Scheduling of unrelated parallel machines: an applicationto PWB manufacturing

LIAN YU1, HELOISA M. SHIH2, MICHELE PFUND3, W. MATTHEW CARLYLE3 and JOHN W. FOWLER3

1Department of Computer Science and Engineering, Arizona State University, PO Box 875406, Tempe, AZ 85287, USAE-mail: lianyu@asu.edu2Department of Industrial Engineering and Engineering Management, Hong Kong University of Science and Technology, Clear WaterBay, Kowloon, Hong Kong3Department of Industrial Engineering, Arizona State University, PO Box 875906, Tempe, AZ 85287, USAE-mail: michele.pfund@asu.edu or mcarlyle@asu.edu or john.fowler@asu.edu

Received August 1998 and accepted October 2001

In this paper, we tackle scheduling the bottleneck operation of a PrintedWiring Board (PWB) manufacturing line. The problemmaybe characterized as unrelated parallel machines and we are interested in several performance measures including: makespan, theaverage finish time, mean flow time, utilization, the number of lots and the total amount of overtime beyond the release interval. Dueto setup reduction techniques employed by the company, setups are negligible, and we construct an Integer Programming (IP) modelwith a special structure – unimodularity. Thus, the corresponding linear programming optimal solution always satisfies the integralityconstraints. In order to account for multiple objectives of the scheduling system, we introduce preference constraints and bring theminto the objective function by Lagrangian Relaxation (LR). We give a heuristic procedure to select constraints to be relaxed, andformulations to determine the Lagrange multipliers. To verify the validity of the proposed method, we compare it with a networkmodel and with a modified FIFOmethod. The proposed method presents very promising results in terms of our measures of interest.

1. Introduction

The trends of shorter life cycles of products and smallerlot sizes have led to the introduction of groups of Nu-merical Control (NC) machines or machine centers to oneor more (or even all) stages of production. The purposeof these machine centers is to achieve higher efficiency(Hitomi, 1992). To fully exploit these powerful and flex-ible machine centers, the integration of a productionmanagement system is necessary. Scheduling is one ofthe key functions for this type of system.The electronics industry is no exception to these trends.

This research is motivated by a Japanese electronicsmanufacturer’s need to improve the scheduling of itsPrinted Wiring Board (PWB) manufacturing line. Thetypical process of PWB manufacturing consists of mul-tiple stages of production, where each board must passthrough a pre-assigned sequence of manufacturing steps.This study addresses the scheduling of the bottleneckoperation in an actual PWB line: the drilling operation(Fig. 1). This operation is performed by a group of par-allel flexible machines, each of which has a differentprocessing speed and operating characteristics. Individuallots may be processed by all or a subset of the machines,

at varying speeds, with different suitability, i.e., attributesof lot match attributes of machine. These flexible ma-chines can switch from one lot to another with negligiblechangeover time. At present, a lot waiting for the drillingoperation is assigned by a human scheduler to one of themachines. About 30% of lots from the upstream opera-tion are sent to an external contractor for the drillingprocess, to overcome the bottleneck problem. However,management realizes that the average utilization of thedrilling machines is less than 70%, with very large vari-ants, i.e., some machines are kept extremely busy, whileothers are left idle. There is a company imperative toimprove the productivity and balance workload on thebank of unrelated machines. At the request of the com-pany, the time for any method to create a schedule for itssystem should be less than 1 minute to allow the manu-facturing schedule to respond to any unpredicted event inthe manufacturing process. Therefore, solving the prob-lem effectively, but quickly, is the objective of this paper.In the next section, we present a survey of the literature

relevant to this research. In Section 3 we describe thescheduling system under consideration. In Section 4 wepresent the mathematical formulation for the schedulingsystem. Section 5 illustrates the solution methodology of

0740-817X � 2002 ‘‘IIE’’

IIE Transactions (2002) 34, 921–931

the formulation with an example. In Section 6, we applythe proposed method to NC drilling scheduling of thecompany’s PWB production and compare it with theother two methods. Finally, in Section 7, we draw ourconclusions and present future research.

