2.3 Production Goals in Manufacturing Activities
2.3.2 Aspiration levels on production goals
Depending upon the objectives of the decision maker, the pro
duction goals may be set to achieve some levels (aspiration levels) on the production rate (or the production time), the production cost, and/or the profit rate. In such cases, the production objectives are expressed in aspiration-level goals (35):
production rate: q > b^ (2.24)
production time: t < b^ (2.25)
production cost: u < b^ (2.26)
profit rate: p > b^ (2.27)
where b , b , b , and b are aspired levels on corresponding measures
q t u p
of performance.^
The production goals are employed according to the manufacturing objectives. An optimal selection from among them should be done from
2 the managerial standpoint.
inequalities (2.24) to (2.27) are represented as three major production goals in the baseline model of goal-programming formulation.
2Methods on priority-selection from among production goals will not be discussed in this study. For priority structure, several references are available, for example, (36).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 3
OPTIMIZATION OF SINGLE-PRODUCT, SINGLE-STAGE MANUFACTURING SYSTEMS 3 *1 Determining Optimal Machining Speeds - Unconstrained Optimization 3.1.1 Characteristics of functions of performance measures
Optimization analysis of the single-stage manufacturing system is fundamental to the optimization of the integrated manufacturing systems.
It is necessary, first, to examine the characteristics of functions of performance measures, i.e., the unit production time (or the production rate), the unit production cost, and the profit rate.
Basic mathematical models of these performance measures as a function of machining speed have been constructed as equations (2.2), (2.4), and (2.6). In this section, some simplified notation is used for simplicity of discussion.1
Definition 3 .1: A controllable continuous variable which specifies the production time and cost is the machining speed, and it is denoted by v. v is feasible if 0 < v < ®.
3.1.1.1 Unit production time. The time function consists of a fixed time and variable (inversely and positively proportional) time elements, and can be expressed simply as
t(v) = a + ^- + cvn (3.1)
where a, b, c, and n _> 1 are positive constants.
Property 3 . 1 ; The time function t(v) is a positive, unimodal twice-differentiable, and strictly convex function.
1 For a more detailed discussion, see reference (37).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is global minimum.
3.1.1.2 Unit production co s t . The cost function (equation (2.4)) consists of a fixed cost and variable (inversely and posi
tively proportional) cost elements, and can be expressed as
u(v) = a 0 + f (3.2)
where a^, 6, y, and m are positive constants.
Property 3 . 2 : The cost function u(v) is a positive, unimodal, twice-differentiable and strictly convex function.
There exists a machining speed at which the cost function is global minimum.
3.1.1.3 Profit. The profit function p(v) is given by a unit revenue r from which is subtracted the cost function.
p(v) = a - ^ - yvm (3.3)
where a = r^ - ag is a positive constant.
This function is strictly concave and is assumed to be positive.
3.1.1.4 Profit rate. The profit-rate function (equation (2.6)) is the magnitude of the gain produced in a unit time, and is expressed as:
r(v) == p (v ) t(v)
6 m
a yv
Property 3 . 3 : The profit-rate function, r(v), is positive, continuous, and has a unique maximum point somewhere between the minimum-cost and minimum-time machining speeds.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
21
3.1.2 Optimal machining speeds under three kinds of evaluation criteria
Optimal machining speeds under three kinds of evaluation criteria for a fixed depth of cut and a fixed feed rate are determined by differentiating these equations with respect to the machining speed, v, and setting them equal to zero.
The maximum-production-rate or minimum-time machining speed is:
v (t) = c / [ ( l / n ) t j n (3.5)
The minimum-cost machining speed is:
v (c) = C [(1/(1/n-l)) (k.+k )/(k.t +k )1 n (3.6)
1 1 m l c t I
The maximum-profit-rate machining speed, v^P \ is determined by:
(1-n) (k.t +r t ) v (p)1/n + A(k -k t ) v (p)1/n-1-nC1/n(r + k t ) = 0
t p n c t m e n m p
(3.7) where r = r - m .
n u c
The maximum-profit-rate machining speed, v^P \ can be explicitly expressed from equation (3.7) for particular values of n, the constant showing the slopes of the Taylor tool-life diagram, i.e., 1, 1/2, 1/3, and 1/4, since this equation becomes the algebraic equations of 1st, 2nd, 3rd, and 4th degrees, respectively. In general, however, an approximate value for v^p ^ should be obtained by numerical computation (8).
