide a maximum production rate. In this case, too high a cutting speed and feed rate can cause a low production rate, since a great deal of time will be wasted in changing tools. On the other hand, too low a cutting speed and feed rate will cause a low production rate. Hence, we have to find the optimal speed and feed that provides the highest production rate. This objective function has been analyzed by Hitomi (1976) and Kaspi (1993), among others. The expected cycle time function is composed of four different terms: tool approach time; load, unload and set-up time; retract time; and tool changing time. In many cases, the cutting operation is not necessarily a bottleneck in the production sequence, so the tool cost must be taken into account. Cutting conditions are usually selected to minimize cost under the assumption that operating at minimum cost conditions will lead to an increase in long-term profits (Taha, 1966; Hitomi, 1976; La Commare et al., 1983; Koulamas et al., 1987; Iakovou et al., 1996, and others). The costs can be divided into two major groups: labor and machining cost (proportional to the cycle time), and tool changing cost. Maximum profit per component (which is equivalent to minimum cost per unit) and maximum production rate are not the only objective functions since these criteria cause, respectively, either high cost or low production rate. Therefore, the maximum profit per unit of time also has to be considered. The profit-rate is calculated by the ratio revenue per unit expected cost per unit : expected cycle time Hitomi and Okushima (1964) were the first to discuss maximum profit criteria. Wu and Ermer (1966) also analyzed the maximum profit cutting speed, and discussed maximum profit conditions, not only in terms of cutting speed, but also in terms of feed rate. This paper differs from most works in the machining economics literature according to the following: 1. In the literature, many papers deal with the machining economics problem, but disregard some of the cutting constraints and the limitations of the machine. We consider those constraints in the analysis of the problem, the modeling, and in the optimal solution procedure. 2. Many works examine the problem under one of the three objective functions. We examine all the three different objective functions. 3. We extend Hitomi’s (1976) definition for the ‘‘efficiency range’’ by representing the ‘‘efficiency range’’ with respect to the cutting constraints.
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In our analysis of the single-machine economics problem, we make the following assumptions: 1. The machine tool permits a stepless variation of speed and feed rate. Thus, all functions described in this paper are continuous. 2. For HSS, m > 2n; mo1 (n and m are the Taylor exponents for the tool life and the feed, respectively). 3. The revenue per component is not dependent on the production rate. 4. After a tool replacement, the process continues from the point at which it stopped. The paper is divided into five sections. In Section 2 we describe the basic notations, the constraints, and suggest an algorithm to optimize the expected production rate. Section 3 presents an algorithm to minimize the expected cost per unit, and in Section 4 we describe an algorithm to maximize the expected profit-rate. A numerical example is given in Section 5. Without loss of generality, we consider a drilling machine; however, adjustments to other cutting machines require only a few modifications. 2. Production rate criterion Using assumption (4), we can define the expected cycle time of the machining operation using the following equation: Tc EðTpÞ ¼ Ta þ Te þ Tt þ Td ; ð1Þ EðTÞ where: EðTpÞ Ta Te Tt Td Tc
expected production time per component (min), tool approach and machining time (min), sum of load, unload and set-up time per component (min), retract time for the drill head (min), tool-changing time (min), drilling time per component (min).
The production time may be expressed in terms of the cutting parameters by the following relationships: L Tc ¼ ; ð2Þ Hf V¼
PDN ; 12
ð3Þ
H ; Hf
ð4Þ
Ta ¼ f ¼
Hf : N
ð5Þ
An extension of the Taylor tool-life equation bonds the tool life to the cutting speed, feed rate, and depth of cut via the following equation: V T% n f m d w ¼ C: We consider the depth of cut as a predetermined parameter. Thus, the Taylor tool-life equation can be represented as follows: PDN 1=n m=n 1=n EðTÞ ¼ T% ¼ C f ; ð6Þ 12
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where: L H N f D Hf
the length of the machining operation (in), the tool distance traveled by the drill head (in), spindle speed (rev/min), feed rate (in/rev), drill diameter (in), feed rate (in/min).
