Optimization of Machining Modes for Multi-head Machines under Group Replacement of Tools

Proceedings of the 13th IFAC Symposium on Information Control Problems in Manufacturing Moscow, Russia, June 3-5, 2009

Optimization of Machining Modes for Multi-head Machines under Group Replacement of Tools Genrikh Levin, Boris Rozin United Institute of Informatics Problems Surganov Str, 6, 220012 Minsk Belarus e-mail:{levin,rozin}@newman.bas-net.by

Abstract: A problem of defining of machining modes for paced multi-station transfer lines composed of workstations with multi-spindle heads is considered. Each spindle head performs some block of operations with its tools. Each tool of a spindle head has its own spindle speed. All tools of a spindle head have the same feed per minute. All spindle heads of all workstations are activated simultaneously. Constraints on ranges of cutting parameters, cutting process characteristics, required productivity, as well as dependences of tools life time on the cutting parameters are given. To decrease the idle time of the equipment the tools are grouped to implement their replacement simultaneously. All tools of each group are replaced after processing the same number of parts, where the number is taken from the given set. The problem consists in finding the cutting parameters and the tools grouping to minimize the unit production cost under the above constraints. The proposed method is based on the combination of fragmentary parameterization concept and the coordinate-wise descent over groups of variables of the parameterized problem. A numerical example is presented. Keywords: Computer-aided design, machining modes, workstation, multi-spindle head, unit-head machines, transfer lines, optimization. 1. INTRODUCTION The problem of defining the machining modes is the significant part of the design problem for paced multi-station transfer line. A considered paced transfer line consists of linearly ordered stations without buffers (see Fig.1). All tools are grouped in spindle heads. All the spindle heads of all stations at all working positions are operated simultaneously and all tools of the same spindle head are operated simultaneously as well. The machining modes of the transfer line are defined by the cutting modes of all spindle heads and the policy of the tools replacement. The cutting modes are defined as the spindle speeds of all tools and the feeds per minute of all spindle heads. Spindle head Tools …

Loading station

…

Workstation

Fig.1. A transfer line layout

978-3-902661-43-2/09/$20.00 © 2009 IFAC

Unloading station

It is assumed that all tools of the spindle head have the same feed per minute. It is also assumed, that formed cutting modes remain invariable during the machining process. The policy of tools replacement should ensure the replacement of each tool during its life time which depends on its cutting modes. The problem consists in determining the cutting modes of transfer line and a policy of tools replacement in the process of functioning to minimize unit production cost under the prescribed ranges of cutting parameters, under the constraints on cutting process characteristics and under the required productivity. Close problems can be divided into three groups: cutting modes optimization for multi-tool machining (see, for example, Levin and Tanaev, 1978; Cakir and Gurarda, 2000; Mukherjee and Ray, 2006); cutting modes and tool replacement optimization for one-tool machining (Liu et al., 1998; Liang et al. 2001; Hui et al. 2001; Shabtay and Kaspi, 2002, 2003); optimization of tool grouping for simultaneous replacement under given machining conditions (Yip-Hoi, Dutta, 2000). In (Levin and Tanaev, 1978) it is assumed that each tool can be replaced at any point during its life time. The problem considered in this work differs from the problem mentioned above in that the replacement of tools can be fulfilled by groups only in given points of time. Such approach allows to

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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

decrease the idle time of the equipment and the unit production cost. This paper considers the cutting modes optimization for multi-tool machining along with optimization of group tools replacement when any replacement can be realized only in given points of time. All tools of the same group are replaced simultaneously during their life time after processing the same number of parts where the number is taken from the given set. The list of used notations is presented in the end of the paper. 2. PROBLEM STATEMENT A transfer line that consists of a number of workstations with multi-spindle heads operating in parallel is considered. The machining of parts is fulfilled simultaneously by the heads from a set I = {1, 2, ..., i }. Each head i ∈ I contains a set J i = {i1, i 2, ..., iji } of tools. Spindle heads i ∈ I are activated simultaneously and are characterized by the given working stroke Li with the feed per minute Si to be defined. Each tool ij ∈ J i , i ∈ I is characterized by spindle speed nij to be

defined, given cutting length l ij , time τ ij = lij / Si of cutting of one part by this tool and by a number y ij (to be defined) of parts machined by this tool between moments of its replacements. Let x ij = ( S i , nij ) be the cutting modes of the tool ij, X i = ( S i , ni1 ,..., nij ) be the cutting modes of the head i, X = ( X 1 , ..., X i ) be the set of the cutting modes of the

transfer line. The functional dependences of life time Tij ( x ij ) and characteristics R pij ( xij ), p ∈ Pij such as force, torsion, power etc. for each tool ij on the cutting modes xij are Tij ( xij ) = min{Tijmax ;1 / biju ( Si α pij β pij

