CHAPTER 4: OPTIMIZATION OF MULTI-PRODUCT, SINGLE-STAGE
4.3 Determining Optimal Machining Speeds and Feed Rates -
4.3.3 Algorithmic procedure for determining optimal
4.3.3.1 Determining optimal feed rates and machining-
ing speed or the motor-power constraint of the machine tool employed.
The aforementioned discussion leads to the following computation procedure for obtaining optimal feed rates and corresponding machin- ing-speed ranges for single-stage manufacturing systems under a GT environment.
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60P Calculate v. . = —
ct. . * 8 . . 9 8k d ^
fij ij ij
i = l ,2,... ,M (4.45)
Then, the maximum allowable machining speed is determined by:
± , i=l,2,...,M (4.46)
Therefore, the range of the machining speed for each job is:
v - • Ê t . , v ij J-dLj* maxijj ’ j=lằ2,... ,N., J i ’ i=l,2,...,M (4.47)
4.3.3.2 Determining optimal machining speeds (Phase II). The problem, then, is determining the set of machining speeds under multi
ple production goals with the machining-speedconstraints; essentially, it is the same as discussed in section 4.2. In this problem, the feed rates, s^_., are no longer decision variables, since they are set to s*. given by equation (4.44); hence, they are constants. Therefore, the decision variables are machining speeds, v „ , j=l,2, — ,N^, i=l,2,___,M. The following is the nonlinear goal-programming formula
tion of this problem.1
^The baseline model of this formulation consists of the one mentioned in section 4.2.4.1, and machining-speed constraints.
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95
Find v „ , j=l ,2,... ,N^, i=l,2,...,M, so as minimize
N 2N
a = { (Ê Pk + Z ^ + P 2N + 1) ằ A } k=l k=N+l k ^N+1
where A is the combination of (P2jj+ 2^ ’ ^p 2N+3^ anc^
sidering priorities and/or weighting factors.
Priority 1 :
G1 - G(2N) (machining-speed constraints):
+ “k - - V J - 1 -2 H i-
G(2N+1) (available-limited-time constraint):
\n .. Z
i=l
_Oij_
+ ^ snM0irlvi>rl)}] + n2N+i.
Priority 2
G(2N+2) (total-production-time goa l ) :
i-i L 1 t-i 1 13 13 r 13 V i*
S ij Vij
•All Jl. + 1
to lexicographically
(4.48)
(nW ' con~
,M, k=I,2,...,N (4.49) ,M, k=N+l,...,2N
(4.50)
P2N+1 b 2N+l (4.51)
P 2N+2 b 2N+2 (4.52)
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m .. + k,L . . cxj 1 pij
+ (k t .. T k ) -r^ - s * y m 0ij-1v . y n0ij 1 1 cij tij CQij ij J ij J
G(2N+4) (profit-rate goal):
M N.
Z z1
1=1 j=l 1=1
\n.
N - f
k S. + Z1 J k S . 1 1 j-i 1 1 :
+ (k, + k ..) - 5 ^ — + (k. t .. + k ..) 7T 1 m ij7 o* 1 cij tij C.
Oij
M F N. f I Xn..
Z S. + Z1 J S . . + 1.. t ..
i=i L 1 j - i i 1J 1 J \ P1J w .ij ij
+ ' 5 s i f i s ô /” o i j ' lv ô / ° o i r j
(4.54) 4.3.4 Numerical example and consideration
According to the computational procedure developed in the previous section, a simplified example for determining multiobjective optimal machining speeds and feed rates is solved.
Example 4 . 3 : Supposing the basic data to be shown in Table 4.3, the computational process of Phase I is flow-charted in Figure 4.4.
Parts are grouped into three part families in terms of dimension and surface-roughness goal. The constraints on the machining speeds, v „ ,
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TABLE 4.3
Basic Data for Constrained Optimization of a Manufacturing System Under a GT Environment
Work parameters
Machining parameters
Group Part Lot size
Machining diameter
Machining length
Depth
of cut a 8
Specific inachining resistance
Nose radius of cutting tool
i j
D ij L u d u k fij Ru
No. No. pcs. mm mm mm kg/mm^
1 50 60 110 1.00 0.85 0.85 500 0.80
1
2 100 80 130 1.00 0.85 0.85 500 0.80
1 30 80 200 1.00 0.85 0.85 500 0.80
2
2 75 90 200 1.00 0.85 0.85 500 0.80
1 80 130 300 1.00 0.85 0.85 500 0.80
2 40 160 350 1.00 0.85 0.85 500 0.80
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TABLE 4.3 (Continued)
Tool-life Time
parameters parameters
Group Part
m 0ij "Oij C0ij
Group set-up time
Job set-up
time
Preparation time
Tool-replacement time
i i Si
s o V j ‘cij
No. No. min min/lot min/pc min/edge
1 1 2
0.50 0.52
0.20 0.20
4.73xlOU 3.65xl0U
15.00 5.00 7.00
1.00 0.75
0.50 0.75
2 1 2
0.55 0.53
0.33 0.25
2.89xl06 1.97xl09
22.00 8.00 11.00
2.00 1.50
1.00 1.00
3 1 2
0.55 0.53
0.33 0.25
1.90xl06 1.14xl09
28.00 18.00 16.00
4.00 3.00
2.00 1.50
TABLE4.3(Continued)
99
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Step 1
MOTOR POWER - 6500 (W) OTHER ASSOCIATED CONSTANTS LISTED IN TABLE 4.3
V y - 400.0 (m/min) v . * 90.0 (m/min)
OPTIMAL FEED RATE s* (mm/rev)
Figure 4.4 Computational Process of Phase I in Example 4.3
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101
90 < v± . < 400, i=l,2,...,N^, j=l,2,...,M (4.55)
0.080 <. s _ < 0.250, 1=1,2,...,!^ j=l,2,...,M (4.56)
P = 6500 (W) (4.57)
Suppose that the following priorities are set from the managerial standpoint.