Inserting Eqs. (2)–(6) into Eq. (1), one obtains H Td L=Hf þ Te þ Tt þ EðTpÞ ¼ : 1=n Hf C ðPDN=12Þ1=n ðHf =NÞm=n
ð7Þ
Eq. (7) is a two-variable function (N; Hf ), so in order to reduce the expected cycle time we have to determine the optimal feed rate and spindle speed, under the following constraints. 2.1. The cutting constraints 1. Feed rate constraint: The maximum feed rate for drilling is restricted by its maximum permissible shear stress. Bhattacherya and Hum (1969) suggested that for standard geometry drills the maximum permissible feed rate, fmax is given by the relationship fmax ¼ Cs KD0:6 ;
ð8Þ
where Cs is a constant dependent upon a tool-workpiece material combination, and K is the ratio of the drill length to the drill diameter. Therefore, Hf NX : ð9Þ fmax Since the selected feed rate of the drill head, Hf ; should be within the drill head feed range available for the machine, Hf has to satisfy the following constraints: Hf pHfmax
ð10Þ
Hf XHfmin :
ð11Þ
and
2. Cutting speed constraints: The Taylor tool-life equation is valid for a wide range of cutting speeds (Taylorized speed range). Let Vtmin and Vtmax represent the limits of this range. Then, Ntmin ¼
12Vtmin ; Pd
ð12Þ
Ntmax ¼
12Vtmax : Pd
ð13Þ
Therefore, NXmaxðNmin ; Ntmin ; Hf =fmax Þ;
ð14Þ
NpminðNmax ; Ntmax Þ;
ð15Þ
where: Nmin
– minimum value of available spindle speed (rev/min),
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Nmax – maximum value of available spindle speed (rev/min), Ntmin – minimum spindle speed within the Taylorized speed range (rev/min), Ntmax – maximum spindle speed within the Taylorized speed range (rev/min). 2.2. Minimum expected cycle time Let K denote the following: L : K¼ 1=n C ðPD=12Þ1=n By inserting Eq. (16) into Eq. (7), we obtain H þ Te þ Tt þ K Td Hf ðmnÞ=n N ð1mÞ=n : EðTpÞ ¼ Hf
ð16Þ
ð17Þ
Proposition 1. For any given feed rate, we have to choose the minimum available spindle speed in order to achieve the minimum expected cycle time. Proof. Differentiating Eq. (17) with respect to N; we obtain qEðTpÞ 1m ¼ N ð1mnÞ=n K Td Hf ðmnÞ=n > 0 8Hf : qN n
ð18Þ
Thus, we are operating at the minimum available spindle speed. We now divide the expected cycle time function into two different functions, each of which is feasible in a different range. Definition. Range 1: maxðNtmin ; Nmin ÞXHf =fmax ; therefore, N ¼ maxðNtmin ; Nmin ; Hf =fmax Þ ¼ max ðNtmin ; Nmin Þ: Increasing the feed rate causes no increase in the spindle speed. Range 2: maxðNtmin ; Nmin ÞpHf =fmax ; therefore, N¼ maxðNtmin ; Nmin ; Hf =fmax Þ ¼ Hf =fmax : Increasing the feed rate directly increases the spindle speed. In order to obtain the optimal feed rate, we now derive the expected cycle time with respect to the feed rate separately for each range. For the first range: H þ Te þ Tt þ K Td Hf ðmnÞ=n N ð1mÞ=n : EðTpÞ ¼ ð19Þ Hf m n qEðTpÞ H ¼ þ N ð1mÞ=n K Td Hf ðm2nÞ=n : qHf n ðHf Þ2 For the second range: H ðm1Þ=n þ Te þ Tt þ K Td Hf ð1nÞ=n fmax ; EðTpÞ ¼ Hf qEðTpÞ H 1 n ðm1Þ=n ð12nÞ=n ¼ þ K Td Hf : fmax qHf n ðHf Þ2
ð20Þ
ð21Þ ð22Þ
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Taking into consideration assumption (2) (1 > m > 2n), it is easy to show that the second derivative of the two functions is positive, meaning that they are both convex, each having one minimum point (not necessarily in the feasible range). Figs. 1a–d describe the four possible options for the expected cycle time function. In each figure, there are two convex functions, each belonging to a different range and appearing in different shades of gray. The feasible function is the one appearing in bold print. Fig. 1a exhibits the situation in which no minimum ðHfp ; Hfp Þ is in its feasible range; therefore the optimal feed rate is Hfp ¼ maxðNmin ; Ntmin Þfmax : Fig. 1b displays the situation in which each minimum ðHfp ; Hfp Þ is in its feasible range; therefore, the optimal feed rate is the one with the smaller expected cycle time. Fig. 1c exhibits the situation in which Hfp is in its feasible range, but Hfp is not; therefore, the optimal feed rate is Hfp : Fig. 1d displays the situation in which Hfp is in its feasible range, but Hfp is not; therefore, the optimal feed rate is Hfp ; where Hfp Hfp Hfp
optimal feed rate for the minimum expected cycle time criteria, optimal feed rate for the first range of the minimum expected cycle time criteria, optimal feed rate for the second range of the minimum expected cycle time criteria.
