CHAPTER 4: OPTIMIZATION OF MULTI-PRODUCT, SINGLE-STAGE
5.3 Determining Optimal Machining Speeds - Unconstrained
5.3.6 Numerical example and consideration
According to the computational procedure developed in the previous sections, numerical examples for determining multiobjective optimal machining speeds to be set at multi-stages of the manufacturing system are solved. The basic data required for optimizing a multi-stage manufacturing system are shown in Table 5.1. The computational process and ranges of machining speeds to be set at production stages of the four-stage manufacturing system, through which raw materials are con
verted into products, are flowcharted in Figure 5.4. The bottleneck
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TABLE 5.1
Basic Data for Optimization of a Four-Stage Manufacturing System
Processing parameter Tool parameter
Stage
number Content Depth
of cut
Feed^
rate Work3 diameter,
tool diameter
Work length
Extra travel of tool in feed
No.
of teeth
1-min tool- life speed
Slope constant
j d 8 D L 1 Z C n
No. mm mm mm mm edge m/min
1
4
Turning 1.00 0.17 110 200 - 1 350 0.25
2 Boring 0.50 0.10 59 130 - 1 300 0.20
3 Milling 5.00 0.05 108 200 100 8 400 0.33
4 Drilling - 0.15 25 95 - 2 350 0.33
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TABLE 5.1 (Continued)
Time parameter Cost parameter
Stage number
Preparation time
Tool replacement
time
Direct labor
cost
Overhead cost
Machining overhead
Tool cost
Machining cost
Gross revenue
j t
P t
c k d k i k
m k t m r
u
No. min/pc min/edge $/min $/min $/min $/edge $/pc $/pc
1 1.50 1.00 0.15 0.35 0.10 2.50 4.00 -
2 2.00 2.50 0.20 0.50 0.25 7.00 - -
3 2.00 2.00 0.15 0.40 0.15 5.00 - -
4 2.50 0.50 0.10 0.15 0.15 6.00 - 20.00
* These data are not required in computation.
^ mm/tooth for No. 3 stage and mm/rev for other stages.
Work diameter for No. 1 and 2 stages and tool diameter for No. 3 and 4 stages.
4
Longitudinal.
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127
STAGE J
V (c) J (m/min)
t(c)J (min/pc)
1 177.8 3.938 (max.)
2 145.8 3.764
3 155.0 3.828
4 112.6 2.949
Step 2
1 K 1 1
Step 3
STAGE t) t(t)
J (m/min) (min/pc)
1 265.9 3.538
2 189.3 3.591 (max.)
3 251.9 3.508
4 348.3 2.713
k(-d + n~2)
177.8 - 231.9 145.8 - 189.3 155.0 - 251.9 112.6 - 348.3
Figure 5.4 Computational Process of Machining-Speed Ranges
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section, first, three performance measures for this manufacturing system are examined, and then a numerical example is shown.
As discussed in section 5.3.4.1, the minimum cycle time is achieved at:
v^ = 231.9 (m/min)
= 189.3 (m/min) v 3 = 197.9 (m/min)
= 112.6 (m/min)
At these machining speeds, the cycle time, the total production cost and the profit rate are 3.591 (min/pc), 14.519 ($/pc), and 1.526 ($/min), respectively.
On the other hand, the minimum total production cost can be obtained by setting the objectives as the following priorities.
Priority 1: Satisfy machining-speed and stage-production-time constraints.
Priority 2: Minimize the total production cost.
The following is a nonlinear goal-programming model formulation to determine optimal machining speeds.
(1) Machining-speed constraints
The machining speed at each stage must satisfy the machining speed constraints.
Gl: y l + n l - Pl = 231.9 (5.59)
G 2 : v 2 + n 2 - P 2 = 189.3 (5.60)
G3: v 3 + n 3 - p 3 = 251.9 (5.61)
G4: v 4 + * 4 - P4 = 348.3 (5.62)
G5: v l + n 5 - p5 = 177.8 (5.63)
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G6: v2 + n fi- p 6 = 145.8 (5.64)
G7: v g + n^ - p^ = 155.0 (5.65)
G8: v^ + n g - p g = 112.6 (5.66)
(2) Stage-production-time constraints
Stage production times at No. 2, 3, and 4 stages should be less than or equal to that at No. 1 (the bottleneck) stage; hence,
-8 3 241.00 0 /.-7Q„i n“ 10„4 _ n Qn „ .
(5.67)
— + 2.710xl0_8v 3 - 254,50 - 7.953x10_6v 2 - 0.50 + n 1(
- p 1Q = 0 O.bl
G11: 406•60 + 2.7 1 0x 10 '8v 2 - - 5.563x10"7v 2 - l.00 + n n
V 1 V4
- p n = 0 (5.6‘
In equations (5."66) to (5.68), the negative deviation variables, n 9~n ll* must zeros in order to satisfy the stage-production-time constraints.
(3) Total-production-cost goal
Since the goal is minimization of the cycle time, the objective value of this goal can be set to 13.00 ($); hence,
v 2 3
+ 1.399x10-7 v 2 ^ + 853-' -9- + 1.219x10-7 v 3 + 7.00
v4 I Vl
p = 13.00 (5 .7 0 )
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means minimizing the total production cost. Also, all decision vari
ables and deviation variables of the above objective functions are non
negative.
