GROUP SCHEDULING PROBLEMS IN FLEXIBLE FLOW SHOPS

Transcription

1 GROUP SCHEDULING PROBLEMS IN FLEXIBLE FLOW SHOPS Rasaratnam Logendran, Sara Carson, and Erik Hanson Department of Industrial and Manufacturing Engineering Oregon State University Corvallis, Oregon Abstract Flexible flow shops are becoming increasingly common in industry practice because of higher workloads imposed by jobs on one or more stages, thus requiring two or more units of the same machine type. We present a methodology for solving this important problem, namely group scheduling, within the context of cellular manufacturing systems in order to minimize the total completion time of all groups of jobs considered in the planning horizon. Two different setup options, namely single setup and multiple-setups, are investigated for jobs within the same group. A combined heuristic solution algorithm, comprised of single- and multiple-pass heuristics, is used for solving the problem. An example problem, consisting of three different problem instances of the same structure, is used to not only demonstrate the applicability of the solution algorithm, but also to identify the setup to average job run time ratios that should simultaneously prevail across all stages represented by two or more identical units in order to select multiple-setups over single setup or vice versa. Keywords Group Scheduling; Flexible Flow Shops 1. Introduction Cellular manufacturing (CM) systems have played a significant role in increasing manufacturing productivity in small to large size parts manufacturing companies. The most quoted advantages of implementing CM systems include reduction in setup times, tooling needs and work-in-process inventories, simplified flow of parts and tools, centralization of responsibility, and improved human relations [1,2]. If perfect disaggregation were accomplished during the design of CM systems, a part family, comprised of similar parts, assigned to a manufacturing cell would be completely processed on a set of dissimilar machines in that cell [3]. Typically, two or more part families are assigned to the same cell in order to increase the utilization of machines. There emerges a problem, known as group scheduling, and it requires determining the sequence in which the parts (jobs) belonging to a family (group) as well as the families (groups) themselves must be processed on the machines to optimize some measure of effectiveness [4]. Unlike in conventional machine scheduling problems, group scheduling is performed at two levels. At the first level, known as the level 1 problem, a sequence for processing the jobs that belong to each group on machines to optimize some measure of effectiveness would need to be determined. At the second level, known as the level 2 problem, a sequence for processing the groups themselves on machines to optimize the same measure of effectiveness would need to be determined. As the jobs belonging to a group are similar, the sequence of operations required of jobs on machines would typically be the same, thus adhering to a flow-line arrangement. For the minimization of total completion time at level 1, heuristic methods based upon optimal and heuristic solution techniques for solving conventional flow-shop scheduling problems have been proposed [5-7]. As the jobs belonging to a group are similar in the level 1 problem, the setup time required to changeover from one job to another is assumed either negligible as compared to the run time or included in the processing time. At level 2, however, a changeover from one group to another is necessary because the processing requirements of jobs in different groups are distinctly different. Realistically, therefore, the setup time required for a group of jobs on a machine should be separated from the run time in the investigation of a group-scheduling problem. Yoshida and Hitomi [8] proposed a polynomial-time optimal algorithm for minimizing the total completion time in a two-machine group-scheduling problem with sequence-independent setups. Allison [9] reported the performance of combining a single-pass heuristic by Petrov (PT) [10] and a multiple-pass heuristic by Campbell, Dudek and Smith (CDS) [11] for completely solving an m-machine group scheduling problem to minimize the total completion time. Motivated by the encouraging performance reported by Logendran and

