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Scheduling and machining of jobs through parallel nonidentical machine cells
Scheduling and Machining of Jobs Through Parallel Nonidentical Machine Cells Edward F. Watson, Pius J. Egbelu, Pennsylvania State University, University Park, Pennsylvania
reasons differ in efficiency, capacity, or unit time production cost. These differences may be the result of age differences, model differences, and differences between makes and machine vendors. For example, milling machines from different manufacturers within a shop will exhibit some performance differences in various aspects. If the machine of one manufacturer, say A, is consistently superior to that of manufacturer B, one can say manufacturer A has a preferred machine. This preference will be recognized in a job assignment whenever machines of the two types are simultaneously idle and a job needs to be assigned to a machine. Similarly, milling machines from the same manufacturer may exhibit performance differences due to age a n d / o r model differences. If one category of machines dominates the other categories, the job allocation decision to machines is a trivial one whenever a job is available and the machines are idle. The decision is simply to assign the j o b to a machine within the dominant group. On the other hand, it is possible that no machine group is superior on all counts to all the other groups. In other words, each machine group has some attributes(s) for which it excels. For example, some machines may generate a better surface finish in a given range of cutting speed, feed rate, and depth of cut than the other groups. Differences may be due to part size, tool size, range of feasible cutting speed, feed, and depth of cut. Differences in operator wages between machine groups may also exist. When these differences are present between machine groups, the job-machine allocation decision is no longer trivial. The difficulty of the problem is intensified when multiple jobs are simul-
Abstract Process planning and scheduling are two functions commonly encountered in any manufacturing system. Despite the well recognized interdependencies between them, research in these areas have progressed separately as if there is virtually no relationship between the two. The result of such separate development in terms of manufacturing systems planning can lead to the adoption of policiesthat are far from optimal. In this paper, a modeling approach that integrates the functions of job scheduling and process planning in cells with parallel and nonidentical machine groups is presented. A heuristic algorithm is developed that allocates the jobs to machines while at the same time generates the best machining parameters to use for each job in order not to violate any job and machine related restrictions. The basis of optimization is the minimization of total manufacturing cost.
Keywords: Scheduling, Process Planning, Optimization, Machining Cells, Parallel Processing.
Introduction The problem of planning for the production of multiple jobs that require similar operation(s) through a machine shop with several machines is not uncommon. Also not uncommon in shops is the presence of machines that perform similar operations but for some
59
Journal of Manu[acturing Systerns Volume 8 / N o . 1
taneously ready for processing through the shop. In this paper, a methodology that simultaneously solves both the machining parameter (i.e., cutting speed, feed, and depth of cut) selection problem and the job-machine assignment problem in a multijob, multimachine system is presented. The preferred solution is selected based on minimum total manufacturing cost. The method of solution is based on a heuristic algorithm with an imbedded integer programming model. The solution from the algorithm specifies:
Background The problem of job scheduling in a manufacturing system has been very well studied as evidenced by the volume of literature in the area. 1.7A typical scheduling problem is described by the number of jobs and machines involved, the arrangement of the machines (i.e., single stage, flow shop, job shop, parallel processors), and the processing times of the jobs on the machines. No discussion is generally given on how the processing times are obtained. The processing times are assumed to be always available. This assumption is not generally true in practice. When jobs arrive at a shop, unless similar jobs had been previously processed in the shop, the processing times of the jobs are not known. What is known about the jobs on arrival are their specifications as finished parts, and the raw material characteristics to be used in fabricating them. If an analyst attempts to schedule the arriving jobs based on the known information, the scheduling task will be difficult to execute. This is because a knowledge of the raw material and the specifications of the finished part is not sufficient to prepare a schedule. What is required is a knowledge of processing times, and when required, some data on costs. Processing time data cannot be generated without a knowledge of the machining parameters, the machine involved, and any setup time or load/unload time requirements. To generate the machining parameters, the planning function of process planning must be invoked. Thus, in practice, there is a strong interaction between scheduling and processing planning. Despite this known relationship, research in scheduling and process planning have progressed independent of each other. This is evidenced by examining the literature in the two areas. 1.4.8-14 The functions of scheduling and process planning when independently executed cannot be expected to yield an overall optimal plan. These functions must be integrated to realize optimal or near optimal plans. Without integration, the machine parameter selection problem is solved always as if the entire planning system involves only one job and one machine. The process planner selects a representative machine and based on this selection, the optimal machining parameters for each job is generated. It is this result, which is generated based on one given machine, that the scheduling analyst is expected to use in making the scheduling decision. Obviously, this approach is based on approximation and will fail to
1. The allocation of the jobs to the machines. 2. The cutting parameters to be used for each job on the assigned machine. 3. The total manufacturing cost associated with the machining/scheduling decision. Only the case of single stage production is considered. The machining/scheduling problem presented here differs in a number of ways from traditional scheduling problems. 1. The processing times of the jobs are not known a priori. What is known are the characteristics of the starting raw material for each job and the design specification of the finished parts. 2. In this problem not only is a decision required on which job should be assigned to a machine, but also the machining parameters (i.e., cutting speed, feed, and depth of cut) to be employed to ensure that the finished part design specifications are satisfied. Machining parameter selection is not considered in the traditional scheduling problem. 3. Machining time is a function of the volume of material to be removed and the machining parameters used. Volume of material to be removed is not explicitly considered in traditional scheduling problems. 4. The state of the finished part (e.g., surface finish) is a function of the machining parameters selected and the machine type used. Conditions such as surface finish requirements are not considered in traditional scheduling problems. Selection of machining parameters for any job is a lower level manufacturing planning function that falls within the domain of process planning. On the other hand, allocation of jobs to machines is a higher level manufacturing planning function within the domain of scheduling. Thus, the problem addressed here is one that integrates the process planning function and the scheduling function in a manufacturing system.
60
Journal of Manufacturing Systems Volume 8/No. 1
any job on any machine, uniform cuts are assumed. • There is only one operation per job.
yield an overall optimal result when the machines under consideration differ in capabilities. A planning technique that integrates the functions of machining parameter selection and machine shop scheduling promises to improve the overall result. The integration involves simultaneous consideration of the special job requirements with the capabilities of each available machine. This is the planning approach adopted here. The value of this approach has been recognized by other researchers. Zimmers ~5 suggests that there should be a responsive interface between the determination of metal cutting parameters and other manufacturing functions such as scheduling. The only paper known to the authors that attempts to integrate the machining parameter optimization problem and the job allocation problem was by Egbelu.~6 Egbelu's paper used production makespan as the criterion of optimization and assumed only one machine per group. The work presented here is motivated by that of Egbelu. 16 However, this work differs from Egbelu's work by considering multiple machines per group and using total manufacturing cost instead of makespan as the criterion of optimization.
The following variables are used in the model developed. l, if job i is processed by machine k in groupj. Xii k =
0, otherwise Lq = Time required to load job i on a machine in groupj. U~i = Time required to unload job i on a machine in groupj. Pqk = A function expressing the total processing time of a job i on machine k in group j. This defines the entire time machine k in g r o u p j is kept busy by job i including L o and Uq. S~ik = Spindle speed for job i on machine k in g r o u p j (revolutions per minute). Fq~ = Feed for machining job i on machine k in groupj (inches per minute). dq~ = Depth of cut for job i on machine k in g r o u p j (inches) T= Length of time required to process all the N jobs through the machines (makespan). N = Number of jobs. G = Number of machine groups or cells. m.= Number of machines in groupj. 1 C~ik = A function expressing the total cost, excluding overhead, of machining job i on machine k of groupj. K = Fixed overhead unit time cost ($ per unit time) that the shop is open. A shop is open as long as there is some uncompleted task. bik = A vector of constants representing the righthand side of constraints on machines in groupj. q~= A vector of constants representing the righthand side of constraints on job i. gqk,hq~ = Constraint functions between job i and machines in group j expressed in terms of the machining parameter values of job i on machines in group j. With the above parameters defined, the generalized mathematical model for simultaneously optimizing the production cost for the machining and the scheduling of the N jobs through the G machine cells can be represented as: Minimize
Model Development The model presented here aims at determining how n jobs are to be scheduled through G machine groups such that the total manufacturing cost is minimized. The total cost is defined as the sum of processing cost and overhead cost. The overhead cost is assumed to be directly proportional to the length of time required to process all jobs available. In the development of the model, the following assumptions are made: • All jobs are available at the start of the planning period. • The initial raw material state and the finished part state of the jobs are known. • All job related constraints such as those on surface finish specifications are known. • Limitations on machines such as the feasible range on spindle rotational frequency, feedrate, and depth of cut are known. • The machining time for each job is a function of the selected cutting speed, feed, and depth of cut for the job.
N
G m.
~ y~ I Cijk (Sijk, Fijk ' dijk ' Lii, Uij) Xijk + K T i:lj:Ik=l (1)
• Whenever multiple tool passes are required for
61
Journal of Manufbcruring Volume 81 No. I
Systems
3.
structural makeup by the of the process. For detailed discussion some of constraints, to Challa Berra* and et al.18 generalized model above is enough to any mix processes. For some of jobs may milling while may require Still others need prosuch as shaping, boring, and planing. ifjob i requires milling, any feasible solution will require that XijL q 0 if j is not a milling machine group and Xjik= 1 for some k in groupj ifj is a milling machine group.
subject to m.
