A COMPARISON OF JOHNSON S RULE AND TOC BASED SCHEDULING METHODOLOGIES

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1 A COMPARISON OF JOHNSON S RULE AND TOC BASED SCHEDULING METHODOLOGIES Jirarat Teeravaraprug and Thassaporn Sakulpipat Faculty of Engineering, Thammasat University, Rangsit Campus, Pathumthani THAILAND tjirarat@engr.tu.ac.th ABSTRACT This paper gives a comparison of Johnson s Rule and TOC based scheduling methodologies. Johnson s Rule scheduling methodology is a step procedure while TOC based scheduling methodology is only concept. This paper shows a comparison of these two concepts of scheduling methodologies. A numerical example is given. Based on the numerical example, it shows that Johnson s Rule is simple but limited; while TOC based scheduling is more difficult but providing more alternatives in job sequencing. KEYWORD Johnson s Rule, Theory of Constraint, Scheduling objectives. The objectives can take many forms. Examples include minimization of the completion time of the last task, minimization of the number of tasks completed after their respective due dates, and minimization of maximum tardiness. This paper gives an attempt to make a comparison of Johnson s Rule and TOC based scheduling methodology though numerical example. Hence, the next two parts shows the basic concept of Johnson s Rule and TOC based scheduling methodologies. Then the use of numerical example is presented. By taking deeply consideration of the numerical example results, comparisons of those two methodologies are given. Finally, conclusion and further research are discussed. 1. Introduction Assume that there is a collection of jobs that must be processed on the machine and that each job has associated with it a processing time and a due date. Scheduling and sequencing are then required. There are many rules of scheduling and sequencing. They include FCFS (Firstcome, First-served), SPT (Shortest processing time), LPT (Longest Processing Time), EDD (Earliest due date), and CR (Critical ratio). In FCFS, jobs are processed in the sequence in which they entered the shop, while SPT, jobs are sequenced in increasing order of their processing times. For EDD, jobs are sequenced in increasing order of their due dates. The job with the earliest due date is first, the job with the next earliest due date is second, and so on. Lastly, CR requires forming the ratio of the processing time of a job divided by the remaining time until the due date, and scheduling the job with the largest ratio next. Even thought those rules seem to be common, they are suitable for scheduling jobs on a single machine only. When two or more machines are considered, these rules cannot be used appropriately. In that case, the process of job scheduling is needed to consider availability and utilization of resources (machine times). Scheduling then deals with the allocation of scarce resources to tasks over time. It is a decision-making process with the goal of optimizing one or more 2. Johnson s Rule Consider the following generic manufacturing environment and the role of its scheduling. Orders that are released in a manufacturing setting have to be translated into jobs with associated due dates. These jobs often have to be processed on the machines in a workcenter in a given order or sequence. The processing of jobs may sometimes be delayed if certain machines are busy. This situation is known as the problem called Johnson problem which is a situation of static arrival two-machine flow shop. Johnson s Rule is a scheduling methodology of solving the problem explained above. The objectives of Johnson s Rule are (1) to minimize the processing time for sequencing a group of jobs through two work centers, (2) to minimize the total idle times on the machines and (3) to minimize the flow time the beginning of the first job until the finish of the last job. It can be said that the main objective of Johnson s Rule is to minimize the completion time for a group of jobs that are to be processed on two machines or at two successive work centers. Even though Johnson s Rule is first used to solve the scheduling problem of only two work centers, it can be extended to some special three-machine problems and it is also a foundation of later heuristics for the m machine problem. Moreover, some limitations are needed to be obtained. First, job time (including setup and processing) must be known and constant for each job at each work