R pij ( xij ) = C pij Si

nij

nij

previously taken from the prescribed set of numbers H = {h1 , h2 , ..., hk }. So all tools with the same number yij are replaced simultaneously. Note, that each such group can include tools from different heads. The sought parameters are related by inequality Dij ( xij ) ≥ yij , where xij , yij η iju μ iju −1

Dij ( xij ) = Tij ( xij ) S i / lij = min{Tijmax S i / lij ; (biju S i

niju )

|

u = 1, ..., uij } is the maximal number of parts that can be

machined by the tool ij during its life time Tij ( x ij ) . The parameters of function Dij(xij) are defined by the parameters of Tij(xij) and the cutting length lij of the tool ij. The set of numbers Y = { yij , i ∈ I , j ∈ J i } define the stop times of the equipment for the tools replacement. Let zν (Y ) = 1 , if hν ∈ Y , zν (Y ) = 0 otherwise. So, if zν (Y ) = 1 then after machining each batch of hν parts the replacement of the respective group of tools should be done. Under assumptions accepted in (Levin and Tanaev, 1978) unit production cost Θ1 ( X , Y ) and unit production time Θ 2 ( X , Y ) are defined as follows: k

Θ q ( X , Y ) = E q t ( X ) + ∑ τ qν zν (Y ) + ∑ ∑ g qij / y ij , q=1, 2, ν =1

i∈I j∈J i

where t ( X ) = max{Li / S i + tbi | i ∈ I } is the cycle time; Li is the working stroke, tbi is the time of auxiliary movements of the head i; E q are given coefficients, that take into consideration (according to value of index q) reliability of the equipment, salary, capital depreciation etc; g qij is the time for replacement of tool ij (q=2) or the cost of its replacement (q=1); τ qν = τ q 0 / hν , τ q 0 is the time (q=2) or the cost (q=1) for one stop of the equipment. Then the mathematical model of defining the optimal machining modes of the considered design problem can be formulated as follows:

described by the functions: ′ μ iju ′ η iju

The replacement of tools is fulfilled by groups after machining by each tool ij of the same group some number of parts yij

) | u = 1,..., uij },

.

Maximal value Tijmax of tool ij life and other parameters of these functions are assumed known for given cutting conditions.

Θ1 ( X , Y ) → min; subject to Θ 2 ( X ,Y ) ≤ t0 ;

The ranges of possible values [σ i , σ i ], [n ij , nij ], [ s ij , sij ] of feed per minute S i , spindle speed nij and feed per revolution Si / nij are also assumed known, as well as maximal allowable

value R pij of the characteristics R pij ( xij ) of the cutting process

σ i ≤ Si ≤ σ i , i ∈ I ;

(2) (3)

n ij ≤ nij ≤ nij , i ∈ I , j ∈ J i ;

(4)

s ij ≤ S i / nij ≤ sij , i ∈ I , j ∈ J i ;

(5)

R pij ( xij ) = C pij S i α pij nij β pij ≤ R pij , p ∈ Pij , i ∈ I , j ∈ J i ;

for a tool ij, p ∈ Pij , and maximal allowable value R pi of the

R pi ( xij ) = ∑ R pij ( xij ) ≤ R pi (xij),

characteristics R pi ( X i ), p ∈ Pi for each i ∈ I .

p ∈ Pi , i ∈ I ;

(6)

j∈ J i

Dij ( xij ) ≥ yij ∈ H, i ∈ I , j ∈ J i .

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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

The objective function (1) is the unit production cost; constraint (2) provides the required productivity rate; constraints (3-7) provide the prescribed machining conditions hold; constraints (8) provide the replacement of each of tools during its life time after machining of the chosen for it number of parts, that not

k

Ω 2 (Y , t ) = E1t + ∑ τ 2ν zν (Y ) + ∑ ∑ g 2ij / yij ≤ t0 ; (10) ν =1

i∈I j∈J i

Li + tbi ≤ t , i ∈ I ; Si

exceed the maximum value of tool ij life Tijmax . The problem

(11)

Dij ( xij ) ≥ yij ∈ H, i ∈ I , j ∈ J i ; X ∈ X.

(1)-(8) will be addressed further as problem A.

