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 q parts of an item be scheduled by a minimization of total actual flow time criterion. The q parts are divided into N batches L[i] (i = 1, 2, ..., N) where the sizes of each batch are Q[i](i = 1, 2, ..., N). If all parts in a batch have been processed completely, then the batch is called as completed batch. If a batch is still containing any part not yet or being processed, it is called as in-process batch.

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 inprocess batch firstly and then for completed batch. The position of batch L[i] in a single machine manufacturing system by a backward approach during a planning horizon is shown in Figure 3.

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

Figure 3. Batch position in a single machine manufacturing system.

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

(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 (Q[i]–1) finished parts.

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

$$f\_1 = c\_1t + c\_12t + c\_13t + \dots + c\_1t\left(Q\_{\left[i\right]} - \mathfrak{Z}\right) + c\_1t\left(Q\_{\left[i\right]} - \mathfrak{Z}\right) + c\_1t\left(Q\_{\left[i\right]} - \mathfrak{Z}\right).$$

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

$$f\_{\mathcal{I}} = c\_2 t Q\_{[\bar{t}]} + c\_2 t \left( Q\_{[\bar{t}]} \mathbf{-1} \right) + c\_2 t \left( Q\_{[\bar{t}]} \mathbf{-2} \right) + \dots + c\_2 \mathbf{3} t + c\_2 \mathbf{2} t + c\_2 t.$$

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

$$\mathbf{f}\_1 = \frac{c\_1}{2} \left[ t Q\_{[i]} \left( Q\_{[i]} - \mathbf{1} \right) \mathbf{d} \mathbf{n} \mathbf{f}\_2 = \frac{c\_2}{2} \left[ t \left( Q\_{[i]} + \mathbf{1} \right) \left( Q\_{[i]} \right) \right] \right]$$

Then, the holding cost of i th in-process batch is the addition of f1 and f2, that is:

$$f\_1 + f\_2 = \frac{c\_1}{2} \left[ t Q\_{[i]} \left( Q\_{[i]} - \mathbf{1} \right) + \frac{c\_2}{2} \left[ t \left( Q\_{[i]} + \mathbf{1} \right) \left( Q\_{[i]} \right) \right] \text{ or }$$

$$f\_1 + f\_2 = \frac{c\_1 + c\_2}{2} t Q\_{[i]}^2 + \frac{c\_2 - c\_1}{2} t Q\_{[i]} \tag{1}$$

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

$$\begin{aligned} \frac{c\_1 + c\_2}{2} t Q\_{[N]}^2 + \frac{c\_2 - c\_1}{2} t Q\_{[N]} + \frac{c\_1 + c\_2}{2} t Q\_{[N-1]}^2 + \frac{c\_2 - c\_1}{2} t Q\_{[N-1]} + \dots \\ \frac{c\_1 + c\_2}{2} t Q\_{[2]}^2 + \frac{c\_2 - c\_1}{2} t Q\_{[2]} + \frac{c\_1 + c\_2}{2} t Q\_{[1]}^2 + \frac{c\_2 - c\_1}{2} t Q\_{[1]} = \frac{c\_1 + c\_2}{2} t \sum\_{i=1}^N Q\_{[i]}^2 + \frac{c\_2 - c\_1}{2} t \sum\_{i=1}^N Q\_{[i]} \end{aligned} \tag{2}$$

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

$$\begin{aligned} &\mathbf{c}\_{1}\left\{\left(t\mathbf{Q}\_{[N-1]}+s\right)+\ldots+\left(t\mathbf{Q}\_{[2]}+s\right)+\left(t\mathbf{Q}\_{[1]}+s\right)\right\}\mathbf{Q}\_{[N]}\\ &+\mathbf{c}\_{1}\left\{\left(t\mathbf{Q}\_{[N-2]}+s\right)+\ldots+\left(t\mathbf{Q}\_{[2]}+s\right)+\left(t\mathbf{Q}\_{[1]}+s\right)\right\}\mathbf{Q}\_{[N-1]}+\ldots\\ &\mathbf{c}\_{1}+\mathbf{c}\_{1}\left\{\left(t\mathbf{Q}\_{[2]}+s\right)+\left(t\mathbf{Q}\_{[1]}+s\right)\right\}\mathbf{Q}\_{[3]}+\mathbf{c}\_{1}\left\{\left(t\mathbf{Q}\_{[1]}+s\right)\right\}\mathbf{Q}\_{[2]}=\mathbf{c}\_{1}\sum\_{i=1}^{N-1}\left\{\sum\_{j=1}^{i}\left(t\mathbf{Q}\_{[j]}+s\right)\right\}\mathbf{Q}\_{[i+1]}.\end{aligned} \tag{3}$$

