Semiconductor Route Planning Under Dual-Fab ...

Semiconductor Route Planning Under Dual-Fab Strategy Muh-Cherng Wu, Chen-Fu Chen and Chan-Fu Shih Department of Industrial Engineering and Management National Chiao Tung University 1001 Dasyue Road, Hsinchu, Taiwan 886-03-5731913

[email protected] Abstract: This paper addresses a route planning problem for a semiconductor company that adopts a dual-fab manufacturing strategy. Under the strategy, two fabs are physically so close in order to share capacity with each other. With the capacity–sharing mechanism, a product may be produced by a cross-fab route. That is, the manufacturing of a product may go through the two fabs. This leads to a routing planning problem, which involves two decisions—determining the cutoff point of the cross-fab route and the route ratio for each product. We proposed an LP-GA approach to solve the problem, where the LP module is for determining the cutoff points and the GA module is for determining the route ratios. Experiment results show that the LP-GA method significantly outperforms existing methods.

INTRUDUCTION To quickly respond to demand booming, a semiconductor company usually adopts a dual-fab strategy in expanding capacity. A large-scale factory space for accommodating two fabs is established in advance; equipments are gradually moved in based on market demand over time. The two fabs, physically so close to each other, are eligible to share capacity with each other. Under the dual-fab strategy, a cross-fab production policy is advocated in order to increase equipment utilization. That is, a job is allowed to be partly manufactured in one fab and partly in the other. This in turn raises a cross-fab route planning problem, which has been rarely examined in literature. Most prior studies, assuming no cross-fab route, focused on effective capacity allocation among fabs (Frederix, 2001; Karabuk ＆ Wu, 2003; Manmohan, 2005; Wu et al., 2005, Lee et al., 2006; Chiang et al., 2007). In this paper, we assume that the sequence of all operations for a product can be cut into two segments, and each segment can be manufactured at either one of the two fabs. As a result, a product has four possible routes. The cross-fab route planning problem thus includes two decisions—determining the route cutoff-point and the route ratio of each product. We proposed an LP-GA approach to solve the problem, where the LP module is for determining the cutoff points and the GA module is for determining the route ratios. Numerical experiments indicate that adopting the cross-fab production policy could significantly increase the throughput of a dual-fab design.

PROBLEM STATEMENT This section intends to describe the cross-fab route planning problem more clearly. We first present the assumptions and proceed to introduce the decision variables, objective function and constraints of the problem. The two fabs hereafter are respectively called Fab_A and Fab_B. Assumption 1: Each of the two fabs is functionally comprehensively and can individually handle the manufacture of each product. Assumption 2: A product has four possible routes. To implement cross-fab production, the manufacturing route of a product is cut into two parts, where the route’s break point is called a cut-off point. The two parts can be manufactured in any one of the two fabs. This yields four possible routes for each product, which can be represented by α , β , α → β , and β → α , where denotes a route at Fab_A and β denotes a route at Fab_B. Assumption 3: The transportation path between any two workstations/buffers is unique, rather than multiple. Theoretically, there may exist multiple paths in transporting a job from a workstation to another; however, to reduce the complexity of traffic control, we predefine a fixed path for such a transport. The route planning problem has two decision variables for each product: the cutoff point and the ratios of its four possible routes

α

(simply called route ratios). Represent the cutoff point and route ratios of product i by (π i , ri ) , where identification code (an integer) of an operation and

πi

denotes the

ri = [ai , bi , ci , di ] is a vector of which each element denotes the

percentage of a particular route. Define Π = [π 1 ,..., π n ] as the cutoff points and R = [ r1.., rn ] as the route ratios for n

166

products. The route planning problem is to determine a (Π , R ) in order to maximize the total throughput of the two fabs, subject to the constraint of meeting a target cycle time. *

*

SOLUTION FRAMEWORK Figure 1 shows the solution framework, which involves two modules. Module 1 attempts to find an optimum set of route cutoff points ( Π ), under the assumption—the transportation time is zero. The module is solved by an iterative use of a linear program (LP) model. The LP model serves as a performance evaluator, which could evaluate a particular Π by computing its minimum *

number of inter-fab transportations. With the LP model, a binary search algorithm is used to identify an optimum Π as the ultimate decision for cutoff points. *