(12) (13)

The problem A is a complicated problem of a mixed-integer programming. In the real life situations there are dozens of spindle heads and hundreds of tools.

The variables of problem B are components of vector ( X , Y , t ). The problem B is equivalent to the problem A from the

In (Levin and Tanaev, 1978) the method for a special case of this problem under simplifying assumption that each tool is replaced independently when its life time is over (for the current xij ) was

then ( X * , Y * ) is the solution of the problem A. Similarly if

proposed. The main distinctions in the problem statement are as follows. Unlike the model in question, H is the set of positive real numbers and consequently yij = Dij ( xij ) for any current value

xij ; besides, components ∑νk =1τ qν zν (Y )

of the

functions Θq ( X ,Y ), q=1, 2, are eliminated but values τ 10 and

τ 20 are added to the parameters g1ij and g2ij respectively for each tool ij. So, the problem in (Levin and Tanaev, 1978) is a significantly simpler geometric programming problem. 3. SOLUTION METHOD The presented method for the problem A solving is elaborated for the routine case: a) parameters C pij , α pij , biju ,ηiju , μiju > 0 for all p ∈ Pij , u = 1, ..., u ij ; b) for each i ∈ I , p ∈ Pi parameters β pij have the same sign (i.e. positive or negative) for all ij ∈ J i ; c) function Dij ( S i , nij* ( S i ))

is decreasing on Si , where

nij* ( Si )

is the

optimal value (in the problem in question) of nij under given S i . Method is based on the combination of fragmentary parameterization concept (Levin and Tanaev, 1978; Levin and Tanaev, 1998) and the coordinate-wise descent over groups of variables of the parameterized problem. This method is the development of the method proposed in the above-mentioned (Levin and Tanaev, 1978) and allows to reduce the solution of the initial problem A to the solution of the finite sequence of simpler interrelated subproblems. For the solution of these subproblems well-known methods such as dynamic programming, geometric programming, etc. can be used.

Denote by X the set of X values satisfying the conditions (3)(7). For solving the problem A, the cycle time t(X) in (1) and (2) is replaced by parameter t and additional constraints Li / S i ≤ t − tbi , i ∈ I are imposed. So the problem A is transformed into the problem B: Ω1 (Y , t ) = E1t +

k

∑ τ 1ν zν (Y ) + ∑ ∑ g1ij / yij → min; (9)

ν =1

i∈I j∈J i

viewpoint that if ( X * , Y * , t * ) is the solution of the problem B ( X * ,Y * ) *

is

*

the

solution

of

the

problem

A

then

*

( X , Y , t ( X )) is a solution of the problem B (see, for example, (Levin and Tanaev, 1998)).

Let X(t ) = { X ∈ X | t ( X ) ≤ t} and t ′ be the cycle time such that X(t ′) ≠ ∅. Denote by ( X ′′(t ′), Y ′′(t ′)) values of the required variables ( X , Y ) that minimize function Ω 2 (Y , t ′) under constraints (11)-(13) for t = t ′. Let t* (t ′) = min{t ( X ) | X ∈ X, Dij ( xij ) ≥ yij′′ (t ′), i ∈ I , j ∈ J i }. Then

a

set

T (Y ′′(t )) = {t ( X ) | X ∈ X , Dij ( xij ) ≥ yij′′ (t ′),

i ∈ I , j ∈ J i } is a segment [t* (t ′), t ′]. Note some properties of the segment [t* (t ′), t ′] : – Y ′′(t ) is invariable in this segment, therefore function ˆ Θ (t ) = Θ ( X ′′(t ), Y ′′(t )) = Ω (Y ′′(t ), t ) is increasing; 2

2

2

ˆ (t ) = min{Ω (Y , t ) | X ∈ X(t ), D ( x ) ≥ y ∈ H} –Θ 1 1 ij ij ij

is

the non-decreasing function of t at the segment [t* (t ′), t ′′(t ′)], ˆ (t ) ≤ t } . where t ′′(t ′) = max{t ∈ [t (t ′), t ′] | Θ *

2

0

3.1 Solution method for problem B The mentioned above properties are used in the iteration method for the solution of the problem B. At each iteration of this method a sequence of the following interrelated easier subproblems is solved. Subproblem B1. Find the value Y2* = Y2* ( X ) of vector Y to minimize function Ω 2 (Y , t ) under conditions ~ ~ yij ≤ yij ( xij ), yij ∈ H, when X, t and yij ( xij ) are fixed, i ∈ I, j ∈ Ji. Subproblem B2. Find the value X (Y ) of vector X to minimize function t ( X ) on the set X under conditions Dij ( xij ) ≥ yij , i ∈ I , j ∈ J i , when vector Y is fixed.