Priority 1: Satisfy the machining-speed constraints and produce all parts in the available limited time (1580 min).
Priority 2: Minimize the total production cost.
The following is a nonlinear goal-programming model formulation to determine multiobjective optimal machining speeds of Phase II, since optimal feed rates have been determined.
Find v „ so as to lexicographically minimize
a = {(p1+P2+P3-^4+p5+P6-Hi7+n8+n9+n10-ta11+n124p13), (pu )}
(4.58) Priority 1 :
(1) Machining-speed constraints
The machining speeds, v ^ , must satisfy the machining-speed constraints which are obtained in Phase I.
61: V 11 + n l - P i =281.5 (4.59)
G 2 :
V 12+ n 2 - P 2 - 281.5 (4.60)
G 3 :
V 21 + n 3 - p 3 =267.7 (4.61)
G4: V 22 + n 4 - P 4 "267.7 (4.62)
G 5 :
V 31+ n5 - P 5 - 258.6 (4.63)
G6: V 32 + n 6 - p 6 =258.6 (4.64)
G 7 :
V 11 + n 7 - P 7 = 90.0 (4.65)
G8: V 12 + n8 - p 8 =90.0 (4.66)
G9: V 21 + n 9 P 9 = 90.0 (4.67)
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611: V 31 + n ll - p ll = 90.0 (4.69)
612: v 32 + n 12 " p 12 = 90.0 (4.70)
(2) Available-limited-time constraint The limited time available is 1580 (min), and all parts have to be done within the limited time; hence,
4587.28 ^ „ ,7, „ in-10 4 ^ 14456.89 .
G13; + 2.477x10 V- - T T J. • / U 1 A JL U V-2 11 v 12
• ^ 3 , 1 2 + ! . 6 2 3 x l o - V - M + I 2 6 7 M 6 + 6 . 0 7 M | ) - 7 3 _
v n 2l v 22 22
• + S . S I W O - 3^ ; 03 + * 2.708X 10-ôv3
v 32 31 v 32 32
+ 737.50 + n 13 - p 13 = 1580 (4.71)
In the case that the positive deviation variable in equation
to 1580 (min), and this constraint is satisfied.
Priority 2 :
(3) Total-production-cost goal.
Since the goal is minimization of the total production cost, the objective values of this goal can be set to 1755 ($); hence, 014; 2523,00 + 5 . 2 0 ^ 0 - 1 0 4 + 8095,86 + 2 .665xI0-9 4
V U 11 v 12 12
+ 358 L 42 + 3 .084x10-4 2.03 + 9 8 9 6 ^ 2 +
V 21 11 V 22
+ 2 3 1 3 2 0 8 + 4 .646xl0-3 2.03 + 1 6 3 2 6 0 3 + 4 .423x10-5 3
V31 31 v 32 32
+ 1229.75 + n 1A - p., = 1755 (4.72)
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103
Minimizing the positive deviation variable in equation (4.72), P ^ > means minimizing the total production cost. Also, all decision and deviation variables of the above objective functions are non
negative.
The following solution is obtained for this problem:*
V 11= 272.3 (m/min)
*
V 12= 245.3(m/min)
*
V 21= 179.3 (m/min)
*
V 22= 244.2 (m/min)
*
V 31= 152.5 (m/min)
*
V 32= 207.5 (m/min)
*
a = (0, 13.269)
In this solution, priority 1 is achieved, i.e., machining- speed constraints are all satisfied, and all parts are able to be machined within the available limited time. Therefore, it may be concluded that the minimum total production time is achieved with the above machining conditions, including the results obtained in Phase I.
At the optimal machining conditions, the total production time is exactly 1580 (min) and the total production cost is 1768.269 ($).
4.4 Conclusions
1. Optimization analysis of multi-product, single-stage manufacturing systems, especially under a GT environment, was conducted for both unconstrained and constrained optimization.
2. The minimum total production time is achieved at the minimum- time machining speeds, and the minimum total production cost is
*See Appendices A and B for computer codes and algorithms.
This solution is the approximate solution to this problem.
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produced. However, the maximum profit rate in a multi-product manufacturing system is not achieved at the maximum-profit-rate machining speed; it may be obtained when the marginal cost is equal to the marginal revenue on each part. A numerical computational procedure is developed for obtaining the solution.
3. A basic nonlinear multiobjective (goal-programming) model was built for multi-product, single-stage manufacturing systems in which parts are grouped into several part families in terms of dimension, shape, and so on. Also, characteristics of three per
formance measures were examined via a numerical example.
4. With the use of the nature of the nondominated solution set, an algorithmic procedure to obtain multiobjective optimal machining speeds and feed rates in constrained optimization was proposed.
5. Numerical examples were shown for solving multiobjective problems of multi-product, single-stage manufacturing systems.
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CHAPTER 5