An operator staffing and assignment model for foundry fabs

DOI 10.1007/s00170-004-2124-5 O R I G I N A L A R T I C L E

Muh-Cherng Wu · Kuo-Cheng Fu

An operator staffing and assignment model for foundry fabs

Received: 7 November 2003 / Accepted: 2 February 2004 / Published online: 12 January 2005 ©Springer-Verlag London Limited 2005

Abstract This paper presents a linear programming (LP) model for simultaneously solving an operator staffing prob-lem in a semiconductor fab. The probprob-lem is to determine the staffing level and assign operators to the staffing positions. The semiconductor fab has the following characteristics: it imposes a no-lay-off policy, but allows overtime; keeps high-quality shop floor operations; aims to control tightly the staffing costs; and faces frequent changes of product mix. The operator staffing problem in such a fab was not addressed in previous literature. The proposed LP model aims to minimize the operator staffing cost. The LP solutions have been examined and were found to be insensitive to stochastic demands modeled by various simulation replicates.

Keywords Foundry fab· Operator assignment · Semiconductor manufacturing· Staffing

1 Introduction

Semiconductor wafer manufacturing is a machine-intensive in-dustry. A typical wafer manufacturing facility (fab) includes about 300–500 machines. These machines are grouped into ap-proximately 100 workstations. A workstation involves several machines that are functionally identical. On the shop floor, a team of operators is assigned to supervise a group of worksta-tions, often called an area, a bay, a cluster or a work center [1].

Two main tasks performed by operators are the loading and unloading of wafers. When a machine is available for processing, operators must load wafers, then unload wafers when the ma-chine has finished an operation. The mama-chine becomes idle when no operator is present to perform loading or unloading tasks. The M.-C. Wu (u) · K.-C. Fu

Department of Industrial Engineering and Management, National Chiao Tung University,

Hsin-Chu, Taiwan, R.O.C. E-mail: mcwu@cc.nctu.edu.tw Fax: +886-03-5720610

idleness of the machine due to the lack of an operator is rather ex-pensive, which might result in the significant loss of capacity [2] and an increase in cycle time [3]. Using more operators can pre-vent machines from being idle, but it also increases staffing costs. Therefore, developing methods to make the operator staffing de-cision is important in determining how many operators should be placed at each work center to prevent machines from being idle while minimizing the staffing cost.

Many studies on the operator staffing decision have been published. Bonal et al. [1] develop a static operator staffing model using data obtained from the manufacturing execution system (MES). The static model calculates, on a weekly ba-sis, the required operator time in a work center and the max-imum available time an operator can provide. A staffing level is obtained by further considering the operators’ availability and absenteeism. The static model assumes that the machine interfer-ence effect can be neglected. Machine interferinterfer-ence occurs when the number of machines that simultaneously call for assistance exceeds the number of available operators, consequently causing production loss [4].

Some studies proposed an M/M/s queuing model [5] to ana-lyze the production loss caused by machine interference [6, 7] by modeling machines as customers and operators as servers. Mey-ersdorf et al. [6] proposed another queuing model to analyze the operator-machine-lot interference, in which wafer lots are taken as customers and operators as servers. The above studies based on queuing theory aimed to compute the minimum weekly de-mand of operators at which the production loss at a work center is tolerable.

The measurement of operator service time has not been thor-oughly explored in the studies based on queuing. Kishimoto et al. [3] proposed a systematic method to estimate the service time of operators. They classified the activities of operators into three types. The first type, such as loading and unloading, is move-based, and the service time is proportional to the number of moves. A move refers to a machine’s finishing of an operation; therefore, each move involves at least a loading and an unloading activity. The second type of activity is time-based; for example, a machine may need to be cleaned for two hours every two days.

The third type is administration-based, and covers the service time for administration. A detailed analysis of the service time of operators led to the development of a heuristic method for real-locating the activities to operators. Such a reallocation, verified by a discrete event simulation model, outperforms the previous allocation of their testing fab.

In semiconductor wafer manufacturing, there are two types of business models: foundry and integrated device

manufactur-ing (IDM). A typical IDM company manufactures only a few

types of ICs (integrated circuits) that are designed by in-house. A foundry fab, by contrast, manufactures a large number types of ICs designed by many customers known as IC design houses. The product mix in a foundry fab may vary frequently, thus changing the staffing needs at each work center, and requiring that a decision be made regarding operator reallocation.