E (Tp)
E (Tp )
Proposition 2. The case described in Fig. 1b is infeasible.
Hf***
Hflim
Hf (inch / min)
Hf*** (c)
Hflim
Hf***
Hf (inch /min )
(b)
E (Tp)
E (Tp)
(a)
Hf**
Hf**
Hf**
Hflim
Hflim
Hf (inch / min)
(d)
Hf *** Hf** Hf (inch / min)
Fig. 1. The expected cycle time as a function of the feed rate: (a) case 1; (b) case 2; (c) case 3; and (d) case 4.
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Proof. Let N ¼ maxðNmin ; Ntmin Þ and Hflim ¼ Nfmax where Hflim is the breakpoint between Ranges 1 and 2. The limit of the differential of the expected cycle time for both sides of Hflim is m n qEðTpÞ H ðm2nÞ=n ð12nÞ=n ¼ lim þ K Td N ; fmax Hf -Hflim qHf Hf 2 n qEðTpÞ H 1 n ðm2nÞ=n ð12nÞ=n ¼ lim þ þ K Td N : fmax Hf 2 n Hf -Hflim qHf It is easy to see that since nomo1; qEðTpÞ qEðTpÞ lim þ > lim : Hf -Hflim qHf Hf -Hflim qHf In Fig. 1b we can see that qEðTpÞ qEðTpÞ lim þ o lim : Hf -Hflim qHf qHf Hf -Hflim Therefore this case is infeasible. In the following we present the optimization algorithm for minimizing the expected cycle time. Algorithm 1. Step 1: Assume that range 1 is feasible. By setting Eq. (20) to zero, we obtain !n=m H Hfp ¼ : N ð1mÞ=n ðm nÞ=n K Td
ð23Þ
If Hfp pHflim ; then Hfp ¼ Hfp (see Fig. 1c), and we go to step 4; otherwise Hfp ¼ Hflim : Step 2: Assume that range 2 is feasible. By setting Eq. (21) to zero, we obtain Hfp ¼
ðm1Þ=n
fmax
H
ð1 nÞ=n K Td
!n :
ð24Þ
If Hfp XHflim ; then Hfp ¼ Hfp (see Fig. 1d), and we go to step 4; otherwise Hfp ¼ Hflim : Step 3: Hfp ¼ Hflim (see Fig. 1a). Step 4: If Hfp violates constraint (10), then Hfp ¼ Hfmax : If Hfp violates constraint (11), then Hfp ¼ Hfmin : Step 5: Np ¼ max Nmin ; Ntmin ; Hfp =fmax ;where Np is the optimal spindle speed for the minimum expected cycle time criteria.
3. Minimum expected cost per unit If the machine is not a bottleneck in the sequence, it is necessary to take into account not only the production rate as a criterion of optimization but also the cutting tool cost. Define: Ct cutting tool cost ($/edge), Co labor and machine cost ($/min).
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The expected cost value is Tc : T% Inserting Eqs. (2)–(6) and (16)–(17) into Eq. (25), yields Co H þ CoðTe þ TtÞ þ K Hf ðmnÞ=n N ð1mÞ=n ðCo Td þ CtÞ: EðcostÞ ¼ Hf EðcostÞ ¼ Co EðTpÞ þ Ct
ð25Þ
ð26Þ
Eq. (26) is a two-variable function (N; Hf ), so we have to determine the optimal feed rate and spindle speed that minimize the expected cost per unit. Proposition 3. For any given feed rate, we have to choose the minimum available spindle speed in order to achieve the minimum expected cost per unit. Proof. Differentiating Eq. (26) with respect to N; we obtain qEðcostÞ 1m ¼ N ð1mnÞ=n K Hf ðmnÞ=n ðCo Td þ CtÞ > 0 qN n
8Hf :
ð27Þ
Thus, we are operating at the minimal feasible spindle speed. In order to obtain the optimal feed rate, we now derive the expected cost per unit with respect to the feed rate separately for each range. For the first range: Co H þ CoðTe þ TtÞ þ K Hf ðmnÞ=n N ð1mÞ=n ðCo Td þ CtÞ; EðcostÞ ¼ ð28Þ Hf qEðcostÞ Co H m n ð1mÞ=n ¼ þ K Hf ðm2nÞ=n ðCo Td þ CtÞ: ð29Þ N qHf n ðHf Þ2 For the second range:
ð1mÞ=n Co H ðmnÞ=n Hf þ CoðTe þ TtÞ þ K Hf ðCo Td þ CtÞ; EðcostÞ ¼ Hf fmax qEðcostÞ Co H 1 n ðm1Þ=n ð12nÞ=n ¼ þK Hf ðCo Td þ CtÞ: fmax qHf n ðHf Þ2