Next, total holding cost (TolC) is computed by adding up the holding cost of inprocess batch in Eq. (2) and completed batch in Eq. (3) to yield Eq. (4).

$$ToI C = c\_1 \sum\_{i=1}^{N-1} \left\{ \sum\_{j=1}^{i} \left( t Q\_{[j]} + s \right) \right\} Q\_{[i+1]} + \frac{c\_1 + c\_2}{2} t \sum\_{i=1}^{N} Q\_{[i]}^2 + \frac{c\_2 - c\_1}{2} t \sum\_{i=1}^{N} Q\_{[i]}.\tag{4}$$

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.

The input-output diagram of the integrated batch production and maintenance

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

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

nonconforming parts, while the maintenance cost consists of PM cost and CM cost. The model will answer how the batch production and maintenance scheduled

3. Integrated batch production and maintenance scheduling for single

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.

The modeling starts with an inventory holding cost for in-process and completed

Holding cost concept is developed from [3]. Let q parts of an item be scheduled by a minimization of total actual flow time criterion. The q parts are divided into N batches L[i] (i = 1, 2, ..., N) where the sizes of each batch are Q[i](i = 1, 2, ..., N). If all parts in a batch have been processed completely, then the batch is called as completed batch. If a batch is still containing any part not yet or being processed, it is

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 inprocess batch firstly and then for completed batch. The position of batch L[i] in a single machine manufacturing system by a backward approach during a planning

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

scheduling to minimize total cost is shown in Figure 2.

Industrial Engineering

schedule, number of CM, and number of nonconforming parts.

minimize the total cost. Drawing influence diagram follows [7].

finished part of inventory holding costs), setup cost, and rework cost for

item processed on a deteriorating machine with a due date

3.1 Inventory holding costs for in-process batch and completed batch

called as in-process batch.

horizon is shown in Figure 3.

Batch position in a single machine manufacturing system.

Figure 3.

32

ΛðÞ¼ t

Integrated Batch Production and Maintenance Scheduling to Minimize Total Production…

Let a system (machine) with a Weibull failure time distribution have a shape parameter of β = 1.69 and a scale parameter α = 2,857.14, then, based on ROCOF cumulative function, the first, the second, and so on estimated failure times that

with scale parameter α and shape parameter β:

<sup>β</sup> <sup>¼</sup> 1 then <sup>t</sup> = 2857.14. If <sup>Λ</sup>ðÞ¼ <sup>t</sup> <sup>t</sup>

<sup>β</sup> <sup>¼</sup> 3 then <sup>t</sup> = 5473.33. If <sup>Λ</sup>ðÞ¼ <sup>t</sup> <sup>t</sup>

could be written as follows can be found:

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

3.3 Estimation of nonconforming parts

for k=2 may be written as follows:

nonconforming parts may be written as

A condition for two production runs and two PMs.

If <sup>Λ</sup>ðÞ¼ <sup>t</sup> <sup>t</sup>

α

If <sup>Λ</sup>ðÞ¼ <sup>t</sup> <sup>t</sup>

Figure 5.

35

α

t α <sup>β</sup>

α

α

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.

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

<sup>M</sup><sup>2</sup> <sup>¼</sup> <sup>p</sup><sup>2</sup> xnumber of parts processed in interval <sup>B</sup>½ � <sup>N</sup><sup>11</sup> � <sup>s</sup> <sup>þ</sup> <sup>α</sup>;C½ � <sup>11</sup> (10)

In the same way for g production runs and g PMs, the number of nonconforming

Mg <sup>¼</sup> <sup>p</sup><sup>2</sup> xnumber of parts processed in interval <sup>B</sup>½ � <sup>N</sup><sup>11</sup> � <sup>s</sup> <sup>þ</sup> <sup>α</sup>;C½ � <sup>11</sup> (11)

E Mð Þ¼ Mk, k ¼ 1, 2, …, g (12)

E Wð Þ¼ cwE Mð Þ (13)

parts will always be of the same form, except if applied to k = 1, 2, …, g, so that

Under an assumption that the probability that the nonconforming part processed under in-control state is p1 = 0, then the expected number of

so that the expected network cost may be computed by

<sup>β</sup> <sup>¼</sup> 2 then <sup>t</sup> = 4305.82.