Most previous studies on fab operator staffing estimate a weekly staffing level at which the production loss or the in-crease in cycle time is acceptable. This implicitly advocates an idea that each work center should be run at a computed staffing level. Staffing level and the number of staffing positions are inter-changeably used in this paper. However, three issues may arise when this idea is implemented in a foundry fab, which aims to tightly control its staffing costs, keep high-quality shop floor op-eration, and impose a no lay-off policy in an environment of a frequently changing product mix.

The first issue involves the fact that the staffing demands of operators at each work center may fluctuate daily. Staffing a work center at a particular level may cause some machines to be idle. One method of reducing machine idleness is by allowing oper-ators work overtime. This implies that the computed demand of staffing level (or the number of staffing positions) may be re-duced by overtime, yet this point has not been considered in previous studies.

The second issue concerns the fact that the available oper-ators may be fewer than the computed demand. A foundry fab may insist on a no lay-off policy because the cost of lay-off is quite high in some non-Western societies. In addition to paying tangible compensation, a company that lays off employees may suffer some intangible but serious losses, such as of the morale of employees and the reputation of the company. A semiconductor company without a lay-off policy tends to hire operators con-servatively. The supply of operators may thus be less than the computed demand when the product mix changes significantly. This issue, as well as the first issue outlined above, points out a decision problem: staffing position allocation; that is, determin-ing how many staffdetermin-ing positions should be allocated to each work center.

The third issue is that the operators in a fab may have limited qualification for taking care of a work center. In a foundry fab, the shop floor operation is quite complex and dynamic. Opera-tors must handle many dynamically changing events such as lot holding, route changing, and among-fab supporting, and famil-iarize with the reporting and coordination practices of handling exceptional cases in a particular work center. To ensure the qual-ity of shop floor operation, a foundry fab may require that an operator receive rehearsal training if he or she has not been

oper-ating in a work center for a certain period of time (for example, six months). Without such training, operators might ignore some of the details of the operating practices, and subsequently cause adverse effects. The assignment of an operator to a work cen-ter may therefore include training cost. This points out another decision problem: operator assignment; that is, how to assign op-erators to the staffing positions such that the training costs are minimized.

This paper proposes a linear programming (LP) model to solve simultaneously the two aforementioned decision problems – staffing position allocation and operator assignment. The LP model is developed for cases in which the available staffing sup-ply is less than the demand, but it can be extended to cases in which the available supply is greater than or equal to the demand. This study also assumes that an operator can only be assigned to one work center. This implies that each work center is an inde-pendent unit in the measurement of performance.

The rest of this paper is organized as follows: Sect. 2 presents a method for computing the staffing cost function of a work center, considering daily demand fluctuation and overtime. The staffing level for which the cost is minimum can be obtained by the staffing cost function. Section 3 describes the proposed LP model for cases in which the available staffing supply is less than the staffing level of minimum cost. Section 4 gives a numerical example to explain the proposed LP model and compares solu-tions in which the staffing demands are generated by simulation with various random seeds. Section 5 describes extensions of the LP model to accommodate cases in which the staffing supply is greater than the staffing level of minimum cost. Concluding remarks are made in Sect. 6.

2 Staffing cost at a work center

This section presents a method for estimating the cost of opera-tors for a work center under various staffing levels in a foundry fab. Without loss of generality, the fab is assumed to implement the following staffing practices.

The shop floor of the fab is operated in a two-shifts-per-day schedule, with 12 hours in each shift. Subtracting the time for two meals and some other breaks in a shift, the availability of an operator is about 75%, or about 9 hours per shift. An operator typically works one shift per day for two consecutive days and then takes a break for the subsequent two days. Accordingly, the fab has four teams of operators, each of which works two shifts every four days.

The daily demand for operator time for the following few days can be estimated to reasonable accuracy using a discrete event simulation program, according to the answers given in an interview with a foundry fab conducted by the authors. Overtime is requested whenever the estimated daily demand exceeds the staffing level. An overtime request should be made on a shift ba-sis; that is, a request of less than one shift of overtime is taken as a request for a whole.

