Best schedule for the example.
Abstract
This chapter discusses an integrated model of batch production and machine maintenance scheduling on a single deteriorating machine and flow shop with a deteriorating machine that produces an item to be delivered at a common due date. The model describes the trade-off between production costs and maintenance costs as the increase of production run length. The objective function of the model is to minimize total cost consisting of in-process and complete inventory holding costs, setup cost, preventive and corrective maintenance costs, and rework cost. The problem is to determine the best production run length and maintenance actions that minimize the total cost.
Keywords
- batch production
- maintenance scheduling
- single item
- deteriorated machine
1. Introduction
The development of today’s manufacturing systems leads to a shorter product life cycle, increasing product varieties and customer demand on the higher quality and timeliness of delivery. Thus, the accuracy and speed of decision-making in the manufacturing system become important.
A machining process industry gets many machining orders from its strategic partner industries in large quantities. The manufacturer processes the order in constant batch size that is set by production section. Meanwhile, Maintenance Section performs machine maintenance only in case of machine failures (reactive maintenance). Delay delivery order to consumers cannot be avoided if the maintenance machine takes a long time and disrupts production activities, and this often happens.
Some root of the problem can be drawn from this description: first, the maintenance has not implemented preventive maintenance system although the machine failure data, the time interval between failures, and the cost of each failure are well recorded. Second, the production section schedules batches in a constant size, whereas according to [1, 2, 3, 4, 5, 6], discussions about nonconstant batch sizes will provide a better shop time. Third, the machine failure occurs when production is in progress, so the machine failure interferes with the productivity of the shop floor. These three issues indicate the independence between production scheduling, and maintenance scheduling may result in the following conditions:
Production scheduling that does not take into consideration the maintenance aspect will cause a machine to be continually operated even though the machine should be maintained. If the maintenance is not conducted, it can result in a breakdown machine during production activities that will certainly interfere with productivity.
Maintenance schedule that does not take into account the production schedule will cause a busy machine to be stopped for maintenance. This also disrupts the pre-arranged production schedule.
2. Model construction
This section discusses how the model is constructed. The linkage between the uncontrollable parameter and the model decision variable in achieving the best total cost is discussed in the influence diagram for integrated batch production scheduling and maintenance scheduling, input-output diagram for the models of batch production and maintenance scheduling, construction of objective function, system constraint, model and algorithm, and then a numerical experience to show how an algorithm works to solve a problem.
2.1 Influence diagram for the model batch production and maintenance scheduling
The integrated batch production and maintenance scheduling developed in this chapter have total cost minimization criteria consisting of inventory holding cost, setup cost, PM cost, CM cost, and rework cost for nonconforming part. In the influence diagram Figure 1, the problem of integrated batch production and maintenance scheduling can be explained as follows. Demand, due date, and machine performances are uncontrollable parameters. Demand and due date will affect batch size (production schedule).
The machine performance will affect the estimated number of PM and number of CM. In the model system, the number of PM and number of CM will affect each other with the production schedule. The number of PM will affect each other with the number of CM, where the increasing number of PM will cause the number of CM to decrease and vice versa. The number of PM and CM also influences each other for the number of nonconforming parts.
2.2 Input-output diagram for the models of batch production and maintenance scheduling
The total cost consists of inventory holding costs, setup cost, PM cost, CM cost and rework cost. The cost of inventory holding costs consists of the work in process (wip) and the finished part of inventory holding costs. In-process inventory holding cost is the cost of inventory for parts in batches during processing are calculated by multiplying the number of parts in the batch by the waiting time for the batch to be processed. The cost of the finished part is the inventory holding cost for parts in the finished batch, calculated by multiplying the number of parts in the batch by unit inventory holding cost for finished part as long as the time of waiting for the batch until the deadline
The input-output diagram of the integrated batch production and maintenance scheduling to minimize total cost is shown in Figure 2.
Figure 2 shows the input parameters model: the number of parts scheduled, due date, Weibull distribution function
The model will address trade-off issues on production costs and maintenance costs, where production costs will consist of inventory holding costs (in-process and finished part of inventory holding costs), setup cost, and rework cost for nonconforming parts, while the maintenance cost consists of PM cost and CM cost. The model will answer how the batch production and maintenance scheduled minimize the total cost. Drawing influence diagram follows [7].
