- Email: [email protected]
Minimizing total weighted completion time at a pre-assembly stage composed of two feeding machines
Int. J. Production Economics 54 (1998) 247—255
Minimizing total weighted completion time at a pre-assembly stage composed of two feeding machines Chang Sup Sung*, Sang Hum Yoon Department of Industrial Engineering, Korea Advanced Institute of Science and Technology, Gusung-dong, Yusung-gu, Taejon, 305-701, South Korea Received 27 September 1996; accepted 19 November 1997
Abstract This paper analyses a deterministic scheduling problem concerned with manufacturing two types of components at a pre-assembly stage which consists of two independent feeding machines each producing its own type of component. Each type represents a unique component which may have variations in its size or quality. Therefore, the completion time of each component depends on both its type and quality (size) variations. Such manufactured components are subsequently assembled into various component-dependent products. The problem has the objective measure of minimizing the total weighted completion time of a finite number of jobs (products) where the completion time of each job is measured by the latest completion time of its two components at the pre-assembly stage. The problem is shown to be NP-complete in the strong sense. A WSPT rule coupled with a machine-aggregation idea is developed for good heuristics which show the error bound of 2. Some empirical evaluations for the average-case performance of the heuristics are also performed. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Scheduling; Pre-assembly; Heuristics
1. Introduction This paper considers a simple manufacturing system where two types of components (or subassemblies) are fabricated first at its pre-assembly stage and then assembled into final products (jobs) at its assembly stage. The pre-assembly stage consists of two independent machines (called feeding machines) each producing its own type of components. Each assembly operation cannot start its process-
* Corresponding author. Tel.: #82 42 869 3131; #82 42 861 3110; e-mail: [email protected].
fax:
ing until each type pair of the associated components are prepared from the pre-assembly stage. This type of manufacturing system is frequently encountered in practice. As pointed out by Lee et al. [1], the body and the chassis, which are the components of a fire engine are manufactured in different (parallel) facilities and then brought into the assembly station. This implies that the final assembly station cannot start its processing until the body and the chassis are delivered to the assembly station. Other similar manufacturing situations including the production of personal computers, the production of multi-plane circuit boards (PCBs) and the just-in-time systems are illustrated
0925-5273/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reserved PII S 0 9 2 5 - 5 2 7 3 ( 9 7 ) 0 0 1 5 1 - 5
248
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
in Potts et al. [2], Duenyas [3] and Lummus [4]. Similarly, in computation work, several programs (or tasks) are independently processed first at their own parallel processors, and then gathered at a main processor for final data-processing where they either wait for all of their siblings to finish processing or are put together when processing is done with all the siblings. In the proposed manufacturing system, this paper is interested in a deterministic scheduling problem concerned with manufacturing two types of components at the pre-assembly stage where the completion time of a product (job) is measured by the latest completion time of its type pair of components at the stage. Minimizing the total weighted completion time of a finite number of products (jobs) is considered as the scheduling measure. In order to respond to a variety of customer needs for products, considerable effort has been made in designing products and manufacturing processes in the direction of assuring agility, flexibility and reconfigurability. For example, the concept of modular component design allows products to be produced flexibly for a variety of different customer demands by using various combinations of standard components, and to have high maintainability with ease of replacing any whole defective module directly by a good one. More importantly, as pointed out by Stoll [5], Boothroyd [6], He and Kusiak [7] and Whitney [8], the modular design concept simplifies the final assembly operation because of its contribution to reducing components to assemble. Moreover, each module can be fully checked prior to installation. These characteristics of the modular design give us a motivation to investigate a scheduling problem for producing components at a pre-assembly stage in a situation where the pre-assembly stage is a bottleneck in a manufacturing line of two stages and so such a modular approach of component fabrication is greatly desired. Lee et al. [1] and Potts et al. [2] have discussed problems similar to the proposed problem of this paper. Specifically, Lee et al. [1] have considered a scheduling problem in the aforementioned assembly system with two feeding machines and one assembly machine under makespan measure. Potts
et al. [2] have extended the problem of Lee et al. [1] to the case of multiple feeding machines. Some other scheduling works for such component productions (referring to Coffman et al. [9], Aneja and Singh [10], and Sung and Park [11]) have treated the whole feeding machines at the pre-assembly stage as an imbedded single machine. The rest of the paper is organized as follows. In Section 2, the problem is formally formulated. In Section 3, the proposed problem is proved to be NP-complete in the strong sense, and then a restricted case which can be solved in polynomial time is discussed. In Section 4, two heuristics incorporating the weighted shortest processing time first (WSPT) rule are suggested, and their worst-case performances are analyzed. In Section 5, some empirical evaluations for the average-case performance of the heuristics are performed.
2. Problem statement There are n jobs (products), denoted by J , J ,2, J , to be processed on two machines 1 2 n M and M . All jobs consist of two different types 1 2 of tasks (components). The tasks of each job J are i assigned to their associated machines such that the task assigned to M must be processed for uninter1 rupted a time units and the other one on M for i 2 uninterrupted b time units. The machines are not i identical but operated in parallel, so both tasks of a job can be handled independently at their respective machines. Each job is processed only when its associated tasks are all available. This implies that the completion time of j under a schedule S is i measured as C (S)"maxMc (S), c (S)N, where i i1 i2 c (S), k"1, 2, denotes the completion time of J on ik i M under S. For each job J , a weight factor w is k i i given, so that the total weighted completion time of the schedule S is expressed as n WC(S)" + w maxMc (S), c (S)N. (1) i i1 i2 i/1 This leads to the objective of this paper to find an optimal schedule S* such that WC(S*))WC(S) for any other possible schedule S.
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
For the given problem, schedules not incurring any idle time on both the machines dominate all other schedules, so that the maximum completion time (makespan) of all jobs is fixed at maxM+n a , +n b N, being independent of schedi/1 i i/1 i ules. It can also be seen that the schedule with minimum total completion time for the problem in the case of w "1 for every J will minimize the i i average number of tasks held in the shop, due to the equivalent relationship between completion time measure and work-in-process inventory (referring to Conway et al. [12]).
3. Analysis of the problem The following property is derived to show that it is sufficient to consider only n! permutations to solve the proposed problem. In the property verification made below, one of job perturbation operations, namely job-insertion operation, will be used. For example, given a sequence on M , an operation k denoted by I(i, j, M ) will be implemented to generk ate a new sequence by inserting J into the position i of J and then shifting each of J and all other jobs j i (possibly, none) located between the positions of J and J to one smaller position (resp., one greater i j position) when J precedes J (resp., J precedes J ) i j j i in the original sequence. Property 1. For the given problem, only permutation schedules need to be considered. Proof. Consider a non-permutation schedule S, which implies that somewhere in S, there must be a pair of jobs J and J in different orders on both i j machines. Without loss of generality, let M be the k machine on which the last one between the completion times of J and J is less than that on the other i j machine. If J precedes J (otherwise, J precedes J ) on M , i j j i k then the operation I(i, j, M ) (resp., I( j, i, M )) does k k not increase the completion time of any job. Similarly, by repeating the job-insertion operation for every other differently positioned job, a permutation schedule can be generated which is not worse than the schedule S. h
249
It will now be shown that the given problem is NP-complete in the strong sense, and hence, even a pseudo-polynomial time algorithm is unlikely to exist for the given problem. The NP-completeness will be proved by showing a reduction of the 3partition problem to an instance of the given problem (referring to Garey and Johnson [13]).