Subproblem B3. Find the value Y1* = Y1* ( X ) of vector Y, to yij ( xij ), minimize function Ω1 (Y , t ) under conditions yij ≤ ~ ~ y ∈ H, Ω (Y , t ) ≤ t , when X, t and y ( x ) are fixed, ij

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2

0

ij

ij

13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

i ∈ I, j ∈ Ji. Algorithm. ~ Step 0. Let t , t , be some upper t max and lower t min bounds

of t ( X * ) , such that X(t max ) ≠∅ and X(t min ) ≠∅. ~ Let X be some approximation Xˆ (t max ) ∈ X(t ) of vector ~ X ′′(t ); t = t ( X ); ~ y (~ x ) = D (~ x ). max

ij

ij

ij

ij

*

Let Ω >> 1. ~ ~ Step 1. Find a vector Y2 = Y2* ( X ) by solving the subproblem B1. ~ Step 2. Find X ′ = X (Y ) by solving the subproblem B2 for ~ ~ ~ ~ t = max [t , t ( X ′)]. Y = Y2 . Let X = X ′ and If ~ ~ Ω 2 (Y , t ) ≤ t 0 , then perform step 3. Otherwise perform step 4. ~ ~ Step 3. Find vector Y1 = Y1* ( X ) by solving the subproblem ~ ~ ~ ~ B3. If Ω1 (Y1 , t ) < Ω* , then let Ω*= Ω1 (Y1 , t ), ( X *, Y * ) = ~ ~ = ( X , Y1 ). ~ Step 4. If {(ij ) | Dij ( ~ xij ) = ~y ( xij ), i ∈ I , j ∈ J i } = ∅ or t = t ,

then stop.

To reduce the calculations and to obtain an approximate solution of the problem B the specified in the algorithm sequential enumeration of subintervals above mentioned can be substituted by their partial enumeration by random choice of some values of parameter t ∈[ t min , t max ] and consequent determination of left extremities of subintervals that contain these values t. The same goal can be achieved by using in the algorithm the following heuristic conditions for the process termination: - a value Ω* is invariable after implementation of a prescribed number of the following one after another iterations; ~ ~ - a value Ω1 (Y1 , t ) achieved at the step 3 of a current iteration is considerably greater than a current Ω*. To find bounds t max and t min for t(X*) the following approach is proposed. Let =max{ timin +tbi| Li / S imax , i∈I.

t max

= min{ timax +tbi| i∈I};

i∈I} , where

timax

=

Li / Simin ,

t min min ti =

To find S imin , i∈I, the following problem is to be solved: Si→ min; (= S imin ), subject to constraints σ i ≤Si≤ σ i ;

(15)

n ij ≤nij≤ nij , j∈Ji;

(16)

s ij ≤Si/nij≤ sij , j∈Ji;

(17)

~ ~ ~ Step 5. Let Y = Y ( X ), where ~ yij ( ~ xij ) = max{hν ∈H | Dij ( ~ xij )> > hν }. Perform step 1.

(14)

Rpij(xij)=Cpij Siα pij nij β pij ≤ R pij ,

The proposed algorithm reduces the solution of the initial problem A to the execution of a finite sequence of iterations. At each iteration j the point t j = t* (t j −1)< t j −1 is defined under fixed ~ ~ yij ( ~ xij ) , as well as the respective parameters X (subproblem ~ ~ ~ B2). Then under fixed X , t ( X ) the value Y2 is defined to ~ minimize the function Ω 2 (Y , t ) (subproblem B1) and if ~ ~ Ω 2 (Y , t ) ≤ t0 , then the value of vector Y1 is defined (subproblem B3) to minimize the function Ω1 (Y , t ). Therefore the parameterization of problem A allows to partition its feasible set into a finite number of subsets so that each subset corresponds to some subinterval of feasible ˆ (t ) and Θ ˆ (t ) are values of parameter t and functions Θ 1 2 non-decreasing and increasing respectively at this subinterval. Thus the solution of problem A is reduced to finding a subinterval of the parameter t with the minimal ˆ (t ) at its left extremity. The value of the function Θ 1 proposed algorithm forms the sequence of subintervals (in general, all) of the parameter t and their estimates in the order of decreasing left extremities of subintervals. The number of obtained subintervals does not exceed m× | H | , where m is the total number of tools.

p∈Pij, j∈Ji;

(18)

Rpi(xi)= ∑ R pij (xij)≤ Rpi , p∈Pi;

(19)

Dij(xij)≥h1∈H, j∈Ji.