ð30Þ ð31Þ
Taking into consideration assumption (2) (1 > m > 2n), it is easy to show that the second derivative of the two functions is positive, meaning that they are both convex and that each has one minimum point (not necessarily within its feasible range). Figs. 2a–d exhibit the four possible options for the expected cost per unit. In each figure there are two convex functions, each belonging to a different range and appearing in different shades of gray. The feasible function is the one appearing in bold print. Fig 2a displays the case in which, neither Hfc nor Hfc is in its feasible range; therefore the optimal feed rate is Hfc ¼ maxðNmin ; Ntmin Þfmax : Fig. 2b exhibits the situation in which each minimum ðHfc ; Hfc Þ is in its feasible range; therefore, the optimal feed rate is the one with the smaller expected cost per unit. Fig. 2c displays the situation in which Hfc is in its feasible range, but Hfc is not; therefore, the optimal feed rate is Hfc : Fig. 2d exhibits the situation in which Hfc is in its feasible range, but Hfc is not; therefore, the optimal feed rate is Hfc ; where: Hfc optimal feed rate for the minimum expected cost per unit criteria; Hfc optimal feed rate for the first range of the minimum expected cost per unit criteria; Hfc optimal feed rate for the second range of the minimum expected cost per unit criteria.
Hf** (a)
Hflim
Hf**
Hf** (b)
Hflim
Hf***
Hf (inch / min)
E (cost)
E (cost)
Hf (inch / min)
Hf *** (c)
221
E (cost)
E (cost)
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Hf **
Hflim
Hflim
Hf (inch / min)
(d)
Hf***
Hf** Hf (inch / min)
Fig. 2. The expected cost per unit as a function of the feed rate: (a) case 1; (b) case 2; (c) case 3; and (d) case 4.
Proposition 4. The case described in Fig. 2b is infeasible. Proof. The limit of the differential of the expected cost per unit for both sides of Hflim is m n qEðcostÞ Co H ðm2nÞ=n ð12nÞ=n ¼ lim þK N ; ðCo Td þ CtÞ fmax 2 Hf -Hflim qHf Hf n qEðcostÞ Co H 1n ðm2nÞ=n ð12nÞ=n ¼ lim þ þK N : ðCo Td þ CtÞ fmax qHf Hf 2 n Hf -Hflim It is easy to see that since nomo1; qEðcostÞ qEðcostÞ lim þ > lim : Hf -Hflim qHf qHf Hf -Hflim In Fig. 2b we can see that qEðcostÞ qEðcostÞ lim þ o lim : Hf -Hf qHf qHf Hf -Hflim lim Therefore, this case is infeasible.
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In the following we present the optimization algorithm for minimizing the expected cost per unit. Algorithm 2. Step 1. Assume that range 1 is feasible. By setting Eq. (29) to zero, we obtain !n=m Co H Hfc ¼ : N ð1mÞ=n ðm nÞ=n K ½Co Td þ Ct If Hfc pHflim ; then Hfc ¼ Hfc (see Fig. 2c) and we go to step 4; otherwise Hfc ¼ Hflim : Step 2: Assume that range 2 is feasible. By setting Eq. (31) to zero, we obtain !n Co H : Hfc ¼ ðm1Þ=n fmax ð1 nÞ=n K½Co Td þ Ct
ð32Þ
ð33Þ
If Hfc XHflim ; then Hfc ¼ Hfc (see Fig. 2d) and we go to step 4; otherwise Hfc ¼ Hflim : Step 3: Hfc ¼ Hflim (see Fig. 2a). Step 4: If Hfc violates constraint (10), then Hfc ¼ Hfmax : If Hfc violates constraint (11), then Hfc ¼ Hfmin : Step 5: Nc ¼ maxðNmin ; Ntmin ; Hfc =fmax Þ; where Nc is the optimal spindle speed for the minimum expected cost per unit criteria.