The first step in computing the staffing cost function is to estimate the daily demand on operator time for a given

master production schedule (MPS) in a quarter. Given the MPS, a discrete event simulation program coded by eM-Plant (http://www.tecnomatix.com/) is used to estimate the daily move of each work center. Assume that the operator service time for each type of move-based activity and time-based activity is known. The daily demand on operator time can be expressed as seen below:

xj(t) =

Mj(t) mj

60 + Sj(t) , where

xj(t) = Demanded manpower in work center j on day t

(man-hours)

Mj(t) = Number of moves completed at work center j on day t

mj = Service time per move at work center j

(man-minutes/move)

Sj(t) = Service time of time-based activities in work center j

on day t (man-hours).

With xj(t), and the staffing policy as stated above, the

opera-tor cost of a work center is a function of the staffing levels:

C(pj) = h(pj2L) + o L
t=1 ytroundup  xj(t) − pj2× 12Ao 12 Ao  .

If xj(t) > (pj2× 12Ao), then yt= 1, else yt= 0, where

pj = Number of staffing positions per shift at work center j

C(pj) = Total staffing costs of work center j with pj positions

($)

h = Normal time rate ($/man-shift) o = Overtime rate ($/man-shift)

Ao = Availability of operators (A0= 0.75 in the fab of inter-est)

L = Total number of days in the decision time horizon. The first term of the equation for C(pj) above denotes the

normal time cost, where the time horizon includes 2L shifts; the second term represents the overtime cost, where (xj(t) − pj2×

12 Ao) denotes the daily demand for overtime in man-hours. This

value, when positive, should be represented in man-shifts (di-vided by 12 Ao) and rounded up to an integer.

Increasing the number of staffing positions ( pj) will increase

the cost of normal time while decreasing the cost of overtime.

Fig. 1. Staffing cost function of a particular work center

The staffing cost function C(pj) is therefore a convex function

with a global minimum point. By iteratively computing C(pj)

at various staffing levels, we can easily determine a p∗_j which gives a minimum in staffing cost. Figure 1 shows a staffing cost function of a work center, where p∗_j = 3.

3 Model development

A linear programming model is proposed to solve the two deci-sion problems – staffing position allocation and operator assign-ment. Let S be the number of operators available per shift, which, over the decision time horizon, cannot be increased according to the corporate staffing policy. Let Dm=

J



j=1p ∗

j represent the

mini-mum cost demand of staffing positions per shift, where J denotes the total number of work centers and p∗_j is the minimum cost de-mand of staffing positions at work center j. Without considering the constraints of operator supply and training requirements, Dm

operators per shift should be supplied. However, when S< Dm,

Dm− S staffing positions should be eliminated to match S (the

number of available operators). Therefore, the staffing position allocation problem is to determine which of the Dmstaffing

pos-itions should be removed.

This study uses dummy operators to solve the problem of re-ducing the number of staffing positions. That is, Dm− S dummy

operators are created to make the operator supply equal Dm. The

linear programming model is then used to assign each staffing position to an operator. The staffing positions assigned to the dummy operators are to be eliminated. The two decision prob-lems, staffing position allocation and operator assignment, can therefore be integrated as an operator assignment problem.

Let wjk represent the kth (1≤ k ≤ p∗j) staffing position at

work center j (1≤ j ≤ J). An encoding function F( j, k) = s is designed such that each staffing positionwjk in the fab can be

sequentially represented by qs (1≤ s ≤ Dm). A decoding

func-tion F−1(s) = j is defined to identify the work center to which qs belongs. As stated, the operators in the fab may have various

qualifications. All the operators are grouped such that each mem-ber of a group has the same qualifications. The dummy operators are placed in a single group. Let d represent the total number of groups and ni represent the number of operators in group Gi

(1≤ i ≤ d). Then, Dm= d



i=1ni.

The linear programming model formulates the two afore-mentioned decision problems as an operator assignment prob-lem. Table 1 presents an example of the formulation, in which the first column represents the staffing positions at each work cen-ter,wjk(1≤ j ≤ J; 1 ≤ k ≤ p∗j); the second column represents

the staffing position qs (1≤ s ≤ Dm); and the first row

repre-sents the groups of operators Gi(1≤ i ≤ d). In Table 1, Dm= 10

and S= 8; two dummy operators (Dm− S) are thus created and

placed in group G6. Group G1includes one operator who is qual-ified only to take care of work centers 2 and 5. The operator must therefore be trained when he or she is assigned to supervise work centers 1, 3, or 4. Let Csirepresent the cost of assigning a staffing

