Decision variables

g: number of production run in model [SISM]

R: number of production run in algorithm [SISM]

<sup>L</sup>½ � ikk : batch scheduled at position <sup>i</sup> in the <sup>k</sup>th production run from due date direction (backward approach), for ik = 1, 2, …, Nk, k = 1, 2, …, g

Q½ � ikk : batch size of L½ � ikk

N: possible total number of batch in a planning horizon

Nk: number of batch in kth production run, k = 1, 2, …, g

B½ � ikk : beginning time for batch L[ik]

C½ � ikk : completion time for batch L[ik]

BPM[k]: beginning time for kth PM

CPM[k]: completion time for kth PM

$$X\_{[i\_k k]} = \begin{cases} \mathbf{1}, \text{if} \\ \mathbf{0}, \text{if} \, \mathbf{Q}\_{[i\_k k]} = \mathbf{0}, \\ \mathbf{0}, \text{if} \, \mathbf{Q}\_{[i\_k k]} = \mathbf{0} \end{cases}, i = \mathbf{1}, \text{2, ..., } N\_{\text{be}} \text{ } k = \mathbf{1}, \text{2, ..., } \mathbf{g}$$

nCM: number of CM minimal repair (restoration)

R: total cost of CM (restoration)

M: number of nonconforming parts

Objective function TC: the total cost consisting of inventory cost in process and complete inventory costs, setup cost, preventive and corrective maintenance cost, and rework cost

The model has some assumptions in formulating the model, as follows:


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

4.The same load force for machine in setup time and in processing time.


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]

3.4 Model formulation

Industrial Engineering

Parameters

cs: unit setup cost

Decision variables

Q½ � ikk : batch size of L½ � ikk

X½ � ikk ¼

and rework cost

machine.

approach).

36

(

t: unit processing time for a part

cPM: unit preventive maintenance cost cr: unit restoration cost (corrective cost)

q: number of parts to be processed

B½ � ikk : beginning time for batch L[ik] C½ � ikk : completion time for batch L[ik] BPM[k]: beginning time for kth PM CPM[k]: completion time for kth PM

> 1, ifQ½ � ikk 6¼ 0, 0, ifQ½ � ikk ¼ 0

R: total cost of CM (restoration) M: number of nonconforming parts

In order to formulate the integrating batch production and maintenance sched-

c2: unit inventory holding cost for the work in process part per unit per time unit

c1: unit inventory holding cost for finished part per unit per time unit

tPM: time interval for preventive maintenance (in constant assumption)

<sup>L</sup>½ � ikk : batch scheduled at position <sup>i</sup> in the <sup>k</sup>th production run from due date

,i= 1, 2, …, Nk,k= 1, 2, …, g

Objective function TC: the total cost consisting of inventory cost in process and complete inventory costs, setup cost, preventive and corrective maintenance cost,

The model has some assumptions in formulating the model, as follows:

1. This integrating model for single item processed on a single deteriorating

3. Batch position number and PM are counted from due date direction (backward

uling into a mathematical model, we use following notations.

cw: unit rework cost per part for nonconforming part

β: shape parameter for the Weibull distribution α: scale parameter for the Weibull distribution p1: probability of defect part on in-control state p2: probability of defect part on out-of-control state

d: an order delivery time (a common due date)

g: number of production run in model [SISM] R: number of production run in algorithm [SISM]

direction (backward approach), for ik = 1, 2, …, Nk, k = 1, 2, …, g

N: possible total number of batch in a planning horizon Nk: number of batch in kth production run, k = 1, 2, …, g

nCM: number of CM minimal repair (restoration)

2. Setup time is not depending on the size of batches.

s: setup time required for any batch processed

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

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

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

$$c\_i \sum\_{k=1}^{\mathcal{S}} N\_k + E(\mathcal{R}) + E(\mathcal{W}) \tag{14}$$

subject to

$$\sum\_{k=1}^{g} \sum\_{i\_k=1}^{N\_k} \mathbf{Q}\_{[i\_k k]} = q \tag{15}$$

$$B\_{\left[i\_1\mathbf{1}\right]} + \sum\_{j\_1=1}^{i\_1} \left( sX\_{\left[j\_1\mathbf{1}\right]} + tQ\_{\left[i\mathbf{1}\right]} \right) - s = d, i\_1 = \mathbf{1}, \dots, N\_1, k = \mathbf{1} \tag{16}$$