2 Nudtasomboon (LN) [7] for solving the level 1 problem, Logendran et al. [4] carried out an investigation to determine how well LN performed in comparison with CDS, if each is combined with PT to completely solve a m- machine group scheduling problem to minimize the total completion time. All of the investigations noted above have assumed that there exists only a unit of a machine in each stage of a group-scheduling problem. Flexible flow shops are becoming increasingly popular in industry, primarily due to large workload requirements imposed by jobs on machines representing one or more stages of a multi-stage (mstage) flow shop-scheduling problem. In general, a flow shop is considered a flexible flow shop should it contain two or more identical units of a machine type representing a stage. We investigate this industry-relevant problem within the context of group scheduling in order to minimize the total completion time of all groups of jobs released in the current planning horizon. 2. Problem Statement Interesting new challenges unfold in the investigation of a group-scheduling problem in a flexible flow shop as compared to a conventional flow shop. In a conventional flow shop, following the setup on a machine it is customary to process the jobs belonging to a group in a prescribed sequence until all of them are completed, as there is only one unit of a machine in each stage. In contrast, in a flexible flow shop, a stage, herein referred to as the next stage, that consists of two or more units of a machine type could lend itself for performing a setup on (say) the second machine if a job in a group has completed its processing requirements in the previous stage at a time when the first machine in the next stage is currently busy processing another job in the same group. We refer to this scenario as performing multiple-setups on two or more identical units of a machine with the intent of processing two or more jobs that belong to the same group. Intuitively it is easy to see that this method may prove to be advantageous if the sequence-independent setup time for the machine representing the next stage is significantly smaller than the run time for the job that requires processing on the machine. On the other hand, if the setup is significantly larger than the run time, it may be advantageous to perform the setup only on one unit of a machine representing the next stage with the intent of processing all jobs that belong to a group. We refer to this scenario as performing a single-setup. An important research question unfolds at this juncture. If the multi-stage flexible flowshop problem consisted of one or more stages with two or more identical units representing a stage, what ratio between the setup and average job run time should simultaneously prevail across all stages that are represented by two or more identical units in order to make it attractive to consider selecting multiple-setups over single setup or vice versa? This paper reports preliminary research findings obtained from addressing this research question through an illustrative example, comprised of three problem instances. Whether it is single or multiple-setup, a suitable solution algorithm is needed to completely solve the groupscheduling problem in a flexible flow shop. For a group-scheduling problem in a conventional flow shop, Logendran et al. [4] reported superior performance by the heuristic LN-PT for minimizing the total completion time at levels 1 and 2, respectively. That is, the use of LN for scheduling jobs within each group and PT for scheduling groups themselves turned out to be best performer for solving problems of all sizes in order to minimize the total completion time. We take advantage of this insight, and in this paper report the findings obtained by using the combined heuristic represented by LN-PT to solve the group-scheduling problem in a flexible flow shop. 3. Steps Associated with PT and LN Heuristics In this section, we present the steps associated with Petrov s (PT) and Logendran and Nudtasomboon s (LN) heuristics for the purpose of illustration. As noted before, the setup required on a machine to change from one job to another that belongs to the same group is negligible. Thus the processing time for jobs in a group is the setup time plus the run time for jobs in that group. Table 1 presents the setup and run time matrix for kth group, consisting of n jobs, on t machines. As noted, the setup time for group k (G k ) on machine j (M j ) is s kj, and the run time for job x (J x ) n on M j is r kxj. The processing time for jobs in group k on M j is p kj = s kj + r kxj. As presented below, the x= 1 underlying concepts behind the application of each heuristic is the same, except that in level 1, when the sequence of jobs in a group is of interest, the run times are used, whereas in level 2, when the sequence of groups is needed, processing times are used.

3 Table 1. Setup and run time matrix for jobs in group k Group k (G k )/Job x (J x ) Machine j (M j ) M 1 M 2 M j M t Setup Time of G k on M j s k1 s k2 s kj s kt J 1 r k11 r k12 r k1j r k1t J 2 r k21 r k22 r k2j r k2t Run Time of J x on M j J x r kx1 r kx2 r kxj r kxt J n r kn1 r kn2 r knj r knt 3.1 Petrov s (PT) Heuristic Step 1: Evaluate the following fictitious run times for job x on a two-machine problem as where u t C D rx = rxj, and rx = rxj j= 1 j= u' u = t/2 and u = (t/2) + 1 for even t, and u = u = (t +1)/2 for odd t. Step 2: Apply Johnson s [12] two-machine algorithm to each of the n jobs, having fictitious processing time on machines C and D, the first and second fictitious machines, respectively. 3.2 Logendran and Nudtasomboon s (LN) Heuristic Step 1: For each job x, compute the average total run time, Txavg = rxj / ω, where ω is the number of machines over all j= 1,...t and r xj > 0 on which job x has r xj > 0. Step 2: Sort the jobs in a descending order of T xavg. Step 3: Select the first two jobs from the sorted list in Step 2, generate the two possible partial schedules, and determine the total completion time of each. Save the schedule that gives the minimum total completion time. Set x = 3, and go to Step 4. Step 4: Select the xth job from the sorted list in Step 2. Generate the x possible partial schedules by inserting the xth job in the schedule saved with x-1 jobs previously and determine the total completion time of each. Save the schedule that gives the minimum total completion time. While performing the insertions of the xth job, follow this order yet making sure that the relative positions of the x-1 jobs with respect to each other is maintained the same. First partial schedule is generated by inserting the xth job at the beginning of the schedule saved with x-1 jobs, second by inserting at the end of the schedule saved with x-1 jobs, and the remaining x-2 partial schedules are generated by inserting the xth job, one by one, in the intermediary positions of the schedule(s) saved with x-1 jobs but starting at the beginning and moving towards the end of the schedule. If at any time the total completion time determined by inserting the xth job is equal to the total completion time of the saved schedule(s) with x-1 jobs, curtail the