3.1. 3-partition problem Given a positive integer B and a set of integers X of 3m elements X"Mx , x ,2, x N such that 1 2 3m + x "mB and B/4(x (B/2 for i"1, 2, xi|X i i 2, 3m, does there exist a partition of X into m disjoint sets X , X ,2, X such that + x" 1 2 m xi|Xk i B for k"1, 2,2, m? Note that each X must conk tain exactly three elements. Theorem 1. ¹he given problem is NP-complete in the strong sense. Proof. Given the 3-partition instance, X"Mx , 1 x ,2, x N and B, consider the following instance 2 3m of the proposed scheduling problem: f number of jobs: n"4m, f job set: N XN where N "MJ , J ,2, J N 1 2 1 1 2 3m and N "MJ ,J , , J N, 2 3m`1 3m`2 2 4m f job processing times: (a , b )"(x , 0) for J 3N i i i i 1 and (a , b )"(1, B#1) for J 3N , where i i 1 2 B/4(x (B/2 for i"1, 2,2, 3m and i +3m x "mB, i/1 i f weight factors: w "x for J 3N and w "m2B i i i 1 i for J 3N i 2 Let us define the following: 3m 3m ¹" + + x x #B(B#1)m3(m#1)/2 i j i/1 j/i #Bm(m!1)/2.
(2)
Note that the first term of Eq. (2) is a lower bound for the total weighted completion time of N and 1 the second term is that for N . Thus, the sum of the 2 first two terms becomes a lower solution (schedule) bound.
250
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
We will now show that the above instance has a schedule S such that WC(S))¹ if and only if X has a 3-partition solution. (a) If X has a 3-partition X , X ,2, X , then it is 1 2 m easy to check that the total weighted completion time of a schedule S with the order MJ Dx 3X N, J , MJ Dx 3X N, J , , i i 1 3m`1 i i 2 3m`2 2 MJ Dx 3X N, J i i m 4m is equal to ¹, recalling that the sum of all the three elements x ’s contained in each X , k"1, 2,2, m is i k B. Note that a /w "1 and b "0 for J 3N , so the i i i i 1 value of the total weighted completion time ¹ is not influenced by any change of orders of the jobs contained in each X . The schedule S satisfying the k relation WC(S)"¹ is depicted in Fig. 1, where ¼ "MJ Dx 3X N for k"1, 2,2, m and z is a job k i i k in N . 2 (b) Now, suppose that it is given a schedule S such that WC(S))¹. In the schedule, the jobs in N should then be finished at M at the sequence of 2 2 times B#1, 2(B#1),2, and m(B#1); otherwise, WC(S)'¹, because any greater time sequence will incur additional completion time penalty of at least m2B due to the weight factor w for J 3N which is i i 2 greater than the third term of Eq. (2). It can also be seen that due to the constant ratio b /w " i i (B#1)/(m2B) for J 3N every such schedule for i 2 N satisfying the time sequence will give the same 2 completion time on M . Moreover, the schedule 2 S requires that at least a jobs in N should be 2 finished on M by the time a(B#1) for 1 a"1, 2,2, m; otherwise, the relation WC(S))¹ can not be satisfied. Such a schedule feature is depicted in Fig. 2. From Fig. 2, it is easy to see that inserting as many as possible jobs in N between the locations 1 of z’s on M will contribute to reduce the total 1 weighted completion time. However, the sum of any four x ’s in N is greater than B, because of the i 1 given range B/4(x (B/2. Thus, the best-possible i schedule is to get jobs in N finished exactly at the 2 sequence of times B#1, 2(B#1),2, and m(B#1) on M . Therefrom, three jobs should be assigned 1 between the locations of those z’s and the sum of the processing times of such three jobs in N (as1 signed between the locations of those z’s) should
Fig. 1. The structure of a schedule S with WC(S)"¹.
equal to B. This implies that the given schedule should constitute a 3-partition solution. h Although the given problem is shown to be NP-complete in the strong sense, we can find the following restricted case which can be solved in polynomial time complexity order O(n log n). For the case, the following property is suggested to decide the order of two jobs in consecutive positions. Property 2. If two jobs J and J are the candidates i j for any two consecutive job positions, and if the jobs satisfy the conditions a /w )a /w and b /w )b /w , (3) i i j j i i j j then the schedules with J followed by J dominate the i j schedules with J followed by J . j i Proof. Let S be a schedule in which J immediately i precedes J on both machines and S@ be the schedj ule generated from S by interchanging the positions of J and J . Then, the sum of the weighted complei j tion times of J and J can be derived as, for each i j schedule +"S, S@, d(+ )"w maxMc (+ ), c (+ )N#w maxMc (+ ), c (+ )N i i1 i2 j j1 j2 w c (+ )#w c (+ ), w c (+ )#w c (+ ) j j1 i i1 j j2 "max i i1 w c (+ )#w c (+ ), w c (+ )#w c (+ ) i i2 j j1 i i2 j j2 (4)
G
H
and the completion time of any other job is not affected by the interchange. Let t and t denote the starting times of J (or J ) 1 2 i j on M and M under S (resp., S@), respectively. 1 2 Then, the completion times of J and J can be i j
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
251
Case 2: d(S@)"w c (S@)#w c (S@). By the similar i i2 j j1 reasoning to Case 1, it can be seen that b *a . i i Thus, under the given condition w b *w b , it is i j j i obvious that
Fig. 2. The structure of a schedule without any additional delay on jobs in N . 2
expressed as c (S)"t #a , i1 1 i
Let us consider two sequences S and S ob1 2 tained by applying the WSPT rule to two singlemachine scheduling problems for the measure of minimizing the total weighted completion time represented by the instance-sets Ma , w D1)i)nN and i i Mb , w D1)i)nN respectively. Then, the following i i corollary holds.