$$\begin{aligned} \mathbf{B}\_{[i\&k]} &+ \sum\_{l=2}^{k} \left[ \sum\_{j\_l=1}^{i\_l} \left( \mathbf{s} \mathbf{X}\_{[j\_l l]} + \mathbf{t} \mathbf{Q}\_{[j\_l l]} + (k-1) \mathbf{t}\_{\text{PM}} \right) \right] - \mathbf{s} + \mathbf{t} \\ &\dots & \dots \end{aligned}$$

$$d\sum\_{i\_1=1}^{N\_1} \left( \mathbf{s} \mathbf{X}\_{[i\_1 1]} + t \mathbf{Q}\_{[i\_1 1]} \right) = d, i\_k = \mathbf{1}, \mathbf{2}, \dots, N\_k \text{ and } k = \mathbf{2}, \mathbf{3}, \dots, \mathbf{g} \tag{17}$$

$$\sum\_{i\_k=1}^{N\_k} \left( t Q\_{[i\_k k]} + s \right) \le d,\\ k = 1 \tag{18}$$

$$\left(\sum\_{i\_k=1}^{N\_k} \left(tQ\_{[i\_k k]} + s\right) \le a, k = 2, 3, \dots, \mathbf{g}\right) \tag{19}$$

Mk ¼ p2xnumber of parts processed in interval

$$\left[B\_{\left[N\_11\right]} - s + a, C\_{\left[11\right]}\right], k = 1, 2, \dots, \text{g} \tag{20}$$

$$E(M) = M\_k, k = 1, 2, \dots, \text{g} \tag{21}$$

$$E(\mathbf{W}) = \mathbf{c}\_{\mathbf{w}} E(\mathbf{M}) \tag{22}$$

$$m\_{\rm CM} = \left\lfloor \left\{ \frac{d - \left( B\_{[N\_1 1]} - s \right)}{a} \right\}^{\beta} \right\rfloor \tag{23}$$

$$E(R) = c\_r n\_{\rm CM} \tag{24}$$

$$B\_{PM\ [I]} = d\_2$$

$$\mathcal{C}\_{\text{PM}[\mathbb{Z}]} = d + t\_{\text{PM}},$$

$$\mathcal{B}\_{\text{PM}[k]} = \mathcal{B}\_{[1\kappa k]} + t \mathcal{Q}\_{[1\kappa k]}, \ k = \mathbf{1}, \ \mathbf{2}, \ldots, \mathbf{g},$$

$$\mathcal{C}\_{\text{PM}[k]} = \mathcal{B}\_{\text{PM}[k]} + t\_{\text{PM}}, \ k = \mathbf{1}, \mathbf{2}, \ldots, \mathbf{g} \tag{25}$$

$$N = \left\lfloor \frac{d - \left\lceil \frac{d}{a} \right\rceil t\_{\rm PM} - tq}{s} \right\rfloor \tag{26}$$

$$\mathbf{g} = \lceil \mathbf{t}q/a \rceil \tag{27}$$

up to g production runs with g PMs. The best algorithm solution is the minimization of all the best total costs for k = 1, 2, …, g. Then, write all decision variables of single item single machine integration problem. Single item single machine integration

Integrated Batch Production and Maintenance Scheduling to Minimize Total Production…

Step-1. Set the length of the first failure time interval after PM as α. Go to Step -2. 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 d, otherwise the problem is unfeasible for a model or if s + tq ≤ d then the problem is feasible; continue to

Step-3. Compute g by Eq. (27), and set Nk = Nb c, being computed by Eq. (26),

Step-7. Substitute the values of g, Nk, p, q, t, s, d, tPM into the model and set Set X½ � ikk = 1 for ik=1 and k=1 dan set X½ � ikk = 0 for other ik and k. Go to Step-8. 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),

=1,2,…,ik and l = 1,2,…,k, and X <sup>j</sup>

= 1,2, …, ik and l = 1,2,…,k+1, and set X <sup>j</sup>

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 TC, write TC[i(k+1)] = TC. Go to Step-20.

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),

<sup>l</sup> ½ �<sup>k</sup> = 0 otherwise.

<sup>l</sup> ½ �<sup>l</sup> = 0 for other <sup>j</sup>

l , l.

and compute a total cost to find TC, write TC½ � <sup>111</sup> = TC. Go to Step-9.

and compute a total cost to find TC, write TC½ � ikk = TC. Go to Step-13.