2. Literature review

There is a very long history of research on schedulingparallel machines (Chen and Sin, 1990; Pinedo, 1995). Iftij denotes the processing time of lot j on machine i, thethree different categories of parallel machine schedulingproblems (Piersma and Van Dijk, 1996) are, in order ofincreasing difficulty (Pinedo, 1995):

1. identical machines: tij ¼ t1j for all i and j;2. uniform machines: tij ¼ tj=vi for all i and j, where vi isthe speed of machine i, and tj is the processing time oflot j at unit speed;

3. unrelated machines: tij arbitrary for all i and j.

Many parallel machine scheduling problems are NP-hard problems (Karp, 1972). In terms of the complexityhierarchy of deterministic scheduling, unrelated machinesscheduling problems are some of the most difficult tosolve. In this research, the drilling of PWBs is performedby a group of unrelated parallel machines, where theprocessing time of each lot may be different for differentmachines, and a machine that has a shorter processingtime for a lot may have a longer processing time for an-other lot. Unrelated machine scheduling problems arisenot only in manufacturing environments but also in theaircraft maintenance process at an airport as reported byKolen and Kroon (1991), which they refer to as classscheduling. A class scheduling problem is relatively moredifficult than a standard parallel machine problem; and adiscussion of the NP-completeness of these problems canbe found in Kolen and Kroon (1991). Further discussionabout class scheduling may be found in Ozdamar andBozyel (1998) and Ozdamar and Sevket (1998).Much research effort has been directed to designing

fast and efficient approximation algorithms with goodtheoretical performance bounds (Lawler et al., 1993).

Some algorithms are based on simple list scheduling rules,like the earliest completion time heuristic of Ibarra andKim (1977). The methods with the best theoretical per-formance bounds are the LP-based heuristics by Potts(1985) and Lenstra et al. (1990). Hariri and Potts (1991)proposed several two-phase heuristics that proceed in thespirit of Potts’ 2-approximation algorithm (Potts, 1985).The first phase is almost the same: linear programming isused to schedule at least part of the jobs. The secondphase proceeds differently: a heuristic is used as a sub-stitute for complete enumeration to schedule the re-maining jobs. The heuristics of Hariri and Potts (1991)dominate other two-phase heuristics in terms of solutionquality but not in terms of speed. Piersma and Van Dijk(1996), as well as Suresh and Chaudhuri (1996) studiedlocal search heuristics for unrelated parallel machinescheduling. These local search methods are often quitegeneral and do not exploit the structure of the problem towhich they are applied.Taking makespan minimization as the objective, Pale-

kar et al. (1991) have proposed several lower boundingprocedures for the problem based on various relaxations.Van de Velde (1993) presents an optimization algorithmand an approximation algorithm, and tests instances with20 to 200 jobs, and two to 20 machines on a Compaq-386/20. He reports that the optimization algorithm obtainedresults in less than 1 minute. Martello et al. (1997) solveoptimally the same problem in less than 5 seconds on aVAX3100/30, which is about two times faster than theCompaq-386/20. They also solve, with an error less than2%, problems with 100 jobs on 15 machines in less than 30seconds on the VAX3100. Exact algorithms for noniden-tical processors for small-sized problems were also devel-oped by Horowitz and Sahni (1976). The computationalburden for their algorithms becomes very large for prob-lem instances with as few as five machines and 50 jobs.Many of the heuristic algorithms reviewed above are

effective for solving complicated scheduling problems.However there is no guarantee that optimal solutions, oreven good solutions, will be achieved. On the other hand,optimality oriented algorithms achieve high-quality so-lutions, but often in an unreasonable amount of time foreven medium-sized problems. Another problem with this

Fig. 1. The process of PWB manufacture.

922 Yu et al.

kind of approach is the rigid structure, which generallydoes not allow adaptations for special structure or man-agement objectives of a particular application.The research mentioned above addresses a single ob-

jective, while in many scheduling systems, as in the pro-blem tackled in this research, multiple criteria must beconsidered. In addition, in most practical situations thereare very strict limitations on the amount of time to get asolution. It is inevitable to trade-off between the solutionquality and time. To fully make use of the effectiveness ofheuristics and the efficiency of optimal algorithms, weconstruct a hybrid methodology. First, a heuristic pro-cedure is utilized to narrow or limit intelligently the so-lution space: then, an IP model is utilized to find anoptimal solution within the targeted space. In this re-search, due to setup reduction techniques employed bythe company, we can assume that setups are negligible.Therefore, we construct an Integer Programming (IP)model, with Lagrangian Relaxation (LR) to absorb somecomplicating constraints into the objective functionleaving a model which has a special structure – unimod-ularity. The corresponding linear programming optimalsolution always satisfies the integrality constraints.