4 enumerations by not performing the remaining insertions and save this schedule with the xth job inserted. The reason being that no other partial schedule generated by inserting the xth job can give a total completion time that will be smaller than the total completion time of the schedule(s) saved with x-1 jobs. Additionally, this heuristic is designed to determine only one best/near optimal solution, and not alternate optimal solutions, for the groupscheduling problem. Set x = x+1 Step 5: If x = n, stop. Otherwise, go to Step Application of the Solution Algorithm with Single and Multiple Setups An Example To demonstrate the application of the algorithm with single and multiple setups, a flexible flow shop problem with four groups and four stages is chosen. To maintain adequate flexibility, two of the four stages (stages 2 and 4) are assumed to have identical-parallel machines as follows. The first stage consists of one unit of machine type 1 (M 11 ), second stage two units of machine type 2 (M 21 and M 22 ), third stage one unit of machine type 3 (M 31 ), and finally the fourth stage consists of three units of machine type 4 (M 41, M 42, and M 43 ). Table 2 presents the set up and run time for jobs in each group on each of the four machine types. In order to address the research question posed in Section 2, we evaluate the ratio between the setup and average run time for each group on every machine, and ratios so evaluated are presented in Table 3. The setup to average run time ratio for this problem instance (1) lies within the range In order to examine the sensitivity exhibited by setup to average run time ratios, two other problem instances (2 and 3) are created. In doing so the run time for jobs in all four groups were maintained the same, while the setup time was progressively increased to evaluate a higher range for the ratio in problem instance 2 and 3 than problem instance 1. The setup to run time ratio for problem instances 2 and 3 is , and For example, in problem instance 2, the setup times for groups G 1, G 2, G 3, and G 4 on machine type 1 (M 1 ), M 2, M 3, and M 4 are changed from those for problem instance 1 in Table 2 to 11, 9, 6, 8; 10, 11, 7, 18; 10, 13, 6, 16; and 12, 12, 6, and 14, respectively. Table 2. Setup and run times for problem instance 1 of example Group G 1 G 2 G 3 G 4 Machine s 1j r 11j r 12j s 2j r 21j r 22j r 23j s 3j r 31j r 32j s 4j r 41j r 42j r 43j M M M M Table 3. Setup to average run time ratios for problem instance 1 Group Machine G 1 G 2 G 3 G 4 M M M M The three problem instances are solved with the combined heuristic LN-PT with single setup and multiple setups. The algorithmic logic is coded in visual basic, and the total completion time with single and multiple setups are evaluated separately. As an example, for problem instance 2, the application of LN for the level 1 problem results in the following job sequences for the single setup and multiple setups, respectively: J 11 - J 12 /J 21 - J 22 - J 23 /J 32 - J 31 /J 43 -

5 J 42 - J 41, and J 11 - J 12 /J 23 - J 22 - J 21 /J 31 - J 32 /J 42 - J 43 - J 41, while the application of PT for the level 2 problem results in the following group sequence: G 2 - G 4 - G 3 - G 1 for both types of setups. The reason for different job sequences in a group with the two types of setup can be further explained as follows. For G 2, J 23, J 22, and J 21 have average run times of 26.75, 23.75, and 21.75, respectively, as in Step 1 of the LN heuristic. Thus, ranks 1, 2, and 3 are associated with J 23, J 22, and J 21, respectively, as in Step 2. Step 3 would consider the partial schedule formed as J 23 - J 22 and J 22 - J 23. With single setup, the partial schedule J 22 - J 23 results in the best completion time of 159 compared with 171 for J 23 - J 22. Contrastingly, with multiple setups, the partial schedule J 23 - J 22 results in the best completion time of 114 compared with 124 for J 23 - J 22, as multiple setups are made possible for jobs in the same group on M 2 and M 4. As the structure of the partial schedule determined during the earlier stages of the application of the LN heuristic is never altered, it explains the reason for identifying the final job schedule for G 2 as J 21 - J 22 - J 23 with single setups, and as J 23 - J 22 - J 21 with multiple setups. Table 4 presents the best total completion times evaluated for problem instances 1, 2, and 3, each representing different ranges of setup time to average job run time ratios. Although the selection of these ranges in this paper is somewhat arbitrary, the findings in this research indicate that given a group scheduling problem instance in a flexible flow shop there is indeed a setup to average job run times ratio that favors the application of multiple setups over single setups or vice versa. The range representing the setup to average run time ratios for problem instances 1, 2, and 3 can loosely be regarded as low, medium, and large. As seen from the results presented in Table 4, when the setup time is small compared to the average job run time, the application of the algorithm favors performing multiple setups on machines with two or more units over single setup for identifying a better total completion time (217), and when the setup time is large compared to the average job run time performing single setup on machines leads to identifying a better total completion time (268). Table 4. Total completion time for problem instances with single setup and multiple setups Total completion time Problem Instance Single setup Multiple setup Conclusions and Future Work Typically, the need for the investigation of group scheduling problems arises following the design of manufacturing cells, and are comprised of scheduling jobs within a group as well as scheduling groups themselves. As one or more stages in a flexible flow shop may consist of two or more identical-parallel machines, two different setup options, namely single setup and multiple-setups, are investigated for jobs within the same group. In accordance with the previous research finding obtained from solving the group-scheduling problem in a conventional flow shop, the combined heuristic LN-PT is used to solve the group-scheduling problem in a flexible flow shop addressed in this paper. However, whether to use LN (or PT) to solve the level 1 problem and PT (or LN) to solve the level 2 problem in a flexible flow shop should rightfully remain an open research question, which is intended to be addressed in future work. To demonstrate the applicability of the solution algorithm and to address the research question with regard to performing single versus multiple setups, an example problem structure, comprised of three problem instances, has been used. To fully apply the findings from this research in industry a broader research question should be addressed. That is, given the structure of a group scheduling problem dictated only by the number of units of machine types representing each stage, which statistically significant setup to average run time ratios would make it attractive to perform single setup over multiple-setups or vice versa, to cover the range of problem instances such as small, medium, and large, determined largely by the number of jobs in each group and the number of groups. Future work will also be aimed at investigating this issue more comprehensively. Acknowledgments This research is funded in part by the National Science Foundation (USA) Research Experiences for Undergraduates (REU) Grant No. DMI Their support is gratefully acknowledged.