c (S)"t #b , i2 2 i
c (S)"t #a #a , c (S)"t #b #b , j1 1 i j j2 2 i j c (S@)"t #a #a , i1 1 j i
c (S@)"t #b #b , i2 2 j i
c (S@)"t #a and c (S@)"t #b . j1 1 j j2 2 j We will now prove the relation d(S@)*d(S) by investigating both the cases separately, corresponding to the situations where d(S@) is obtained as the maximum out of four terms in Eq. (4). Considering the cases of the first and forth terms in Eq. (4), and referring to the fact that the WSPT rule guarantees the optimum for the single machine case, the following are obvious: w c (S@)#w c (S@)*w c (S)#w c (S) i i1 j j1 i i1 j j1 and w c (S@)#w c (S@)*w c (S)#w c (S). i i2 j j2 i i2 j j2 Now, let us consider the two remaining cases separately. Case 1: d(S@)"w c (S@)#w c (S@). This case i i1 j j2 shows that c (S@)*c (S@) and c (S@)*c (S@), that i1 i2 j2 j1 t #a #a *t #b #b and t #b *t #a , 1 j i 2 j i 2 j 1 j and so a *b . Under the given condition of i i w a *w a , we have i j j i w c (S@)#w c (S@)"w (t #a #a )#w (t #b ) i i1 j j2 i 1 j i j 2 j *w (t #a )#w (t #b #a ) i 1 i j 2 j i *w (t #a )#w (t #b #b ) i 1 i j 2 j i "w c (S)#w c (S). i i1 j j2
w c (S@)#w c (S@)"w (t #b #b )#w (t #a ) i i2 j j1 i 2 j i j 1 j *w (t #b )#w (t #a #b ) i 2 i j 1 j i *w (t #b )#w (t #a #a ) i 2 i j 1 j i "w c (S)#w c (S). (6) i i2 j j1 The two cases show that d(S@)*d(S). h
(5)
Corollary 1. If S and S are the same, then the 1 2 sequence is optimal. Proof. If S and S are the same (denoted by the 1 2 schedule S), then it must be sure that any two jobs satisfy condition (3), and furthermore the relation be transitive for all jobs. This implies that the first job in S must be the job with minMa /w D1)i)nN i i and minMb /w D1)i)nN. Let J denote the first i i k job. Then, by Property 2, J should be processed k first in an optimal schedule. Thereafter, exclude J from further consideration. Likewise, the second k job in S should be the second job in an optimal schedule since the job has the minimal values (a /w i i and b /w ) of the remaining jobs. If the procedure is i i similarly repeated with each other job, then it will surely be ended up with the conclusion that S is an optimal schedule. h
4. Two simple heuristics The result of Theorem 1 gives us the motivation of exploiting heuristics for the proposed problem. In this section, two simple O(n log n) heuristics are proposed and their worst-case performances are analyzed. The underling idea of the heuristics is
252
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
that the job processing times on the two feeding machines are aggregated into a single sequence of information for the WSPT rule to be applied to generate a permutation schedule for each of the feeding machines. The machine aggregation idea was initially introduced by Lee et al. [1] and Potts et al. [2] such that considering the two-stage assembly line scheduling problem for minimizing the makespan, the machine aggregation was used to simplify the overall line to be a two-stage flowshop and then to apply the well-known Johnson’s rule for the resulting flowshop makespan problem. The first heuristic Max-agg is briefly described as follows: first, compute p "maxMa , b N for each job i i i J , and then, generate the WSPT schedule for the * single machine instanceMp , w Di"1, 2,2, nN. Let i i S and S* denote the schedule generated by MaxH1 agg and an optimal schedule, respectively. Theorem 2. WC(S )/WC(S*))2. H1 Proof. Without loss of generality, let jobs be ordered in S such that p /w )p /w ),2, H1 1 1 2 2 )p /w , where p "maxMa , b N. Let C (S ) denote n n i i i i H1 the completion time of J under S . It is then easy i H1 to check that i C (S )) + p for i"1, 2,2, n. i H1 j j/1
(7)
Thus, the total weighted completion time of S is H1 derived as i n n WC(S )" + w C (S )) + w + p . j H1 i i H1 i i/1 i/1 j/1
(8)
Now, consider the feature of S*. Let [i] denote the subscript to indicate job position under S*. The completion time of J under S* is *i+
A
B
i i i C (S*)"max + a , + b * + (a #b )/2. *i+ *j+ *j+ *j+ *j+ j/1 j/1 j/1 (9) It follows that n i WC(S*)* + w + (a #b )/2 *i+ *j+ *j+ i/1 j/1
n i * + w + max(a , b )/2 *i+ *j+ *j+ i/1 j/1 n i * + w + p /2. (10) i j i/1 j/1 The last inequality of Eq. (10) is based on the fact that the WSPT rule guarantees an optimal schedule for the single machine case. From Eq. (8), the result is confirmed. h As pointed out by Baker and Powell [14] and Baker et al. [15], it may be noted that an assembly station loaded from several feeding machines may adapt two classified loading mechanisms, simultaneous loading and independent loading. Under the independent loading, any completed component is moved to the input buffer of the assembly station, independently of the status of any other component. However, under the simultaneous loading, the assembly station does not accept any completed component until all the associated components are ready. This implies that a feeding machine can be blocked when its output buffer is not allowed to receive any completed component. If the incorporated pre-assembly stage of this paper adapts the simultaneous loading and any output buffer is not allowed, then it can easily be seen that the schedule generated by Max-agg guarantees the optimality for the given problem. The second heuristic Sum-agg is now briefly described as follows: first, compute p "(a #b ) for i i i each job J , and then, generate the WSPT schedule * for the single machine instance Mp , w Di"1, 2, i i 2, nN. Let SH2 be the heuristic schedule generated by Sum-agg. Theorem 3. WC(S )/WC(S*))2. H2 Proof. Without loss of generality, let jobs be ordered in S such that p /w )p /w ),2, H2 1 1 2 2 )p /w , where p "(a #b ). Then, the total n n i i i weighted completion time of S is derived as H2 n n i WC(S )" + w C (S )) + w + p . (11) H2 i i H2 i j i/1 i/1 j/1 Then, by using Eq. (9) of Theorem 2, the result can easily be confirmed. h
253
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
Theorem 4. If it is satisfied that (a #b )/2*Da !b D i i i i for all jobs, then the bound of S can be reduced to 3. 2 H2 Proof. Without loss of generality, let jobs be ordered as in S . The completion time of J under H2 i S can be expressed as H2 i i C (S )"max + a , + b j j i H2 j/1 j/1 i i i " + (a #b )!min + a , + b j j j j j/1 j/1 j/1 for i" 1, 2,2, n. (12) Then, from the relation
A
A
B A
B
B
A
i i i i max + a , + b !min + a , + b j j j j j/1 j/1 j/1 j/1 i ) + Da !b D for i"1, 2,2, n, j j j/1 it is clear that
A
B
From Theorem 3, it can be expressed as WC(S ))2WC(S*)!WC(S ) H2 H2 n i # + w + Da !b D, i j j i/1 j/1 and further simplified as
(15)
WC(S )/WC(S*) H2 n i )1#(1/2) + w + Da !b D/WC(S*) . i j j i/1 j/1 From the given condition, it is clear that
A
B
i n WC(S*)* + w + Da !b D, and thus, j j i i/1 j/1 WC(S )/WC(S*))3/2. h H2
(16)
(17)
(13) 5. Computational experience
B
i i i C (S )) + (a #b )!max + a , + b j j i H2 j j j/1 j/1 j/1 i # + Da !b D for i" 1, 2,2, n. (14) j j j/1
This section describes the computational tests which are made to evaluate and compare the effectiveness of the developed heuristics, Max-agg and Sum-agg. The heuristics are programmed in C-language, and the tests are performed on 586-PC. The