Table 1. Staffing position allocation and operator assignment matrix wij qs G1 G2 G3 G4 G5 G6(Dummy) Demand w11 q1 T 0 T T T C(p∗1− 1) − C(p∗1) 1 w12 q2 T 0 T T T C(p∗1− 2) − C(p∗1− 1) 1 w13 q3 T 0 T T T M 1 w21 q4 0 T T T 0 C(p∗2− 1) − C(p∗2) 1 w22 q5 0 T T T 0 M 1 w31 q6 T T T T 0 C(p∗3− 1) − C(p∗3) 1 w32 q7 T T T T 0 M 1 w41 q8 T 0 0 0 T C(p∗4− 1) − C(p∗4) 1 w42 q9 T 0 0 0 T M 1 w51 q10 0 T 0 0 T M 1 Supply 1 2 3 1 1 2 10

position qsto an operator in group Gi. For the non-dummy oper-ator groups (1≤ i ≤ d − 1), Csi= T if training is needed, while Csi= 0 otherwise. The matrix of Csi(1≤ i ≤ d −1, 1 ≤ s ≤ Dm) therefore denotes the qualification profile of operators in the fab. Note that the example in Table 1 is simple; in fact, the training cost Csi is not necessarily a constant, when the operators have different capabilities in supporting a work center [9].

Assigning a dummy operator in Gd to a staffing position means that the position should be removed. The cost of removing the first staffing position from work center j can be represented by Cxd= C(p∗_j − 1) − C(p∗_j) where x = F( j, 1). Likewise, the cost of removing the second staffing position from work cen-ter j can be represented by Cxd= C(p∗j− 2) − C(p∗j− 1), where x= F( j, 2). Notice that the formulation C(p∗_j− 2) − C(p∗_j− 1)

implies that the first staffing position has been removed. There-fore, there exists an inherent priority in removing the staffing positions from a work center j. That is, staffing positionwj,k+1

cannot be removed ifwjkhas not been removed.

Moreover, the staffing position of a work center cannot be all removed. Such a removal implies that no operator will be as-signed to the work center. The work center will subsequently become idle all the time. Removing the last staffing position from work center j is thus not allowed. The cost of doing so is represented by Cxd = M where x = F( j, p∗j) and M is a very large positive number.

The operator assignment problem can be formulated as the following linear program. Let Qsi represent the binary decision variable; that is, Qsi= 1 if the staffing position qsis assigned to an operator in group Gi, otherwise Qsi= 0.

Minimize TC= Dm
s=1 d
i=1 CsiQsi (1)

subject to the following constraint sets: d
i=1 Qsi= 1 for 1≤ s ≤ Dm (2) Dm
s=1 Qsi= ni for 1≤ i ≤ d (3) Qsd≥ Q(s+1) d for 1≤ s ≤ Dmwhere F−1(s) = F−1(s + 1) (4) Qsi= 0 or 1 for 1≤ s ≤ Dm, 1≤ i ≤ d (5)

The objective function Eq. 1 minimizes the total assignment costs, which are of two types. One is the cost of training Csi (1≤ i ≤ d − 1) which applies when real operators are assigned. The other is the cost of eliminating staffing position Csd (1≤ s≤ Dm), which is incurred when dummy operators are assigned. Constraint set Eq. 2 specifies that each staffing position qs be assigned to a single operator; and constraint set Eq. 3 requests that each operator in each group must be assigned to a staffing position.

Constraint set Eq. 4 denotes that the elimination of staffing positions in a work center should be performed in a prede-fined order. For example, in work center j, let qs and qs+1 represent the first two staffing positions; that is, F( j, 1) = s and

F( j, 2) = s + 1. If only one staffing position is to be removed

from work center j, then qs is removed. Skipping qsand jump-ing to remove qs+1 is prohibited because C_(s+1)d denotes the marginal cost of removing the second staffing position. There-fore, Qsd≥ Q(s+1)dif qs and qs+1are in the same work center; that is, F−1(s) = F−1(s + 1).

The above LP formulation, an extension of a generalized as-signment problem (GAP) [10], is distinguished by the inclusion of constraint set Eq. 4. Studies on the real-life applications of GAP include those of [11–17]. To the authors’ knowledge, no previous GAP study has examined the staffing problem at a semi-conductor fab with the features presented in this paper.