<sup>β</sup> <sup>¼</sup> 4 then <sup>t</sup> = 6489.03.

(9)

Figure 4.

Batch position in a single machine manufacturing system.

Eq. (4) and Figure 3 are to be developed into a formulation of holding cost for g production run and PM interval inserted sequentially as shown in Figure 4.

By considering any changes taking place in each production run and total PM for g production runs and g PM intervals, total holding cost will become Eq. (5).

$$c\_1 \sum\_{i=1}^{N-1} \left\{ \sum\_{j=1}^i \left( t Q\_{\left[i\_1\right]} + s \right) \right\} Q\_{\left[ (i+1)\_1 1 \right]} + \frac{c\_1 + c\_2}{2} t \sum\_{i\_1=1}^{N\_1} Q\_{\left[ i\_1 1 \right]}^2 + \frac{c\_2 - c\_1}{2} t \sum\_{i\_1=1}^{N\_1} Q\_{\left[ i\_1 1 \right]} +$$

$$\sum\_{k=2}^g \left[ c\_1 \sum\_{i\_k=1}^{N\_k-1} \left\{ \sum\_{j=1}^i \left( t Q\_{\left[ i\_k k \right]} + s \right) \right\} Q\_{\left[ (i+1)\_k k \right]} + \frac{c\_1 + c\_2}{2} t \sum\_{i\_k=1}^{N\_k} Q\_{\left[ i\_k k \right]}^2 +$$

$$\frac{c\_2 - c\_1}{2} t \sum\_{i\_k=1}^{N\_k} Q\_{\left[ i\_k k \right]} + c\_1 \sum\_{i\_k=1}^{N\_k} Q\_{\left[ i\_k k \right]} \left( (k-1) t\_{PM} + \sum\_{j\_k=1}^{N\_{\left[ k-1 \right]}} \left( t Q\_{\left[ j\_k k \right]} + s \right) \right) \right]. \tag{5}$$

#### 3.2 ROCOF function

Rate of occurrence of failures (ROCOF) is a concept that is useful in the modeling of failures over time and the effect of PM (and CM) actions [8]. The ROCOF characterizes the probability that a failure occurs in the interval [t,t + δt]. The ROCOF is given by an intensity function

$$\lambda(t) = \lim\_{\delta t \to 0} \frac{P\{N(t + \delta t) - N(t) \ge 1\}}{\delta t} \tag{6}$$

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 + δt] is zero as δt–›0, we have the intensity function equal to the derivative of the conditional expected number of failures, so that

$$
\lambda(t) = \frac{d}{dt} E\{N(t)\}. \tag{7}
$$

When the failures are minimally repaired and the time to repair is negligible, then ROCOF function λðÞ¼ t r tð Þ, the failure rate function. The cumulative ROCOF function is given by

$$\Lambda(t) = \int\_0^t \lambda(t)dt\tag{8}$$

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

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

$$A(t) = \left(\frac{t}{a}\right)^{\beta} \tag{9}$$

with scale parameter α and shape parameter β:

Let a system (machine) with a Weibull failure time distribution have a shape parameter of β = 1.69 and a scale parameter α = 2,857.14, then, based on ROCOF cumulative function, the first, the second, and so on estimated failure times that could be written as follows can be found:

If <sup>Λ</sup>ðÞ¼ <sup>t</sup> <sup>t</sup> α <sup>β</sup> <sup>¼</sup> 1 then <sup>t</sup> = 2857.14. If <sup>Λ</sup>ðÞ¼ <sup>t</sup> <sup>t</sup> α <sup>β</sup> <sup>¼</sup> 2 then <sup>t</sup> = 4305.82. If <sup>Λ</sup>ðÞ¼ <sup>t</sup> <sup>t</sup> α <sup>β</sup> <sup>¼</sup> 3 then <sup>t</sup> = 5473.33. If <sup>Λ</sup>ðÞ¼ <sup>t</sup> <sup>t</sup> α <sup>β</sup> <sup>¼</sup> 4 then <sup>t</sup> = 6489.03.