(20)

j∈J i

To find S imax , i∈I, the following problem is to be solved: Si→ max; (= S imax ), subject to constraints σ i ≤Si≤ σ i ;

(21)

n ij ≤nij≤ nij , j∈Ji;

(23)

s ij ≤Si/nij≤ sij , j∈Ji;

(24)

(22)

Rpij(xij)=Cpij Siα pij nij β pij ≤ R pij , p∈Pij, j∈Ji;

(25)

Rpi(xi)= ∑ R pij (xij)≤ Rpi , p∈Pi;

(26)

Dij(xij)≥h1∈H, j∈Ji.

(27)

j∈J i

Both problems (14)-(20) and (21)-(27) are the convex problems in variables ln Si, ln nij. In fact these problems are the problems of finding the lower and upper bounds of value Si in the projection of the subset of feasible X of the set X on the axis of variable Si. If any of these problems is ill-posed problem then the initial problem A is also ill-posed.

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4. NUMERICAL EXAMPLE

Characteristics

So as the paper volume is restricted, below a proposed method is illustrated on a part of the real example of machining of a basic part of some unit of the tractor. In the complete example there are 60 tools united in 15 spindle heads. In the illustration part eight tools allocated in three spindle heads execute in parallel at several positions operations of drilling, core drilling and chamfering of different diameters.

R1ij , R2ij , R3ij denote respectively power

(Kwatt), force (n), torsion (n⋅m) for tool ij. The set of possible numbers of parts to be machined by different tools between moments of their replacements is H={20,35,50,100,150,200}, Tijmax = 60 min for all tools.

Table 3. An optimal solution for heads i 1 2 3

The input data for the spindle heads and tools are presented in Tables 1, 2.

Si 61.49 70.7 53.25

R1i 0.657 0.38 0.741

R2i 3138.65 1433.54 3488.4

R3i 15.39 11.56 18.42

Table 1. Spindle heads and their parameters

σi

σi

i

Li

1 2 3

84 29.9 81.3 0.2 0.7 4000 19 4 18.4 86.1 0.15 0.4 1450 12 78 33.5 78.8 0.2 0.8 4000 19

tbi

R1i

R2i

Table 4. An optimal solution for tools

R3i

11 12 ij nij 386.08 516.9

Table 2. Tools and their parameters i 1 2 3 ij 11 12 13 21 22 31 32 33 g1ij 8000 8000 4000 3000 3000 6000 6000 2000 g 2ij

20

20

20

10

10

30

30

30

lij

75

20

15

1.5

1.5

75

20

20

n ij

295.3 422.5 95.5

160

169

290

430

190

nij

386.2 552.5 318.5

295

290

380

550

310

s ij

0.08

0.06

0.31

0.11 0.11

sij

0.21

0.16

0.83

0.3

0.3

0.08 0.21

0.07 0.18

R1ij 0.426 0.24 0.028 0.19 0.19 0.426 0.25 0.13

C3ij 45.88 α3ij 0.8 β3ij -0.8 R3ij 10.55 bij

ηij μij

2940

2940

8036

5880

2352

1 -1

1 -1

0.8 -0.8

0.8 -0.8

1 -1

1096.6 64.97

705.6

735

1960

980

882

0.67 1.67

2.2 4.8

2.2 4.8

3.4 3.6

3.4 3.6

33

103.9

295

285.2

334.2

500

190

200

200

35

150

150

0.19

0.384

0.25

0.107

728.92 1849.3

980

659.15

3.75

4.116

yij

35

150

200

R1ij

0.426

0.222

0.0085

0.19

R2ij 1977.05 1096.6 64.97

704.6

R3ij

5.68

10.55

4.096

0.746

5.88

10.55

i is the total number of spindle heads in the transfer line; I = {1, 2, ..., i } is the set of spindle heads in the line;

4.096 0.755 5.68 5.88 10.55 4.096 5.39 3.4 3.6

32

6. LIST OF NOTATIONS

22.48 1.088 23.72 23.72 45.86 22.54 14.7 1 0.8 0.8 1 0.8 0.72 1 -0.08 -0. 27 -1 -1 -0.8 -0.8 -1

ji is the total number of tools in the head i ∈ I ;

3.7E-18 8.5E-21 1.1E-8 2.1E-21 2.1E-21 3.7E-18 8.5E-20 6.4E-21

3.4 3.6

31

A problem of optimization of the cutting modes and group policy of tools replacement for multi-tool parallel machining on multi-position multi-station transfer line is considered. It is a complex problem of mixed-integer programming. To solve the problem the special iteration decomposition procedure is proposed. The procedure reduces the solution of the initial problem to the solution of a sequence of interrelated easier subproblems. The parallel software for the proposed decomposition procedure is implemented. The elaborated programs for the solution of the problem in question can be run on computers both with single and multiple processors.