4 Examples and comparison

This section first describes a numerical example to explain the proposed methods for estimating staffing costs and assigning op-erators. Next, the operator assignment results are compared with values of daily demanded manpower generated by simulation using various random seeds.

4.1 Example

The fab in the example includes five work centers. The nor-mal time rate is h= $ 1200/shift and the overtime rate is o = $ 2400/shift. The training cost for an operator is T = $ 3600. The time horizon for the decision is five days.

The daily moves of each work center can be determined using a discrete event simulation program coded by eM-Plant (http://www.tecnomatix.com/). Table 2 presents the daily de-manded operator hours at each work center given the service time rates for move-based and time-based activities. Accord-ingly, Table 3 presents the total cost over the decision time hori-zon for staffing each work center using different number of op-erators, where NOP denotes the number of staffing positions per shift.

Table 3 shows that p∗₁= 3, p∗₂ = 2, p∗₃= 2, p∗₄= 2 and

Table 2. Daily demand of operator hours at each work center

WS Day 1 Day 2 Day 3 Day 4 Day 5

W1 43 37.4 45.8 40.2 43

W2 35 27 33 29 27

W3 36.2 33 29.8 29.8 33

W4 29.8 29.8 29.8 33.4 26.2

W5 17.8 17.8 20.2 22.6 20.2

Table 3. Staffing costs at each work center

W1 W2 W3 W4 W5

NOP Cost NOP Cost NOP Cost NOP Cost NOP Cost 1 50 400 1 31 200 1 38 400 1 33 600 1 19 200 2 38 400 2 24 000 2 26 400 2 24 000 2 24 000 3 36 000 3 36 000 3 36 000 3 36 000 3 36 000 4 48 000 4 48 000 4 48 000 4 48 000 4 48 000 5 60 000 5 60 000 5 60 000 5 60 000 5 60 000 6 72 000 6 72 000 6 72 000 6 72 000 6 72 000

Table 4. Result of operator assignment

G1 G2 G3 G4 G5 G6 Demand W1-1 3600 0 3600 3600 3600 2400 1 W1-2 3600 0 3600 3600 3600 12 000 1 W1-3 3600 0 3600 3600 3600 100 000 1 W2-1 0 3600 3600 3600 0 7200 1 W2-2 0 3600 3600 3600 0 100 000 1 W3-1 3600 3600 3600 3600 0 12 000 1 W3-2 3600 3600 3600 3600 0 100 000 1 W4-1 3600 0 0 0 3600 9600 1 W4-2 3600 0 0 0 3600 100 000 1 W5-1 0 3600 0 0 3600 100 000 1 Supply 1 2 3 1 1 2 10 per shift is Dm= 5  j=1

p∗_j = 10. Suppose that the supply of

staffing operators per shift is S= 8. Then, two dummy op-erators must be created. Table 4 displays the cost matrix as-sociated with operator assignment as well as the results ob-tained using the proprietary software package LINGO 5.0 (http://www.lindo.com/table/lingot.html/), where M (a large positive real number) is set to $ 100 000 and G6 represents the group of dummy operators. The total assignment cost is $ 13 200= $ 2400 + $ 7200 + $ 3600. Table 4 reveals that two staffing positions are eliminated, one from work center 1 and the other from work center 2.

4.2 Comparison

The foregoing discussion assumes that the daily demanded man-power profile was obtained by simulation. One question may be raised: how sensitive is the operator assignment solution to the change of the daily demand profile, which is stochastic in the real world? This study therefore compares the assignment solutions

Table 5. Percentile distribution of
ij(1≤ i ≤ 30, 1 ≤ j ≤ 30, i = j) 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 0 0 0 1.41 1.41 1.41 1.41 1.41 2.45

for various daily demand profiles, obtained by simulation with various random seeds.

A typical fab, data set 6 provided by the MASM Lab (http:// www.eas.edu/∼masmlab/home.htm), is used as a test bed to compare the operator assignment results. The fab includes 228 machines grouped into 104 workstations, which are grouped into 10 work centers. Nine types of products are produced in the fab. In generating the daily demand profile, the simulation as-sumes that each machine is down at random, and that the down-time and repair down-time follow exponential distributions. Therefore, the operator assignment results associated with 30 simulation replicates are compared. In these experiments, the number of available operators (S) is 35.