Step-14. Write TC[k]\*= TC½ � ikk and writeall of TC\*-related decisionvariables, go to Step-15.

problem algorithm fully is as SISM algorithm.

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

Step-4. For R = 1, 2, …, g. Go to Step-5.

Step-3. If s + tq> d, the problem is unfeasible, stop.

Algorithm [SISM]

k = 1, 2,…, g. Go to Step-4.

Step-5. Set R = 1. Go to Step-6. Step-6. Set g = R. Go to Step-7.

Step-9. Set k=1. Go to Step-10. Step-10. Set ik = 2. Go to Step-11.

<sup>l</sup> ½ �<sup>k</sup> = 1 for <sup>j</sup>

l

Step-13. Observe whether TC½ � ikk <sup>&</sup>lt;TC ð Þ <sup>i</sup>�<sup>1</sup> <sup>k</sup> ½ �<sup>k</sup> , - If TC½ � ikk <sup>&</sup>lt;TC ð Þ <sup>i</sup>�<sup>1</sup> <sup>k</sup> ½ �<sup>k</sup> , observe whether ik=Nk, - If ik=Nk, go to Step -14.



l

Step-20. Observe whether TC <sup>i</sup>½ � ð Þ <sup>k</sup>þ<sup>1</sup> ð Þ <sup>k</sup>þ<sup>1</sup> <sup>&</sup>lt;TC[k]\*,



and writeall of TC[k]\* -related decision variables, go to Step-15.

Step-15. Observe whether k = g,

Step-16. Set k = k + 1, go to Step-17. Step-17. Set ik= 2, go to Step-18.

<sup>l</sup> ½ �<sup>l</sup> = 1 for <sup>j</sup>

Step-11. Set X <sup>j</sup>

Go to Step-12.

Step-18. Set X <sup>j</sup>

Go to Step-19.

39

$$X\_{[i\_k k]} = \begin{cases} \mathbf{1}, \text{if } \mathbf{Q}\_{[i\_k k]} \neq \mathbf{0},\\ \mathbf{0}, \text{if } \mathbf{Q}\_{[i\_k k]} = \mathbf{0} \end{cases}, i\_k = \mathbf{1}, \mathbf{2}, \dots, \mathbf{N}\_k, k = \mathbf{1}, \mathbf{2}, \dots, \mathbf{g} \tag{28}$$

$$Q\_{[i\_k k]} \ge 0, i\_k = 1, 2, \dots, N\_k, k = 1, 2, \dots, \mathbf{g} \tag{29}$$

$$Q\_{[i\_k k]} \le X\_{[i\_k k]} \mathbf{q}, i\_k = \mathbf{1}, \mathbf{2}, \dots, N\_k, k = \mathbf{1}, \mathbf{2}, \dots, \mathbf{g} \tag{30}$$

$$N\_k \ge \mathbf{1}, \mathbf{k} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{g} \tag{31}$$

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 d sequentially. Eqs. (18) and (19) state the length of first run and the next runs, respectively. Eq. (20) states the estimation of nonconforming parts for each run. Eqs. (21) and (22) state the estimation of total nonconforming parts and total rework cost for nonconforming parts, respectively. Eqs. (23) and (24) state the possible number of CM action with cumulative Weibull ROCOF and the expected cost of CM action, respectively. Eq. (25) represents a set of constraints for the beginning and the next of the PM times, with the assumption that first PM in schedule or the last PM in processing (backward approach) after all batches has been completed at a common due date d to ensure the machine in as good as new condition for the next order. Eq. (26) states the possible number of batches in a planning horizon. Eq. (27) states the possible number of production runs in a planning horizon. Equation (28) states a binary constraint that each batch will have: X½ � ik<sup>k</sup> ¼ 1 for non-empty batches and X½ � ik<sup>k</sup> ¼ 0 for empty batches. Eq. (29) states non-negativity of batch size. Eq. (30) states batch size less or equal with all parts that will be scheduled. Eq. (31) states the existence of the number of batches in each run.

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

up to g production runs with g PMs. The best algorithm solution is the minimization of all the best total costs for k = 1, 2, …, g. Then, write all decision variables of single item single machine integration problem. Single item single machine integration problem algorithm fully is as SISM algorithm.