G
c c’ xii,=
1
(2)
'i
j=l k=l N
C Piit (‘ii,, Fiia, diin, Lii, ‘ii) Xii, ~ T ViV p (3) i-l gii,
(Siik Fiik dii,) Xii, d bi, 3
7
hii, (Silk Fiik diik) Xii, ’ qi 3
Xiik
r,
q
0
,
7
1
vi, vi, v,
(4)
vi+& Vi,
v,
(6)
Solution Approach
Fiik, diik 2 0
The above mathematical model is a mixed integer nonlinear programming problem. Generally the cost functions C,, in the objective function are nonlinear.‘6J9J0 The processing time functions of constraint type (3) are highly nonlinear. Depending on the type of constraints that are applicable to the particular machining process, constraints types (4) and (5) may also be nonlinear. Analysis of the structure of the above model even for small values of N, G, and mj, j = 1, 2, .. G, suggests that the model is very complex. A closed form solution method to the model will be difficult to realize. Thus, we recommend the use of a heuristic algorithm to obtain a good solution. As previously mentioned, the model incorporates two subproblems, namely, scheduling in a parallel processor system and machine parameter selection. When treated independent of one another, each of the problems is difficult to optimally resolve. Therefore, it is logical to expect an even harder problem to result from the integration of these two areas. However, by exploiting the special structure of the model, a heuristic algorithm can be developed. Examination of the constraint sets (4) and (5) reveals that for any job-machine pair, the constraints contain no interaction effect between the jobs. There is also no interaction between machine groups and also between machines within a group. This lack of interaction makes it possible to isolate each job-machine pair and solve for the machining parameters that optimize the cost function Cijt subject to constraints (4) and (5). The solution to the isolated job-machine pair automatically yields the value of the processing time function Piik. With the optimal values of C,,and Piikknown for all job-machine pairs, the functions
An
of each set above in order. set (2) that a is scheduled processed on one machine. constraint of (3) gives sum of processing time machine k groupj, and sum must less than equal to makespan, T. sets (4) (5) ensure the operational of any must not violated when any job, that if machine is to machine job, the specifications of a job satisfied. Constraint (6) captures binary restriction the variables Constraint sets and (5) address different of concern the machines the jobs. the machines, function can constraintsr6 1. Cutting 2. Spindle 3. Feed 4. Depth cut 5. force (i.e., force that be resisted the tool 6. Power (i.e., power at the at the feed, and of cut 7. Machine and chatter Tool life Size of that can produced by particular machine part width, length, and height) 10. temperature. For jobs, the function may sent design on: 1. finish condition Dimensional accuracy tolerance
62
Journal of Manufacturing Systems Volume 8/No.
Cqk (Sir,, Fqk, d,rk, Lqk, Uq)
in Eq. (1)
Park(Sirk, Fqk' dark, Zq, Uq)
in Eq. (3)
The method of solution to steps 2 and 9 of the algorithm require some further discussion. If the processes required by the jobs is diverse (i.e., nonsimilar operations such as milling, drilling, grinding, etc.), some of the job-machine pair optimization problems will be infeasible. For example, a job i' that requires milling cannot be processed on a drilling machine group,j'. The job-machine pair optimization problem would be infeasible. For infeasible pairs, the processing time Prr" and the cost, Crr. can be set to an arbitrarily large number, say + ~. However, for other job-machine pairs, several numerical search techniques can be used to solve the problem. Among the search algorithms that have been employed are sequential unconstraint minimization technique (SUMT), Hooke-Jeeves (pattern search), Davidon-Fletcher-Powell, Rosen's hill climbing algorithm, and gradient methods such as the steepest descent algorithm, and cyclic coordinate method. 17.z°-z3 Details regarding these algorithms can be found in most nonlinear programming texts.21-23 Kimbler et al. z0 have characterized basic machine parameter optimization problems by the form of constraints present. Their work provides a good guide to the selection of the applicable algorithms. The problem of step 9 is a basic mixed integer linear program having integer restrictions on the Xq~.s. The only continuous variable is the makespan indicator, T. Several algorithms exist today for solving integer programming problems. Notable among these are branch and bound, cutting plane methods, and partitioning methods. 24 The branch and bound method has been incorporated in some commercially available mathematical programming softwares such as MPSX z5 and Lindo. z6 Using these softwares, the solution to the problem of step 9 can be readily obtained. However, for large N, G, and mr, heuristic algorithms will prove more efficient than the integer programming algorithms, since these algorithms (i.e., integer programming) are very computationally intensive.
and can be substituted by their optimal values obtained by solving the machine parameter selection problem for each job-machine pair. The substitution of constant values for these functions eliminate the nonlinear problem. The proposed algorithm for solving the problem is presented below: Firstly, let C..II = cost of processing job i on any machine in group
y.
Par= processing time for job i on any machine in group j. 1. Set i - 1,j= 1. 2. Solve the following machine parameter optimization problem for job i on machine 1 in group j. Minimize C..II (8) Subject to: Cql (Sq,, Fql, dql, Lip Uq) -Cq: 0 (8a)
Pari (Sail, Fii !, dq z, Lq, Uq) - Par= 0 gai I (Sir I' Fir 1, dlr, ) ~ br = hq, (Sql, Fql, darI) <, q, Sir 1, ~r i, dq i >~0 3. Set Cirk = Car, for k = 1, 2 .... , m r Pq, = Pii, for k = 1, 2 ..... m r 4. Set j = j + l .
(8b) (8C) (8d) (8e)
5. Ifj~< G, go to 2. 6. Set j = 1. 7. S e t / = / + l . 8. Ifi~
N
1
G m.
Minimize Z E EJ C*IrkXq,+ K T i = l j = l k=l
Illustrative Problem
G m. E
2 I Xiik-" 1,
To illustrate the use of the model presented in this paper, consider a scenario involving the machining of 25 jobs that require a slab milling operation. There are four classes of milling machines in the shop. Machine group 1 consists of 3 machines, group 2 has 2 machines, there is only one machine in group 3,
Vi
j:l k:l N P*ijk girk - T<~ 0,
V r Vk
i=1
X.rk : 0 or 1.
Va, Vj, V k
63
mrnal q/‘Mamfacluring olume 81 No. I
SJsrems
nd 2 machines in group 4. Table I describes the mate.a1characteristics of each job. The raw material for all arts are rectangular shaped blocks already cut to esired length and width. Column Band D provide the ngth and the width of each raw workpiece. Column C the depth of material to be machined off from one of re faces defined by the workpiece length and width. :olumn E describes the specific cutting energy for each material, while column F defines the maximum design uface roughness allowed for each job. Table 2 gives data on expected loading/ unloading mes for each job on the various machine groups. Table provides the cost data and the mechanical characteriscs of each machine group. Given the described data, le problem is to allocate the jobs to the machines in rder to minimize total manufacturing cost without iolating the design requirements of each job and the mechanical constraints of each machine. The explicit job allocation/ machining model for
the problem is as shown in Appendix A. The machine parameter optimization problem that is generated from step 2 of the algorithm for a given job i and machine groupj is given in Appendix B. The machining parameter problem in this example was solved using the cyclic coordinate method described in Bazara.2’ The results obtained from solving the model of Appendix B for every job-machine pair are summarized in Tables 4 and 5. The data from Tables 4 and 5 constitute the inputs to the model of step 9 of the algorithm. The model was solved using the mathematical programming software, MPSX.25 The job-machine assignment results are summarized in Table 6. A mark of ‘X’ in Table 6 indicates that the job from that row is assigned to the machine from that column. For example, job 2 is assigned to machine 1 in group 1, while job 15 is assigned to machine 2 in group 4. The total manufacturing cost is calculated to be $192.82. All the jobs can
Table 1
Table 2
Job Characteristics and Requirements
Machine Load and Unload Times
*
PART
DMTBR
LENGTH
WIDTH
& (inches)
SPECIFIC CUT. ENERGY (hp-min per cubic inch)
PART
MAX SURFACE ROUGHNESS (RdCl-0
inches)
GROUP load (min.)
1
unload (min.)
GROUP load (min.)
2
unload (min.)
il
(inches)
(inches)
1
20.0
0.36
7.0
120.0
300.0
1
6.0
6.0
7.0
8.0
0.28
5.0
170.0
450.0
2
8.0
7.0
9.0
6.0
15.0
z-
~~ 10.0
"
GROUP load (min.) 8.0
3
GROUP
unload load (min 1 (min.)
4
unload (min )
10.0
5.0
7 0
10.0
11.0
7.0
80
3
16.0
0.25
8.0
180.0
400.0
3
15.0
17.0
17.0
18.0
16.0
12.0
14.0
4
15.0
0.34
10.0
140.0
350.0
4
70
9.0
90
11 0
10.0
9.0
60
70
5
18.0
0.37
9.0
120.0
300 0
5
7.0
8.0
8.0
10.0
9.0
5.0
5.0
6.0
6
18.0
0.25
17.0
180.0
300.0
6
6.0
7.0
7.0
6.0
90
5.0
5.0
60
7
11.0
0.35
8.0
200.0
500.0
7
9.0
8.0
11.0
9.0
13.0
11.0
0.30
4.0
120.0
400.0
6
5.0
6.0
6.0
80
8 I
6.0
6.0
5.0
8.0
7.0
40
50
9
13.0
0.25
6.0
180.0
450 0
9
15.0
14.0
16.0
14.0
16.0
17.0
14.0
12.0
10
9.0
0.40
8.0
150.0
350.0
10
16.0
17.0
18.0
16 0
20.0
19.0
15.0
16.0
11
12.0
0.36
12.0
170.0
500.0
11
7.0
6.0
8.0
7.0
9.0
12.0
140.0
300.0
12
9.0
6.0
11 0
10.0
12.0
12
I
13
11.0
6.0
100
90
7.0
16.0
0.34
15.0
0.26
5.0
200.0
450.0
13
8.0
5.0
9.0
8.0
10.0
9.0
7.0
6.0
150.0
300.0
14
9.0
6.0
10.0
9.0
11.0
11.0
7.0
50
10.0
14
14.0
0.35
10.0
15
15.0
0.40
90
180.0
500.0
15
9.0
8.0
10.0
9.0
12.0
12.0
8.0
9.0
15.0
0.36
9.0
120.0
400.0
16
14.0
12.0
16.0
14 0
18.0
17.0
13.0
I1 0
6.0
16
0.28
5.0
120.0
500.0
17
14.0
12.0
16.0
15.0
17.0
15.0
12 0
14.0
15.0
0.26
5.0
120.0
450.0
18
7.0
6 0
60
60
8.0
70
6.0
7 0
16.0
0.34
12.0
200.0
300.0
19
6.0
7.0
9.0
7.0
8.0
7.0
70
a.0
0.37
12.0
140.0
350.0
20
6.0
5.0
7 0
50
70
90
17
18 z
1
r12.0
15.0
140
21
9.0
0.25
8.0
180.0
4oo;o
21
11.0
6.0
13 0
11.0
19.0
18.0
9.0
6 0
22
13.0
0.35
6.0
120.0
500.0
22
15.0
12.0
17.0
16.0
14.0
16 0
14 0
11.0
23
6.0
0.30
4.0
170.0
450.0
23
10.0
9.0
12.0
14.0
11.0
13 0
8.0
6.0
24
11 0
0.25
7.0
120.0
300 0
24
9.0
11.0
10.0
8.0
50
4.0
6.0
7.0
18.0
0.40
17.0
120.0
300.0
25
9.0
7.0
ll.C
9.0
12.0
13.0
5.0
7."
