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

Integrated Batch Production and Maintenance Scheduling to Minimize Total Production… DOI: http://dx.doi.org/10.5772/intechopen.85004

#### Figure 1.

progress, so the machine failure interferes with the productivity of the shop floor. These three issues indicate the independence between production scheduling, and

1. 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

2. 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

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

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

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

2.2 Input-output diagram for the models of batch production and maintenance

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

2.1 Influence diagram for the model batch production and maintenance

maintenance scheduling may result in the following conditions:

interfere with productivity.

algorithm works to solve a problem.

batch size (production schedule).

other for the number of nonconforming parts.

2. Model construction

Industrial Engineering

scheduling

scheduling

30

pre-arranged production schedule.

Influence diagram for the model of batch production and maintenance scheduling.

until the deadline d. The setup cost is calculated by multiplying the number of scheduled batches by unit setup cost. The rework cost is calculated by multiplying the number of nonconforming parts by unit rework cost. The PM cost is calculated by multiplying the number of PM by unit PM cost, and the CM cost is calculated by multiplying the number of CM by unit CM cost.

Figure 2. Input-output diagram for the models of batch production and maintenance scheduling.

The input-output diagram of the integrated batch production and maintenance scheduling to minimize total cost is shown in Figure 2.

(0,t) in the in-process batch L[i], there are Q[i] work-in-process parts (parts not yet or being processed). In interval (t, 2 t) there are (Q[i]-1) work-in-process parts and l finished part, until in interval ((Q[i]–1)t,Q[i]t), there are I work-in-process parts and

Integrated Batch Production and Maintenance Scheduling to Minimize Total Production…

The amount of holding cost for the finished part, designated f1, is as follows:

The amount of holding cost for the in-process part, designated f1, is as follows:

By summation f1 and f1, f2 and f2 in a reverse order, a simpler result is found, i.e.:

c2

½ �<sup>i</sup> þ

Based on Eq. (1), the total holding cost of the in-process batch of all batches may

½ � <sup>N</sup>�<sup>1</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup> � <sup>c</sup><sup>1</sup>

½ � <sup>1</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup> � <sup>c</sup><sup>1</sup>

Next, total holding cost (TolC) is computed by adding up the holding cost of in-

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

c<sup>1</sup> þ c<sup>2</sup> <sup>2</sup> <sup>t</sup> <sup>∑</sup> N i¼1 Q2 ½ �<sup>i</sup> þ

� i or <sup>h</sup> <sup>h</sup>

� � <sup>þ</sup> c2t Q½ �<sup>i</sup> –<sup>2</sup>

� idanf <sup>2</sup> <sup>¼</sup> <sup>c</sup><sup>2</sup>

� i <sup>þ</sup>

<sup>2</sup> tQ<sup>2</sup>

h h

<sup>2</sup> tQ½ �<sup>i</sup> <sup>Q</sup>½ �<sup>i</sup> � <sup>1</sup>

<sup>f</sup> <sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>2</sup> <sup>¼</sup> <sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup>

<sup>2</sup> tQ<sup>2</sup>

<sup>2</sup> tQ<sup>2</sup>

The holding cost of completed batch may be formulated as follows:

� � <sup>þ</sup> tQ½ � <sup>1</sup> <sup>þ</sup> <sup>s</sup>

� � <sup>þ</sup> tQ½ � <sup>1</sup> <sup>þ</sup> <sup>s</sup> n o � � <sup>Q</sup>½ � <sup>N</sup>�<sup>1</sup> <sup>þ</sup> …

process batch in Eq. (2) and completed batch in Eq. (3) to yield Eq. (4).