3. Problem setting

In this section, we give a general description of the sched-uling system and in the next section we present a mathe-matical formulation for the problem under consideration.

3.1. Lots

Lots are conveyed from upstream operations in batches,with a regular arrival interval of T time units betweensubsequent releases of batches (herein called release in-terval). Depending on its attributes, such as lot size,layout of the board, minimum/maximum hole size, andother specifications, a lot has to be assigned to a machinethat matches its attributes. Each lot has Q attributes todefine specific processing requirements.The scheduling system is triggered every time a batch of

lots arrives (herein called a round), that is at intervals oflength T time units. If a lot will be finished at a timegreater than T (either because it has a long processingtime or has been assigned to a machine with a longqueue), its processing will be extended into the nextround. In order to guarantee production quality, each lotis processed by the same machine until its completion,i.e., no pre-emption is allowed.

3.2. Machines

There are m unrelated NC machines in parallel, respon-sible for the drilling operation. Each of the NC machines

has Q kinds of attributes corresponding to the Q kinds oflots attributes. The machines will be released only aftercompletion of the remaining lots assigned to them in theprevious scheduling rounds.

3.3. System performance measures

There are six system performance measures of interest tous:

1. to minimize the makespan;2. to minimize the average and standard deviation of thefinish time, i.e., the time for the last lot completed oneach machine per round;

3. to minimize the mean flow time;4. to balance the utilization of machines;5. to minimize the number of lots that need more than Ttime units of completion time per round; and

6. to minimize the total overtime per round, for all ma-chines, beyond T .

If these performance measures could be improved, then itmight be possible to reduce the amount of subcontractedwork of the company.

3.4. Assumptions

The formulation in the next section is subject to the fol-lowing assumptions:

1. for the convenience of management, splitting of lots isnot permitted;

2. since setup times are quite small (about 2 minutes)relative to the processing time (varying from 40 min-utes to more than 6 hours), and are quite uniform fordifferent machines, the setup time for all machine-lotpairs is negligible and not included in our model;

3. in a real workplace, it is inevitable that some unpre-dictable events happen, such as machine breakdownsor urgent order arrivals. In this model, neither of theseis taken into account.

4. Mathematical formulation

Our problem can be stated: for a given release interval, T ,assign a set of lots to unrelated parallel machines to op-timize the system performance measures listed in Section3 subject to scheduling feasibility constraints.The performance measures in Section 3.3, although

similar in nature, cannot always be simultaneously opti-mized. The approach used to address this problem is touse a surrogate objective function that ‘‘encourages’’ so-lutions that are good for all of these measures. Thissurrogate objective will start out as a sort of a weightedcompletion time measure. This initial surrogate does not

Scheduling of unrelated parallel machines 923

do a very good job in terms of some of the original per-formance measures. However, if constraints are added todiscourage ‘‘bad’’ solutions, and then these constraintsare dualized through the addition of Lagrangian terms tocreate a modified surrogate objective, the resulting opti-mal solutions perform much better for all six perfor-mance measures of interest.The following is the notation used in the formulas.

xijk = processing lot j on machine i in the kth positionwhere

xijk ¼1 if let j is scheduled as the kth to last lot

on machine i,0 otherwise;

8<:

aij = suitability for machine i to process lot j (see Sec-tion 4.1.1);

k = position corresponding to kth to last lot processing;tij = processing time of lot j on machine i;

cijk = cost of processing lot j on machine i in the kthposition and defined as cijk ¼ aijktij;

n = number of lots to schedule;m = number of parallel machines;K = a superset, each of its elements containing three

pieces of information: (i) lot b that is assigned tomachine a; (ii) machine a that is assigned lot b;and (iii) position k1th to last lot for lot b to be inthe waiting queue on machine a;

P = a superset, each of its elements containing twopieces of information: (i) machine c on which thetotal processing time assigned is longer than Ttime units; and (ii) position k2th to last lot that isthe first available position of machine c.