25
,
I UMTBH
=
“epth
of Haterlal
Removed
through
to be Machining
64
Journal of Manufacturing Systems Volume 8/No. I
the milling of 25 part types through four groups of milling machines. The machine groups differ in population size and performance. The approach is easy to use and can be applied for the planning of production under various machining processes.
be processed within 66.87 minutes measured from some arbitrary starting point. The optimal machining parameters and the cutting parameters required to achieve the total cost of $192.82 are given in Table 6. Table 6, in essence, is a complete production plan necessary to achieve low cost manufacturing for all the 25 jobs.
Table 4 Machine Parameter Solution LJo~I
Concluding Remarks The benefits of integrated models in the planning of manufacturing systems have long been recognized. Such recognition has ushered notable advances such as computer integrated manufacturing and materials requirements planning. In this paper, a technique to integrate production scheduling and machine parameter selection problems has been developed. The technique is based on mathematical programming. However, because the resulting model is highly nonlinear in both the objective function and the constraints, a heuristic algorithm was developed to solve the model. An application of the model was demonstrated in planning
oR#up . . . . .
P2
4 I cat
J
(b)
(c)
(a)
cb)
i
i
o#o.P 2
(c)
ca)
cb)
(c)
(a)
1387 2
e.14
34.5
266,6
0.14
40,0
365.9
8.13
43,9
8.18 6.16
34.5 34.5
266.6
0.10
40.8
358,0
9.10
42.8
200,0 178.9
0.1~
2 1287.3 3:287.2
e.to
40,0 35.8
251.7 266.5
0.10 0,10
39.7 43.9
33.8
34.5
330,6 365.8
8.18
0.14
37.8 40.0
169.0
4 1287.2
8.10 0.12
2~.8
266.6
U.14
40.0
365,8
0.10
43.9
200.0
8.10 0.10
40.9 40.D
6 ,287.2
8.18
34.5
251.7
0.18
~3_.3
3.3~.6
33~2
169.1
0.10
33.8
? !287.3 8 [287,2
0.10 0.16
34.5 34.5
226,6
34.8
297.5
35.7
266.6
0.18 8.14
40.0
365.6
~ 0.10 0.10
43.9
152.0 2ge.#
8.10 0.10
30.8 40.0
9 1287.3 ~ 0 ~287.2
0.10 0.10
34.5 34.5
251.?
0.10
37.8
0.10
39.7
169.1
43.9
2ee.o
3~,Q 48.0
34.5
48.8 48.8
0,11
0.10
8.11 0.10
0.10 0.10
11
266.6 266.6
330.6 365.8 350.0
e.1#
42.0
173.9
0.18
33.8
200,8
0.18
40.0
152.0
0.10
30.4
300.0 169.0
O.tO 0.10
40.0 33.8
3OO.O 4 3 . 9 i2OO.#
0.12
4O,O
8.18
4e.O
43.9
21m.#
0.10
40.0
35.7 43.9
152.0
0.10
38.4
2~.O
0.18
48.8
39.7
169,1
8.10
33.8
43.8
2im. O
e.12
40.0
42.0
178.9
43,9 i2ee.#
0.10 0.18
35.8 48.0
43.9 : 2 ~ . 8
0.10
48.0
287.2
14
287.2
15
287.210,tO
8.12
34.5
266.6
34.5
251.?
266.6 226.6
24
287.3
0.14
8.18 0.10
40.0 37.0
0.14 0.10
365.8
0.10
43.9
338.6
0.10
39,? 43.9
365,8
40.8 34.0
34,5
(a)
Nomenclature:
= Calculated Cb) : C a l c u l a t e d
Table 3
(c)
M a c h i n e Characteristics Machine
Range of Spindle
2
50-500
Resulting
spindle ~otatlonal teequenc9 d e p t h o f ct~t C i n c h e s )
feed
Range
40-600
Table 5
of
5-45
5-40
5-50
5-40
.25
.40
.30
.40
(ipm)
job
Upper L i m i t o n A l l o w a b l e Depth Cut
(inch)
Range
of C u t t i n g (sfpm)
50-600
75-500
50-800
50-1000
Velocity Fixed
Feed
(ipr)
Maximum Width of C u t based on T o o l on Machine
0.12
0.15
3.5
0.12
4
0.2
3.5
4
(inch)
Diameter of Current Tool on Machine
3
3.5
3
4
(inch)
Time R e q u i r e d Tool
Change
per
Allowable Power (Hp)
Machine Overhead
5.4
6
5,4
5.4
(min)
Max
3000
4000
3000
4000
0.3
0.3
0.3
0.25
Rate
(S/hour)
Direct Labor R a t e (S/hour) Cost Tool
of Current (S/tool)
Machining Inefficiency Factor
(inches/min.)
Results on Processing Times and Processing Costs
(rpm)
on Feed
Rate
Pate
(~ev./mi..)
4
50-600
Rotational
Frequency
0.10
Group 3
50-600
8.10
297.5 8 . 1 8
365.3
I
GROOP 4 (h) (c)
0.25
0.25
6
1.2
0.25
10 1.25
0.25
6
1.2
~
5
1.15
GROUP I
#
Ca)
(b)
1
3.87
2
1.22
3 4
3.15
5 6
Ca)
(hi
15.57
5.56
2.33
16.53
16.42
4.?7
2.74 33.13
9.78
19.65
3.6? 19.26
GROUP 3 (el
(a)
5.25
2.68 28.93
e.93 18.00
4.94
1.Re 2 2 . 1 3
2 . 2 8 36.36
18.18
6.62
2.77 23.88
6.89
2 . 7 8 21.81
6,53 28.57
8,68
7
2.68 20.10
6.47
3
0.54 11.62
3.28
8.47
9
1.68 38.95
8.65
1.35
(bl
GROUP 4 (c)
(a)
6.66
2.61
(El
(¢)
14.79
4.88
6.85
1.84 16.18
4.48
36.63
18.32
2.46 28.68
3.38
7.88
3.44 22.94
7.68
3.33 16.55
5.64
6.59
4.82
18.68
6.83
3.87
15.14
5.54
5.23 18,62
7.86
5.67 28.25
7.84
5.85 17.18
6.88
2.39 23.53
6.67
2.59
7.90
2.63
17.31
5.56
11.58
3.09
9.67
14.76
4.GP5 8 . 6 5
31.45
8.48
1.46
36.61
q.8?
1.75 35.98
9.81
1.89
2.33
26.79
9.69
2.73
1.51 2?.68
7.55
41.17 11.33
1.92
33.~
9.14
17.26
6.g7
18
2.53
35.93
1R.35
11
4.74 28.49
7.69
3.59
18.88
6.44
3.89
34.3?
8.18
4.82
12
4.82
4.24 23.53
7.38
19.66
7.89
3.54 24.84
7.91
4.39
27.83
9.18
13
1.8e 15.89
4.75
1.60 18.78
5.38
1.74 88.38
6.81
1 . 8 8 16.89
4.97
14
3.53 19.89
6.68
3.19 22.46
?.15
3.31 25.79
8.25
3 . 2 8 15.42
5.38
15
4.64 22.37
3.18
3.71
22.99
7.46
4.83 28.44
9.88
4.16
16
9.~
39.47
8.99
2.27
32.46
9.28
2.53 37.98
17
8.51
86.68
6.93
8.41
31.45
8.86
18
1.16
16.34
4.?8
8.88
14.95
4.16
19
5.59 21.48
8.48
4.98
21.28
7.49
28
3.32
14.85
5.51
3.49
15.79
5.62
3.68 33.15
21 22
1 . 5 8 18.83 t . ? e 29.96
5.56 8.16
1.27 1.28
25.36 34.39
23
9.71
19.82
5.34
24
1.28 21.48
25
7.58 24.69
Ho~nclature:
65
GROUP 2 (cl
(a)
21.39
7.13
18.85
2.54 26.72
7.33
8.68 33.68
8.49
8.58 26.62
6.92
1.26
16.45
4.88
1.22 14.38
4.13
5.48 28.82
?.66
5.53 28.86
?.53
18.26
3.58 19.74
6.52
6.93 9.21
1.37 38.31 18.38 1.55 31.77 8.73
1.42 16.58 1.58 26.68
4.73 ?.33
8.64 26.78
6.98
@.?e 2 4 . 7 8
6.56
8.72 14.77
4.81
6.OO
8.91
18.98
5.19
1.31
3.34
1.87
16.36
4.67
18.24
5.6?
26.15
9.26
7.92 28.46
8.78
= Calculated
~achining
18.5R
3.31.34.39 time
(b)
Cal©ulated
total
processing
time
(c)
= Calculated
total
pPocesslng
cost
13.86
Journal
o/‘Manujkturing
Volume E/No.
Sjutems
I
Rai= Worst permissible inches) for job i
Table 6
Job-Machine Assignment Results
surface
roughness
(micro
PS, = Specific cutting energy (Hp-min/ cubic inch) for job i gjk = Power drive inefficiency rating on machine k in group j P#
Available electrical power (rating) for machine k in group j
q
= Lower Sikmh,Sik max
and upper limits on the speed of machine k in group j
= Lower and upper limits of the feed on machine k in group j
.f;.~J&
di,&,di, mllr= Lower and upper limits of the depth of cut on machine k in group j.