� � <sup>þ</sup> c1t Q½ �<sup>i</sup> –<sup>2</sup>

� � <sup>þ</sup> c1t Q½ �<sup>i</sup> –<sup>1</sup>

� � <sup>þ</sup> … <sup>þ</sup> c23t <sup>þ</sup> c22t <sup>þ</sup> c2t:

� � <sup>Q</sup>½ �<sup>i</sup>

th in-process batch is the addition of f1 and f2, that is:

� � <sup>Q</sup>½ �<sup>i</sup>

� i:

<sup>2</sup> tQ½ �<sup>i</sup> (1)

½ �<sup>i</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup> � <sup>c</sup><sup>1</sup>

<sup>2</sup> <sup>t</sup><sup>∑</sup> N i¼1 Q½ �<sup>i</sup>

(2)

(3)

Q½ �<sup>i</sup> : (4)

<sup>2</sup> t Q½ �<sup>i</sup> <sup>þ</sup> <sup>1</sup>

<sup>2</sup> t Q½ �<sup>i</sup> <sup>þ</sup> <sup>1</sup>

c<sup>2</sup> � c<sup>1</sup>

<sup>2</sup> tQ½ � <sup>N</sup>�<sup>1</sup> <sup>þ</sup> …

<sup>2</sup> tQ½ � <sup>1</sup> <sup>¼</sup> <sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup>

n o � � <sup>Q</sup>½ � <sup>2</sup> <sup>¼</sup> c1 <sup>∑</sup>

<sup>2</sup> <sup>t</sup><sup>∑</sup> N i¼1 Q2

N�1 i¼1

∑ i j¼1

c<sup>2</sup> � c<sup>1</sup> <sup>2</sup> <sup>t</sup> <sup>∑</sup> N i¼1

tQ½ �<sup>j</sup> þ s � � ( )Q½ � <sup>i</sup>þ<sup>1</sup> :

� �:

f <sup>1</sup> ¼ c1t þ c12t þ c13t þ … þ c1t Q½ �<sup>i</sup> –3

<sup>2</sup> tQ½ �<sup>i</sup> <sup>Q</sup>½ �<sup>i</sup> � <sup>1</sup>

f <sup>2</sup> ¼ c2tQ½ �<sup>i</sup> þ c2t Q½ �<sup>i</sup> –1

DOI: http://dx.doi.org/10.5772/intechopen.85004

f1 <sup>¼</sup> <sup>c</sup><sup>1</sup>

<sup>f</sup> <sup>1</sup> <sup>þ</sup> f2 <sup>¼</sup> <sup>c</sup><sup>1</sup>

<sup>2</sup> tQ½ � <sup>N</sup> <sup>þ</sup> <sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup>

<sup>2</sup> tQ½ � <sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup>

n o � � <sup>Q</sup>½ � <sup>N</sup>

n o � � <sup>Q</sup>½ � <sup>3</sup> <sup>þ</sup> c1 tQ½ � <sup>1</sup> <sup>þ</sup> <sup>s</sup>

tQ½ �<sup>j</sup> þ s � � ( )Q½ � <sup>i</sup>þ<sup>1</sup> <sup>þ</sup>

Then, the holding cost of i

be written as follows:

½ � <sup>N</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup> � <sup>c</sup><sup>1</sup>

½ � <sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup> � <sup>c</sup><sup>1</sup>

� � <sup>þ</sup> … <sup>þ</sup> tQ½ � <sup>2</sup> <sup>þ</sup> <sup>s</sup>

� � <sup>þ</sup> tQ½ � <sup>1</sup> <sup>þ</sup> <sup>s</sup>

N�1 i¼1

∑ i j¼1

(in-process batch) in one production run.

� � <sup>þ</sup> … <sup>þ</sup> tQ½ � <sup>2</sup> <sup>þ</sup> <sup>s</sup>

c<sup>1</sup> þ c<sup>2</sup> <sup>2</sup> tQ<sup>2</sup>

<sup>þ</sup> <sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup> <sup>2</sup> tQ<sup>2</sup>

c1 tQ½ � <sup>N</sup>�<sup>1</sup> þ s

þc1 tQ½ � <sup>N</sup>�<sup>2</sup> þ s

þc1 tQ½ � <sup>2</sup> þ s

ToIC ¼ c1 ∑

33

(Q[i]–1) finished parts.

Figure 2 shows the input parameters model: the number of parts scheduled, due date, Weibull distribution function f(t), unit processing time, PM interval length, inventory holding cost, rework cost, setup time between batches, and the probability of nonconforming part on the machine in the status of in-control and out of control. The output of the model are the size of batches and the schedule, PM schedule, number of CM, and number of nonconforming parts.

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].