2 center. Second, job times must be independent of the job sequence. Third, all jobs must follow the same two-setup work sequence. Last, job priorities cannot be used. Johnson s Rule can be applied in a simple procedure as shown below. Step 1: Schedule the group of jobs U that are shorter on the first machine than the second; U={j/P j1 <P j2 } as the first job priority group. Step 2: Schedule the group of jobs V that are shorter on the second machine; V={j/ P j2 P j1 } as the second priority group. Step 3: Schedule within U by SPT on the first machine. Step 4: Schedule within V by LPT on the second machine. where P j1 and P j2 are the process time of job j on Machines 1 and 2, respectively. The application of this method through a numerical example is given in the next two sections. 3. TOC based Scheduling Methodology About the theory of constraints (TOC) management philosophy, its development began in 1979 with the introduction of optimized production timetables scheduling methodology later renamed Optimized Production Technology or OPT [1-2]. Due to its simple yet robust methodology, application of TOC techniques have been discussed in the academic literature and popular press across a variety of operations management disciplines [3-6]. At its introduction, OPT schedules were controversial because they kept some machines or work centers efficiently busy while others were idle at times. This contradicted the performance measurement system in place at most plants. These actions created unsynchronized material flows through the plant, scrambling the schedule and endangering the success of OPT itself. As part of the initial efforts to combat the traditional measure, Goldratt released the nine OPT rules. First, balance flow, not capacity. Second, level of utilization of a non-bottleneck is determined not by its own potential but by some other constraint in the system. Third, Utilization and activation of a resource are not synonymous. Forth, an hour lost a bottleneck is an hour lost for the total system. Fifth, an hour saved at a non-bottleneck is just a mirage. Sixth, bottlenecks govern both throughput and inventory in the system. Seventh, a transfer batch may not, and many times should not, be equal to the process batch. Eighth, the process batch should be variable, not fixed. Ninth, schedules should be established by looking at all of the constraints simultaneously. Lead times are a result of a schedule and cannot be predetermined. Based on the rules, the principal tenet of TOC is that within each system at least one constraint exists that limits the ability of the system to achieve higher levels of performance relative to its goal. Maximum utilization of the constraint therefore should lead to maximum output from the system. The situation considered in this paper is the situation of two-machine flow shop and there is nothing stating about the constraint. Based on the concept of TOC, there must be at least one constraint in the system. Therefore, one of the machines needs to be the constraint. Two alternatives are considered in the numerical example. The first one is, the first machine is the constraint and the second is, the second machine is the constraint. Detailed computation is shown in the next section. 4. Numerical Example Consider two-machine flow shop makespan problem with dynamic availability but static arrivals. Process times on both machines are shown in Table 1 [7]. Table 1: Numerical Example Job Machine Machine Based on the concept of Johnson s Rule, these jobs can be scheduled in two alternatives as shown in Table 2. Those two alternatives give the makespan as 37 time period. Even though Johnson s Rule gives two alternative of job sequence, there may be some other job sequences that provide the makespan equally.