Appendix
From the problem description in Tabtes Z-3, the following constraints must be considered in the model: power, surface finish, spindle speed, feed rate, and depth of cut. These conditions are specified in constraints 5.4 through 5.6. Using the above notations, the mathematical programming formulation for the N-job and G-machine cell milling problem is represented below:
A
To present the model formulation, the following notation in addition to previous notation is presented: li = Length of workpiece
of job i (inches)
wi= Width of workpiece of job i (inches) di
q
Depth of material to be removed (inches)
Min
from job i
{‘ijk
-
k,=
( [ 1i + (dijk(Dj - dijk))“’ I Fijk wdj)+ . (dJ dJ+}) X,, = 0
64.2)
for any ICijk
km= Machining overhead cost (dollars/ minute) for any machine in any group k%Q= Cost per tool (dollars/tool)
(A.11
in ‘Cwil
Direct labor cost (dollars/minute) machine in any group
KT
subject to
wdj = Width of cutting tool on machines in groupj (inches) Di= Diameter of the cutting tool in machines groupj (inches)
25 4m. C C C’ C,, X++ i=lj=l k=l
-
(‘ii + Lij)kd + Pijk(kd + km) (t’) (Pijk) [(Sij, Dj)
lc’l (“n’(k,I
for machine k in
group .i tfik= Time to replace tool edge (minutes/edge) machine k in group j
Pij,[CSijk Di) / ~‘1(I’“” fiw) on [(di/ d,,)’ (wi/ wdi)+ [ 1i + dijn(Dj - dijJ”2]
n ‘ik q Tool life constant (slope of tool life curve) for machine k in group j
kdl /‘@IXijk= 0
(A.3)
cfik = Tool life constant (one-minute tool life speed) for machine k in group j
to.0642 (Bill 1 Dj) (Fijt 1 SiiJ2 } Xi+ < Rai
(A.41
inefficiency Bit = Surface roughness machine k in group j
{PS . dijk . wdj. Fijk. gjl} X,, d Pjk
(A.51
factor
for
66
Journal o f Manufacturing Systems Volume 8/No. 1
s,,xlt, <- si,, x,, <. s,,. xi,, F,, i Xit,
F, tk X,t,
F,,. Xi,,
(Oj- 4jk) ]'/2)k,] / 60 = 0
P,s-{ [Li +(disk(DFd,sk)) ]'12(w,I wdi)+ (d,I disk)+ lF,ik : 0
(A.6)
0.0642(~,.,XF, Id S,O, <~ Ra,
4,.xit, < air, Xit, N
Y~
i=1
xi,,
Sj,.= <<. S,t ~ <<. S~.=
Pit* Xitk <" T
a
PS. d,j,. wd.~. F,t k . S,, <<.ej~ (A.7)
F.,, ,,~, <~ Fit * <~ F.%, x m.
X E ' X irk = 1, j = l k=l
Vl
4, .-< d,,,-< 4,o=
(A.8)
Xit k = 0 or 1 for all (i,j,k) combinations
T, Sit ,, Fqk, dit k >1 0 for i = 1..... N
T, Siik, Fiik, dii k > Ofor i = 1,2..... N (A.9)
j = 1,2 ..... G k = any machine in groupj
(A.10)
j = 1..... G k = 1..... m.1
The above model contains the notation (dl/dltk) + and (%/wdi)+. This notation ( d / d q , ) + implies the smallest integer greater than or equal to (d/ditk). This adjustment ensures that only integer numbers of tool passes are allowed. Similarly, the notation ( w i / w ~ ) + ensures that the number of passes across the width of the part is integer.
Equation (A.2) is the processing time constraint for job i on machine k in groupj. Note that the indicator variable Xitk equals 0 if j o b / d o e s not get assigned to machine k in groupj. Similarly, Eq. (A.3) represents the constraint for the total cost. Equations (A.4) through (A.6) represent the surface roughness constraint, power constraint and machine parameter constraint, respectively. Equation (A.7) ensures that the sum of the processing times on any given machine is less than or equal to the makespan. Equation (A.8) limits each j o b to only one machine. Finally, Eq. (A.9) represents the indicator variable constraint and Eq. (A.10) is the usual nonnegativity.
References 1. K.R. Baker. Introduction to Sequencing and Scheduling, John Wiley and Sons, 1974. 2. R.W. Conway, W.L. Maxwell, L.W. Miller. Theory of Scheduling, Addison-Wesley Publishing Company, 1967. 3. J.E. Day, M.P. Hottenstein. "Review of Sequencing Research", Naval Research Logistics Quarterly, Vol. 17, 1970. 4. S. French. Sequencing and Scheduling: A n Introduction to Mathematics of the Job-Shop, John Wiley and Sons, 1982. 5. M.J. Gonzalez, Jr. "Deterministic Processor Scheduling", Computing Surveys, Vol. 9, No. 3, September 1977, pp. 173-204. 6. R.L. Graham, R.L. Lawler, J.K. Lenstr, A. H.G. Rinooy Kan. "Optimization and Approximation in Deterministic Sequencing and Scheduling: A Survey", Annals of Discrete Mathematics, Vol. 5, 1979, pp. 287-326. 7. K. Phillips. "Aspects of Job Scheduling", Journal of Engineering for Industry, Vol. 101, February 1979. 8. K. Challa, P.B. Berra. "Automated Planning and Optimization of Machining Processes: A Systems Approach", Computers and Industrial Engineering, Vol. I, 1976, pp. 15-46. 9. T. Chang, R.A. Wysk. "An Introduction to Automated Process Planning Systems", Prentice-Hall, Inc., 1985. 10. M.P. Groover, A.M. Gunda, R.J. Johnson. "Determination of Machining Conditions by a Self-Adaptive Procedure", Proc. 4th North American Metalworking Research Conference, 1976. 11. S.K. Hati, S.S. Rao. "Determination of Optimum Machining Conditions: Deterministic and Probabilistic Approaches", Transactions of the ASME, February 1976, pp. 354-359. 12. G.M. Hayes, R.P. Davis. "A Discrete Variable Approach to Machine Parameter Optimization", AIIE Transactions, June 1979. 13. A. Houtzeel. "Computer Assisted Process Planning - A First Step Towards Integration", Proc. A UTOFA CT WEST, Society of Manufacturing Engineers, Anaheim, California. November 1980, pp. 801-808. 14. S.S. Rao, S.K. Hati. "Computerized Selection of Optimum Machining Conditions for a Job Requiring Multiple Operations", Transactions of the ASME, Vol. 100, August 1978.
Appendix B The model below is the decomposed machine parameter optimization for a given j o b / a n d machinej. Some variables not defined in the main text used in the model below are defined in Appendix A. Minimize C..q subject to:
Cit-(Uit + Lit) ka+ plt ~ (kd+ k,,) +0') (Pltk) [ Sitk DtJ I c" ] ,,I.~ (k~) +Pitt [(Sit* Dr) I c" ] "/"~ (kw) +[(d)i/ dlt,)+ ( w / w d ) + [1 i+ (dit , 67
Journal of Manufacturing Systems Volume 8/No. I
15. E.W. Zimmers, Jr. "Practical Applications for Computer Augmented
20. D.L. Kimbler, R.A. Wysk, R.P. Davis. "Alternative Approaches to the Machining Parameter Optimization Problem", Computers and Industrial Engineering, Vol. 2, 1978, pp. 195-202. 21. M.S. Bazaraa, C.M. Shetty. Nonlinear Programming: Theory and Applications, John Wiley and Sons, 1979. 22. C.S. Beightler, D.T. Phillips, D.J. Wilde. "Foundations of Optimization", Prentice-Hall, International Series of Industrial and Systems Engineering, Prentice-Hall, New Jersey, 1979. 23. D. Himmelblau. Applied Nonlinear Programming, McGraw-Hill Book Company, Inc., New York, 1972. 24. H.M. Salkin. Integer Programming, Addison-Wesley Publishing Company, 1975. 25. Mathematical Programming Systems Extended (MPSX) Mixed lnteger Programming (MIP), Program No, 5734-XMA. 26. L. Schrage. "Introduction to LINDO", Graduate School of Business, University of Chicago, Chicago, Illinois, November 1981.
Systems for Determination of Metal Removal Parameter and Production Standards in a Job Shop", SME Technical Paper No. MS71-136, 1971. 16. P.J. Egbelu. "Planning for Machining in a Multijob Multimachine Manufacturing Environment", Journal of Manufacturing Systems, Vol. 5, No. I, 1985, pp. 1-13. 17. Hooke and Jeeves. "Direct Search Solution of Numerical and Statistical Problems", J. Assoc. Comp. Mach., Vol. 8, No. 2, April 1961, pp. 212-229. 18. P.J. Egbelu, R.P. Davis, R.A. Wysk, J.M.A. Tanchoco. "An Economic Model for the Machining of Cast Parts", Journal of Manufacturing Systems, Vol. I, No. 2, 1984, pp. 207-213. 19. R.P. Davis, R.A. Wysk, M.H. Agee. "Characteristics of Machine Parameter Optimization Models", Applied Mathematical Modelling, Vol. 2, December 1978.
Author(s) Biography Edward F. Watson received his Masters degree in Industrial Engineering and Operations Research and is currently working on his Doctorate in Industrial Engineering at the Pennsylvania State University, University Park, Pennsylvania. He received his Bachelors degree in Industrial Engineering at Syracuse University. His interests are in manufacturing systems analysis, operations research, and artificial intelligence. His professional affiliations include SME, liE, ASEE, and ORSA. He is also a member of Tau Beta Pi and Alpha Pi Mu. Pius J. Egbelu is an Assistant Professor of Industrial and Management Systems Engineering at Pennsylvania State University. He was formerly with the Industrial Engineering Department at Syracuse University. He received a B.S.I.E. degree from Louisiana Tech University and a M.S.I.E. and Ph.D. from Virginia Polytechnic Institute and State University. Dr. Egbelu's teaching and research interests are in the area of automated material handling, manufacturing systems analysis, robotics, and production planning and control. His professional affiliations include SME, liE, Tau Beta Pi, Alpha Pi Mu and Phi Kappa Phi.