3. Integrated batch production and maintenance scheduling for single item processed on a deteriorating machine with a due date
The modeling starts with an inventory holding cost for in-process and completed batches, modeling of system constraints of the problem, model of the problem, algorithm, and an example to show how the algorithm works to solve the model.
3.1 Inventory holding costs for in-process batch and completed batch
Holding cost concept is developed from [3]. Let
In an assumption raw materials arrive just in the time when they are required, i.e., in the beginning of a batch processing, the holding cost is only for in-processed batch and completed batch. The formulation of holding cost is conducted for in-process batch firstly and then for completed batch. The position of
An assumption for in-process batch is that parts in a batch shall wait in the batch until all parts in the batch have already been processed. Therefore, in the interval (
The amount of holding cost for the finished part, designated
The amount of holding cost for the in-process part, designated
By summation
Then, the holding cost of
Based on Eq. (1), the total holding cost of the in-process batch of all batches may be written as follows:
The holding cost of completed batch may be formulated as follows:
Next, total holding cost (
The first term of Eq. (4) is total holding cost in completed batch, and the second and third terms are the total holding cost, while the part is being processed in batch (in-process batch) in one production run.
Eq. (4) and Figure 3 are to be developed into a formulation of holding cost for
By considering any changes taking place in each production run and total PM for
3.2 ROCOF function
where N(t) is the number of failures in the interval [0,t). Since the probability of two or more failures in the interval [t,t +
When the failures are minimally repaired and the time to repair is negligible, then ROCOF function
A ROCOF function that has been used extensively is the Weibull ROCOF. The cumulative ROCOF (or the expected total number of failures) is given by the function
with scale parameter
Let a system (machine) with a Weibull failure time distribution have a shape parameter of β = 1.69 and a scale parameter
If
From the calculation above, the time interval between machine failure times can be estimated, where the time between failures of a machine is diminishing over time. It indicates that the machine has increasing failure rate distribution.
3.3 Estimation of nonconforming parts
This research developed a policy in that PM is carried out before an expected first failure time based on cumulative ROCOF function. An example of a condition for a case of two production runs and two PMs is shown in Figure 5. In the second production run, there is no nonconforming part, because the out-of-control state takes place in the first production run, so that the number of nonconforming parts for
In the same way for
Under an assumption that the probability that the nonconforming part processed under in-control state is
so that the expected network cost may be computed by
3.4 Model formulation
In order to formulate the integrating batch production and maintenance scheduling into a mathematical model, we use following notations.
R: number of production run in algorithm [SISM]
Objective function
The model has some assumptions in formulating the model, as follows:
This integrating model for single item processed on a single deteriorating machine.
Setup time is not depending on the size of batches.
Batch position number and PM are counted from due date direction (backward approach).
The same load force for machine in setup time and in processing time.
The machine cannot be interrupted as long as production runs.
Batch size value is in real positive.
Using those defined notations and based on those assumptions, the integrating batch production and maintenance scheduling to minimize production and maintenance costs on a deteriorating machine in just in time environment (model [SISM]) can be expressed as mixed-integer nonlinear programming as follows:
Model [SISM]
subject to
Eq. (14) declares an objective function to minimize total costs consisting of inventory costs, setup cost, preventive maintenance cost, corrective maintenance cost, and rework cost. Eq. (15) states the balance of the material in the shop, where the number of parts in all batches must be equal to the number of parts that will be scheduled. Eqs. (16) and (17) state the beginning time of each batch on the first run and the next runs, respectively. All batches are scheduled tight to a common due date
3.5 Algorithm
The algorithm developed begins with problem solving without involving restoration cost (CM) and rework cost for nonconforming parts or without the constraints of Eqs. (20)–(24). It begins with one batch in one production run with one PM. After having obtained a production schedule, estimate the number of nonconforming parts by Eq. (20) and the number of restoration (CM) by Eq. (23). Next, compute estimated rework cost by Eq. (22) and estimated restoration cost by Eq. (24), and then compute total cost. This step is done for two batches until an increased total cost is found. Write the best total cost for one production run and one PM. This process is carried out for two production runs with two PMs until the best total cost is found for two production runs with two PMs. Continue the process up to
Step-1. Set the length of the first failure time interval after PM as
Step-2. A problem is said as feasible if and only if the total time of process with one setup doesn’t exceed the due date of delivering
Step-3. Compute
Step-4. For R = 1, 2, …, g. Go to Step-5.