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

Eq. (4) and Figure 3 are to be developed into a formulation of holding cost for g

By considering any changes taking place in each production run and total PM for

c<sup>1</sup> þ c<sup>2</sup> <sup>2</sup> <sup>t</sup> <sup>∑</sup> N<sup>1</sup> i1¼1 Q2 ½ � <sup>i</sup><sup>11</sup> þ

> c<sup>1</sup> þ c<sup>2</sup> <sup>2</sup> <sup>t</sup> <sup>∑</sup> Nk ik¼1 Q2 ½ � ikk þ

> > Nð Þ <sup>k</sup>�<sup>1</sup> j <sup>k</sup>¼1

� � !#

tQ <sup>j</sup> <sup>k</sup> ½ �<sup>k</sup> <sup>þ</sup> <sup>s</sup>

<sup>δ</sup><sup>t</sup> (6)

dt ENt f g ð Þ : (7)

λð Þt dt (8)

<sup>Q</sup> ð Þ <sup>i</sup>þ<sup>1</sup> <sup>k</sup> ½ �<sup>k</sup> <sup>þ</sup>

Q½ � ikk ð Þ k � 1 tPM þ ∑

Rate of occurrence of failures (ROCOF) is a concept that is useful in the modeling

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 + δt] is zero as δt–›0, we have the intensity function equal to the derivative of the conditional expected number of failures, so that

d

When the failures are minimally repaired and the time to repair is negligible, then ROCOF function λðÞ¼ t r tð Þ, the failure rate function. The cumulative ROCOF

ðt

0

A ROCOF function that has been used extensively is the Weibull ROCOF. The

PNt f g ð Þ� þ δt N tð Þ≥1

of failures over time and the effect of PM (and CM) actions [8]. The ROCOF characterizes the probability that a failure occurs in the interval [t,t + δt]. The

λðÞ¼ t

ΛðÞ¼ t

cumulative ROCOF (or the expected total number of failures) is given by the

c<sup>2</sup> � c<sup>1</sup> <sup>2</sup> <sup>t</sup> <sup>∑</sup> N<sup>1</sup> i1¼1

Q½ � <sup>i</sup><sup>11</sup> þ

: (5)

production run and PM interval inserted sequentially as shown in Figure 4.

g production runs and g PM intervals, total holding cost will become Eq. (5).

<sup>Q</sup> ð Þ <sup>i</sup>þ<sup>1</sup> <sup>1</sup> ½ �<sup>1</sup> <sup>þ</sup>

c<sup>1</sup> ∑ N�1 i¼1

Figure 4.

Industrial Engineering

∑ g

"

k¼2

c<sup>2</sup> � c<sup>1</sup> 2

3.2 ROCOF function

function is given by

function

34

∑ i j¼1

c1 ∑ Nk�1 ik¼1

> t ∑ Nk ik¼1

tQ <sup>j</sup> <sup>1</sup> ½ �<sup>1</sup> <sup>þ</sup> <sup>s</sup> � � ( )

Batch position in a single machine manufacturing system.

∑ i j¼1

ROCOF is given by an intensity function

tQ <sup>j</sup> <sup>k</sup> ½ �<sup>k</sup> <sup>þ</sup> <sup>s</sup> � � ( )

> Nk ik¼1

λðÞ¼ t lim δt!0

Q½ � ikk þ c<sup>1</sup> ∑

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 k=2 may be written as follows:

$$M\_2 = p\_2 \text{number of parts processed in interval} \left[ B\_{[N\_11]} - \mathfrak{s} + a, C\_{[11]} \right] \tag{10}$$

In the same way for g production runs and g PMs, the number of nonconforming parts will always be of the same form, except if applied to k = 1, 2, …, g, so that

$$M\_{\mathfrak{g}} = p\_2 \text{ number of parts processed in interval} \left[ B\_{[N\_1 1]} - \mathfrak{s} + a, C\_{[11]} \right] \tag{11}$$

Under an assumption that the probability that the nonconforming part processed under in-control state is p1 = 0, then the expected number of nonconforming parts may be written as

$$E(M) = M\_k, k = 1, 2, \dots, \text{g} \tag{12}$$

so that the expected network cost may be computed by

$$E(\mathcal{W}) = \mathbf{c}\_{\mathbf{w}} E(\mathcal{M}) \tag{13}$$

Figure 5. A condition for two production runs and two PMs.