4.1. Objective function

The objective function is to minimize the total cost ofactivities that n lots are processed on m unrelated parallelmachines.

MinimizeXmi¼1

Xnj¼1

Xnk¼1

cijkxijk ¼Xmi¼1

Xnj¼1

aijXnk¼1

ktijxijk: ð1Þ

The objective function is equivalent to the sum ofweighted completion times. This surrogate objective at-tempts to represent many objectives. We will add ap-propriate constraints to improve the decisions for theperformance measures of our interest.The parameter aij in the objective function is discussed

below.

4.1.1. Suitability aijaij is the aggregate suitability of lot j processed on ma-chine i related to the Q kinds of attributes, and defined as:

aij ¼YQq¼1

aijðqÞ: ð2Þ

The aijðqÞ, suitability of lot j processed on machine i interms of attribute q is defined as:

aijðqÞ ¼r1 � aijðqÞ � r2 if machine i is able to

process lot j, r1 � 0, r2 <1,1 otherwise.

(

The values of r1 and r2 may be determined either byconvention or in a statistical manner. In this application,we were provided the values of r1 and r2 which were de-termined conventionally by the company. To run the LPprogram, instead of using 1 in the formulation, we uti-lized a large finite number, say 1000 times of rQ2 .

4.2. Constraints

Two kinds of constraints are included in the model:scheduling feasibility constraints and preference con-straints. Scheduling feasibility constraints must be satis-fied. Preference constraints help improve the performancemeasures of interest.

4.2.1. Scheduling feasibility constraintsXnj¼1

xijk � 1 ði ¼ 1; 2; . . . ;m; k ¼ 1; 2; . . . ; nÞ; ð3Þ

Xmi¼1

Xnk¼1

xijk ¼ 1 ðj ¼ 1; 2; . . . ; nÞ; ð4Þ

xijk ¼ 0 or 1 ð8 i; j; kÞ: ð5ÞConstraint (3) ensures that each position on each machineis taken by at most one lot. Constraint (4) ensures thateach lot is scheduled exactly once. Constraint (5) simplyensures that xijk has zero–one integer values.

4.2.2. Preference constraints

To improve the performance measures (other than themean flow time) listed in Section 3.3, the following sets ofconstraints are added to the model:

xabk1 � 1 fabk1g 2 K; ð6ÞXnj¼1

Xnk¼k2

xcjk � 0 fck2g 2 P: ð7Þ

Constraint (6) forces lot b, which has a Long ProcessingTime on all Machines (LPTM) compared to the releaseinterval T to be assigned to a more favorable machine a atposition k1. This helps reduce the makespan. Constraint(7) prevents other lots from being further assigned to themachines which have been occupied by lots with LPTM,or having a total processing time longer than T . Thishelps reduce finish times of lots, balance utilization ofmachines, reduce the number of lots processed beyond Tand the total processing times longer than T .The reader can refer to a formulation for unrelated

parallel machines in a textbook by Pinedo (1995), where

924 Yu et al.

the objective function is the sum of completion times,without preference constraints (6) and (7).

5. Solution methodology

In this research, we propose a Lagrangian RelaxationHeuristic (LRH) method to solve our problem. There aretwo key issues highlighted about the use of LagrangianRelaxation:

• strategic issue: choose to relax the set of constraints;• tactical issue: find numerical values for themultipliers.

The two issues are not usually easy to deal with.However, in our problem they are pretty straightforward.The problem without constraints (6) and (7) is an integerprogram with a special structure – unimodularity. It isknown from the theory of networks that the integralityconstraints on the xijk may be replaced by non-negativityconstraints without changing the feasible set (Schrijver,1986; Murty, 1992). Thus by using Lagrangian Relax-ation, we bring into the objective function constraints (6)and (7) in Section 5.1. We end up with an IP, whosecorresponding linear programming optimal solution al-ways satisfies the integrality constraints.