68
reasons differ in efficiency, capacity, or unit time production cost. These differences may be the result of age differences, model differences, and differences between makes and machine vendors. For example, milling machines from different manufacturers within a shop will exhibit some performance differences in various aspects. If the machine of one manufacturer, say A, is consistently superior to that of manufacturer B, one can say manufacturer A has a preferred machine. This preference will be recognized in a job assignment whenever machines of the two types are simultaneously idle and a job needs to be assigned to a machine. Similarly, milling machines from the same manufacturer may exhibit performance differences due to age a n d / o r model differences. If one category of machines dominates the other categories, the job allocation decision to machines is a trivial one whenever a job is available and the machines are idle. The decision is simply to assign the j o b to a machine within the dominant group. On the other hand, it is possible that no machine group is superior on all counts to all the other groups. In other words, each machine group has some attributes(s) for which it excels. For example, some machines may generate a better surface finish in a given range of cutting speed, feed rate, and depth of cut than the other groups. Differences may be due to part size, tool size, range of feasible cutting speed, feed, and depth of cut. Differences in operator wages between machine groups may also exist. When these differences are present between machine groups, the job-machine allocation decision is no longer trivial. The difficulty of the problem is intensified when multiple jobs are simul-
Abstract Process planning and scheduling are two functions commonly encountered in any manufacturing system. Despite the well recognized interdependencies between them, research in these areas have progressed separately as if there is virtually no relationship between the two. The result of such separate development in terms of manufacturing systems planning can lead to the adoption of policiesthat are far from optimal. In this paper, a modeling approach that integrates the functions of job scheduling and process planning in cells with parallel and nonidentical machine groups is presented. A heuristic algorithm is developed that allocates the jobs to machines while at the same time generates the best machining parameters to use for each job in order not to violate any job and machine related restrictions. The basis of optimization is the minimization of total manufacturing cost.
Keywords: Scheduling, Process Planning, Optimization, Machining Cells, Parallel Processing.
Introduction The problem of planning for the production of multiple jobs that require similar operation(s) through a machine shop with several machines is not uncommon. Also not uncommon in shops is the presence of machines that perform similar operations but for some
59
Journal of Manu[acturing Systerns Volume 8 / N o . 1
taneously ready for processing through the shop. In this paper, a methodology that simultaneously solves both the machining parameter (i.e., cutting speed, feed, and depth of cut) selection problem and the job-machine assignment problem in a multijob, multimachine system is presented. The preferred solution is selected based on minimum total manufacturing cost. The method of solution is based on a heuristic algorithm with an imbedded integer programming model. The solution from the algorithm specifies:
Background The problem of job scheduling in a manufacturing system has been very well studied as evidenced by the volume of literature in the area. 1.7A typical scheduling problem is described by the number of jobs and machines involved, the arrangement of the machines (i.e., single stage, flow shop, job shop, parallel processors), and the processing times of the jobs on the machines. No discussion is generally given on how the processing times are obtained. The processing times are assumed to be always available. This assumption is not generally true in practice. When jobs arrive at a shop, unless similar jobs had been previously processed in the shop, the processing times of the jobs are not known. What is known about the jobs on arrival are their specifications as finished parts, and the raw material characteristics to be used in fabricating them. If an analyst attempts to schedule the arriving jobs based on the known information, the scheduling task will be difficult to execute. This is because a knowledge of the raw material and the specifications of the finished part is not sufficient to prepare a schedule. What is required is a knowledge of processing times, and when required, some data on costs. Processing time data cannot be generated without a knowledge of the machining parameters, the machine involved, and any setup time or load/unload time requirements. To generate the machining parameters, the planning function of process planning must be invoked. Thus, in practice, there is a strong interaction between scheduling and processing planning. Despite this known relationship, research in scheduling and process planning have progressed independent of each other. This is evidenced by examining the literature in the two areas. 1.4.8-14 The functions of scheduling and process planning when independently executed cannot be expected to yield an overall optimal plan. These functions must be integrated to realize optimal or near optimal plans. Without integration, the machine parameter selection problem is solved always as if the entire planning system involves only one job and one machine. The process planner selects a representative machine and based on this selection, the optimal machining parameters for each job is generated. It is this result, which is generated based on one given machine, that the scheduling analyst is expected to use in making the scheduling decision. Obviously, this approach is based on approximation and will fail to
1. The allocation of the jobs to the machines. 2. The cutting parameters to be used for each job on the assigned machine. 3. The total manufacturing cost associated with the machining/scheduling decision. Only the case of single stage production is considered. The machining/scheduling problem presented here differs in a number of ways from traditional scheduling problems. 1. The processing times of the jobs are not known a priori. What is known are the characteristics of the starting raw material for each job and the design specification of the finished parts. 2. In this problem not only is a decision required on which job should be assigned to a machine, but also the machining parameters (i.e., cutting speed, feed, and depth of cut) to be employed to ensure that the finished part design specifications are satisfied. Machining parameter selection is not considered in the traditional scheduling problem. 3. Machining time is a function of the volume of material to be removed and the machining parameters used. Volume of material to be removed is not explicitly considered in traditional scheduling problems. 4. The state of the finished part (e.g., surface finish) is a function of the machining parameters selected and the machine type used. Conditions such as surface finish requirements are not considered in traditional scheduling problems. Selection of machining parameters for any job is a lower level manufacturing planning function that falls within the domain of process planning. On the other hand, allocation of jobs to machines is a higher level manufacturing planning function within the domain of scheduling. Thus, the problem addressed here is one that integrates the process planning function and the scheduling function in a manufacturing system.
60
Journal of Manufacturing Systems Volume 8/No. 1
any job on any machine, uniform cuts are assumed. • There is only one operation per job.
yield an overall optimal result when the machines under consideration differ in capabilities. A planning technique that integrates the functions of machining parameter selection and machine shop scheduling promises to improve the overall result. The integration involves simultaneous consideration of the special job requirements with the capabilities of each available machine. This is the planning approach adopted here. The value of this approach has been recognized by other researchers. Zimmers ~5 suggests that there should be a responsive interface between the determination of metal cutting parameters and other manufacturing functions such as scheduling. The only paper known to the authors that attempts to integrate the machining parameter optimization problem and the job allocation problem was by Egbelu.~6 Egbelu's paper used production makespan as the criterion of optimization and assumed only one machine per group. The work presented here is motivated by that of Egbelu. 16 However, this work differs from Egbelu's work by considering multiple machines per group and using total manufacturing cost instead of makespan as the criterion of optimization.
The following variables are used in the model developed. l, if job i is processed by machine k in groupj. Xii k =
0, otherwise Lq = Time required to load job i on a machine in groupj. U~i = Time required to unload job i on a machine in groupj. Pqk = A function expressing the total processing time of a job i on machine k in group j. This defines the entire time machine k in g r o u p j is kept busy by job i including L o and Uq. S~ik = Spindle speed for job i on machine k in g r o u p j (revolutions per minute). Fq~ = Feed for machining job i on machine k in groupj (inches per minute). dq~ = Depth of cut for job i on machine k in g r o u p j (inches) T= Length of time required to process all the N jobs through the machines (makespan). N = Number of jobs. G = Number of machine groups or cells. m.= Number of machines in groupj. 1 C~ik = A function expressing the total cost, excluding overhead, of machining job i on machine k of groupj. K = Fixed overhead unit time cost ($ per unit time) that the shop is open. A shop is open as long as there is some uncompleted task. bik = A vector of constants representing the righthand side of constraints on machines in groupj. q~= A vector of constants representing the righthand side of constraints on job i. gqk,hq~ = Constraint functions between job i and machines in group j expressed in terms of the machining parameter values of job i on machines in group j. With the above parameters defined, the generalized mathematical model for simultaneously optimizing the production cost for the machining and the scheduling of the N jobs through the G machine cells can be represented as: Minimize
Model Development The model presented here aims at determining how n jobs are to be scheduled through G machine groups such that the total manufacturing cost is minimized. The total cost is defined as the sum of processing cost and overhead cost. The overhead cost is assumed to be directly proportional to the length of time required to process all jobs available. In the development of the model, the following assumptions are made: • All jobs are available at the start of the planning period. • The initial raw material state and the finished part state of the jobs are known. • All job related constraints such as those on surface finish specifications are known. • Limitations on machines such as the feasible range on spindle rotational frequency, feedrate, and depth of cut are known. • The machining time for each job is a function of the selected cutting speed, feed, and depth of cut for the job.
N
G m.
~ y~ I Cijk (Sijk, Fijk ' dijk ' Lii, Uij) Xijk + K T i:lj:Ik=l (1)
• Whenever multiple tool passes are required for
61
Journal of Manufbcruring Volume 81 No. I
Systems
3.
structural makeup by the of the process. For detailed discussion some of constraints, to Challa Berra* and et al.18 generalized model above is enough to any mix processes. For some of jobs may milling while may require Still others need prosuch as shaping, boring, and planing. ifjob i requires milling, any feasible solution will require that XijL q 0 if j is not a milling machine group and Xjik= 1 for some k in groupj ifj is a milling machine group.
subject to m.