Module 1

Cutoff Point

Linear program (LP)

Genetic algorithm (GA) Route Ratio

Module 2 Queueing network Figure 1: Solution Framework

With Π having been obtained, Module 2 aims to find an optimal R under a more relaxed assumption—transportation time is not zero. The higher the traffic intensity on a path, the longer is the required transportation time. This module involves two sub-modules. The first one provides a queueing model for evaluating the performance of a particular (Π , R ) . That is, the *

*

throughput of (Π , R ) , subject to a target mean cycle time, can be efficiently evaluated by the queueing model. The second *

sub-module provides a search algorithm to find R . In summary, the solution space of the

cross-fab

routing

planning

problem

can

be

described

by

S

= {(Π , R ) | Π ∈ Π _ Set , R ∈ R _ Set} . The objective is to find an optimum (Π , R ) from S, in terms of maximizing the throughput subject to a target cycle time. Since S can be very huge, this problem is decomposed into two sub-problems. The first *

one is to find an optimum Π

*

*

by assuming no transportation time, and the second one is to find an optimum R

*

by taking

Π as known and relaxing the transportation assumption. *

MODULE 1－FINDING Π* The solution Π for Module 1 is obtained through an iterative use of an LP model. We first describe the LP model and then present the iterative search algorithm. *

Indices and Parameters i: index of product g : index of workstation in Fab_A h: index of workstation in Fab_B n: total number of products π i : cutoff point for defining the cross-fab routes of product i

167

Π : Π = [π i ], 1 ≤ i ≤ n , a vector for describing the cut-off points of all products Q : an estimated total throughput of the two fabs while in high utilization (in lots), which is used as the target throughput in the LP model. n

∑ P =1, 0 ≤ P ≤ 1

Pi : percentage of product i in the product mix,

i

i =1

i

C g : available machine hours of workstation g in Fab_A C h : available machine hours of workstation h in Fab_B ma : total number of workstations in Fab_A mb : total number of workstations in Fab_B Wiga : total processing time per lot required on workstation g in Fab_A, while product i is manufactured by route α Wigc total processing time per lot required on workstation g in Fab_A,, while product i is manufactured by route α → β Wigd : total processing time per lot required on workstation g in Fab_A, while product i is manufactured by route β → α

Wihb : total processing time per lot required on workstation h in Fab_B, while product i is manufactured by route β Wihc : total processing time per lot required on workstation h in Fab_B, while product i is manufactured by route α → β Wihd total processing time per lot required on workstation h in Fab_B, while product i is manufactured by route β → α Decision Variables

ai : percentage of using route bi : percentage of using route ci : percentage of using route d i percentage of using route

α in producing product i β in producing product i α → β in producing product i β → α in producing product i

LP Model The LP program is to evaluate a particular Π (a decision for the route cutoff points). The objective function of the LP program is denoted by Z( Π ). n

Min

Z (Π ) = ∑ Q ⋅ Pi ⋅ (ci + di ) i =1

s. t.

ai + bi + ci + di = 1 n

1≤ i ≤ n

(1)

∑ Q ⋅ P ⋅ (a ⋅W

a ig

+ di ⋅Wigd + ci ⋅Wigc ) ≤ C g

1 ≤ g ≤ ma

(2)

∑ Q ⋅ P ⋅ (b ⋅W

b ih

+ di ⋅Wihd + ci ⋅Wihc ) ≤ Ch

1 ≤ h ≤ mb

(3)

i =1 n

i =1

i

i

i

i

The objective function is to minimize the number of cross-fab production lots. The rationale for defining this objective is that cross-fab production requires longer transportation time than within-fab production. Subject to a target cycle time, an attempt to minimize cross-fab production lots tends to increase total throughput. Constraint (1) describes the relationship among the route ratios. Constraints (2) and (3) ensure that the capacity used in each workstation should be lower than its available supply.