A speed range between the maximum-production-rate machining speed, v^C \ and the minimum-cost machining speed, v^C\ is called the high-efficiency (Hi-E) (speed) range (38,39), or non-inferior range, which means that any machining speed in this range is preferable from the managerial standpoint.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the high-efficiency range, i.e., v^C/‘ < < v ^ . * The high-efficiency range is described by Siekman (39) as
Tool engineers are generally interested in machining parts at the lowest possible cost or at the greatest possible production rate . . . . The Hi-E principle can tell them the range of machining speeds in which they should operate and can help them select the optimum speed within that range . . . . The exact speed you select will depend on whether you are emphasizing low cost or high rate of production.
Any speed within the Hi-E range will be a compro
mise between the two factors, whereas any speed outside the range will sacrifice both of them.
(pp. 97-98)
The unit production time, the unit production cost, and the profit rate given by equations (2.2), (2.4), and (2.6), respectively, are functions of machining speed with minimal or maximal points at the optimal machining speeds. Figure 3.1 shows such curves versus machining speed for the production data (shown in Table 3.1). The time and cost curves are fairly flat at their minimal point. Hence, increase in unit production time or cost is small even if the machining speed deviates from the optimal values. However, deviation of machining speed should be directed towards the inside of the high- efficiency range so that the increase in unit production cost or time can be kept lower. This is why the high-efficiency range may be called a non-inferior range. Machining speeds in this range are the high-efficiency (or non-inferior) machining speeds and are to be employed in preference to machining speeds outside the range.
1See reference (37) for a proof of this theorem.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
23
TABLE 3.1
Basic Data for Numerical Computation of Machining Speeds for the Single-Stage Manufacturing System.
Tool-life parameters
slope constant, n 0.23
1-min tool-life machining speed, C (m/min) 430.00 Machining parameters ^
depth of cut, d (mm) 1.00
feed rate, s (mm/rev) 0.20
Work parameters
work diameter, D (mm) 50.00
work length, L (mm) 200.00
Time parameters
preparation time, tp (min/pc) tool-replacement time, t£ (min/edge)
0.75 1.50 Cost parameters
direct labor cost, k ($/jnin) overhead cost, k^ ($/min)
0.15 0.35
machining overhead cost, k ($/min) 0.05
tool cost, k ($/edge) m 2.50
material cost, m ($/pc) 2.00
gross revenue, r^ ($/pc) 7.00
^This information is not required in the computation process.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.0
2.0
1.0
0
100 150 200 250 300 350
MacMnino Speed v m/min
Figure 3.1 Unit Production Cost, Unit Production Time and Profit Rate versus Machining Speed
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25
Therefore, machining speeds in the high-efficiency speed range are to be preferred solutions in multi-objective optimization consider
ing three production goals. The machining speeds in the high- efficiency speed range construct nondominated (Pareto optimal, or efficient) solution sets.*
Based upon the previous discussion, the examples for determining optimal machining speeds are solved. For the examples in sections 3 .1.2 and 3.1.3, the data used for numerical computation of the optimal machining speeds for a single-stage manufacturing system are summa
rized in Table 3.1.
Based upon the data, the optimal machining speeds are calculated by equations (3.5), (3.6), and (3.7):
The minimum-cost machining speed (m/min) v (c) = 216
The maximum-profit-rate machining speed (m/min) v ^ = 271 The maximum-production-rate machining speed v ^ = 296 (m/min)
Figure 3.2 shows the curves of three performance measures and the first derivatives of them with respect to the machining speed in the high-efficiency speed range (216 <_ v <_ 296). Obviously, the first derivatives of the functions of the unit production cost, the profit rate, and the production time vanish at the minimum-cost machining speed, the maximum-profit-rate machining speed, and the minimum-time machining speed, respectively.
*See reference (30), pp. 186-187, for a definition of the nondomi- nated solution set.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2.84 2.82 2.80 2.78 2.76 2.74 2.72 2.70 2..68 .66 .64
5
3 2 1 0
■1 -2 -3
Figure 3.2 Functions of Performance Measures and Their Derivatives in the High-Efficiency Speed Range
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
27
3.1.3 Optimal machining speeds under multiple objectives As discussed in section 2.3, when optimizing machining speeds under multiple objectives in this study, the conflicting objectives of the production rate (or the production time), the production cost, and/or the profit rate are ranked and/or weighted. Together they compose an integrated production goal. The following two examples might show typical situations of this problem.