G
c c’ xii,=
1
(2)
'i
j=l k=l N
C Piit (‘ii,, Fiia, diin, Lii, ‘ii) Xii, ~ T ViV p (3) i-l gii,
(Siik Fiik dii,) Xii, d bi, 3
7
hii, (Silk Fiik diik) Xii, ’ qi 3
Xiik
r,
q
0
,
7
1
vi, vi, v,
(4)
vi+& Vi,
v,
(6)
Solution Approach
Fiik, diik 2 0
The above mathematical model is a mixed integer nonlinear programming problem. Generally the cost functions C,, in the objective function are nonlinear.‘6J9J0 The processing time functions of constraint type (3) are highly nonlinear. Depending on the type of constraints that are applicable to the particular machining process, constraints types (4) and (5) may also be nonlinear. Analysis of the structure of the above model even for small values of N, G, and mj, j = 1, 2, .. G, suggests that the model is very complex. A closed form solution method to the model will be difficult to realize. Thus, we recommend the use of a heuristic algorithm to obtain a good solution. As previously mentioned, the model incorporates two subproblems, namely, scheduling in a parallel processor system and machine parameter selection. When treated independent of one another, each of the problems is difficult to optimally resolve. Therefore, it is logical to expect an even harder problem to result from the integration of these two areas. However, by exploiting the special structure of the model, a heuristic algorithm can be developed. Examination of the constraint sets (4) and (5) reveals that for any job-machine pair, the constraints contain no interaction effect between the jobs. There is also no interaction between machine groups and also between machines within a group. This lack of interaction makes it possible to isolate each job-machine pair and solve for the machining parameters that optimize the cost function Cijt subject to constraints (4) and (5). The solution to the isolated job-machine pair automatically yields the value of the processing time function Piik. With the optimal values of C,,and Piikknown for all job-machine pairs, the functions
An
of each set above in order. set (2) that a is scheduled processed on one machine. constraint of (3) gives sum of processing time machine k groupj, and sum must less than equal to makespan, T. sets (4) (5) ensure the operational of any must not violated when any job, that if machine is to machine job, the specifications of a job satisfied. Constraint (6) captures binary restriction the variables Constraint sets and (5) address different of concern the machines the jobs. the machines, function can constraintsr6 1. Cutting 2. Spindle 3. Feed 4. Depth cut 5. force (i.e., force that be resisted the tool 6. Power (i.e., power at the at the feed, and of cut 7. Machine and chatter Tool life Size of that can produced by particular machine part width, length, and height) 10. temperature. For jobs, the function may sent design on: 1. finish condition Dimensional accuracy tolerance
62
Journal of Manufacturing Systems Volume 8/No.
Cqk (Sir,, Fqk, d,rk, Lqk, Uq)
in Eq. (1)
Park(Sirk, Fqk' dark, Zq, Uq)
in Eq. (3)
The method of solution to steps 2 and 9 of the algorithm require some further discussion. If the processes required by the jobs is diverse (i.e., nonsimilar operations such as milling, drilling, grinding, etc.), some of the job-machine pair optimization problems will be infeasible. For example, a job i' that requires milling cannot be processed on a drilling machine group,j'. The job-machine pair optimization problem would be infeasible. For infeasible pairs, the processing time Prr" and the cost, Crr. can be set to an arbitrarily large number, say + ~. However, for other job-machine pairs, several numerical search techniques can be used to solve the problem. Among the search algorithms that have been employed are sequential unconstraint minimization technique (SUMT), Hooke-Jeeves (pattern search), Davidon-Fletcher-Powell, Rosen's hill climbing algorithm, and gradient methods such as the steepest descent algorithm, and cyclic coordinate method. 17.z°-z3 Details regarding these algorithms can be found in most nonlinear programming texts.21-23 Kimbler et al. z0 have characterized basic machine parameter optimization problems by the form of constraints present. Their work provides a good guide to the selection of the applicable algorithms. The problem of step 9 is a basic mixed integer linear program having integer restrictions on the Xq~.s. The only continuous variable is the makespan indicator, T. Several algorithms exist today for solving integer programming problems. Notable among these are branch and bound, cutting plane methods, and partitioning methods. 24 The branch and bound method has been incorporated in some commercially available mathematical programming softwares such as MPSX z5 and Lindo. z6 Using these softwares, the solution to the problem of step 9 can be readily obtained. However, for large N, G, and mr, heuristic algorithms will prove more efficient than the integer programming algorithms, since these algorithms (i.e., integer programming) are very computationally intensive.
and can be substituted by their optimal values obtained by solving the machine parameter selection problem for each job-machine pair. The substitution of constant values for these functions eliminate the nonlinear problem. The proposed algorithm for solving the problem is presented below: Firstly, let C..II = cost of processing job i on any machine in group
y.
Par= processing time for job i on any machine in group j. 1. Set i - 1,j= 1. 2. Solve the following machine parameter optimization problem for job i on machine 1 in group j. Minimize C..II (8) Subject to: Cql (Sq,, Fql, dql, Lip Uq) -Cq: 0 (8a)
Pari (Sail, Fii !, dq z, Lq, Uq) - Par= 0 gai I (Sir I' Fir 1, dlr, ) ~ br = hq, (Sql, Fql, darI) <, q, Sir 1, ~r i, dq i >~0 3. Set Cirk = Car, for k = 1, 2 .... , m r Pq, = Pii, for k = 1, 2 ..... m r 4. Set j = j + l .
(8b) (8C) (8d) (8e)
5. Ifj~< G, go to 2. 6. Set j = 1. 7. S e t / = / + l . 8. Ifi~
N
1
G m.
Minimize Z E EJ C*IrkXq,+ K T i = l j = l k=l
Illustrative Problem
G m. E
2 I Xiik-" 1,
To illustrate the use of the model presented in this paper, consider a scenario involving the machining of 25 jobs that require a slab milling operation. There are four classes of milling machines in the shop. Machine group 1 consists of 3 machines, group 2 has 2 machines, there is only one machine in group 3,
Vi
j:l k:l N P*ijk girk - T<~ 0,
V r Vk
i=1
X.rk : 0 or 1.
Va, Vj, V k
63
mrnal q/‘Mamfacluring olume 81 No. I
SJsrems
nd 2 machines in group 4. Table I describes the mate.a1characteristics of each job. The raw material for all arts are rectangular shaped blocks already cut to esired length and width. Column Band D provide the ngth and the width of each raw workpiece. Column C the depth of material to be machined off from one of re faces defined by the workpiece length and width. :olumn E describes the specific cutting energy for each material, while column F defines the maximum design uface roughness allowed for each job. Table 2 gives data on expected loading/ unloading mes for each job on the various machine groups. Table provides the cost data and the mechanical characteriscs of each machine group. Given the described data, le problem is to allocate the jobs to the machines in rder to minimize total manufacturing cost without iolating the design requirements of each job and the mechanical constraints of each machine. The explicit job allocation/ machining model for
the problem is as shown in Appendix A. The machine parameter optimization problem that is generated from step 2 of the algorithm for a given job i and machine groupj is given in Appendix B. The machining parameter problem in this example was solved using the cyclic coordinate method described in Bazara.2’ The results obtained from solving the model of Appendix B for every job-machine pair are summarized in Tables 4 and 5. The data from Tables 4 and 5 constitute the inputs to the model of step 9 of the algorithm. The model was solved using the mathematical programming software, MPSX.25 The job-machine assignment results are summarized in Table 6. A mark of ‘X’ in Table 6 indicates that the job from that row is assigned to the machine from that column. For example, job 2 is assigned to machine 1 in group 1, while job 15 is assigned to machine 2 in group 4. The total manufacturing cost is calculated to be $192.82. All the jobs can
Table 1
Table 2
Job Characteristics and Requirements
Machine Load and Unload Times
*
PART
DMTBR
LENGTH
WIDTH
& (inches)
SPECIFIC CUT. ENERGY (hp-min per cubic inch)
PART
MAX SURFACE ROUGHNESS (RdCl-0
inches)
GROUP load (min.)
1
unload (min.)
GROUP load (min.)
2
unload (min.)
il
(inches)
(inches)
1
20.0
0.36
7.0
120.0
300.0
1
6.0
6.0
7.0
8.0
0.28
5.0
170.0
450.0
2
8.0
7.0
9.0
6.0
15.0
z-
~~ 10.0
"
GROUP load (min.) 8.0
3
GROUP
unload load (min 1 (min.)
4
unload (min )
10.0
5.0
7 0
10.0
11.0
7.0
80
3
16.0
0.25
8.0
180.0
400.0
3
15.0
17.0
17.0
18.0
16.0
12.0
14.0
4
15.0
0.34
10.0
140.0
350.0
4
70
9.0
90
11 0
10.0
9.0
60
70
5
18.0
0.37
9.0
120.0
300 0
5
7.0
8.0
8.0
10.0
9.0
5.0
5.0
6.0
6
18.0
0.25
17.0
180.0
300.0
6
6.0
7.0
7.0
6.0
90
5.0
5.0
60
7
11.0
0.35
8.0
200.0
500.0
7
9.0
8.0
11.0
9.0
13.0
11.0
0.30
4.0
120.0
400.0
6
5.0
6.0
6.0
80
8 I
6.0
6.0
5.0
8.0
7.0
40
50
9
13.0
0.25
6.0
180.0
450 0
9
15.0
14.0
16.0
14.0
16.0
17.0
14.0
12.0
10
9.0
0.40
8.0
150.0
350.0
10
16.0
17.0
18.0
16 0
20.0
19.0
15.0
16.0
11
12.0
0.36
12.0
170.0
500.0
11
7.0
6.0
8.0
7.0
9.0
12.0
140.0
300.0
12
9.0
6.0
11 0
10.0
12.0
12
I
13
11.0
6.0
100
90
7.0
16.0
0.34
15.0
0.26
5.0
200.0
450.0
13
8.0
5.0
9.0
8.0
10.0
9.0
7.0
6.0
150.0
300.0
14
9.0
6.0
10.0
9.0
11.0
11.0
7.0
50
10.0
14
14.0
0.35
10.0
15
15.0
0.40
90
180.0
500.0
15
9.0
8.0
10.0
9.0
12.0
12.0
8.0
9.0
15.0
0.36
9.0
120.0
400.0
16
14.0
12.0
16.0
14 0
18.0
17.0
13.0
I1 0
6.0
16
0.28
5.0
120.0
500.0
17
14.0
12.0
16.0
15.0
17.0
15.0
12 0
14.0
15.0
0.26
5.0
120.0
450.0
18
7.0
6 0
60
60
8.0
70
6.0
7 0
16.0
0.34
12.0
200.0
300.0
19
6.0
7.0
9.0
7.0
8.0
7.0
70
a.0
0.37
12.0
140.0
350.0
20
6.0
5.0
7 0
50
70
90
17
18 z
1
r12.0
15.0
140
21
9.0
0.25
8.0
180.0
4oo;o
21
11.0
6.0
13 0
11.0
19.0
18.0
9.0
6 0
22
13.0
0.35
6.0
120.0
500.0
22
15.0
12.0
17.0
16.0
14.0
16 0
14 0
11.0
23
6.0
0.30
4.0
170.0
450.0
23
10.0
9.0
12.0
14.0
11.0
13 0
8.0
6.0
24
11 0
0.25
7.0
120.0
300 0
24
9.0
11.0
10.0
8.0
50
4.0
6.0
7.0
18.0
0.40
17.0
120.0
300.0
25
9.0
7.0
ll.C
9.0
12.0
13.0
5.0
7."