6 References 1. Hyer, N.K., and Wemmerlov, U., 1989, Group Technology in U.S. Manufacturing Industry: A Survey of Current Practices, International Journal of Production Research, 27, Suresh, N.C., and Kay, J.M., (eds.), 1998, Group Technology and cellular Manufacturing: State-of-the Art Synthesis of Research and Practice, Kluwer Academic Publishers, Boston, MA. 3. Logendran, R, and Sirikrai, V., 2000, Machine Duplication and Part Subcontracting in the Presence of Alternative Cell Locations in Manufacturing Cell Design, Journal of the Operational Research Society, 51, Logendran, R., Mai, L., and Talkington, D., 1995, Combined Heuristics for Bi-Level Group Scheduling Problems, International Journal of Production Economics, 38, Radharamanan, R., 1986, A Heuristic Algorithm for Group Scheduling, Proc. of the International Industrial Engineering Conf., Al-Qattan, I., 1988, Designing GT Cells Enhanced by Group Scheduling, Proc. of the IIE Integrated Systems Conf., Logendran, R., and Nudtasomboon, N., 1991, Minimizing the Makespan of a Group Scheduling Problem: A New Heuristic, International Journal of Production Economics, 22, Yoshida, T., and Hitomi, K., 1979, Optimal Two-Stage Production Scheduling with Setup Times Separated, AIIE Transactions, 11, Allison, J.D., 1990, Combining Petrov s Heuristic and the CDS Heuristic in Group Scheduling Problems, Proc. of the 12 th Annual Conference on Computers and Industrial Engineering, Orlando, FL, Petrov, V.A., 1966, Flow Line Group Production Planning, Business Publications, London. 11. Campbell, H.G., Dudek, R.A., and Smith, M.L., 1970, A Heuristic Algorithm for the n Job, m Machine Sequencing Problem, Management Science, 16, B630-B Johnson, S.M., 1954, Optimal Two- and Three-Stage Production Schedules with Set-up Times Included, Naval Research Logistics Quarterly, 1, Rasaratnam Logendran is an Associate Professor in the Department of Industrial and Manufacturing Engineering at Oregon State University. Dr. Logendran s primary areas of research interests are cellular manufacturing, scheduling, and integration of design and manufacturing. He is currently serving on the Editorial Board of the IIE Transactions on Design and Manufacturing, a senior member of IIE, a member of INFORMS, SME, and ASEE, and the Faculty Advisor of the Alpha Pi Mu Student Chapter at Oregon State University. Sara Carson is currently a senior in the Department of Industrial and Manufacturing Engineering with a specialization in Manufacturing Engineering at Oregon State University. Sara is currently a member of IIE and holds the position as treasurer for the Society of Women Engineers. She is also the former Student Alumni Organization Alumni Chair. Erik Hanson is currently a senior in the Department of Industrial and Manufacturing Engineering with a specialization in Manufacturing Engineering at Oregon State University. Erik is currently the President of the Alpha Pi Mu Student Chapter at Oregon State University. He is also a member of the Tau Beta Pi Engineering Honor Society and a student member of IIE.