scheduling for dedicated machine constraint using integer programming huy nguyen anh pham, arthur...

Scheduling for Dedicated Machine Scheduling for Dedicated Machine Constraint Using Integer Constraint Using Integer

ProgrammingProgramming

Huy Nguyen Anh Pham, Arthur Shr, and Peter P. ChenHuy Nguyen Anh Pham, Arthur Shr, and Peter P. ChenLouisiana State University, Baton Rouge, LA, U.S.A.Louisiana State University, Baton Rouge, LA, U.S.A.

Alan LiuAlan LiuNational Chung Cheng University, Chia-Yi, TaiwanNational Chung Cheng University, Chia-Yi, Taiwan

OutlineOutline

Dedicated Machine Constraint in Semiconductor Dedicated Machine Constraint in Semiconductor

ManufacturingManufacturing

Related WorkRelated Work

Integer Programming ApproachInteger Programming Approach

ExperimentsExperiments

Conclusion Conclusion

Dedicated Machine Constraint in Dedicated Machine Constraint in Semiconductor ManufacturingSemiconductor Manufacturing

The operations of The operations of semiconductor semiconductor manufacturing repeatedly manufacturing repeatedly make an IC product layer make an IC product layer by layer. by layer.

The The dedicated machine dedicated machine constraintconstraint is in the is in the photolithographyphotolithography process process area area

Photolithography ProcessPhotolithography Process

The most important process for the yield of IC– “Blue Print” for each photo layer

– Alignment of layers

Photo machines could be used in different layers.

Different process time for photo each layer

Dedicated Photolithography Dedicated Photolithography Machine ConstraintMachine Constraint

Other Stages

No constraint:

Machine A Machine B Machine CMachine X Machine Y Machine Z

Busy

Photolithography Stages

Dedicated Machine Constraint:

idle

Wafer lots

With this constraint, the wafer lots dedicated to machine X need to wait With this constraint, the wafer lots dedicated to machine X need to wait for machine X, even if there is no wafer lot waiting for machine Y. for machine X, even if there is no wafer lot waiting for machine Y.

Without this constraint, the wafer lots can be scheduled to A, B, or C.Without this constraint, the wafer lots can be scheduled to A, B, or C.

Issues and Current SolutionsIssues and Current Solutions

– Photo machines might become load unbalanced if Photo machines might become load unbalanced if the wafers are dispatched randomly to the photo the wafers are dispatched randomly to the photo machines at the first photo layer.machines at the first photo layer.

– Some photo machines will be “lost” for a while and Some photo machines will be “lost” for a while and the others will be “run” with heavy loadthe others will be “run” with heavy load

– Switch from the highly congested machines to the Switch from the highly congested machines to the idle machines. idle machines.

– Rely on experienced engineers to manually handle Rely on experienced engineers to manually handle alignment problems of the wafer lotsalignment problems of the wafer lots

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Related Work

Kimms (1996) proposed a mixed-integer programming model for optimally scheduling machines.

(Miwa, 2005) proposed a load balance allocation function between machines and used this function for a dynamic programming method to schedule them.

=> Their problems without dedicated machine constraint.

(Shr 2008a; 2008b) proposed a heuristic Load Balancing scheduling method.

=> The approaches did not attempt to optimize the cost for the overall productivity with the dedicated machine constraint.

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Modeling the Dedicated Machine Modeling the Dedicated Machine Constraint-RSEMConstraint-RSEM

Stepsj

Wafer lots i

1 2 3 4 5 6 7 8 9 10

W1 R3 R3

W2 R3 R3

W3 R2 R2 R2 R2

W4 R4 R4

W5 R1 R1 R1

A Resource Schedule and Execution Matrix (RSEM):

Modeling the Dedicated Machine Modeling the Dedicated Machine Constraint-Breakdown (BD) MatrixConstraint-Breakdown (BD) Matrix

Steps jMachines k

1 2 ..

R1 100 ..

R2 467 ..

R3 838 ..

R4 100 ..

A Breakdown Matrix:

Formulation of the ProblemFormulation of the Problem Variables:

Xi,j,k is a binary variable.

The penalty cost for switching:

(1)

(2)

i,j s j,ir

i,jWjis

ki,jWjir

i,j,km

machine.another toassigned is step if seconds, ,

, machine toassigned is step if seconds, ,

Formulation of the Problem – Cont’dFormulation of the Problem – Cont’d

The production cost for wafer i at step j assigned to machine k:

Minimize the production cost of the assignments:

Minimize the queue time for wafer lots waiting at machines

The objective function:

(4)

(3)

j k ikjiX

2

,, minimize

i j k

kjikji CX ,,,, minimize

(5)

j k ikji

i j kkjikji XCXf

2

,,,,,, minimize (6)


Constraints:– Wafer i at step Wi,j is assigned to exactly one machine:

– Assume that each wafer lot Wi has Pi photo steps. The dedicated machine constraint for wafer i at step Wi, j :

for either k = 1, 2, …, or Q.

(7)

(8)


– Because of requirements for a canonical form of an IP formulation, by using the Big M method Equation (8) is transformed into:

(9)

},1,0{,...,for

,1

,

..........

,

,

.,..,1,,..,1,,..,1,1,

,

,,,

2,2,,

1,1,,

QkMjNiQii

kki

jQiQji

jiji

jiji

Y Y

Y

MYMX

MYMX

MYMX

The Integer Formulation:


(10)

},1,0{,...,,for

,1 ,1

,

..........

,

,

minimize

.,..,1,,..,1,,..,1

:Subject to

,1,,,

,,,

,,,

2,2,,

1,1,,

2

,,,,,,,

QkMjNiQiikji

kki

kkji

jQiQji

jiji

jiji

j k ikji

i kki

j kkjikji

YY X

YX

MYMX

MYMX

MYMX

XYCXf

Since the Big M method

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100, 200, and 300 wafer lots with 10 steps are used.

The number of machines is set equal to 3.

BD[j, k] is randomly given in the range [0, 1000].

Each experiment uses 30 examples

Experiments – Cont’dExperiments – Cont’d

01020304050607080

1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

Examples

Perc

en

tag

e

122

124

126

128

130

132

134

Execu

tio

n t

ime

Percentage of the cost saved in seconds Execution time in seconds

Experiment of 100 wafer lots, 10 steps and 3 machines.

• The average execution time: 127 seconds for each example.

• The average percentage cost saved: approximately 41.1%.

Experiments – Cont’dExperiments – Cont’dExperiment of 200 wafer lots, 10 steps and 3 machines.

• The average execution time: 1,186.6 seconds for each example.


0

10

20

30

40

50

60

70

1 2 3 4 5 6 7 8 9101112131415161718192021222324252627

Examples

Perc

en

tag

e

1140115011601170118011901200121012201230

Execu

tio

n t

ime


Experiments – Cont’dExperiments – Cont’dExperiment of 300 wafer lots, 10 steps and 3 machines.

• The average execution time: 2,781 seconds for each example.


0

10

20

30

40

50

1 2 3 4 5 6 7 8 9 10 11 12

Examples

Perc

en

tag

e

2400

2500

2600

2700

2800

2900

3000

Execu

tio

n t

ime


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ConclusionConclusion

Provide an Integer Programming (IP) approach to model the dedicated machine constraint

Future work– Improving the performance of IP for on-line scheduling

problems– Improving the technique to model the constraint

Thank You

scheduling for dedicated machine constraint using integer programming huy nguyen anh pham, arthur...

Documents

machine x

layer slide

related work kimms

taiwan slide

scheduling machines

congested machines

idle machines

wafer lot