Let Wk= [wk1, wk2, . . . , wk10] represent an operator assign-ment solution, where k denotes kth simulation replicate. An indi-cator
ij=Wi− Wj =  10  k=1(w ik− wjk)2is defined to measure the distance between two solutions. Table 5 shows the percentile distribution of
ij (1≤ i ≤ 30, 1 ≤ j ≤ 30, i = j), and indicates that 40% of
ij is zero, and 90% is less than or equal to 1.41. Notably,
ij= 1.41 =

√

2 means that the two solutions (Wiand Wj) only differ in the assignment of one operator. This implies that the proposed method for operator assignment is insensitive to the stochastic variation of daily demand profile. That is, only one simulation replicate is required to solve the staffing problem with 90% confidence that the computed solution differs by no more than one operator from the best solution.

Note that the cases for S= Dmare special cases of the above formulation, and can be easily solved.

5 Model extensions

The linear programming model described above can be extended to solve the operator assignment problems in cases in which

S> Dm. For all Dmstaffing positions, those in work center j are modeled as a group Wj0, where the number of staffing positions is p∗_j. When Dmpositions are allocated, the fab may require that overtime be worked. As mentioned earlier, the fab imposes a no-lay-off policy. Therefore, when S> Dm, we may need to increase the number of staffing positions to match S. In doing so, the over-time cost may be reduced.

Let V(pj) represent the overtime cost function of work center j, which can be expressed as below, with reference to Sect. 2. V(pj) = o L
t=1 ytroundup _x j(t) − pj2× 12Ao 12 Ao 

At work center j with p∗_j staffing positions, the overtime cost is reduced by V(p∗_j+ 1) − V(p∗_j) when one staffing pos-ition is added. Extra staffing pospos-itions can be added until no further reduction in overtime cost applies. That is, let ej repre-sent the total number of newly added positions at work center j. Then, V(p∗_j + ej+ 1) − V(p∗j+ ej) = 0 and V(p∗j+ ej) − V(p∗j+ ej− 1) > 0. Let Wjk(1≤ k ≤ ej) represent the kth newly added staffing position at work center j. Remember that Wjk (k= 0) represents the p∗_j staffing positions at work center j. The total number of staffing positions at work center j is thus p∗_j+ ej.

Let D represent the total number of staffing positions includ-ing the original staffinclud-ing level (Dm=

J



j=1

p∗_j) and the newly added staffing positions ( J  j=1 ej); that is, D= Dm+ J  j=1 ej. Note that D refers to the minimum cost staffing level without overtime.

The operator assignment decision is to allocate S operators to D positions, which is to be addressed in the following two cases:

S> D > Dmand Dm< S < D.

5.1 S> D > Dm

When S> D, we need to create a dummy work center Wd0 (d= J + 1), which includes S − D staffing positions so that the supply and demand are equal. Table 6 shows an example, where

S= 15, D = 13 and Dm= 10. The table includes D − Dm= 3 newly added staffing positions (W11, W12, W21), and S− D = 2 dummy positions in work center W60.

The staffing positions (Wjk, 1≤ j ≤ J + 1, 0 ≤ k ≤ ej) are sequentially represented by bs through an encoding function s= Y( j, k). Let Rsdenote V(p∗_j+ k) − V(p∗_j+ k − 1), where s = Y( j, k), which is the marginal overtime cost reduction when the kth extra staffing position is added to worker center j. A

decod-ing function j= Y−1(s) is defined accordingly.

In Table 6, the available operators are classified into groups (Gi) such that each operator in a group has the same qualifi-cations. Let Csi represent the cost incurred when one staffing position in bs is assigned to an operator in Gi. For the staffing group bs representing Wj0, Csi= T if the operator assignment needs a training; otherwise, Csi= 0. For the staffing group bs representing Wjk (k= 0), Csi= T − Rs if the operator

assign-Table 6. Assignment cost matrix for S> D > Dm

G1 G2 G3 G4 Demand W10 b1 T 0 T T 3 W11 b2 T− R2 −R2 T− R2 T− R2 1 W12 b3 T− R3 −R3 T− R3 T− R3 1 W20 b4 0 T T T 2 W21 b5 −R5 T− R5 T− R5 T− R5 1 W30 b6 T T T T 2 W40 b7 T 0 0 0 2 W50 b8 0 T 0 0 1 W60 b9 M M M M 2 Supply 5 4 3 3 15

ment requires a training; otherwise, Csi= −Rs. For the staffing group bs representing dummy staffing positions (Wd0, d= J + 1), Csi= M, where M is a large positive number. Assigning an operator to a dummy work center implies that no job will be assigned to the operator. The cost of such an assignment (M) should therefore be much higher than T and Rs.