Example 3 . 1 : Suppose that the production goals are set as the following priorities from the managerial standpoint:
Priority 1: Achieve a profit rate of at least 2.75 $/min.
Priority 2: Achieve a unit production cost of no more than 2.95
$/pc.
Priority 3: Minimize the production time.
The machining speeds in the high-efficiency speed range are enough to be considered in order to obtain the multiobjective optimal machining speeds. The high-efficiency speed range is 216 < v < 296. In priority i, first, when v = 235 m/sec, the profit rate is 2.75 $/min, as indicated in Figure 3.3. The profit rate at v = 296 m/sec (i.e., minimum-time machining speed) is more than 2.75
$/min. Hence, the speed range satisfying priority 1 is 235 < v < 296.
Next, when v = 273 m/sec, the unit production cost is 2.95 $/pc.
Hence, the speed range satisfying both priorities 1 and 2 is 235 <
v < 273 (Figure 3.4).
According to priority 3, the machining speed should be maximized in order to minimize the production time. Therefore, the multi
objective optimal solution is v* = 273 m/min
Figure 3.5 shows this optimal solution.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
220 230 240 250 260 270 280 290 Machining Speed v m/rain
Figure 3.3 Speed Range Satisfying Priority 1 in Example 3.1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
min/pc
29
|
Machining Speed
Figure 3.4 Speed Range Satisfying Priorities 1 and 2 in Example 3.1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
min/pc
$/pc 3.06 3.04 3.02
J 3.00
| 2.98
I 2-96 I 2.94 2.92 2.90 2.88 2.86
Figure 3.
.84 .82 .80 .78 .76 .74 .72 .70 .68 .66 .64
220 230 240 250 260 270 280 290 Machining Speed v m/min
The Multiobjective Optimal Solution in Example 3.1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
min/pc
31
At the optimal machining speed, the profit rate is 2.802 $/min, the unit production cost is 2.951 $/pc, and the production time is 1.445 min/pc. This satisfies the profit-rate and production-cost goals and minimizes the production time.
Example 3 . 2 : Suppose that the production goals were determined as the following priority from the managerial point of view:
Priority 1: Achieve a profit rate of at least 2.74 $/min.
Priority 2: Achieve minimization of the production cost and maximization of the production rate with the same weights.
The high-efficiency speed range, again, is 216 <_ v <_ 296. In this range, the speed range satisfying priority 1 is 232 <_ v <_ 296.
In considering priority 2, the following two production goals can be set in order to minimize the production cost and maximize the production rate.
Production-Cost Goal: Achieve a production cost of no more than 2.893 $/pc. (This goal can be achieved at the minimum-cost machining speed.) Production-Time Goal: Achieve a production time of no more than
1.437 min/pc. (This goal can be achieved at the minimum-time machining speed.) It is obvious that these two production goals cannot be attained simultaneously. Hence, in order to find the optimal solution, the following problem can be formulated:
Find v so as to minimize
Z = (u(v)-2.892) + (t(v)-1.437) (3.8)
(u(v)-2.892) is the deviation of the unit production cost from the objective value (2.892 $/pc), and (t(v)-1.437) is the deviation of the production time from the objective value (1.437 min/pc). The value of Z is minimum at v = 253, i.e.,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The machining speed, 253 m/min, is between 232 and 296 m/min.
Therefore, the optimal machining speed is 253 m/min. At this machining speed, the production cost is 2.917 $/pc, and the production time is 1.463 min/pc, that is, the deviations from the objective values are 0.025 $/pc and 0.026 min/pc, respectively. This multi
objective optimal solution is a solution making a compromise between these two production goals.
3.1.4 Sensitivity of single-stage manufacturing systems In this section, the sensitivity of changing some paramenters in a single-stage manufacturing system is examined. In particular, analyses of time and cost parameters are made to explore their impact on optimal machining speeds under three kinds of evaluation criteria.
Because the maximum-production-rate machining speed and the minimum- cost machining speed determine the high-efficiency speed range or nondominated solution set, it is important to examine which parameters have an impact on the optimal machining speed.