5.1. Lagrangian Relaxation formulation

Let k > 0 and p > 0 be the Lagrange multipliers whichrelax constraints (6) and (7). The relaxed problem isformulated as follows:

MinimizeXmi¼1

Xnj¼1

Xnk¼1

cijkxijk þX

fabk1g2K

kfabk1gð1� xfabk1gÞ

þX

fck2g2P

pfck2gXnj¼1

Xnk¼k2

xcjk; ð8Þ

subject toXnj¼1

xijk � 1 ði¼ 1;2; . . . ;m; k ¼ 1;2; . . . ;nÞ; ð9Þ

Xmi¼1

Xnk¼1

xijk ¼ 1 ðj¼ 1;2; . . . ;nÞ; ð10Þ

0� xijk � 1 ð8 i; j;kÞ: ð11ÞAs we relax our preference constraints, we will alwaysobtain a feasible scheduling solution, which is guaranteedby constraints (9)–(11). To further determine the twosupersets K and P in the objective function above, weprovide the following heuristic procedure.

5.1.1. Heuristic procedure

In the heuristic procedure, lots that can use just a fewmachines and have long processing times on all machines

(�T ) will be assigned first, then lots that can use just a fewmachines, and finally lots that have long processing times(�T ).The notation used in the heuristic procedure is as fol-

lows:

M = set of all unrelated parallel machines;ML = a matrix which contains all machines in

consideration in column 1, and column 2(3)contains the processing time of the last(next to last) lot that is to be processed onthe corresponding machine, and so on;

B = set of lots that are selected;L = set of all lots to be scheduled at the begin-

ning of a round;Oi = overtime beyond T on machine i in the

previous round;TAi = total time assigned to machine i;

MT ða; bÞ = total processing time after selecting lot b onmachine a.

Step 1. (Initial conditions)Set K ¼ ;, P ¼ ;, B ¼ ;.

Step 2. (Assign lots which can only be processed by afew machines)Scan L to find out lots which only can processedby a few machines (say two or fewer out of 14machines), and put them into set B. Set L ¼L� B.2.1 If there is a lot b 2 B whose minimum pro-cessing time is larger than T (ties are broken ar-bitrarily), assign to machine a that lot such that:

MT ða; bÞ ¼ mini2M

ðOi þ TAi þ tibÞ:

Put lot b before the previous lot, if any, on ma-chine a. Set a, b and the position k1 to superset K.Update set ML. B ¼ B� fbg, if B ¼ ; goto Step3, otherwise repeat 2.1.2.2 If there is a lot b 2 B with minimum pro-cessing time (ties are broken arbitrarily), assignto machine a that lot such that:

MT ða; bÞ ¼ mini2M

ðOi þ TAi þ tibÞ:

Put lot b before the previous lot, if any, onmachine a. Set a, b and the position k1 to supersetK.Update setML.B ¼ B� fbg, repeat2.2 tillB ¼ ;.

Step 3. (Assign lots whose processing times are largerthan T on all machines)Scan L to find out lots whose processing timesare larger than T on all machines and put theminto set B, set L ¼ L� B.3.1 Find a lot b 2 B (ties are broken arbitrarily)such that:

MT ða; bÞ ¼ mini2M ;j2B

ðOi þ TAi þ tijÞ:

Scheduling of unrelated parallel machines 925

Put lot b before the previous lot, if any onmachinea. Set a, b and the position k1 to superset K. Up-date setML. B ¼ B� fbg, repeat 3.1 till B ¼ ;.

Step 4. (Check the machines whose assigned total pro-cessing times are larger than T )Scan ML to check the machines whose assignedtotal processing times are larger than T . Put thenumber of machines and the position k2 beforethe previous lot on that machine to superset P.

Step 5. (Check the feasibility)Scan L. If there is any lot which can only beprocessed by one of machines c 2 P, remove themachine from superset P.

Now that we have determined the two supersets K and P,we know exactly which set of constraints to relax. In thefollowing section, we determine the values of the Lag-range multipliers.