The linear program for S> D > Dm is formulated below, where K represents the total number of operator groups and

d represents the total number of staffing groups, including the

dummy one. Let ni represent the number of operators in op-erator group Gi, and ms represent the number of positions in staffing group bs. The term Esi(1≤ s ≤ d; 1 ≤ i ≤ K) represents the cost of assigning an operator in Gi to staffing group bs. The non-negative integer ysi(1≤ s ≤ d; 1 ≤ i ≤ K) is a decision vari-able, which represents the number of operators in Giassigned to staffing group bs. Minimize TC= d
s=1 K
i=1 Esiysi (6) such that d
s=1 ysi= ni 1≤ i ≤ K (7) K
i=1 ysi= ms 1≤ s ≤ d (8) ysi∈ Z 1≤ s ≤ d ; 1 ≤ i ≤ K (9)

The objective function in Eq. 6 models the total assignment costs. Constraint set Eq. 7 denotes that each operator in Gi should be assigned to a position, and set Eq. 8 specifies that each position in bsshould be assigned an operator. Note that this model is similar to the LP model presented in Sect. 3 except in the following two ways: first, the decision variable is not binary; second, constraint set Eq. 4 has been removed.

5.2 Dm< S < D

When Dm< S < D, a dummy group of operators must be cre-ated. Table 7 presents an example, where S= 8, D = 10 and

Dm= 7. Note that the assignment cost for a dummy operator is M, and the other assignment costs are as discussed in Sect. 5.1.

Table 7. Assignment cost matrix for Dm< S < D

G1 G2 G3 GK Demand W10 b1 T 0 T M 3 W11 b2 T− R2 −R2 T− R2 M 1 W12 b3 T− R3 −R3 T− R3 M 1 W20 b4 0 T T M 2 W21 b5 −R5 T− R5 T− R5 M 1 W30 b6 T T T M 2 Supply 3 2 3 2 10

With the assignment cost matrix, a linear program model for

Dm< S < D can be formulated as follows: note that operator group K in the model denotes the dummy operator group.

Minimize TC= d
s=1 K
i=1 Esiysi (10) such that d
s=1 ysi= ni 1≤ i ≤ K (11) K
i=1 ysi= ms 1≤ s ≤ d (12) ysi∈ Z for 1≤ s ≤ d; 1 ≤ i ≤ K (13) ysK≤ ys+1,K for Y−1(s) = Y−1(s + 1) (14) This model is similar to the model presented in Sect. 5.1 ex-cept in that it includes constraint set Eq. 14, which is to ensure the appropriate assignments of dummy operators. With reference to Table 7, b2and b3are two newly added positions created in se-quence. Therefore, b2 cannot be assigned to a dummy operator while b3is assigned to a real operator.

The cases including S= D or D = Dmare special cases of the formulated models and can be easily solved.

6 Concluding remarks

This paper formulates the operator staffing problem faced by a foundry fab that aims to control tightly staffing costs, main-tains a no lay-off policy, and ensures high-quality operation prac-tices. A fab with these features cannot directly use previously described methods to staff work centers.

The staffing problem of such a fab is formulated as an LP model. The model, an extension of the model for solving the tra-ditional GAP, is distinct in that it models some new constraints pertaining to the problem of interest. Much research on the appli-cation of GAP has been published. Yet, none has addressed the application of GAP to the problem of staffing a semiconductor fab.

The sensitivity of the LP solutions to the demand variations, modeled by various simulation replicates, has been examined. The study reveals that the operator assignment solution obtained by the proposed method is insensitive to stochastic variations of the daily demand profile. That is, only one simulation replicate is required to solve the staffing problem with 90% confidence that the computed solution differs by no more than one operator from the best solution.

The proposed LP method does not consider the machine interference effect. A future study will attempt to justify the pro-posed solutions, using queuing models or simulation models to evaluate the machine interference. That is, production loss due to machine interference must be evaluated for a work center that includes bottleneck machines. Minor adjustment of the proposed solution can therefore be made.

Acknowledgement The Taiwan Semiconductor Manufacturing Co. is ac-knowledged for providing this research problem and its shop floor manage-ment practices.

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