Time and cost parameters which have an impact on optimal machin
ing speeds under three kinds of evaluation criteria are:
time parameters
tool-replacement time, t^ (min/edge) Figure 3.6(a) preparation time, t^ (min/pc) Figure 3.6(b) cost parameters
direct labor cost and overhead, k^ ($/min) Figure 3.6(c) machining overhead, k^ ($/min) Figure 3.6(d)
computer program for obtaining this solution is shown in Appendix B.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
33
tool cost, k t ($/edge) Figure 3.6(e) gross revenue, r^ ($/pc) Figure 3.6(f) The data used for the numerical computation of three performance measures are shown in Table 3.2. The impact of changes of time and cost parameters on optimal machining speeds is summarized in Table 3.3. The tool-replacement time, t£ (min/edge), is the only parameter that has an impact on the maximum-production-rate (or minimum-time) machining speed; this can be obtained from equation (3.5). The direct labor cost and overhead, ($/min), the machining overhead, km ($/min), and the tool cost, k fc ($/edge) have an impact on the minimum-cost machining speed; this can be obtained from equation (3.6).
The preparation time, t^ (min/pc), and the gross revenue, ru ($/pc), have no impact on either the maximum-production-rate or the minimum- cost machining speed, namely, the high-efficiency speed range. How
ever, every time and cost parameter except direct labor and overhead cost has an impact on the maximum-profit-rate machining speed.
3.2 Determining Optimal Machining Speeds and Feed Rates - Constrained Optimization
3.2.1 Introduction
In section 3.1, the optimization analysis of machining speed is made. In practice, determining both optimal machining speed and optimal feed rate is required. A unique optimal set of machining speed and feed rate, however, cannot be determined for the uncon
strained minimization cf production time and cost (31). Hence, the optimization must be performed by considering the manufacturing constraints, such as machining-speed constraints, feed-rate con
straints, the power constraint, the roughness constraint, and so on.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
withpermissionof thecopyright owner. Further reproduction prohibitedwithout permission.
TABLE 3.2
Basic Data for Numerical Computation on Sensitivity.
Parameter Changed Parameters
1 Tool- Relacement Time
2 Prepara
tion Time
3 Direct Labor &
Overhead 4 Machining Overhead
5 Tool Cost
6 Gross Revenue
Tool-Life Parameters slope constant
430 1-min tool-life
machining speed,
C, m/min 0.23
Machining Parameters feed rate,
s , mm/rev 0.20
Work Parameters
work dia, D, mm 50
work length,
L, mm 200
Reproducedwithpermissionof thecopyright owner. Further reproductionprohibitedwithout permission.
TABLE 3.2 (Continued)
Parameter Changed Parameters
1 Tool- Replacement Time
2 Prepara
tion Time
3 Direct Labor &
Overhead 4 Machining Overhead
5 Tool Cost
6 Gross Revenue
Time Parameters tool-replace- ment time, tc, min/edge
0.5, 1.5
2.5, 3.5 1.5 1.5 1.5 1.5 1.5
preparation
time, tp, min/pc 0.75
0.50, 0.75
1.00, 1.25 0.75 0.75 0.75 0.75
Cost Parameters direct labor &
overhead, k . ,
$/min 0.5 0.5
0.3, 0.5
0.7, 0.9 0.5 0.5 0.5
machining over
head, k , $/min
m 0.05 0.05 0.05
0.05, 0.15
0.25, 0.35 0.05 0.05
tool cost, k
$/edge 2.5 2.5 2.5 2.5
1.5, 3.5 5.5, 7.5 2.5 material cost,
m , $/pc 2.0 2.0 2.0 2.0 2.0 2.0
gross revenue
ru , $/pc 5.0 5.0 5.0 5.0 5.0
4.0, 5.0 6.0, 7.0
m1n/pc$/pc
□ maximum-production-rate machining speed A maxim um-profit-rate machining speed O minimum-cost machining speed
“ “ h ig h -e ffic ie n c y speed range ^,u(tc=3.5) x ' V u(V z-5) r ' ''' ^u ( t c=1.5) ' ^ ' ' ^ ' u ( t c*0.5)
ô ! !>!<
150 200 250 300 350 400
Machining Speed v m/m'n
Figure 3.6(a) Change in (tool-replacement time, min/edge)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
$/pc
37
A maxim um-profit-rate machining speed O minimum-cost machining speed
h ig h -e ffic ie n c y speed range
0 — I'--- 1--- 1--- 1--- 1___________ 1___________ I_
150 200 250 300 350 dOO
Machining Speed v m/min
Figure 3.6(b) Change in tp (preparation time, min/pc)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
min/pc $/pc
O maximum-production-rate machining speed A maxim um-profit-rate machining speed O minimum-cost machining speed
^ h ig h -e ffic ie n c y speed range
„ u(k1=0.7)
u(k^=0.5)
u(k- =0.3)
.0
ptk^O.3) ptk^O.5)
p( k-| *0.9)
Machining Speed
Figure 3.6(c) Change in (direct labor cost and overhead, $/min)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
min/pc $/pc
39
0
u(kmô.35).