The achievement function of this problem is expressed as:
lexico-min a = { (p1+p2+ P 3+ P 4+ n 5+ n 6+ n 7+ n 8+ n g+ n 10+ n 11) , (p12)} (5.71)
The following solution is obtained on this problem:*
v^ = 200.7 (m/min)
= 148.3 (m/min)
*
v^ = 166.3 (m/min)
*
v^ = 112.6 (m/min) a* = (0, 0.898)
At these optimal machining speeds, the cycle time, the total production cost, and the profit rate are 3.745 (min/pc), 13.898 ($/pc), and 1.630 ($/min). respectivply.
In order to examine performance measures in the cycle time between the two situations mentioned earlier, several sets of optimal solutions are obtained by setting the objectives as the following priorities.
Priority 1: Satisfy machining-speed and stage-production-time constraints.
Priority 2: Achieve the cycle time of less than b£ (min/pc).
Priority 3: Minimize the total production cost.
As the objective values of b fc, 3.62, 3.64, 3.66, 3.68, 3.70, and 3.72 (min/pc) are set in priority 2. Results are summarized in Figures 5.5
*See Appendix A for a discussion of computation algorithms and related problems.
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131
and 5.6. Figure 5.5 shews the machining speed at each stage, and Figure 5.6 shows the total production cost and the profit rate in relation to the cycle time. At No. 4 stage, the machining speed obtained is always the minimum-cost machining speed, and waiting time exists. At the other stages, there is no waiting time. The total production cost reduces as the cycle time increases, and the maximum profit rate is obtained some- vrtiere in this cycle-time range in this manufacturing system.
Example 5 . 1 : Suppose that the production goals are set as the following priorities from.the managerial standpoint.
Priority 1: Satisfy machining-speed and stage-production-time constraints.
Priority 2: Achieve the profit rate of at least 1.62 ($/min).
Priority 3: Produce a unit piece at a cost of less than 14.00 ($/pc).
Priority 4: Minimize the cycle time.
G1 through Gil (i.e., (1) machining-speed constraints, and (2) stage-production-time constraints) are the same as stated earlier in this section.
(3) Cycle-time (production-time) goal
Since the goal is minimization of the cycle time, the objective value of this goal can be set to 3.59 (min/pc); hence,
G 1 2i 4 0 6 + 2.710x10"8vJ + 1.50 + n 12 - p 12 = 3.59 (5.72) V l
Minimizing the positive deviation variable in equation (5.72), P ^ ’ means minimizing the cycle time.
(4) Total-production-cost goal
The objective value of this goal is 14.00 ($/pc); hence,
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220 210 200 190 180 170 160 150 140 130 120 110
3.60 3.62 3.64 3.66 3.68 3.70 3.72 3.74 Cycle Time tCT min/pc
Figure 5.5 Optimal Machining Speeds at Corresponding Cycle Time
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14.6 14.5 14.4 14.3 14.2 14.1 14.0 13.9 13.8
profit rate
total production cost
3.60 3.62 3.64 3.66 3.68 3.70 3.72 3.74 1.65 1.64 1.63 1.62 =
.E 1.61 I 1.60 * i.5 9 | 1.58 1.57 1.56 1.55 1.54 1.53 1.52
Figure 5.6 Total Production Cost and Profit Rate at Corresponding Cycle Time
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(5) Profit-rate goal
The objective value of this goal is 1.62 ($/min); hence,
G14: 16.00 - 406.60 1.666xl0-10v^ j - 241.00 | + 2.88lxlO-1^ j - 254.50 | + 7.813xl0~8v 3J - 47.79
406,60 + 2.710x10~8v^ + 1.50 1 .399x10_7v?
4
4 U / L V1
- 2.00 + n u - p u = 1.62 (5.74)
In the case that the negative deviation variable in equation (5.74), n^4 , is zero, the profit rate is more than or equal to 1.62 ($/min), and this goal (the aspiration level for the profit rate) is satisfied.
Also, all decision variables and deviation variables of the above ob
jective functions are non-negative.
The achievement function of this problem is expressed as:
lexico-min a = { ( p ^ p ^ p ^ ^ ^ g + n ^ g + n g + n ^ + n j ^ J , (n14) , (p13) , (P12)>
(5.75)
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135
The following solution is obtained for this problem:
v^ = 213.0 (m/min)
= 159.6 (m/min)
= 178.9 (m/min)
*
= 112.6 (m/min) S * = (0, 0, 0, 0.081)
In this multiobjective optimal solution, priorities 1 through 3 are completely achieved. In other words, the optimal solution satisfies machining-speed constraints, and unit production times at the slack stages are equal to or less than that at the bottleneck stage, i.e., No. 1 stage. Furthermore, this solution achieves both the profit-rate goal and the total production-cost goal. Therefore, it may be con
cluded that this optimal solution minimizes the cycle time of the multi-stage machining system,satisfying the other production goals and constraints. Actually, at the optimal machining speeds, the profit rate, the total production cost, and the cycle time are 1.64 ($/min), 14.00 ($/pc), and 3.67 (min/pc), respectively. Also, in this case slack time exists at No. 4 stage.