Table 1 Computational results of percentage deviation from the optimal solution (top: class 1, middle: class 2, bottom: class 3) n
5
6
7
8
9
10
Max-agg
Sum-agg
Mean
Maximum
Median
Mean
Maximum
Median
1.40 0.54 0.20 1.69 1.32 0.24 1.56 1.24 0.33 2.96 2.19 0.16 3.08 1.79 0.25 3.44 1.66 0.39
8.18 3.49 3.29 9.36 8.75 3.27 11.65 11.13 3.65 9.94 6.20 1.76 12.11 7.50 2.80 11.21 5.46 3.39
0.14 0.00 0.00 0.59 0.23 0.00 0.37 0.21 0.00 2.03 1.75 0.00 2.52 1.25 0.00 2.92 1.15 0.03
2.07 1.51 1.76 2.26 2.18 1.82 2.03 2.09 1.61 3.29 2.75 1.89 2.94 2.43 2.36 3.78 2.79 2.52
12.44 11.95 7.39 8.95 9.84 10.86 8.35 11.77 7.83 11.84 8.11 8.55 12.10 7.64 7.29 15.82 7.56 7.97
0.89 0.34 0.44 1.60 1.44 0.63 1.19 1.43 0.89 2.55 2.18 1.57 2.36 1.86 1.91 3.49 2.43 2.41
254
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
test problems are generated with the number of jobs ranging from 5 to 50. For all the test problems, the weight factors (w ) are generated from the disi crete uniform distribution with range [1, 20]. How hard to solve such a problem is likely to depend on whether or not there is any workload balance between the machines. To indicate each problem difficulty, the job processing times are separately generated in three classes. In class 1, the processing times on both machines are commonly generated from the discrete uniform distribution with range [1, 30], which implies that the workloads between the machines are balanced. In class 2, the processing times on the first machine are generated from the uniform distribution with range [5, 30] but those on the second machine from the uniform distribution with range [1, 25], which implies that the workload on the first machine is slightly larger than that on the second machine. To represent the extremely unbalanced case, in class 3, the processing times on the first machine are generated from the uniform distribution with range [10, 30] and those on second machine from the uniform distribution with range [1, 20]. For each combination of the number of jobs and the class of
the processing times, 50 problems are generated to gather some statistics for the performance of the heuristics. The tests are carried out in two phases; first, 900 problems considering 5, 6, 7, 8, 9 and 10 jobs are generated and solved by use of the heuristics, and also solved (optimally) in full enumeration. Then, all the heuristic results are compared with the corresponding optimal solutions. The results of such an evaluation are reported in Table 1 which shows mean, maximum and median percentage deviations from the optimal solution. Each of the percentage deviations from the optimal solution is defined as 100](heu!opt)/opt, where heu is the heuristic objective value and opt is the optimal value. In the table, the top, middle and bottom of each cell represent the results of classes 1, 2 and 3, respectively. The second phase of the tests is carried out for 900 large-size problems having 13, 15, 20, 30, 40 and 50 jobs. The solutions to such problems searched by the heuristics are compared with the lower bound which is obtained as follows: LB"maxMWSPT , WSPT N, 1 2
Table 2 Computational results of percentage deviation from the lower bound (top: class 1, middle: class 2, bottom: class 3) n
13
15
20
30
40
50
Max-agg
Sum-agg
Mean
Maximum
Median
Mean
Maximum
Median
10.62 5.62 0.76 11.42 6.62 0.65 11.98 6.11 0.64 13.14 5.35 0.54 12.91 5.26 0.55 13.09 4.97 0.59
25.67 25.53 4.33 24.40 19.23 3.36 28.27 18.21 4.15 26.98 14.61 4.01 23.80 12.95 2.44 24.14 10.98 2.43
9.28 4.35 0.18 10.26 4.65 0.17 11.56 5.79 0.36 12.48 4.34 0.34 11.44 4.54 0.43 12.26 5.03 0.51
10.62 6.68 2.81 11.36 7.85 2.69 11.94 6.89 2.86 11.89 6.53 3.19 12.34 6.49 3.13 12.07 6.47 3.38
29.17 22.33 8.65 23.22 20.19 6.69 22.64 17.93 6.78 26.72 15.06 6.26 21.41 10.98 5.54 23.37 11.54 5.96
9.35 5.76 2.59 10.25 6.53 2.41 11.18 6.44 2.74 11.17 5.69 3.22 11.71 6.23 3.01 10.69 6.62 3.28
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
where WSPT , k"1, 2, denotes the total weighted k completion time obtained by applying the WSPT rule to M . The lower bound is easily derived by k considering Eq. (1) and using the property that the WSPT rule provides an optimal solution for each machine. The results of the second phase are shown in Table 2 where mean, maximum and median percentage deviations from the lower bound are listed. Each of the percentage deviations from the lower bound is defined as 100](heu!LB)/LB.
6. Conclusions This paper considers a scheduling problem with the objective measure of minimizing the total weighted completion time at a pre-assembly stage composed of two feeding machines. It is shown that the problem is strongly NP-complete. Some solution properties are characterized for a restricted case which can be solved in O(n log n). Based on the properties, two O(n log n) heuristics incorporating the WSPT rule are derived, and their worst-case performances are analyzed as giving the error bound of 2. The properties may also be used to search for solutions in a branch and bound method. The problem is important in its own right, but it is also significant in a sense that the results of this paper will provide the underlined scheme and insight for any extended problem such as the preassembly stage with m(m'2) feeding machines.
References [1] C.Y. Lee, T.C.E. Cheng, B.M.T. Lin, Minimizing the makespan in the 3-machine assembly-type flowshop scheduling problem, Management Science 39 (1993) 616—625.
255
[2] C.N. Potts, S.V. Sevast’janov, V.A. Strusevich, L.N. Wassenhove, C.M. Zwaneveld, The two-stage assembly scheduling problem: complexity and approximation, Operations Research 43 (1995) 346—355. [3] I. Duenyas, Estimating the throughput of a cyclic assembly system, International Journal of Production Research 32 (1994) 1403—1410. [4] R.R. Lummus, A Simulation analysis of sequencing alternatives for JIT lines using kanbans, Journal of Operations Management 13 (1995) 183—191. [5] H.W. Stoll, Design for manufacture: An overview, Applied Mechanics Review 39 (1986) 1356—1364. [6] G. Boothroyd, Assembly Automation and Product Design, Marcell Dekker, New York, 1992. [7] D.W. He, A. Kusiak, Performance analysis of modular products, International Journal of Production Research 34 (1996) 253—272. [8] D.E. Whitney, Nippondenso Co. Ltd: a case study of strategic product design, Research in Engineering Design 5 (1993) 1—20. [9] E.D. Coffman, A. Nozari, M. Yannakakis, Optimal scheduling of products with two subassemblies on a single machine, Operations Research 37 (1989) 426—436. [10] Y.P. Aneja, N. Singh, Scheduling problem for common components at a single facility, IIE Trans. 22 (1990) 234—237. [11] C.S. Sung, C.K. Park, Scheduling of products with common and product-dependent components manufactured at a single facility, Journal of the Operational Research Society 44 (1993) 773—784. [12] R.W. Conway, W.L. Maxwell, L.W. Miller, Theory of Scheduling, Addison-Wesley, Reading, MA, 1967. [13] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, San Francisco, 1979. [14] K.R. Baker, S.G. Powell, A predictive model for the throughput of simple assembly systems, European Journal of Operational Research 81 (1995) 336—345. [15] K.R. Baker, S.G. Powell, D.F. Pyke, Optimal allocation of work in assembly systems, Management Science 39 (1993) 101—106.