3.0 r(km..05)
2.0
1.0
0
150 200 250 300 350 400
Machining Speed v m/min
Figure 3.6(d) Change in (machining overhead, $/min)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Q maximum-production-rate machining speed A maximum-profit-rate machining speed
h ig h -e ffic ie n c y speed range
150 200 250 300 350 4(
Machining Speed v m/min
Figure 3.6(e) Change in k (tool cost, $/edge)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
41
□ maximum-production-rate machining speed A maxim um-profit-rate machining speed O minimum-cost machining speed
— h ig h -e ffic ie n c y speed range
150 200 250 300 350
Machining Speed v m/min
Figure 3.6(f) Change in r (gross revenue, $/pc)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
withpermissionof thecopyright owner. Further reproductionprohibitedwithout permission.
TABLE 3.3
Impact of Changes of Time and Cost Parameters on Optimal Machining Speeds Under Three Kinds of Evaluation Criteria.
Optimal Machining Speeds Minimum-Cost
Criterion
Maximum-Profit- Rate Criterion
Maximum-
Parameters Change Production-
Rate Criterion tool-replacement
time, t , min/edge more less less less
preparation time,
t p , mon/pc more — less —
direct labor and
overhead, k p $/min more more — —
machining overhead,
k , $/min m more more more —
tool cost,
k fc, $/edge more less less -
gross revenue
ru , $/P c more more
— : no change
43
In the following sections, first, the characteristics of per
formance measures, including the nondominated solution set, are discussed. Next, the basic nonlinear multiobjective (goal-programming) model of a single-stage manufacturing system is developed. Lastly, optimal machining conditions, especially optimal machining speeds and feed rates, are analyzed under multicriteria achievement functions.
3.2.2 Basic measures of performance.
Basic mathematical models for single-stage manufacturing as a function of machining speed, v (m/sec), and feed rate, s (mm/rev), are constructed as:
unit production time, t (min/pc):
t . t + ^0 + t io (3.9)
p sv c C 0
production rate, q (pc/min):
1 (3.10)
q =
unit production cost, u ($/pc) :
(3.11)
profit rate, p ($/min):
r ■[
i/mo-! ^ V 1
i v
'u P =
(3.12)
where: k 1 = (=k<i+ki ) direct labor cost and overhead ($/min)
km = machining overhead ($/min)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
mc = material cost ($/pc)
= gross revenue ($/pc)
t£ = tool-replacement time (min/edge)
t = machining time (min/pc) m
tp = preparation tiem (min/pc)
T = tool life (min/edge) X = machining constant
The data used for obtaining figures of performance measures are shown in Table 3.4.
Based upon the above data, the basic measures of performance are calculated by equations (3.9) through (3.12), and Figures 3.7(a) through 3.7(d) are obtained:
the unit production time: Figure 3.7(a) the production rate: Figure 3.7(b) the unit production cost: Figure 3.7(c)
the profit rate: Figure 3.7(d)
In these figures, curves are contour lines of each performance measure, the dotted lines are the v-minimum lines, which express the locus of the minimum value of v at a fixed s-value for the unit production time and the unit production cost, and the v-maximum lines for the produc
tion rate and the profit rate. Contour lines of each of the perform
ance measures are similar to one another. The relative location of the dotted lines (i.e., v-minimum or v-maximum lines) remain unchanged without intersecting each o t her.1
1See also Figures 3.9(a) and 3.9(b), and (40).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.