3.1. Efficiency range Hitomi (1976) showed that the efficiency range is the range of the economic cutting feed rate (or cutting speed). Within this range, there exists an optimal feed rate that satisfies the economic needs of the factory. In this paper we consider the efficiency range regarding the machining constraints (which was disregarded in Hitomi’s paper). Definition 1. The efficiency range is defined as 8 > ½Hfp ; Hfc > < E ¼ Hfp > > : ½Hf ; Hf c
p
if Hfp oHfc ; if Hfp ¼ Hfc ; if
Hfp
>
ð34Þ
Hfc :
Proposition 5. 0oHfc pHfp oN; so that E ¼ ½Hfc ; Hfp : Proof. Let us note that n=m Co Td Hfc ¼ Hfp ; Co Td þ Ct and Hfc
¼
Hfp
Co Td Co Td þ Ct
for Range 1;
n ;
for Range 2:
Since, Co Td=ðCo Td þ CtÞo1; and n is positive, then Hfc oHfp :
ð35Þ
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If Hfc > Hflim ; then from Eq. (35) we know that Hfp > Hflim ; therefore, Hfc ¼ Hfp ¼ maxðNmin ; Ntmin Þfmax ; otherwise Hfc oHfp : Thus, we can conclude that Hfc pHfp : Using the same analysis, we can see that Hfc pHfp : The conclusion is that Hfc pHfp ; i.e., E ¼ ½Hfc ; Hfp : Let us assume that the cutting machine is a machine in a multi-machine production line. If this machine is not a bottleneck in the sequence when operating at the optimal feed rate for the minimum expected cycle time criteria (Hfc ), we can find the bordered feed rate for which the machine becomes a bottleneck machine in the sequence—Hfb : To operate the system economically, we choose the feed rate as Hf ¼ maxðHfc ; Hfb Þ: 4. Maximum expected profit-rate A natural criterion for the selection of the optimal cutting conditions is the maximum profit-rate. We define r as the revenue per component ($/unit). Due to assumption (3), the revenue per component is not dependent on the production rate. Thus, the expected profit-rate is EðOÞ ¼
r EðcostÞ r Ct KN ð1mÞ=n Hf ðmnÞ=n ¼ Co: EðTpÞ H=Hf þ Te þ Tt þ Td KN ð1mÞ=n Hf ðmnÞ=n
ð36Þ
Eq. (36) is a function of two decision variables ðN; Hf Þ; so we have to determine the optimal feed rate and spindle speed that maximizes the expected profit-rate. Proposition 6. For any given feed rate, we choose the minimum feasible spindle speed available in order to maximize the expected profit-rate. Proof. Differentiating Eq. (36) with respect to N; we obtain
qEðOÞ 1m H 1 ð1mnÞ=n ðmnÞ=n ¼ K þ Te þ Tt þ Td r Hf Ct o0 N qN n Hf ½EðTpÞ 2
8Hf > 0:
ð37Þ
Thus, we are operating at the minimal available spindle speed. In order to obtain the optimal feed rate, let us derive the expected profit-rate with respect to the feed rate separately for each range. For the first range, the numerator from the derivative of Eq. (36) with respect to Hf is m m n Hr ð1mÞ=n ðm3nÞ=n K Ct H Hf ð38Þ N KN ð1mÞ=n Hf ðm2nÞ=n ½ðTe þ TtÞCt þ rTd : n n ðHf Þ2 For the second range, it is ðm1Þ=n
EðOÞ ¼
r Ct Kfmax
Hf ð1nÞ=n ðm1Þ=n
H=Hf þ Te þ Tt þ Td Kfmax
Hf ð1nÞ=n
:
The numerator from the derivative of Eq. (39) with respect to Hf is Hr 1 ðm1Þ=n 1n ð13nÞ=n ðm1Þ=n K Ct H Hf Hf ð12nÞ=n ½ðTe þ TtÞCt þ rTd ; f Kfmax n max n ðHf Þ2
ð39Þ
ð40Þ
since: (a) The expected profit-rate function and Eqs. (38) and (40) are continuous; (b) Eqs. (38) and (40) become smaller as the feed rate grows; and (c) if Hf ) 0; Eqs. (38) and (40) tend to infinity, and if Hf ) N; Eqs. (38) and (40) tend to negative infinity. Therefore, we conclude that if we set Eqs. (38) and (40) to zero to find the local maximum, there is only one solution to each equation.