Minimizing total weighted completion time at a pre-assembly stage composed of two feeding machines Chang Sup Sung*, Sang Hum Yoon Department of Industrial Engineering, Korea Advanced Institute of Science and Technology, Gusung-dong, Yusung-gu, Taejon, 305-701, South Korea Received 27 September 1996; accepted 19 November 1997
Abstract This paper analyses a deterministic scheduling problem concerned with manufacturing two types of components at a pre-assembly stage which consists of two independent feeding machines each producing its own type of component. Each type represents a unique component which may have variations in its size or quality. Therefore, the completion time of each component depends on both its type and quality (size) variations. Such manufactured components are subsequently assembled into various component-dependent products. The problem has the objective measure of minimizing the total weighted completion time of a finite number of jobs (products) where the completion time of each job is measured by the latest completion time of its two components at the pre-assembly stage. The problem is shown to be NP-complete in the strong sense. A WSPT rule coupled with a machine-aggregation idea is developed for good heuristics which show the error bound of 2. Some empirical evaluations for the average-case performance of the heuristics are also performed. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Scheduling; Pre-assembly; Heuristics
1. Introduction This paper considers a simple manufacturing system where two types of components (or subassemblies) are fabricated first at its pre-assembly stage and then assembled into final products (jobs) at its assembly stage. The pre-assembly stage consists of two independent machines (called feeding machines) each producing its own type of components. Each assembly operation cannot start its process-
* Corresponding author. Tel.: #82 42 869 3131; #82 42 861 3110; e-mail: [email protected].
fax:
ing until each type pair of the associated components are prepared from the pre-assembly stage. This type of manufacturing system is frequently encountered in practice. As pointed out by Lee et al. [1], the body and the chassis, which are the components of a fire engine are manufactured in different (parallel) facilities and then brought into the assembly station. This implies that the final assembly station cannot start its processing until the body and the chassis are delivered to the assembly station. Other similar manufacturing situations including the production of personal computers, the production of multi-plane circuit boards (PCBs) and the just-in-time systems are illustrated
0925-5273/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reserved PII S 0 9 2 5 - 5 2 7 3 ( 9 7 ) 0 0 1 5 1 - 5
248
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
in Potts et al. [2], Duenyas [3] and Lummus [4]. Similarly, in computation work, several programs (or tasks) are independently processed first at their own parallel processors, and then gathered at a main processor for final data-processing where they either wait for all of their siblings to finish processing or are put together when processing is done with all the siblings. In the proposed manufacturing system, this paper is interested in a deterministic scheduling problem concerned with manufacturing two types of components at the pre-assembly stage where the completion time of a product (job) is measured by the latest completion time of its type pair of components at the stage. Minimizing the total weighted completion time of a finite number of products (jobs) is considered as the scheduling measure. In order to respond to a variety of customer needs for products, considerable effort has been made in designing products and manufacturing processes in the direction of assuring agility, flexibility and reconfigurability. For example, the concept of modular component design allows products to be produced flexibly for a variety of different customer demands by using various combinations of standard components, and to have high maintainability with ease of replacing any whole defective module directly by a good one. More importantly, as pointed out by Stoll [5], Boothroyd [6], He and Kusiak [7] and Whitney [8], the modular design concept simplifies the final assembly operation because of its contribution to reducing components to assemble. Moreover, each module can be fully checked prior to installation. These characteristics of the modular design give us a motivation to investigate a scheduling problem for producing components at a pre-assembly stage in a situation where the pre-assembly stage is a bottleneck in a manufacturing line of two stages and so such a modular approach of component fabrication is greatly desired. Lee et al. [1] and Potts et al. [2] have discussed problems similar to the proposed problem of this paper. Specifically, Lee et al. [1] have considered a scheduling problem in the aforementioned assembly system with two feeding machines and one assembly machine under makespan measure. Potts
et al. [2] have extended the problem of Lee et al. [1] to the case of multiple feeding machines. Some other scheduling works for such component productions (referring to Coffman et al. [9], Aneja and Singh [10], and Sung and Park [11]) have treated the whole feeding machines at the pre-assembly stage as an imbedded single machine. The rest of the paper is organized as follows. In Section 2, the problem is formally formulated. In Section 3, the proposed problem is proved to be NP-complete in the strong sense, and then a restricted case which can be solved in polynomial time is discussed. In Section 4, two heuristics incorporating the weighted shortest processing time first (WSPT) rule are suggested, and their worst-case performances are analyzed. In Section 5, some empirical evaluations for the average-case performance of the heuristics are performed.
2. Problem statement There are n jobs (products), denoted by J , J ,2, J , to be processed on two machines 1 2 n M and M . All jobs consist of two different types 1 2 of tasks (components). The tasks of each job J are i assigned to their associated machines such that the task assigned to M must be processed for uninter1 rupted a time units and the other one on M for i 2 uninterrupted b time units. The machines are not i identical but operated in parallel, so both tasks of a job can be handled independently at their respective machines. Each job is processed only when its associated tasks are all available. This implies that the completion time of j under a schedule S is i measured as C (S)"maxMc (S), c (S)N, where i i1 i2 c (S), k"1, 2, denotes the completion time of J on ik i M under S. For each job J , a weight factor w is k i i given, so that the total weighted completion time of the schedule S is expressed as n WC(S)" + w maxMc (S), c (S)N. (1) i i1 i2 i/1 This leads to the objective of this paper to find an optimal schedule S* such that WC(S*))WC(S) for any other possible schedule S.
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
For the given problem, schedules not incurring any idle time on both the machines dominate all other schedules, so that the maximum completion time (makespan) of all jobs is fixed at maxM+n a , +n b N, being independent of schedi/1 i i/1 i ules. It can also be seen that the schedule with minimum total completion time for the problem in the case of w "1 for every J will minimize the i i average number of tasks held in the shop, due to the equivalent relationship between completion time measure and work-in-process inventory (referring to Conway et al. [12]).
3. Analysis of the problem The following property is derived to show that it is sufficient to consider only n! permutations to solve the proposed problem. In the property verification made below, one of job perturbation operations, namely job-insertion operation, will be used. For example, given a sequence on M , an operation k denoted by I(i, j, M ) will be implemented to generk ate a new sequence by inserting J into the position i of J and then shifting each of J and all other jobs j i (possibly, none) located between the positions of J and J to one smaller position (resp., one greater i j position) when J precedes J (resp., J precedes J ) i j j i in the original sequence. Property 1. For the given problem, only permutation schedules need to be considered. Proof. Consider a non-permutation schedule S, which implies that somewhere in S, there must be a pair of jobs J and J in different orders on both i j machines. Without loss of generality, let M be the k machine on which the last one between the completion times of J and J is less than that on the other i j machine. If J precedes J (otherwise, J precedes J ) on M , i j j i k then the operation I(i, j, M ) (resp., I( j, i, M )) does k k not increase the completion time of any job. Similarly, by repeating the job-insertion operation for every other differently positioned job, a permutation schedule can be generated which is not worse than the schedule S. h
249
It will now be shown that the given problem is NP-complete in the strong sense, and hence, even a pseudo-polynomial time algorithm is unlikely to exist for the given problem. The NP-completeness will be proved by showing a reduction of the 3partition problem to an instance of the given problem (referring to Garey and Johnson [13]).
