CHAPTER 4: OPTIMIZATION OF MULTI-PRODUCT, SINGLE-STAGE
4.1 Multi-Product, Single-Stage Manufacturing Systems
4.1.2 Optimal machining speeds under fundamental
turing systems under fundamental evaluation criteria for a fixed depth of cut and a fixed feed rate are determined or characterized as follows.
Lemma 4 . 1 ; The minimum total production time is achieved at the minimum-time machining speeds on all parts (or jobs) to be produced.
Proof: Let N be the number of parts to be processed. The total production time is given by:
1Parts to be made are called "jobs."
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N
(4.1)
Now, t.(v.), i=l,2,...,N, are unimodal, convex functions, and twice—differentiable functions. Also, they are minimum at the mini
mum-time machining speeds. Therefore, the minimum total production time is achieved at the minimum-time machining speed on all parts (or jobs).
In a similar fashion, the following lemma can be obtained on the minimum total production cost.
Lemma 4 . 2 : The minimum total production cost is achieved at the minimum-cost machining speeds on all parts to be produced.
Lemma 4 . 3 : In the case when the set of machining speeds in multi-product, single-stage manufacturing systems under maximum- profit-rate criteria is obtained, the marginal cost is equal to the marginal revenue on each part.
P r oof: The profit rate is given by:
t(v) = the total production time (min)
The set of machining speeds under maximum-profit-rate criteria is determined by taking partial derivatives of p(v) (equation (4.2)) with
N
p(v) ---
t(v) (4.2)
u(v) = the total production cost ($)1
*In this section, u(v) stands for the total production cost exclusive of material cost.
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respect to the machining speed, i=l,2 ,...,n and setting them equal to zero:
= 0, i=l,2,... ,N (4.3)
3 I r /t(v) 3 u(v)/t(v) i=l 111 ______________
i=l,2,...,N
3t_
- 1 = ^ , i=l,2,... ,N (4.5)
whe r e : N
rT = .J‘1rni^t ^v ^ : the total revenue per the unit time interval ($/min)
uT = u(v)/t(v): the total production cost per the1 unit time interval ($/min) In equation (4.4), the left term is the marginal cost and the right term is the marginal revenue on each product.*
Therefore, in general, the maximum profit rate in a multi-product manufacturing system is not achieved at the maximum-profit-rate
This is equivalent to the fundamental economic principle, i.e., "the maximum profit requires that the marginal cost be equal to the marginal revenue" (42). In the field of economics of manufacturing, the case of single-product manufacturing systems has been discussed by Wu and Ermer (7), and Hitomi (8). The above statement is the generalized extension to multi-product manufacturing systems for obtaining the max
imum profit rate. This is a necessary condition; however, it might also be a sufficient condition for the existence of the maximum profit rate in this problem.
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of machining speeds satisfying equation (4.4).
Example 4 . 1 : A simplified example is shown for illustrating this problem. Suppose that two different kinds of jobs (Jj^ and J2>
are to be processed. The data used for numerical computation are summarized in Table 4.1. At the maximum-profit-rate machining speeds (v(p) = 158 (m/min) and = 214 (m/min)), the production cost and the production time are:
Production cost Production time ____________($/pc)____________________ (min/pc)_____
Jx 1.0485 2.6342
J2 4.9904 8.4639
Therefore, the profit rate is obtained from equation (4.2), where p (p) = 0.8344 ($/min).
On the other hand, the set of machining speeds which provides the maximum profit rate is obtained by searching machining speeds satisfy
ing equation (4.4)*:
As discussed in section 3.1.2, in general, the machining speeds pro
viding the maximum profit rate cannot be expressed explicitly. In this section, a numerical computation procedure is developed for ob
taining the solution, as follows. , s
Step I: Calculate the minimum-cost machining speed, v^ (by equation (3,6)), and the maximum-production-rate machining speed, v. ' (by equation (3.5)).
Step II: ^5et i=l.
Step III: Fix machining speeds of other jobs for integer values in the high-efficiency speed range. ^ , Step IV: Select an initial machining speed, v ^ greater than y} ,
and also an appropriate machining speed increment Av/T^O) (for example, 1 m/min).
Step V: Find the machining speed satisfying 4 <cT ).. < 4 (rT ). and 4fcT )j+1 > 4 <rT )j+1
where: A (cT ) . = the increment of the total production cost
^ per the unit time interval at the j th iteration ($/min).
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TABLE 4.1
Basic Data for Numerical Computation of Example 4.1
Job 1______ Job 2 Material
parameter
Work diameter Length
D L
mm 130.00
300.00 10.00 63.00 Machining
parameter Feed rate s mm/rev .15 .10
Tool-life parameter
Slope of Taylor tool-
life curve n .33 .33
1-min tool- life machining
speed C 400.00 300.00
Time parameter
Preparation
time t
P min/pc 3.50 2.50
Tool-replace
ment time Cc min/pc 2.00 .50
Cost parameter
Direct labor
cost k d $/min .16 .10
Overhead cost k.
i $/min .24 .25
Machining cost
km $/min .15 .15
Tool cost
k t $/edge • 1.80 6.00
Revenue r -m
u c $/pc 10.30 5.00
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The production cost and the production time are:
Production cost Production time ($/pc)___________________ (min/pc)_____
J x 1.0331 2.6475
J 2 5.0141 8.4318
Then, the profit rate is obtained:
p* = 0.8351 > 0.8344 = p (p)
Therefore, the maximum profit rate is obtained not at the maxi- mum-profit-rate machining speeds, but at the set of machining speeds satisfying equation (4.4).
The profit rate in relation to machining speeds is shown in Figure 4.1. Curves stand for contour lines of the profit rate, and the dotted lines represent machining speeds which satisfy equation (4.4), fixing another machining speed. From the contour lines of the profit rate, and the shape is fairly flat around the maximum point.
A (r ). = the increment of the total revenue per the unit -1 time interval at the jth iteration ($/min) Compute the profit rate.
Step VI: Repeat Steps IV and V for all combinations of appropriate fixed machining speeds in the high-efficiency speed range on other jobs.
Step VII: Set i=i+l. If i>n, go to Step VIII. Otherwise, go to Step III.
Step VIII: Identify the set of machining speeds which provides the maximum profit rate. Stop.
The above numerical computation procedure yields the set of machining speeds which provides a near-maximum profit rate in multi
product manufacturing systems.
*At these machining speeds, the marginal costs (or the marginal revenues) of and j£ are 0.11x10”^ ($/min) and 0.75xl0- ^ ($/min), respectively. (Av^=Av2=l m/min)
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MachiningSpeed ofJj
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320 310 300 290 280 270 260 250 240 230 220
■0.8330 $ln 210
200 190 180 170
.p-0.8360 S/mir 160
150 140 130 120 110 100 90 80
0 160 170 180 190 200 210 220 230 240 250 260 270 280 Machining Speed of Jj m/nin
Figure 4.1 Profit Rate in Two-Product Manufacturing
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