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Figs. 3a–d display the four possible options for the optimization process. In each figure there are two functions, each belonging to a different range and appearing in different shades of gray. The feasible function is the one appearing in bold print. Fig. 3a exhibits the situation in which no maximum ðHfO ; HfO Þ is in its feasible range; therefore the optimal feed rate is HfO ¼ maxðNmin ; Ntmin Þfmax : Fig. 3b displays the situation in which each maximum ðHfO ; HfO Þ is in its feasible range; therefore the optimal feed rate is the one for which the expected profitrate is bigger. Fig. 3c displays the situation in which HfO is in its feasible range, but HfO is not; therefore the optimal feed rate is HfO : Fig. 3d exhibits the situation in which HfO is in its feasible range, but Hfp is not; therefore the optimal feed rate is HfO ; where: HfO the optimal feed rate for the maximum expected profit-rate criteria; HfO the optimal feed rate for the first range of the maximum expected profit-rate criteria; and HfO the optimal feed rate for the second range of the maximum expected profit-rate criteria. Proposition 7. The case described in Fig. 3b is infeasible. Proof. The limit of the differential of the expected profit-rate for both sides of Hflim is lim
Hf -Hflim
¼
qEðOÞ qHf
m n Hr ðm3nÞ=n ð13nÞ=n ðm2nÞ=n ð12nÞ=n Kðm=nÞCt Hfmax N K N ½ðTe þ TtÞCt þ rTd fmax 2 n ðNfmax Þ ðmnÞ=n
ðH=Nfmax þ Te þ Tt þ K Td fmax
lim
þ Hf -Hflim
qEðOÞ qHf 2
Hr=ðNfmax Þ
ðm3nÞ=n ð13nÞ=n Kð1=nÞCt Hfmax N
¼
N ð1nÞ=n Þ2
;
1n ðm2nÞ=n ð12nÞ=n K N ½ðTe þ TtÞCt þ r Td fmax n ðmnÞ=n
ðH=Nfmax þ Te þ Tt þ K Td fmax
N ð1nÞ=n Þ2
:
It is easy to see that since nomo1; lim
þ Hf -Hflim
qEðOÞ qEðOÞ o lim : Hf -Hflim qHf qHf
In Fig. 3b, we can see that lim
þ Hf -Hflim
qEðOÞ qEðOÞ > lim : Hf -Hflim qHf qHf
Hence, this case is infeasible. In the following we present the optimization algorithm that maximizes the expected profit-rate. Algorithm 3. Step 1: Assume that range 1 is feasible. Set Eq. (38) to zero and find HfO using any enumeration method. If HfO pHflim ; then HfO ¼ HfO (see Fig. 3c), and we go to step 4; otherwise HfO ¼ Hflim :
225
E ()
E ()
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Hf***
(a)
Hf ** Hflim
Hflim Hf**
(b)
Hf ***
Hf (inch / min)
E ()
E ()
Hf (inch / min)
Hf ** Hf ***
(c)
Hflim
Hf (inch / min)
Hflim Hf ***
(d)
Hf** Hf (inch / min)
Fig. 3. The expected profit-rate as a function of the feed rate: (a) case 1; (b) case 2; (c) case 3; and (d) case 4.
Step 2: Assume that range 2 is feasible. Set Eq. (40) to zero and find HfO using any enumeration method. If HfO XHflim ; then HfO ¼ HfO (see Fig. 3d), and we go to step 4; otherwise HfO ¼ Hflim : Step 3: HfO ¼ Hflim (see Fig. 3a). Step 4: If HfO violates constraint (10), then HfO ¼ Hfmax : If HfO violates constraint (11), then HfO ¼ Hfmin : Step 5:
Hf NO ¼ max Nmin ; Ntmin ; O ; fmax
where NO is the optimal spindle speed for the maximum expected profit-rate criteria. Proposition 8. If r ) N; then the optimal solution for the maximum profit-rate criteria is the same as the optimal solution for the maximum production rate criteria. Proof. For the first range, if we assign r ) N to Eq. (38), we obtain m n qEðOÞ Hr 1 ð1mÞ=n ðm2nÞ=n ¼ Hf r Td : KN 2 qHf limr)N ðHf Þ n ½E-ðTpÞ 2
ð41Þ