3.1. 3-partition problem Given a positive integer B and a set of integers X of 3m elements X"Mx , x ,2, x N such that 1 2 3m + x "mB and B/4(x (B/2 for i"1, 2, xi|X i i 2, 3m, does there exist a partition of X into m disjoint sets X , X ,2, X such that + x" 1 2 m xi|Xk i B for k"1, 2,2, m? Note that each X must conk tain exactly three elements. Theorem 1. ¹he given problem is NP-complete in the strong sense. Proof. Given the 3-partition instance, X"Mx , 1 x ,2, x N and B, consider the following instance 2 3m of the proposed scheduling problem: f number of jobs: n"4m, f job set: N XN where N "MJ , J ,2, J N 1 2 1 1 2 3m and N "MJ ,J , , J N, 2 3m`1 3m`2 2 4m f job processing times: (a , b )"(x , 0) for J 3N i i i i 1 and (a , b )"(1, B#1) for J 3N , where i i 1 2 B/4(x (B/2 for i"1, 2,2, 3m and i +3m x "mB, i/1 i f weight factors: w "x for J 3N and w "m2B i i i 1 i for J 3N i 2 Let us define the following: 3m 3m ¹" + + x x #B(B#1)m3(m#1)/2 i j i/1 j/i #Bm(m!1)/2.
(2)
Note that the first term of Eq. (2) is a lower bound for the total weighted completion time of N and 1 the second term is that for N . Thus, the sum of the 2 first two terms becomes a lower solution (schedule) bound.
250
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
We will now show that the above instance has a schedule S such that WC(S))¹ if and only if X has a 3-partition solution. (a) If X has a 3-partition X , X ,2, X , then it is 1 2 m easy to check that the total weighted completion time of a schedule S with the order MJ Dx 3X N, J , MJ Dx 3X N, J , , i i 1 3m`1 i i 2 3m`2 2 MJ Dx 3X N, J i i m 4m is equal to ¹, recalling that the sum of all the three elements x ’s contained in each X , k"1, 2,2, m is i k B. Note that a /w "1 and b "0 for J 3N , so the i i i i 1 value of the total weighted completion time ¹ is not influenced by any change of orders of the jobs contained in each X . The schedule S satisfying the k relation WC(S)"¹ is depicted in Fig. 1, where ¼ "MJ Dx 3X N for k"1, 2,2, m and z is a job k i i k in N . 2 (b) Now, suppose that it is given a schedule S such that WC(S))¹. In the schedule, the jobs in N should then be finished at M at the sequence of 2 2 times B#1, 2(B#1),2, and m(B#1); otherwise, WC(S)'¹, because any greater time sequence will incur additional completion time penalty of at least m2B due to the weight factor w for J 3N which is i i 2 greater than the third term of Eq. (2). It can also be seen that due to the constant ratio b /w " i i (B#1)/(m2B) for J 3N every such schedule for i 2 N satisfying the time sequence will give the same 2 completion time on M . Moreover, the schedule 2 S requires that at least a jobs in N should be 2 finished on M by the time a(B#1) for 1 a"1, 2,2, m; otherwise, the relation WC(S))¹ can not be satisfied. Such a schedule feature is depicted in Fig. 2. From Fig. 2, it is easy to see that inserting as many as possible jobs in N between the locations 1 of z’s on M will contribute to reduce the total 1 weighted completion time. However, the sum of any four x ’s in N is greater than B, because of the i 1 given range B/4(x (B/2. Thus, the best-possible i schedule is to get jobs in N finished exactly at the 2 sequence of times B#1, 2(B#1),2, and m(B#1) on M . Therefrom, three jobs should be assigned 1 between the locations of those z’s and the sum of the processing times of such three jobs in N (as1 signed between the locations of those z’s) should
Fig. 1. The structure of a schedule S with WC(S)"¹.
equal to B. This implies that the given schedule should constitute a 3-partition solution. h Although the given problem is shown to be NP-complete in the strong sense, we can find the following restricted case which can be solved in polynomial time complexity order O(n log n). For the case, the following property is suggested to decide the order of two jobs in consecutive positions. Property 2. If two jobs J and J are the candidates i j for any two consecutive job positions, and if the jobs satisfy the conditions a /w )a /w and b /w )b /w , (3) i i j j i i j j then the schedules with J followed by J dominate the i j schedules with J followed by J . j i Proof. Let S be a schedule in which J immediately i precedes J on both machines and S@ be the schedj ule generated from S by interchanging the positions of J and J . Then, the sum of the weighted complei j tion times of J and J can be derived as, for each i j schedule +"S, S@, d(+ )"w maxMc (+ ), c (+ )N#w maxMc (+ ), c (+ )N i i1 i2 j j1 j2 w c (+ )#w c (+ ), w c (+ )#w c (+ ) j j1 i i1 j j2 "max i i1 w c (+ )#w c (+ ), w c (+ )#w c (+ ) i i2 j j1 i i2 j j2 (4)
G
H
and the completion time of any other job is not affected by the interchange. Let t and t denote the starting times of J (or J ) 1 2 i j on M and M under S (resp., S@), respectively. 1 2 Then, the completion times of J and J can be i j
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
251
Case 2: d(S@)"w c (S@)#w c (S@). By the similar i i2 j j1 reasoning to Case 1, it can be seen that b *a . i i Thus, under the given condition w b *w b , it is i j j i obvious that
Fig. 2. The structure of a schedule without any additional delay on jobs in N . 2
expressed as c (S)"t #a , i1 1 i
Let us consider two sequences S and S ob1 2 tained by applying the WSPT rule to two singlemachine scheduling problems for the measure of minimizing the total weighted completion time represented by the instance-sets Ma , w D1)i)nN and i i Mb , w D1)i)nN respectively. Then, the following i i corollary holds.