5.2. Determination of Lagrange multipliers

There are two well-known approaches to deciding valuesof Lagrange multipliers: subgradient optimization, andmultiplier adjustment (Beasley, 1995). However, it is notthat complicated in our problem. If we give Lagrangemultipliers large enough numerical values, the LP willencourage xfabk1g ¼ 1 ðfabk1g 2 KÞ, and xifck2g ¼ 0 ðfck2g2 PÞ. However, there is a bound over the numericalvalues, i.e., they can not be bigger than M1

ij , a largenumber in the formulation of LP with Lagrangian Re-laxation, which indicates that lot j can not be processedby machine i. Let amax ¼ maxi;j aij (for feasible process-ing), and tmax ¼ maxi;j tij. Let M1 ¼ M1

ij (for all i, j).Then M1 can be valued as:

M1 ¼x � dn� amax � tmaxe; ð12Þwhere x is a value ranging from 100 to 1000, and dze isceiling of real number z, i.e., it is the smallest integer � z.Let k ¼ kfabk1g (for all fabk1g), p ¼ pifck2g (for all

ifck2g). Then the values of Lagrange multipliers of k, pare defined as follows:

k ¼50%M1; ð13Þp ¼80%k: ð14Þ

5.3. Example

In this section, we expound the insight of the LRHmethod by a small example. Assume there are six lots

processed on three unrelated parallel machines (Table 1)in a round. Let release interval T be equal to 3 time units,otiði ¼ 1,2,3Þ be overtimes of previous round on the threemachines. Without loss of generality, assume that

Suitability aij of

Lot j on machine i ¼

1 if machine i is able

to process Lot j,

M1 (value to be determined)

otherwise.

8><>:

Machine M1 is somewhat inefficient; it has less flexibilityand longer processing times than the other machines. LotL1 needs some special processing and only can assignedto Machine M3. Lot L5 can be worked on two machines(M2, M3) with long processing times on all machines(� T ). Let ot1 ¼ ot2 ¼ 0, ot3 ¼ 0:5.Applying the heuristic procedure, Lot L1 is assigned to

Machine M3 (the last lot to be processed) at positionk1 ¼ 1, Lot L5 to Machine M2 at position k1 ¼ 1,therefore K ¼ fð3,1,1Þ, ð2,5,1Þg. Since Machine M2 hasmore than T units of work to do, lots are not expected tobe assigned on it further, i.e., k2 ¼ 2 thus P ¼ fð2; 2Þg.According to Equation (12), we set x ¼ 100, so M1 ¼2000. According to Equations (13) and (14), k ¼ 50%M1 ¼ 1000, p ¼ 80%k ¼ 800. The formulation to thisinstance is as follows:

MinimizeX3i¼1

X6j¼1

aijX6k¼1

ktijxijk ð15Þ

þ 1000ð1� x311Þ þ 1000ð1� x251Þ ð16Þ

þ 800X6j¼1

X6k¼2

x2jk; ð17Þ

subject to

X6j¼1

xijk � 1 ði ¼ 1; 2; 3; k ¼ 1; 2; . . . ; 6Þ; ð18Þ

X3i¼1

X6k¼1

xijk ¼ 1 ðj ¼ 1; 2; . . . ; 6Þ; ð19Þ

0 � xijk � 1 ð8 i; j; kÞ: ð20Þ

Figure 2(a) shows the results of the above small ex-ample using LRH method; Fig. 2(b) is the results usinga simple dispatching rule, where lots are ranked in orderthat they arrive, and then assigned to the machines with

Table 1. The processing time on the machines

Machine Lot L1 Lot L2 Lot L3 Lot L4 Lot L5 Lot L6

M1 1 1.7 2.4 1.3 1 0.5M2 1 0.5 2.2 0.8 3.2 0.4M3 1 0.7 2.2 0.8 3.1 0.4

926 Yu et al.

shortest processing time. Table 2 lists the six perfor-mance measures used to evaluate the two methods.The LRH method outperforms the simple heuristicmethod for all measures bar the average finish time. Itis observed that although the simple heuristic methodachieves a lower average finish time, it has a largerstandard deviation among the three machines, i.e.,Machines M2, and M3 have much work to do, whilstMachine(s) M1 is totally idle. The table clearly illus-trates the myopic nature of the dispatching rule and theadvantages of the global optimization of the LRHmethod.

6. Applications

In this section, we apply the LRH method to schedule thedrilling operation of a PWB Manufacturing line in aJapanese electronic company, which provided us with allnecessary data. One hundred and sixty rounds of simu-lation are conducted, each round corresponding to aPWB production release. The plant operates 24 hours aday with a release interval of 3 hours, having thereforeeight releases for the drilling operation per day. As theplant operates 5 days per week, the 160 rounds of simu-lation correspond to roughly 1 month of operation. Each

Fig. 2. (a) The schedule of the six lots on Machines M1–M3 using the LRH method; and (b) the schedule of the six lots onMachines M1–M3 using FIFO with shortest processing time.