25
,
I UMTBH
=
“epth
of Haterlal
Removed
through
to be Machining
64
Journal of Manufacturing Systems Volume 8/No. I
the milling of 25 part types through four groups of milling machines. The machine groups differ in population size and performance. The approach is easy to use and can be applied for the planning of production under various machining processes.
be processed within 66.87 minutes measured from some arbitrary starting point. The optimal machining parameters and the cutting parameters required to achieve the total cost of $192.82 are given in Table 6. Table 6, in essence, is a complete production plan necessary to achieve low cost manufacturing for all the 25 jobs.
Table 4 Machine Parameter Solution LJo~I
Concluding Remarks The benefits of integrated models in the planning of manufacturing systems have long been recognized. Such recognition has ushered notable advances such as computer integrated manufacturing and materials requirements planning. In this paper, a technique to integrate production scheduling and machine parameter selection problems has been developed. The technique is based on mathematical programming. However, because the resulting model is highly nonlinear in both the objective function and the constraints, a heuristic algorithm was developed to solve the model. An application of the model was demonstrated in planning
oR#up . . . . .
P2
4 I cat
J
(b)
(c)
(a)
cb)
i
i
o#o.P 2
(c)
ca)
cb)
(c)
(a)
1387 2
e.14
34.5
266,6
0.14
40,0
365.9
8.13
43,9
8.18 6.16
34.5 34.5
266.6
0.10
40.8
358,0
9.10
42.8
200,0 178.9
0.1~
2 1287.3 3:287.2
e.to
40,0 35.8
251.7 266.5
0.10 0,10
39.7 43.9
33.8
34.5
330,6 365.8
8.18
0.14
37.8 40.0
169.0
4 1287.2
8.10 0.12
2~.8
266.6
U.14
40.0
365,8
0.10
43.9
200.0
8.10 0.10
40.9 40.D
6 ,287.2
8.18
34.5
251.7
0.18
~3_.3
3.3~.6
33~2
169.1
0.10
33.8
? !287.3 8 [287,2
0.10 0.16
34.5 34.5
226,6
34.8
297.5
35.7
266.6
0.18 8.14
40.0
365.6
~ 0.10 0.10
43.9
152.0 2ge.#
8.10 0.10
30.8 40.0
9 1287.3 ~ 0 ~287.2
0.10 0.10
34.5 34.5
251.?
0.10
37.8
0.10
39.7
169.1
43.9
2ee.o
3~,Q 48.0
34.5
48.8 48.8
0,11
0.10
8.11 0.10
0.10 0.10
11
266.6 266.6
330.6 365.8 350.0
e.1#
42.0
173.9
0.18
33.8
200,8
0.18
40.0
152.0
0.10
30.4
300.0 169.0
O.tO 0.10
40.0 33.8
3OO.O 4 3 . 9 i2OO.#
0.12
4O,O
8.18
4e.O
43.9
21m.#
0.10
40.0
35.7 43.9
152.0
0.10
38.4
2~.O
0.18
48.8
39.7
169,1
8.10
33.8
43.8
2im. O
e.12
40.0
42.0
178.9
43,9 i2ee.#
0.10 0.18
35.8 48.0
43.9 : 2 ~ . 8
0.10
48.0
287.2
14
287.2
15
287.210,tO
8.12
34.5
266.6
34.5
251.?
266.6 226.6
24
287.3
0.14
8.18 0.10
40.0 37.0
0.14 0.10
365.8
0.10
43.9
338.6
0.10
39,? 43.9
365,8
40.8 34.0
34,5
(a)
Nomenclature:
= Calculated Cb) : C a l c u l a t e d
Table 3
(c)
M a c h i n e Characteristics Machine
Range of Spindle
2
50-500
Resulting
spindle ~otatlonal teequenc9 d e p t h o f ct~t C i n c h e s )
feed
Range
40-600
Table 5
of
5-45
5-40
5-50
5-40
.25
.40
.30
.40
(ipm)
job
Upper L i m i t o n A l l o w a b l e Depth Cut
(inch)
Range
of C u t t i n g (sfpm)
50-600
75-500
50-800
50-1000
Velocity Fixed
Feed
(ipr)
Maximum Width of C u t based on T o o l on Machine
0.12
0.15
3.5
0.12
4
0.2
3.5
4
(inch)
Diameter of Current Tool on Machine
3
3.5
3
4
(inch)
Time R e q u i r e d Tool
Change
per
Allowable Power (Hp)
Machine Overhead
5.4
6
5,4
5.4
(min)
Max
3000
4000
3000
4000
0.3
0.3
0.3
0.25
Rate
(S/hour)
Direct Labor R a t e (S/hour) Cost Tool
of Current (S/tool)
Machining Inefficiency Factor
(inches/min.)
Results on Processing Times and Processing Costs
(rpm)
on Feed
Rate
Pate
(~ev./mi..)
4
50-600
Rotational
Frequency
0.10
Group 3
50-600
8.10
297.5 8 . 1 8
365.3
I
GROOP 4 (h) (c)
0.25
0.25
6
1.2
0.25
10 1.25
0.25
6
1.2
~
5
1.15
GROUP I
#
Ca)
(b)
1
3.87
2
1.22
3 4
3.15
5 6
Ca)
(hi
15.57
5.56
2.33
16.53
16.42
4.?7
2.74 33.13
9.78
19.65
3.6? 19.26
GROUP 3 (el
(a)
5.25
2.68 28.93
e.93 18.00
4.94
1.Re 2 2 . 1 3
2 . 2 8 36.36
18.18
6.62
2.77 23.88
6.89
2 . 7 8 21.81
6,53 28.57
8,68
7
2.68 20.10
6.47
3
0.54 11.62
3.28
8.47
9
1.68 38.95
8.65
1.35
(bl
GROUP 4 (c)
(a)
6.66
2.61
(El
(¢)
14.79
4.88
6.85
1.84 16.18
4.48
36.63
18.32
2.46 28.68
3.38
7.88
3.44 22.94
7.68
3.33 16.55
5.64
6.59
4.82
18.68
6.83
3.87
15.14
5.54
5.23 18,62
7.86
5.67 28.25
7.84
5.85 17.18
6.88
2.39 23.53
6.67
2.59
7.90
2.63
17.31
5.56
11.58
3.09
9.67
14.76
4.GP5 8 . 6 5
31.45
8.48
1.46
36.61
q.8?
1.75 35.98
9.81
1.89
2.33
26.79
9.69
2.73
1.51 2?.68
7.55
41.17 11.33
1.92
33.~
9.14
17.26
6.g7
18
2.53
35.93
1R.35
11
4.74 28.49
7.69
3.59
18.88
6.44
3.89
34.3?
8.18
4.82
12
4.82
4.24 23.53
7.38
19.66
7.89
3.54 24.84
7.91
4.39
27.83
9.18
13
1.8e 15.89
4.75
1.60 18.78
5.38
1.74 88.38
6.81
1 . 8 8 16.89
4.97
14
3.53 19.89
6.68
3.19 22.46
?.15
3.31 25.79
8.25
3 . 2 8 15.42
5.38
15
4.64 22.37
3.18
3.71
22.99
7.46
4.83 28.44
9.88
4.16
16
9.~
39.47
8.99
2.27
32.46
9.28
2.53 37.98
17
8.51
86.68
6.93
8.41
31.45
8.86
18
1.16
16.34
4.?8
8.88
14.95
4.16
19
5.59 21.48
8.48
4.98
21.28
7.49
28
3.32
14.85
5.51
3.49
15.79
5.62
3.68 33.15
21 22
1 . 5 8 18.83 t . ? e 29.96
5.56 8.16
1.27 1.28
25.36 34.39
23
9.71
19.82
5.34
24
1.28 21.48
25
7.58 24.69
Ho~nclature:
65
GROUP 2 (cl
(a)
21.39
7.13
18.85
2.54 26.72
7.33
8.68 33.68
8.49
8.58 26.62
6.92
1.26
16.45
4.88
1.22 14.38
4.13
5.48 28.82
?.66
5.53 28.86
?.53
18.26
3.58 19.74
6.52
6.93 9.21
1.37 38.31 18.38 1.55 31.77 8.73
1.42 16.58 1.58 26.68
4.73 ?.33
8.64 26.78
6.98
@.?e 2 4 . 7 8
6.56
8.72 14.77
4.81
6.OO
8.91
18.98
5.19
1.31
3.34
1.87
16.36
4.67
18.24
5.6?
26.15
9.26
7.92 28.46
8.78
= Calculated
~achining
18.5R
3.31.34.39 time
(b)
Cal©ulated
total
processing
time
(c)
= Calculated
total
pPocesslng
cost
13.86
Journal
o/‘Manujkturing
Volume E/No.
Sjutems
I
Rai= Worst permissible inches) for job i
Table 6
Job-Machine Assignment Results
surface
roughness
(micro
PS, = Specific cutting energy (Hp-min/ cubic inch) for job i gjk = Power drive inefficiency rating on machine k in group j P#
Available electrical power (rating) for machine k in group j
q
= Lower Sikmh,Sik max
and upper limits on the speed of machine k in group j
= Lower and upper limits of the feed on machine k in group j
.f;.~J&
di,&,di, mllr= Lower and upper limits of the depth of cut on machine k in group j.