168

Bi-section Search Algorithm We proposed a bi-section search algorithm to find an optimum solution Π . The algorithm is an iterative process. In an iteration, each product has only two possible cutoff points to select. Taking a product route as a line, the two cutoff points are respectively on the first and the third quartiles (Fig 2). By evenly cutting the route into two segments, each cutoff point is in the middle of a particular segment. Of the two evenly divided segments, the one where a cutoff point stays is called the r-segment of the point. *

Figure 2: Process of the cutoff point In each iteration i, the size of the space {∏} is 2 manner (i.e., 2

n

n

if there are n products. By solving the LP program in an exhaustive

times), we can obtain the best solution in this iteration--denoted by Π i . For each product, the *

γ

-segment (i.e.,

remaining segment) is the output of iteration i and will be the input of iteration i+1. The bi-section search algorithm continues until the γ -segment cannot be divided anymore.

MODULE 2—FINDING R* *

Module 2 attempts to find an optimal route ratio decision ( R ) in order to maximize the total throughput of the two fabs subject that the corresponding average cycle time is less than CT0 (the target cycle time). This problem is essentially a search problem in a solution space H = {R} = {[ r1 ,...r n ] | ri = ( ai , bi , ci , d i )} . A genetic algorithm (GA) is proposed to solve the problem. In the GA, the performance of a solution R is evaluated by a queueing network model. We first introduce the queueing network model and proceed to the genetic algorithm.

Queueing Network The queueing network model is an extension of the model developed by Connors et al. (1996). The input/output function of their model can be briefly formulated as follow: CT = f (TH , R, Π ) . That is, given a target total throughput (TH), a route ratio decision (R), and a cutoff point decision ( Π ), the queueing model (f) can be used to compute the two fabs’ mean cycle time (CT). However, Connor et al. (1996) did not consider the effect of transportation among workstations. We extended their model to include the effect of transportation based on two assumptions. First, we assume that the transportation path between any two stations is unique. Second, each transportation path between any two stations is modeled as a “conveyor machine” with only one unit of capacity. Such an extension makes the queueing model closer to a fab in the real world. Likewise, the extended queuing model can also be described as CT = f (TH , R, Π ) . The objective function in Module 2 is to maximize throughput (TH) subject to a target cycle time (CT0). To evaluate the objective function, we used a bi-section search technique to find the total throughput (TH) for a particular route ratio (R); that is

TH = f ( R, Π* , CT0 ) where Π* denotes the cutoff point decision obtained in Module 1 and CT0 is the target cycle time. Genetic Algorithm *

The genetic algorithm (GA) is to identify an optimal solution R from the space {R}. As stated, the performance of R is obtainable by the enhanced queueing model. A possible solution R (or called a chromosome) is represented by a vector

R = [r1 ,...r n ] where ri = (ai , bi , ci , di ) . We call ri a gene-segment and each of its element a gene, and the gene values are

169

imposed by the following constraints:

ai + bi + ci + di = 1 and 0 ≤ ai , bi , ci , di ≤ 1 .

The GA is an iterative algorithm which can be briefly described as follows.

Procedure GA Step 1: Initialization z t = 0, Status = ‘Not-terminate’ z Randomly generate N p valid chromosomes to form a population P0 Step 2: Genetic Search While (Status = ‘Not-Terminate’) do z Use cross-over operator to create z

N c new chromosomes Use mutation operator to create N m new chromosomes

z z

Form a pool by including Pt and the set of newly created chromosomes Set t = t + 1; select the best N p chromosomes from the pool to form Pt

z Check if termination condition is met; if yes, set Status = “Terminate” Endwhile *

Step 3: Output the best chromosome R in Pt The crossover operation is to create two new chromosomes from two existing ones. Let each gene-segment i in R1 and R2 be respectively represented by ri1 and ri 2 . A one-point crossover operation (Binh & Lan, 2007) is imposed on ri1 and ri 2 to create two new ones ri 3 and ri 4 . Consider an example where the 2nd gene is chosen as the cross-over point for mixing

ri1 = (ai1 , bi1 , ci1 , di1 )

and

ri 2 = (ai 2 , bi 2 , ci 2 , d i 2 )

ri 3 = (ai1 , bi 2 , ci1 ,1 − ai1 − bi 2 − ci1 ) and ri 4

.

By

the

crossover

operation,

we

would

obtain

(a i 2 , bi1 , ci 2 ,1 - ai 2 - bi1 - ci 2 ) .