c (S)"t #b , i2 2 i
c (S)"t #a #a , c (S)"t #b #b , j1 1 i j j2 2 i j c (S@)"t #a #a , i1 1 j i
c (S@)"t #b #b , i2 2 j i
c (S@)"t #a and c (S@)"t #b . j1 1 j j2 2 j We will now prove the relation d(S@)*d(S) by investigating both the cases separately, corresponding to the situations where d(S@) is obtained as the maximum out of four terms in Eq. (4). Considering the cases of the first and forth terms in Eq. (4), and referring to the fact that the WSPT rule guarantees the optimum for the single machine case, the following are obvious: w c (S@)#w c (S@)*w c (S)#w c (S) i i1 j j1 i i1 j j1 and w c (S@)#w c (S@)*w c (S)#w c (S). i i2 j j2 i i2 j j2 Now, let us consider the two remaining cases separately. Case 1: d(S@)"w c (S@)#w c (S@). This case i i1 j j2 shows that c (S@)*c (S@) and c (S@)*c (S@), that i1 i2 j2 j1 t #a #a *t #b #b and t #b *t #a , 1 j i 2 j i 2 j 1 j and so a *b . Under the given condition of i i w a *w a , we have i j j i w c (S@)#w c (S@)"w (t #a #a )#w (t #b ) i i1 j j2 i 1 j i j 2 j *w (t #a )#w (t #b #a ) i 1 i j 2 j i *w (t #a )#w (t #b #b ) i 1 i j 2 j i "w c (S)#w c (S). i i1 j j2
w c (S@)#w c (S@)"w (t #b #b )#w (t #a ) i i2 j j1 i 2 j i j 1 j *w (t #b )#w (t #a #b ) i 2 i j 1 j i *w (t #b )#w (t #a #a ) i 2 i j 1 j i "w c (S)#w c (S). (6) i i2 j j1 The two cases show that d(S@)*d(S). h
(5)
Corollary 1. If S and S are the same, then the 1 2 sequence is optimal. Proof. If S and S are the same (denoted by the 1 2 schedule S), then it must be sure that any two jobs satisfy condition (3), and furthermore the relation be transitive for all jobs. This implies that the first job in S must be the job with minMa /w D1)i)nN i i and minMb /w D1)i)nN. Let J denote the first i i k job. Then, by Property 2, J should be processed k first in an optimal schedule. Thereafter, exclude J from further consideration. Likewise, the second k job in S should be the second job in an optimal schedule since the job has the minimal values (a /w i i and b /w ) of the remaining jobs. If the procedure is i i similarly repeated with each other job, then it will surely be ended up with the conclusion that S is an optimal schedule. h
4. Two simple heuristics The result of Theorem 1 gives us the motivation of exploiting heuristics for the proposed problem. In this section, two simple O(n log n) heuristics are proposed and their worst-case performances are analyzed. The underling idea of the heuristics is
252
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
that the job processing times on the two feeding machines are aggregated into a single sequence of information for the WSPT rule to be applied to generate a permutation schedule for each of the feeding machines. The machine aggregation idea was initially introduced by Lee et al. [1] and Potts et al. [2] such that considering the two-stage assembly line scheduling problem for minimizing the makespan, the machine aggregation was used to simplify the overall line to be a two-stage flowshop and then to apply the well-known Johnson’s rule for the resulting flowshop makespan problem. The first heuristic Max-agg is briefly described as follows: first, compute p "maxMa , b N for each job i i i J , and then, generate the WSPT schedule for the * single machine instanceMp , w Di"1, 2,2, nN. Let i i S and S* denote the schedule generated by MaxH1 agg and an optimal schedule, respectively. Theorem 2. WC(S )/WC(S*))2. H1 Proof. Without loss of generality, let jobs be ordered in S such that p /w )p /w ),2, H1 1 1 2 2 )p /w , where p "maxMa , b N. Let C (S ) denote n n i i i i H1 the completion time of J under S . It is then easy i H1 to check that i C (S )) + p for i"1, 2,2, n. i H1 j j/1
(7)
Thus, the total weighted completion time of S is H1 derived as i n n WC(S )" + w C (S )) + w + p . j H1 i i H1 i i/1 i/1 j/1
(8)
Now, consider the feature of S*. Let [i] denote the subscript to indicate job position under S*. The completion time of J under S* is *i+
A
B
i i i C (S*)"max + a , + b * + (a #b )/2. *i+ *j+ *j+ *j+ *j+ j/1 j/1 j/1 (9) It follows that n i WC(S*)* + w + (a #b )/2 *i+ *j+ *j+ i/1 j/1
n i * + w + max(a , b )/2 *i+ *j+ *j+ i/1 j/1 n i * + w + p /2. (10) i j i/1 j/1 The last inequality of Eq. (10) is based on the fact that the WSPT rule guarantees an optimal schedule for the single machine case. From Eq. (8), the result is confirmed. h As pointed out by Baker and Powell [14] and Baker et al. [15], it may be noted that an assembly station loaded from several feeding machines may adapt two classified loading mechanisms, simultaneous loading and independent loading. Under the independent loading, any completed component is moved to the input buffer of the assembly station, independently of the status of any other component. However, under the simultaneous loading, the assembly station does not accept any completed component until all the associated components are ready. This implies that a feeding machine can be blocked when its output buffer is not allowed to receive any completed component. If the incorporated pre-assembly stage of this paper adapts the simultaneous loading and any output buffer is not allowed, then it can easily be seen that the schedule generated by Max-agg guarantees the optimality for the given problem. The second heuristic Sum-agg is now briefly described as follows: first, compute p "(a #b ) for i i i each job J , and then, generate the WSPT schedule * for the single machine instance Mp , w Di"1, 2, i i 2, nN. Let SH2 be the heuristic schedule generated by Sum-agg. Theorem 3. WC(S )/WC(S*))2. H2 Proof. Without loss of generality, let jobs be ordered in S such that p /w )p /w ),2, H2 1 1 2 2 )p /w , where p "(a #b ). Then, the total n n i i i weighted completion time of S is derived as H2 n n i WC(S )" + w C (S )) + w + p . (11) H2 i i H2 i j i/1 i/1 j/1 Then, by using Eq. (9) of Theorem 2, the result can easily be confirmed. h
253
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
Theorem 4. If it is satisfied that (a #b )/2*Da !b D i i i i for all jobs, then the bound of S can be reduced to 3. 2 H2 Proof. Without loss of generality, let jobs be ordered as in S . The completion time of J under H2 i S can be expressed as H2 i i C (S )"max + a , + b j j i H2 j/1 j/1 i i i " + (a #b )!min + a , + b j j j j j/1 j/1 j/1 for i" 1, 2,2, n. (12) Then, from the relation
A
A
B A
B
B
A
i i i i max + a , + b !min + a , + b j j j j j/1 j/1 j/1 j/1 i ) + Da !b D for i"1, 2,2, n, j j j/1 it is clear that
A
B
From Theorem 3, it can be expressed as WC(S ))2WC(S*)!WC(S ) H2 H2 n i # + w + Da !b D, i j j i/1 j/1 and further simplified as
(15)
WC(S )/WC(S*) H2 n i )1#(1/2) + w + Da !b D/WC(S*) . i j j i/1 j/1 From the given condition, it is clear that
A
B
i n WC(S*)* + w + Da !b D, and thus, j j i i/1 j/1 WC(S )/WC(S*))3/2. h H2
(16)
(17)
(13) 5. Computational experience
B
i i i C (S )) + (a #b )!max + a , + b j j i H2 j j j/1 j/1 j/1 i # + Da !b D for i" 1, 2,2, n. (14) j j j/1
This section describes the computational tests which are made to evaluate and compare the effectiveness of the developed heuristics, Max-agg and Sum-agg. The heuristics are programmed in C-language, and the tests are performed on 586-PC. The
Table 1 Computational results of percentage deviation from the optimal solution (top: class 1, middle: class 2, bottom: class 3) n