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226
By setting Eq. (41) to zero, we obtain HfO ¼
H N ð1mÞ=n ðm nÞ=n K Td
!n=m ¼ Hfp ;
ð42Þ
which is exactly what we obtained for the first range of the minimum expected cycle time criteria (Eq. (23)). Using the same analysis method, it is easy to show that for the second range, if we assign r ) N to Eq. (40) we obtain !n H HfO ¼ ¼ Hfp ; ð43Þ ðm1Þ=n fmax ðð1 nÞ=nÞK which is exactly what we obtained for the second range of the minimum expected cycle time criteria (Eq. (24)). This proposition is quite intuitive since if the item-selling price is much bigger than any item cost, the item should be produced at the maximum production rate. 4.1. Efficiency range and profit-rate function It has long been recognized that the optimal feed rate to operate a machining system is selected from the efficiency range, fEg: Later on Hitomi and Okushima (1964) suggested a procedure to maximize the expected profit-rate. Hitomi (1976) proved that the optimal cutting speed for the maximum expected profitrate exists within the efficiency range. The following proposition extends Hitomi (1976) definition to the case in which the cutting constraints are considered. Proposition 9. EðOÞ is maximized by HfO which is taken from the efficiency range fEg: Proof. In fEg; qEðcostÞ=qHf X0 and qEðTpÞ=qHf p0: Also, we know that qEðcostÞ ¼0 qHf
and
qEðTpÞ o0; qHf Hfc ;Hfc
ð44Þ
qEðcostÞ > 0 and qHf Hfp ;Hfp
qEðTpÞ ¼ 0: qHf Hfp ;Hfp
ð45Þ
Hfc ;Hfc
and
From Eqs. (36) and (44), we conclude that qEðOÞ=qHf jHfc ;Hfc > 0; and from Eqs. (36) and (45), we conclude that qEðOÞ=qHf jHfp ;Hfp o0: Hence, since EðOÞ is a continuous function, the optimal point is within fEg:
5. Numerical example In this section we examine a numerical example that illustrates the optimal algorithms demonstrated in the previous sections. Numerical data: Te ¼ 0:5 min; Tt ¼ 0:1 min; Td ¼ 6 min; D ¼ 1 in; C ¼ 4:77; L ¼ 4 in; H ¼ 5 in; n ¼ 0:125; m ¼ 0:75; Hfmin¼ 1 in/min; Hfmax ¼ 100 in/min; fmax ¼ 0:055 in/rev; maxðNtmin ; Nmin Þ ¼ 100 rev/min; Ct ¼ 10 $; Co ¼ 0:2 $/min; and r ¼ 4 $/unit. The tool parameters were taken from Hoffman and Haavisto (1962). First we calculate K; which is used as a parameter over the various optimization
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227
algorithms: K¼
L C 1=n ðPD=12Þ1=n
¼
4 4:771=0:125 ðP1=12Þ1=0:125
¼ 3:293 1010 :
The following presents three optimization algorithms for the three objective functions. 5.1. Minimum expected cycle time Hflim ¼ fmax maxðNtmin ; Nmin Þ ¼ 5:5 in=min: Step 1: n=m 0:125=0:75 H 5 Hf ¼ ¼ N ð1mÞ=n ððmnÞ=nÞTd K 100ð10:75Þ=0:125 ðð0:750:125Þ=0:125Þ6 3:293 1010 ¼ 6:082 in=min; Hfp ¼ 6:082 > 5:5; then Hfp ¼ Hflim ¼ 100 0:055 ¼ 5:5 in=min: Step 2: ! 0:125 H 5 Hf ¼ ðm1Þ=n ¼ 0:05ð0:751Þ=0:125 ðð1 0:125Þ=0:125Þ6 3:293 1010 fmax ðð1 nÞ=nÞTd K ¼ 5:687 in=min; Hfp
¼ 5:687 > 5:5; then Hfp ¼ Hfp ¼ 5:687 in=rev; and go to step 4.
Step 4: Hfmin oHfp oHfmax : Step 5: Np
5:687 ¼ max 100; ¼ 103:39 rev=min; 0:055
E ðTpÞ ¼
5 þ 0:5 þ 0:1 þ 3:293 1010 6ð5:687Þð0:750:125Þ=0:125 ð103:39Þð10:75Þ=0:125 ¼ 1:605 min : 5:687
5.2. Minimum expected cost per unit Step 1: Hfc
n=m Co H ¼ N ð1mÞ=n ððm nÞ=nÞK½Co Td þ Ct 0:125=0:75 0:2 5 ¼ 100ð10:75Þ=0:125 ðð0:75 0:125Þ=0:125Þ3:293 1010 ½0:2 6 þ 10 ¼ 4:191 in=min:
Hfc ¼ 4:191o5:5; then Hfc ¼ Hfc ¼ 4:191 in=min; and go to step 4.
