3.4 Model formulation

In order to formulate the integrating batch production and maintenance scheduling into a mathematical model, we use following notations.

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

Integrated Batch Production and Maintenance Scheduling to Minimize Total Production…

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])

> c<sup>1</sup> þ c<sup>2</sup> 2

t ∑ 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

5. The machine cannot be interrupted as long as production runs.

can be expressed as mixed-integer nonlinear programming as follows:

tQ <sup>j</sup> <sup>1</sup> ½ �<sup>1</sup> <sup>þ</sup> <sup>s</sup> � � ( )<sup>Q</sup> ð Þ <sup>i</sup>þ<sup>1</sup> <sup>1</sup> ½ �<sup>1</sup> <sup>þ</sup>

> cs ∑ g

> > k¼1

∑ g

k¼1 ∑ Nk ik¼1

<sup>1</sup> ½ �<sup>1</sup> <sup>þ</sup> tQ½ � <sup>i</sup><sup>1</sup>

sX <sup>j</sup>

ik¼<sup>1</sup> tQ½ � ikk <sup>þ</sup> <sup>s</sup>

<sup>l</sup> ½ �<sup>l</sup> <sup>þ</sup> tQ <sup>j</sup>

Mk ¼ p2xnumber of parts processed in interval

nCM <sup>¼</sup> <sup>d</sup> � <sup>B</sup>½ � <sup>N</sup><sup>11</sup> � <sup>s</sup> � �

BPM 1½ � ¼ d,

α

ik¼<sup>1</sup> tQ½ � ikk <sup>þ</sup> <sup>s</sup>

<sup>l</sup> ½ �<sup>l</sup> <sup>þ</sup> ð Þ <sup>k</sup> � <sup>1</sup> tPM � � " # � <sup>s</sup><sup>þ</sup>

� � <sup>¼</sup> d, ik <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, Nk and <sup>k</sup> <sup>¼</sup> <sup>2</sup>, <sup>3</sup>, …, g (17)

∑ ik j <sup>k</sup>¼1

Q½ � ikk ð Þ k � 1 tPM þ ∑

tQ <sup>j</sup> <sup>k</sup> ½ �<sup>k</sup> <sup>þ</sup> <sup>s</sup> � � ( )<sup>Q</sup> ð Þ <sup>i</sup>þ<sup>1</sup> <sup>k</sup> ½ �<sup>k</sup> <sup>þ</sup>

> 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>

Nk þ E Rð Þþ E Wð Þ (14)

Q½ � ikk ¼ q (15)

� � !# <sup>þ</sup> gcPM<sup>þ</sup>

� � � <sup>s</sup> <sup>¼</sup> d, i<sup>1</sup> <sup>¼</sup> <sup>1</sup>, …, N1, k <sup>¼</sup> 1 (16)

� � <sup>≤</sup>d, k <sup>¼</sup> <sup>1</sup> (18)

� � <sup>≤</sup>α, k <sup>¼</sup> <sup>2</sup>, <sup>3</sup>, …, g (19)

<sup>B</sup>½ � <sup>N</sup><sup>11</sup> � <sup>s</sup> <sup>þ</sup> <sup>α</sup>;C½ � <sup>11</sup> � �, k <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, g (20)

E Mð Þ¼ Mk, k ¼ 1, 2, …, g (21) E Wð Þ¼ cwE Mð Þ (22)

� �<sup>β</sup> \$ % (23)

E Rð Þ¼ crnCM (24)

6.Batch size value is in real positive.

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

N1�1 i1¼1

g

"

Nk ik¼1

k¼2

Q½ � <sup>i</sup><sup>11</sup> þ ∑

Q½ � ikk þ c<sup>1</sup> ∑

B½ � <sup>i</sup><sup>11</sup> þ ∑ i1 j <sup>1</sup>¼1

> B½ � ikk þ ∑ k l¼2

<sup>i</sup>1¼<sup>1</sup> sX½ � <sup>i</sup><sup>11</sup> <sup>þ</sup> tQ½ � <sup>i</sup><sup>11</sup>

∑ i1 j <sup>1</sup>¼1

c1 ∑ Nk�1 ik¼1

sX <sup>j</sup>

∑ il j l ¼1

∑Nk

∑Nk

Model [SISM]

t ∑ Nk ik¼1

subject to

∑<sup>N</sup><sup>1</sup>

37

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

c<sup>2</sup> � c<sup>1</sup> 2

Minimize TC ¼ c1 ∑