Table 2. A comparison of the LRH with a simple dispatching rule

Performance measurement LRH Simple dispatch rule

1. Makespan 3.2 4.62. Average finish time (deviation among machines) 3.03 (0.15) 2.87 (2.51)3. Mean flow time 1.9 2.684. Utilization (deviation among machines) 0.99 (0.018) 0.67 (0.58)5. Number of lots of overtime 1 36. Total overtime 0.2 2.5

Scheduling of unrelated parallel machines 927

round simulates the real operation and carries forwardthe remaining unfinished lots from the previous round.

6.1. Configuration

There are 14 (m ¼ 14) unrelated machines in parallel forthe drilling operation. Each of them has six kinds of at-tributes (Q ¼ 6), such as number of boards in a lot, layoutof the board, minimum/maximum size of the holes and soon. There are 90 types of lots, divided into 13 groups dueto similarities in attributes (Table 3). Lots have also six(Q) kinds of attributes correspondingly. aijðqÞ, suitabilityof lot j processed on machine i relative to attribute q isgiven by the plant, ranging from one to 10. The smallerthe value is, the more suitable it is for machine i to pro-cess lot j relative to attribute q. A release of lots is con-veyed via an AGV from the upstream operation every 3hours (i.e., T ¼ 3). The production mix and the numberof lots (n) vary from release to release. The productionmix of every release is randomized according to the his-toric percentage of the 13 groups of lots. The number oflots in a production release is modeled as a normal dis-tribution (l ¼ 30, r ¼ 1:5 rounded up to the next highestinteger). The scheduling programs were developed in theC language and run on a Sun Sparc computer.

6.2. Solution procedure

Figure 3 is the flowchart of the solution procedure of thisapplication.

Step 1. Based on simulation data provided by the plant,apply Equation (2) to calculate aijði ¼ 1; . . . ;14; j ¼ 1; . . . ; 30� rÞ; calculate cijk ¼ aijktij.When lot j cannot be processed on machine i,cijk ¼ M1, which is 4000 according to Equation(12).

Step 2. Determine values of the Lagrange multipliers k,p, per Equations (13) and (14).

Step 3. Invoke heuristic procedure to determine the twosupersets K, P.

Step 4. Run the LP, Equations (8)–(11).Step 5. Calculate overtimes of current round on each

machine, modify processing times of lots, whichmay use the machines with overtime processing.Collect statistical results for each performancemeasure. Goto Step 3 until 160 rounds have beencreated.

Step 6. Terminate the simulation.

6.3. Results and discussion

To determine the effectiveness of the proposed method-ology, it was compared to two other solution methodol-ogies. One is the model consisting of Equations (1)–(5),which we refer to as the network model. This model hasno preference constraints of Equations (6) and (7) and

Table 3. The 90 types of lots divided into 13 groups accordingto their similarities

Group Number of types(Percentage)

Alternative machines(Favored machines)

1 30 (33.33%) 1�4, 9�14: (1, 2, 3, 4)2 13 (14.44%) 1�14, 1�13: (1, 2, 3, 4)3 10 (11.11%) 3, 4, 9�14: (3, 4, 9, 10, 11)4 9 (10%) 7, 8, 12, 13, 14: (7, 8)5 6 (6.67%) 5�8, 12�14: (5, 6, 7, 8)6 5 (5.56%) 1�4, 9�13: (3, 4)7 5 (5.56%) 3, 4, 7�12: (3, 4)8 3 (3.33%) 3, 4, 9�139 2 (2.22%) 3, 4, 9�12, 14: (3, 4)10 2 (2.22%) 5�8, 12, 13: (5, 6, 7, 8)11 2 (2.22%) 3, 4, 9�12: (3, 4)12 2 (2.22%) 3, 4, 7�12, 14: (3, 4)13 1 (1%) 3�13: (3, 4)

Fig. 3. The solution procedure.