Step-5. Set R =
Step-6. Set
Step-7. Substitute the values of
Step-8. Solve SISM Model without the constraint of Eqs. (20)–(24). Compute estimated rework cost by Eq. (22) and estimated restoration cost by Eq. (24), and compute a total cost to find TC, write
Step-9. Set
Step-10. Set
Step-11. Set
Step-12. Solve SISM Model without the constraint of Eqs. (20)–(24). Compute estimated rework cost by Eq. (22) and estimated restoration cost by Eq. (24), and compute a total cost to find TC, write
Step-13. Observe whether
- If
- If
- If
- If
and writeall of
Step-14. Write
Step-15. Observe whether
- if
- if
Step-16. Set k = k + 1, go to Step-17.
Step-17. Set
Step-18. Set
Step-19. Solve SISM Model without the constraint of Eqs. (20)–(24). Compute estimated rework cost by Eq. (22) and estimated restoration cost by Eq. (24), and compute a total cost to find
Step-20. Observe whether
- If
- If
- If
- If
Step-21. Write
Step 22. Observe whether R
- If R
- If R
Step 23. Write {
Step-24. The minimal solution is
Min {
Step-25. Write all values of decision variables.
3.6 Numerical experience
To clarify how the proposed algorithm works, the following example is given. Consider an integrating problem with parameters as follows: number of parts
The computational steps to solve the problem are the followings.
Step-1. Yields
Step-2. Yields 30
Step-3.Yields
Step-4. Yields R = 1, 2, 3.
Step-5 to Step-21, for R = 1, yield the best solution
1st looping
Step-5 to Step-21, for R = 2, yield the best solution
2nd looping
Step-5 to Step-21, for R = 3, yield the best solution
Step-23. Yields a set of the best solutions for R
Step-24. Yields a minimal
Step-25. Yield the complete solution as shown in Table 1 and Figure 6.
42.46 | 9150.77 | 201124.80 | |||
39.46 | 8331.54 | ||||
36.46 | 7572.31 | ||||
33.46 | 6873.08 | ||||
30.46 | 6233.85 | ||||
27.46 | 5654.62 | ||||
24.46 | 5135.39 | ||||
21.46 | 4676.15 | ||||
18.46 | 4276.92 | ||||
15.46 | 3937.69 | ||||
6.46 | 3718.46 | ||||
3.46 | 3619.23 | ||||
0.46 | 3580.00 |
3.7 Conclusion
This chapter proposes a model of integrating batch scheduling and maintenance scheduling by criterion of minimization in holding cost, setup cost, PM cost, rework cost, and restoration cost (CM). The criterion of scheduling used is the minimization of the total actual flow time. The first preventive maintenance (PM) scheduling policy (from the direction of due date) is made precisely on due date. The second PM and so on would be done before the time of first deterioration according to Weibull ROCOF cumulative function.
In the model developed, searching for a solution begins with problem resolving without involving restoration cost (CM) and rework cost for nonconforming part. It begins with one batch in one production run and one PM. After finding a production schedule, estimate the number of nonconforming parts and total restoration (CM). Next, compute estimated rework cost and estimated restoration cost, and then compute total cost. This step is done for two batches and so on until an increase on total cost is found. Write the best total cost for one production run and one PM. This process is carried out for two production runs with two PMs until the best total cost is found for two production runs with two PMs. Continue the process until g production runs with g PMs. The best solution of algorithm is a minimization of all of the best total cost for
The model makes a trade-off in the following two things. An increase in the number of batch (length of production run) up to a certain limit will minimize the total inventory holding cost. Meanwhile, an increase in the length of production run will imply on an increase in the number of nonconforming parts and in a number of restoration (CM).
For readers who want to learn more about the integration of batch production and machine maintenance scheduling, they can read Zahedi et al. [4, 5, 6].
3.8 Exercises
Try the next cases to understand the method that was developed at this chapter.
Let four jobs from single item with job size 25 parts with processing time 500 min. The other parameters of this problem are the same as the problem above. Schedule the jobs with minimization total cost criteria.
Let four jobs from single item consist of Job 1 = 40 parts (800 min), Job 2 = 30 parts (600 min), Job 3 = 20 parts (400 min), Job 4 = 10 parts (200 min), and due date d = 3000, and the other parameters of this problem are the same as the problem above. Schedule the jobs with minimization total cost criteria
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