0.3

0.8 0.67 -0.08 -0.67

22

5. CONCLUSION

0.08

6022.1 92.32

21

An optimal solution is presented in Tables 3, 4. The running time of Algorithm on PC Pentium IV 2.4 GHz is 0.0238 sec. For complete example it takes 1.4297 sec. to achieve the solution.

C1ij 4.8E-3 2.4E-3 1.2E-4 2.7E-3 2.7E-3 5.0E-3 3.0E-3 2.0E-3 1 0.8 0.8 1 α1ij 0.8 0.8 0.72 1 0 0.2 0.2 0 β1ij 0.2 0.2 0.28 0 C2ij 8596.6 α2ij 0.8 β2ij -0.8 R2ij 1979.6

13

2.2 4.8

The parameters of the objective function are: E1 = 100, τ10 = 1000. The parameters of productivity constraint are: E 2 = 0.2, τ 20 = 10, t 0 = 3.5 min .

J i = {i1, i 2, ..., i ji } is the set of tools in the head i ∈ I ; ij is the index of the tool j of the head i ; Si is the feed per minute (to be defined) for the head i ; [σ i , σ i ] is the given range of admissible values of Si ; nij is the spindle speed (to be defined ) for the tool ij ∈ J i ;

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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

[n ij , nij ] is the given range of admissible values of nij ;

E1, E2, τ 1ν , τ 2ν , g1ij , g 2ij are the given parameters of the

[ s ij , sij ] is the given range of admissible values of the feed per

functions Θ1 ( X ,Y ) and Θ2 ( X ,Y ) .

revolution S i / nij ;

ACKNOWLEDGMENT

xij = ( Si , nij ) is the vector of cutting modes (to be defined) for

the tool ij ∈ J i ; X i = ( S i , ni1 ,..., nij ) is the vector of cutting modes (to be

The authors thank I. Oreschenkov for implementation of the proposed method.

defined) for all tools of the head i ∈ I ; X = ( X 1, ..., X i ) is the collection of cutting modes (to be

REFERENCES

defined) for all tools of the line; yij is the number (to be defined) of parts machined by the tool ij ∈ J i between moments of its replacements; H = {h1 , h2 , ..., hk } is the given integer set of possible values of yij ; Y = { yij , i ∈ I , j ∈ J i } is the set of numbers yij (to be

defined) for all tools; zν (Y ) is the auxiliary function that zν (Y ) = 1 if hν ∈ Y and zν (Y ) = 0 otherwise ( zν (Y ) = 1 if the line stops for replacement of some tool after machining of hν parts); lij is the given cutting length for the tool ij ∈ J i per one part;

Li is the given working stroke for the head i ; τ ij = lij / Si is the cutting time of one part by the tool ij ∈ J i ; Tij ( xij ) is the life time of the tool ij under fixed values of xij ; Tijmax is the given maximal possible value of the life time for

tool ij; Pij is the set of indices of the considered cutting process characteristics for the tool ij; R pij ( xij ) is the value of the characteristic p ∈ Pij for the tool ij under fixed values xij ; C pij ,α pij , β pij , η 'iju , μ 'iju , biju , uij are given parameters of the

functions Tij ( xij ) and R pij ( xij ) ; R pij

is the given maximal allowable value of the characteristic R pij ( xij ) for possible values of xij ;

Pi is the set of indices of the considered aggregated cutting process characteristics for all tools of the head i ; R pi ( X i ) is the value of the aggregated characteristic p ∈ Pi for

the tools of the head i under fixed values of X i ; R pi

is the given maximal allowable value of the

characteristic R pi ( X i ) for possible values of X i ; Dij ( xij ) is the maximal number of parts that can be

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machined by the tool ij during its life time Tij ( xij ) ; t ( X ) is the line cycle time under fixed X ;

Θ1 ( X ,Y ) is the unit production cost at the line under fixed values X, Y; Θ2 ( X ,Y ) is the unit production time at the line under fixed values X, Y; 1125