Appendix
From the problem description in Tabtes Z-3, the following constraints must be considered in the model: power, surface finish, spindle speed, feed rate, and depth of cut. These conditions are specified in constraints 5.4 through 5.6. Using the above notations, the mathematical programming formulation for the N-job and G-machine cell milling problem is represented below:
A
To present the model formulation, the following notation in addition to previous notation is presented: li = Length of workpiece
of job i (inches)
wi= Width of workpiece of job i (inches) di
q
Depth of material to be removed (inches)
Min
from job i
{‘ijk
-
k,=
( [ 1i + (dijk(Dj - dijk))“’ I Fijk wdj)+ . (dJ dJ+}) X,, = 0
64.2)
for any ICijk
km= Machining overhead cost (dollars/ minute) for any machine in any group k%Q= Cost per tool (dollars/tool)
(A.11
in ‘Cwil
Direct labor cost (dollars/minute) machine in any group
KT
subject to
wdj = Width of cutting tool on machines in groupj (inches) Di= Diameter of the cutting tool in machines groupj (inches)
25 4m. C C C’ C,, X++ i=lj=l k=l
-
(‘ii + Lij)kd + Pijk(kd + km) (t’) (Pijk) [(Sij, Dj)
lc’l (“n’(k,I
for machine k in
group .i tfik= Time to replace tool edge (minutes/edge) machine k in group j
Pij,[CSijk Di) / ~‘1(I’“” fiw) on [(di/ d,,)’ (wi/ wdi)+ [ 1i + dijn(Dj - dijJ”2]
n ‘ik q Tool life constant (slope of tool life curve) for machine k in group j
kdl /‘@IXijk= 0
(A.3)
cfik = Tool life constant (one-minute tool life speed) for machine k in group j
to.0642 (Bill 1 Dj) (Fijt 1 SiiJ2 } Xi+ < Rai
(A.41
inefficiency Bit = Surface roughness machine k in group j
{PS . dijk . wdj. Fijk. gjl} X,, d Pjk
(A.51
factor
for
66
Journal o f Manufacturing Systems Volume 8/No. 1
s,,xlt, <- si,, x,, <. s,,. xi,, F,, i Xit,
F, tk X,t,
F,,. Xi,,
(Oj- 4jk) ]'/2)k,] / 60 = 0
P,s-{ [Li +(disk(DFd,sk)) ]'12(w,I wdi)+ (d,I disk)+ lF,ik : 0
(A.6)
0.0642(~,.,XF, Id S,O, <~ Ra,
4,.xit, < air, Xit, N
Y~
i=1
xi,,
Sj,.= <<. S,t ~ <<. S~.=
Pit* Xitk <" T
a
PS. d,j,. wd.~. F,t k . S,, <<.ej~ (A.7)
F.,, ,,~, <~ Fit * <~ F.%, x m.
X E ' X irk = 1, j = l k=l
Vl
4, .-< d,,,-< 4,o=
(A.8)
Xit k = 0 or 1 for all (i,j,k) combinations
T, Sit ,, Fqk, dit k >1 0 for i = 1..... N
T, Siik, Fiik, dii k > Ofor i = 1,2..... N (A.9)
j = 1,2 ..... G k = any machine in groupj
(A.10)
j = 1..... G k = 1..... m.1
The above model contains the notation (dl/dltk) + and (%/wdi)+. This notation ( d / d q , ) + implies the smallest integer greater than or equal to (d/ditk). This adjustment ensures that only integer numbers of tool passes are allowed. Similarly, the notation ( w i / w ~ ) + ensures that the number of passes across the width of the part is integer.
Equation (A.2) is the processing time constraint for job i on machine k in groupj. Note that the indicator variable Xitk equals 0 if j o b / d o e s not get assigned to machine k in groupj. Similarly, Eq. (A.3) represents the constraint for the total cost. Equations (A.4) through (A.6) represent the surface roughness constraint, power constraint and machine parameter constraint, respectively. Equation (A.7) ensures that the sum of the processing times on any given machine is less than or equal to the makespan. Equation (A.8) limits each j o b to only one machine. Finally, Eq. (A.9) represents the indicator variable constraint and Eq. (A.10) is the usual nonnegativity.
References 1. K.R. Baker. Introduction to Sequencing and Scheduling, John Wiley and Sons, 1974. 2. R.W. Conway, W.L. Maxwell, L.W. Miller. Theory of Scheduling, Addison-Wesley Publishing Company, 1967. 3. J.E. Day, M.P. Hottenstein. "Review of Sequencing Research", Naval Research Logistics Quarterly, Vol. 17, 1970. 4. S. French. Sequencing and Scheduling: A n Introduction to Mathematics of the Job-Shop, John Wiley and Sons, 1982. 5. M.J. Gonzalez, Jr. "Deterministic Processor Scheduling", Computing Surveys, Vol. 9, No. 3, September 1977, pp. 173-204. 6. R.L. Graham, R.L. Lawler, J.K. Lenstr, A. H.G. Rinooy Kan. "Optimization and Approximation in Deterministic Sequencing and Scheduling: A Survey", Annals of Discrete Mathematics, Vol. 5, 1979, pp. 287-326. 7. K. Phillips. "Aspects of Job Scheduling", Journal of Engineering for Industry, Vol. 101, February 1979. 8. K. Challa, P.B. Berra. "Automated Planning and Optimization of Machining Processes: A Systems Approach", Computers and Industrial Engineering, Vol. I, 1976, pp. 15-46. 9. T. Chang, R.A. Wysk. "An Introduction to Automated Process Planning Systems", Prentice-Hall, Inc., 1985. 10. M.P. Groover, A.M. Gunda, R.J. Johnson. "Determination of Machining Conditions by a Self-Adaptive Procedure", Proc. 4th North American Metalworking Research Conference, 1976. 11. S.K. Hati, S.S. Rao. "Determination of Optimum Machining Conditions: Deterministic and Probabilistic Approaches", Transactions of the ASME, February 1976, pp. 354-359. 12. G.M. Hayes, R.P. Davis. "A Discrete Variable Approach to Machine Parameter Optimization", AIIE Transactions, June 1979. 13. A. Houtzeel. "Computer Assisted Process Planning - A First Step Towards Integration", Proc. A UTOFA CT WEST, Society of Manufacturing Engineers, Anaheim, California. November 1980, pp. 801-808. 14. S.S. Rao, S.K. Hati. "Computerized Selection of Optimum Machining Conditions for a Job Requiring Multiple Operations", Transactions of the ASME, Vol. 100, August 1978.
Appendix B The model below is the decomposed machine parameter optimization for a given j o b / a n d machinej. Some variables not defined in the main text used in the model below are defined in Appendix A. Minimize C..q subject to:
Cit-(Uit + Lit) ka+ plt ~ (kd+ k,,) +0') (Pltk) [ Sitk DtJ I c" ] ,,I.~ (k~) +Pitt [(Sit* Dr) I c" ] "/"~ (kw) +[(d)i/ dlt,)+ ( w / w d ) + [1 i+ (dit , 67
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15. E.W. Zimmers, Jr. "Practical Applications for Computer Augmented
20. D.L. Kimbler, R.A. Wysk, R.P. Davis. "Alternative Approaches to the Machining Parameter Optimization Problem", Computers and Industrial Engineering, Vol. 2, 1978, pp. 195-202. 21. M.S. Bazaraa, C.M. Shetty. Nonlinear Programming: Theory and Applications, John Wiley and Sons, 1979. 22. C.S. Beightler, D.T. Phillips, D.J. Wilde. "Foundations of Optimization", Prentice-Hall, International Series of Industrial and Systems Engineering, Prentice-Hall, New Jersey, 1979. 23. D. Himmelblau. Applied Nonlinear Programming, McGraw-Hill Book Company, Inc., New York, 1972. 24. H.M. Salkin. Integer Programming, Addison-Wesley Publishing Company, 1975. 25. Mathematical Programming Systems Extended (MPSX) Mixed lnteger Programming (MIP), Program No, 5734-XMA. 26. L. Schrage. "Introduction to LINDO", Graduate School of Business, University of Chicago, Chicago, Illinois, November 1981.
Systems for Determination of Metal Removal Parameter and Production Standards in a Job Shop", SME Technical Paper No. MS71-136, 1971. 16. P.J. Egbelu. "Planning for Machining in a Multijob Multimachine Manufacturing Environment", Journal of Manufacturing Systems, Vol. 5, No. I, 1985, pp. 1-13. 17. Hooke and Jeeves. "Direct Search Solution of Numerical and Statistical Problems", J. Assoc. Comp. Mach., Vol. 8, No. 2, April 1961, pp. 212-229. 18. P.J. Egbelu, R.P. Davis, R.A. Wysk, J.M.A. Tanchoco. "An Economic Model for the Machining of Cast Parts", Journal of Manufacturing Systems, Vol. I, No. 2, 1984, pp. 207-213. 19. R.P. Davis, R.A. Wysk, M.H. Agee. "Characteristics of Machine Parameter Optimization Models", Applied Mathematical Modelling, Vol. 2, December 1978.
Author(s) Biography Edward F. Watson received his Masters degree in Industrial Engineering and Operations Research and is currently working on his Doctorate in Industrial Engineering at the Pennsylvania State University, University Park, Pennsylvania. He received his Bachelors degree in Industrial Engineering at Syracuse University. His interests are in manufacturing systems analysis, operations research, and artificial intelligence. His professional affiliations include SME, liE, ASEE, and ORSA. He is also a member of Tau Beta Pi and Alpha Pi Mu. Pius J. Egbelu is an Assistant Professor of Industrial and Management Systems Engineering at Pennsylvania State University. He was formerly with the Industrial Engineering Department at Syracuse University. He received a B.S.I.E. degree from Louisiana Tech University and a M.S.I.E. and Ph.D. from Virginia Polytechnic Institute and State University. Dr. Egbelu's teaching and research interests are in the area of automated material handling, manufacturing systems analysis, robotics, and production planning and control. His professional affiliations include SME, liE, Tau Beta Pi, Alpha Pi Mu and Phi Kappa Phi.
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