5
6
7
8
9
10
Max-agg
Sum-agg
Mean
Maximum
Median
Mean
Maximum
Median
1.40 0.54 0.20 1.69 1.32 0.24 1.56 1.24 0.33 2.96 2.19 0.16 3.08 1.79 0.25 3.44 1.66 0.39
8.18 3.49 3.29 9.36 8.75 3.27 11.65 11.13 3.65 9.94 6.20 1.76 12.11 7.50 2.80 11.21 5.46 3.39
0.14 0.00 0.00 0.59 0.23 0.00 0.37 0.21 0.00 2.03 1.75 0.00 2.52 1.25 0.00 2.92 1.15 0.03
2.07 1.51 1.76 2.26 2.18 1.82 2.03 2.09 1.61 3.29 2.75 1.89 2.94 2.43 2.36 3.78 2.79 2.52
12.44 11.95 7.39 8.95 9.84 10.86 8.35 11.77 7.83 11.84 8.11 8.55 12.10 7.64 7.29 15.82 7.56 7.97
0.89 0.34 0.44 1.60 1.44 0.63 1.19 1.43 0.89 2.55 2.18 1.57 2.36 1.86 1.91 3.49 2.43 2.41
254
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
test problems are generated with the number of jobs ranging from 5 to 50. For all the test problems, the weight factors (w ) are generated from the disi crete uniform distribution with range [1, 20]. How hard to solve such a problem is likely to depend on whether or not there is any workload balance between the machines. To indicate each problem difficulty, the job processing times are separately generated in three classes. In class 1, the processing times on both machines are commonly generated from the discrete uniform distribution with range [1, 30], which implies that the workloads between the machines are balanced. In class 2, the processing times on the first machine are generated from the uniform distribution with range [5, 30] but those on the second machine from the uniform distribution with range [1, 25], which implies that the workload on the first machine is slightly larger than that on the second machine. To represent the extremely unbalanced case, in class 3, the processing times on the first machine are generated from the uniform distribution with range [10, 30] and those on second machine from the uniform distribution with range [1, 20]. For each combination of the number of jobs and the class of
the processing times, 50 problems are generated to gather some statistics for the performance of the heuristics. The tests are carried out in two phases; first, 900 problems considering 5, 6, 7, 8, 9 and 10 jobs are generated and solved by use of the heuristics, and also solved (optimally) in full enumeration. Then, all the heuristic results are compared with the corresponding optimal solutions. The results of such an evaluation are reported in Table 1 which shows mean, maximum and median percentage deviations from the optimal solution. Each of the percentage deviations from the optimal solution is defined as 100](heu!opt)/opt, where heu is the heuristic objective value and opt is the optimal value. In the table, the top, middle and bottom of each cell represent the results of classes 1, 2 and 3, respectively. The second phase of the tests is carried out for 900 large-size problems having 13, 15, 20, 30, 40 and 50 jobs. The solutions to such problems searched by the heuristics are compared with the lower bound which is obtained as follows: LB"maxMWSPT , WSPT N, 1 2
Table 2 Computational results of percentage deviation from the lower bound (top: class 1, middle: class 2, bottom: class 3) n
13
15
20
30
40
50
Max-agg
Sum-agg
Mean
Maximum
Median
Mean
Maximum
Median
10.62 5.62 0.76 11.42 6.62 0.65 11.98 6.11 0.64 13.14 5.35 0.54 12.91 5.26 0.55 13.09 4.97 0.59
25.67 25.53 4.33 24.40 19.23 3.36 28.27 18.21 4.15 26.98 14.61 4.01 23.80 12.95 2.44 24.14 10.98 2.43
9.28 4.35 0.18 10.26 4.65 0.17 11.56 5.79 0.36 12.48 4.34 0.34 11.44 4.54 0.43 12.26 5.03 0.51
10.62 6.68 2.81 11.36 7.85 2.69 11.94 6.89 2.86 11.89 6.53 3.19 12.34 6.49 3.13 12.07 6.47 3.38
29.17 22.33 8.65 23.22 20.19 6.69 22.64 17.93 6.78 26.72 15.06 6.26 21.41 10.98 5.54 23.37 11.54 5.96
9.35 5.76 2.59 10.25 6.53 2.41 11.18 6.44 2.74 11.17 5.69 3.22 11.71 6.23 3.01 10.69 6.62 3.28
C.S. Sung, S.H. Yoon/Int. J. Production Economics 54 (1998) 247—255
where WSPT , k"1, 2, denotes the total weighted k completion time obtained by applying the WSPT rule to M . The lower bound is easily derived by k considering Eq. (1) and using the property that the WSPT rule provides an optimal solution for each machine. The results of the second phase are shown in Table 2 where mean, maximum and median percentage deviations from the lower bound are listed. Each of the percentage deviations from the lower bound is defined as 100](heu!LB)/LB.
6. Conclusions This paper considers a scheduling problem with the objective measure of minimizing the total weighted completion time at a pre-assembly stage composed of two feeding machines. It is shown that the problem is strongly NP-complete. Some solution properties are characterized for a restricted case which can be solved in O(n log n). Based on the properties, two O(n log n) heuristics incorporating the WSPT rule are derived, and their worst-case performances are analyzed as giving the error bound of 2. The properties may also be used to search for solutions in a branch and bound method. The problem is important in its own right, but it is also significant in a sense that the results of this paper will provide the underlined scheme and insight for any extended problem such as the preassembly stage with m(m'2) feeding machines.
References [1] C.Y. Lee, T.C.E. Cheng, B.M.T. Lin, Minimizing the makespan in the 3-machine assembly-type flowshop scheduling problem, Management Science 39 (1993) 616—625.
255
[2] C.N. Potts, S.V. Sevast’janov, V.A. Strusevich, L.N. Wassenhove, C.M. Zwaneveld, The two-stage assembly scheduling problem: complexity and approximation, Operations Research 43 (1995) 346—355. [3] I. Duenyas, Estimating the throughput of a cyclic assembly system, International Journal of Production Research 32 (1994) 1403—1410. [4] R.R. Lummus, A Simulation analysis of sequencing alternatives for JIT lines using kanbans, Journal of Operations Management 13 (1995) 183—191. [5] H.W. Stoll, Design for manufacture: An overview, Applied Mechanics Review 39 (1986) 1356—1364. [6] G. Boothroyd, Assembly Automation and Product Design, Marcell Dekker, New York, 1992. [7] D.W. He, A. Kusiak, Performance analysis of modular products, International Journal of Production Research 34 (1996) 253—272. [8] D.E. Whitney, Nippondenso Co. Ltd: a case study of strategic product design, Research in Engineering Design 5 (1993) 1—20. [9] E.D. Coffman, A. Nozari, M. Yannakakis, Optimal scheduling of products with two subassemblies on a single machine, Operations Research 37 (1989) 426—436. [10] Y.P. Aneja, N. Singh, Scheduling problem for common components at a single facility, IIE Trans. 22 (1990) 234—237. [11] C.S. Sung, C.K. Park, Scheduling of products with common and product-dependent components manufactured at a single facility, Journal of the Operational Research Society 44 (1993) 773—784. [12] R.W. Conway, W.L. Maxwell, L.W. Miller, Theory of Scheduling, Addison-Wesley, Reading, MA, 1967. [13] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, San Francisco, 1979. [14] K.R. Baker, S.G. Powell, A predictive model for the throughput of simple assembly systems, European Journal of Operational Research 81 (1995) 336—345. [15] K.R. Baker, S.G. Powell, D.F. Pyke, Optimal allocation of work in assembly systems, Management Science 39 (1993) 101—106.