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Step 4: Hfmin oHfc oHfmax : Step 5:
4:191 Nc ¼ max 100; ¼ 100 rev=min; 0:055 E ðcostÞ ¼
0:2 5 þ 0:2ð0:5 þ 0:1Þ 4:191 þ 3:293 1010 ð4:191Þð0:750:125Þ=0:125 ð100Þð10:75Þ=0:125 ½0:2 6 þ 10
¼ 0:406 $=unit: As a result of the optimization of the expected cycle time and the expected cost per unit, the efficiency range of economic cutting feed rates is E ¼ ½4:191; 5:687 : 5.3. Maximum expected profit-rate Step 1: Using an enumeration method we solve Eq. (38) for our numerical example, and obtain: HfO ¼ 5:567 in=min; HfO ¼ 5:567 > 5:5: Thus, HfO ¼ Hflim ¼ 100 0:055 ¼ 5:5 in=min : Step 2: Using an enumeration method we solve Eq. (40) for our data, and obtain HfO ¼ 5:323 in=min; HfO ¼ 5:523o5:5; then HfO ¼ Hflim ¼ 100 0:055 ¼ 5:5 in=min : Step 3: HfO ¼ Hflim ¼ 100 0:055 ¼ 5:5 in=rev : Step 4: Hfmin oHfO oHfmax : Step 5: NO
5:5 ¼ max 100; ¼ 100 rev=min; 0:055
EðOÞ ¼ ¼
r Ct KN ð1mÞ=n Hf ðmnÞ=n Co H=Hf þ Te þ Tt þ Td KN ð1mÞ=n Hf ðmnÞ=n 4 10 3:293 1010 100ð10:75Þ=0:125 5:5ð0:750:125Þ=0:125 0:2 ¼ 2:184 $=min : 5=5:5 þ 0:5 þ 0:1 þ 6 3:293 1010 100ð10:75Þ=0:125 5:5ð0:750:125Þ=0:125
A summary of the numerical results is presented in Table 1. Table 1 shows that Hfc is 4.191 in/min while Hfp is 5.687 in/min. These values define the efficiency range E ¼ ½4:191; 5:687 : Although the expected cycle time and the expected cost per unit functions are insensitive to small changes in the feed rate within the close neighborhood of the optimal feed rate, in cases where the efficiency range is wide, operating in high speeds can cause a very significant increase in the production costs even within the efficiency range. In our example, the expected cost is 0.53 $/unit if operating in the optimal cutting conditions for the expected cycle time criterion while the expected cost is 0.406 $/unit if operating in the optimal cutting conditions for the expected cost per unit criterion. The main cause for this difference is the tool break costs since when operating under the optimal cutting conditions for the expected
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Table 1 The optimal cutting conditions for the three objective functions under the FRS Objective parameter
Ex. cycle time
Ex. cost per unit
Ex. profit-rate
Hf (in/min) N (rev/min) Cycle time (min) Cost ($/unit) Profit ($/min)
5.687 103.39 1.605 0.53 2.162
4.191 100 1.819 0.406 1.976
5.5 100 1.609 0.488 2.184
cycle time criterion a tool gets broken after 47.8 cycles on average, while when operating under the optimal cutting conditions for the expected cost criterion a tool gets broken only after 234.68 cycles on average. In our numerical example the maximal expected profit-rate is positive, and for such a case it has been found that HfO is within the efficiency range (HfO ¼ 5:5 in=min). Because the revenue per unit (4 $/unit) is much larger than the expected cost within the efficiency range (about 0.5 $/unit) the optimal cutting conditions for the expected profit-rate criterion are quite similar to those of the minimum expected cycle time criterion. The expected profit-rate function is insensitive to changes in the feed rate within the close neighborhood of the optimal solution. Thus, selecting the mid-efficiency range usually provides a reasonable solution for the expected profit-rate criterion. In our numerical example the expected profit-rate for operating in the mid-efficiency range (at 4.94 in/min) is 2.137 $/min, which is only 2.1% less than the optimal expected profit-rate even though HfO is close to the right side of the efficiency range rather than to the middle of the efficiency range.
6. Conclusions The optimal cutting conditions are usually determined by an economic model that optimizes one of the following criteria: minimum expected cycle time, minimum expected cost per unit, or maximum expected profit-rate. In this paper, we optimized the machining economics problem under all three criteria taking into consideration the cutting constraints. In addition, we evaluated the relationships between the optimal solutions by defining the efficiency range of the feed rate. The optimization algorithms are demonstrated using a numerical example, in which the optimal cutting conditions for the expected profit function are selected from the efficiency range. This function is not very sensitive to small changes in feed rate within the efficiency range. Therefore, defining the efficiency range is very important not only from a theoretical perspective but also from a practical point of view since any feed-rate selection from this range will provide a practical solution for the machining economics problem.
Acknowledgements This research has been partially supported by the Paul Ivanier Center for Robotics and Production Management, Ben-Gurion University of the Negev.
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