In the mutation operation, a new chromosome is created by swaping two particular gene-segments in an existing one. For

i* is chosen for modification; and the 2nd and 4th genes are chosen to swap for ri*1 = (ai1 , bi1 , ci1 , di1 ) , then ri* 2 = (ai1 , di1 , ci1 , bi1 ) , which in turn yield a new chromosome R2 = [r11 ,..ri* 2 ,...r n1 ] from

example, if gene-segment

R1 = [r11 ,..ri* 1 ,...r n1 ] . Two termination conditions are defined for the GA. First, the best solution in Pt has been no change for over a certain period. Second, population Pt has evolved over a certain number of iterations.

EXPERIMENTS Numeric experiments were carried out to justify the effectiveness of the proposed LP-GA method. The benchmark to compare is called M-GA, which denotes that the cutoff point of each route is just on the middle of the route. Data sets for the experiments are adapted from an HP-fab in literature (Wein, 1988). Of the two fabs, one involves 93 machines and the other involves 72 machines. Three types of products are produced. One product involves 150 operations; the other two both involve 172 operations but different in processing times. The comparisons are carried out in a scenario with product mixes RA = (3:2:5). By the queueing model, we obtain a throughput level (QA = 128 lots) that would keep the two fabs in high utilization. Given a target cycle time CT0 =11,081 min, we compare the throughput of each method. Table 1 shows the cutoff points of each route obtained by the LP-GA method, which indicates that the cutoff points suggested by the LP-GA are different from that of M-GA. Table 2 compares the throughput, which indicates that the LP-GA method outperforms the M-GA, about 2.3% higher. This implies that optimal planning of cross-fab production is positive in increasing throughput.

170

Product 1 Product 2 Product 3 Total Step Number RA

172

172

150

85th step 85th step 129th step

Table 1: Cutoff points obtained by the LP program RA (CT0=11081 min) Algorithm

Throughput (lots) Gap (%)

LP-GA

128

0%

M-GA

125

2.34 %

Table 2: A comparison of computing times of throughput of different algorithms

CONCLUSION This paper presents an approach to solve the cross-fab route planning problem for a semiconductor company adopting the dual-fab manufacturing strategy. The route planning problem involves two decisions—determining the cutoff point and the route ratio for each product. An LP-GA method is proposed to solve the route planning problem, where the LP is for making the cutoff point decisions, the GA is for making the decision of route ratio. Experiment results show that the LP-GA method significantly outperforms the benchmark method M-GA.

Acknowledgement This research is financially supported by National Science Council, Taiwan, under a contract NSC-96-2628-E009-026-MY3.

171

References Binh Q.D. and Lan P. N., 2007, Application of a genetic algorithm to the fuel reload optimization for a research reactor, Applied Mathematics and Computation, 187, 977-988. Chiang D., Guo R.S., Chen A., Cheng M.T. and Chen C.B., 2007, Optimal supply chain configurations in semiconductor manufacturing, International Journal of Production Research, 45(3), 631–651. Connors D.P., Feigin G.E., Yao D.D., 1996, A queueing network model for semiconductor manufacturing, IEEE Transactions on Semiconductor Manufacturing, 9(3), 412-427. Frederix F., 2001, An extended enterprise planning methodology for the discrete manufacturing industry, European Journal of Operational Research, 129, 317-325. Karabuk S. and Wu S.D., 2003, Coordinating strategic capacity planning in the semiconductor industry, Operations Research, 51, 839-849. Lee Y.H., Chung S., Lee B., Kang K.H., 2006, Supply chain model for the semiconductor industry in consideration of manufacturing characteristics, Production Planning ＆ Control, 17(5), 518-533. ManMohan S.S., 2005, Managing demand risk in tactical supply chain planning for a global consumer electronics company, Production and Operations Management, 14(1), 69-79. Wein L. M., 1988, Scheduling semiconductor wafer fabrication, IEEE Transactions on Semiconductor Manufacturing, 1(3), 115-130. Wu S.D., Erkoc M., Karabuk S., 2005, Managing capacity in the high-tech industry: a review of literature, The Engineering Economist, 50, 125-158.

172