3 Alternative Table 2: Scheduled Job Sequence based on Johnson s Rule Machine 1 Machine Alternative Machine 1 Machine Using TOC based scheduling methodology, constraint is needed to identify. First, assume that Machine 1 is the constraint of this process. Since Machine 1 is the constraint, all jobs are scheduled in order to keep Machine 1 busy as possible. Eight jobs are applied to the TOC based. It can be seen that the results given by Johnson s Rule become a subset of TOC based alternatives because Machine 1 is always busy except after finishing all jobs. Some other alternatives are shown in Table 3. The reason that all alternatives of TOC based scheduling cannot be presented is, the size of the alternative numbers are large. The number of alternatives in this case is 8! (=40,320). Therefore, only examples of TOC based are shown. It can be seen that the total times used in Machine 1 are 36 equally for all alternatives. Total task time of each alternative shown in Table 3 is different and it is always greater or equal to the total task time provided by Johnson s Rule. Considering deeply, to make the total task time becomes 37, Job 8 needs to be at last and at the time Job 8 finishes Machine 1, Machine 2 is available. The reason is the process time of Job 8 on Machine 2 is only 1. Combining the time of all jobs finishing Machine 1 which is 36, the total task time is 37. Even though this paper gives an explanation about total task time of TOC based scheduling, it is not a focus of TOC. TOC focuses only on the constraint which is Machine 1 only and all alternatives gives total used on Machine 1 equally 36. Therefore, based on TOC based scheduling, all alternatives can be used. The second option is assuming that Machine 2 is the constraint of this process. Therefore, the schedule is then set based on Machine 2. Based on the concept, some of the alternatives can be shown as Table 4. All alternatives give their total task times equal 37 while their idle times equal 6. Based on those alternatives, it can be seen that Job 8 needs to be the last job sequence since its processing time on Machine 2 is the minimum one. The first job sequence tends to be one of Jobs 1, 2, 5, and 6, while Jobs 1, 7 or 4 tends to be prior to the last sequence. 5. Conclusions and Discussions Based on the concepts of Johnson Rule and TOC based scheduling, their considerable objectives are different. Johnson s Rule objective is to find the sequence making the total task time shortest while TOC based scheduling objective is to maximize the utilization of constraint. Even though Johnson s Rule gives an attempt to find the job sequence of minimum total task time, Johnson s Rule does not provide all possible alternatives based on the objective. TOC based scheduling has a constant step procedure. Therefore, possible alternatives can be called as its result. However, based of that reason, it is difficult to obtain all possible alternatives. Moreover, since the considered problem is only two machine flow shop, it may be difficult to find the different of total task time between those two methodologies. However, using TOC based scheduling methodology; a constraint needs to be set in the system. In the case of setting Machine 1 as the constraint, only Machine 1 is given an attention and it may obtain higher level of total task time. This situation may be relaxed if considering more jobs or continuously. In the case of using Machine 2 as the constraint, total task times of this method in all alternatives equal to what Johnson s Rule is given but more alternatives are found by using this method. Therefore, it can be concluded that in this case, Johnson s Rule gives good results but limited, while TOC based scheduling methodology gives many alternative results. However, Johnson s Rule seems to be simpler and easier to obtain the results. Johnson s Rule can then be extended to three or more machines but there are some limitations. TOC based scheduling methodology is generally used and can be extended in a lot more situations.

4 References [1] E.M. Goldratt, Optimized production timetables: a revolutionary program for industry, APICS 23 rd Annual International Conference, APICS, Falls Church, [2] E.M. Goldratt and J. Cox, The Goal (North River Press, Great Barrington, MA, 1984). [3] L.P. Leach, Critical chain project management improves project performance, Project Management, 1999, 10 (2), 39. [4] M. Umble and E. Umble, Manage your projects for success: an application of the theory of constraints, Production and Inventory Management Journal, 2000, 41(2), 27. [5] H. Steyn, An investigation into the fundamentals of critical chain project scheduling, International Journal of Project Management, 2001, 19(6), 363. [6] I. Cohen, A. Mandelbaum, and A. Shtub, Multiproject scheduling and control: a process based comparative study of the critical chain methodology and some alternatives, Project Management Journal, 2004, 35(2), [7] T. E. Morton and D. W. Pentico, Heuristic Scheduling Systems (Wiley Series in Engineering & Technology Management, John Wiley & Sons, New York, 1993). Table 3: Examples of TOC based Scheduling (Constraint = Machine 1) 1 5 Job Job P P P P Machine 1 begin Machine 1 begin end end Machine 2 begin Machine 2 begin end end Job Job P P P P Machine 1 begin Machine 1 begin end end Machine 2 begin Machine 2 begin end end Job Job P P P P Machine 1 begin Machine 1 begin end end Machine 2 begin Machine 2 begin end end Job Job P P P P Machine 1 begin Machine 1 begin end end Machine 2 begin Machine 2 begin end end

5 Table 4: Examples of TOC based Scheduling (Constraint = Machine 2) Job sequence Alternative Alternative Alternative Alternative Alternative Alternative Alternative Alternative Alternative Alternative Alternative Alternative Alternative Alternative Alternative Alternative Alternative