928 Yu et al.

does not use Lagrangian Relaxation. The model is run onthe same machine as the LRH method. Another is amodified FIFO method that is often used in practicalsituations and is similar to the current method at thecompany. This method assigns the lot at the front of aFIFO ranked queue to the first available machine whichis suitable to process it; and when a machine becomesavailable, it chooses from the queue the first lot that it canprocess (regardless of the processing time). The modifiedFIFO method is programmed with SLAM II and run ona Sun Sparc-20. The run times of the LRH method andthe network model for each round are less than 5 seconds,which certainly satisfies the company’s requirement ofless than 1 minute, and the modified FIFO method needsless than 1 second per round.Table 4 shows results of the six performance measures

for each of the three methods: LRH method, networkmodel and modified FIFO. From Table 4, it is observedthat the LRH method performed better than the networkmodel for all performance measures but utilizationswhich are almost similar. The LRH and the networkmodel outperform the modified FIFO method for all butthe number of lots of overtime. Specific comparisonsbetween the LRH method and the network model and themodified method are discussed below.Compared to the network model, the LRH method

reduces makespan by 0.41 hours every scheduling roundof 3 hours. This is due to the fact that the LRH methodavoids assigning to an already busy machine a lot withlong processing time, i.e., choosing the ‘‘favorite’’ ma-chine for that lot, when the last finished time on thatmachine is already over 3 hours. The network model re-alizes the optimal schedule that does not necessarily haveto be nondelay, i.e., even if a machine is idle a lot may beassigned to another very busy but more suitable machine.The total overtime of all machines is decreased by 1.04

hours per round. Average finishing time and mean flowtime are also reduced, while utilization is almost equiva-lent for the two methods. Therefore the LRH methodtries to schedule the lots that favor the already busymachines to other available ones, in an optimal wayrather than to blindly move.The LRH method prevents the assignment of further

lots to a machine that has been assigned lot(s) with longprocess time(s), when there is(are) other available ma-

chine(s) to assign that(those) lot(s) to. This leads to adecrease in the number of lots that need processing be-yond 3 hours. On the average, the total number of thoselots is decreased by 0.58 lots per round. The LRHmethoddirects some lots that would favor machines with rela-tively short processing times to other machines with rel-atively long processing times if technically possible,aiming at shrinking makespan.Even though the LRH method increases the objective

function, notice that it improves our performance mea-sures of interest. This illustrates the utility of using theLRH formulation to model/account for ‘‘more compli-cated’’ objectives.Compared to the modified FIFO method, it is noted

that the modified FIFO method has a smaller number ofovertime lots per round, but has a much bigger totalovertime. This means that the modified FIFO methoddumps fewer lots on fewer machines and occupies themfor a very long time, while the LRH method spreads lotsevenly on suitable machines and avoids delays that arelong relative to the release interval.Using the current human scheduling procedure in the

drilling operation, each release has about 20 lots drilledin-house, while about 10 other lots are drilled by ex-ternal contractors. By increasing the release size from 20to 30 lots, the plant expects to produce the 30% ofproduction by themselves rather than to send that toexternal contractors. The simulation results show thatthe average finish time of last lot on each machine bythe LRH method is about 2.8 hours with a standarddeviation 1.07, while the modified FIFO method gets 3.4hours with standard deviation 1.67. That is the LRHmethod has a higher probability to process 30 lotswithin the 3 hour-release interval. In addition, due toits intelligent assignment, the utilization of the LRHmethod is lower than the simple heuristic method. Thus,there is considerable potential for the LRH method toincrease throughput further than the modified FIFOmethod. Therefore, it is possible through the schedulingmethod proposed in this research to increase the releasesize to 30 lots, and as a result, to accommodate the lotsthat are currently sent to external contractors for drill-ing process without the addition of new machines. Theplant is preparing to implement the proposed system ontheir production line.

Table 4. A comparison of the results obtained after 160 rounds

Performance measure LRH Network model Modified FIFO

1. Makespan 5.03 5.44 6.662. Average and standard deviation of finish time 2:82� 1:07 2:90� 1:17 3:36� 1:673. Mean flow time 2.37 2.52 4.734. Utilization and standard deviation 0:81� 0:08 0:81� 0:07 0:86� 0:105. Number of lots of overtime per round 7.78 8.36 6.826. Total overtime per round 5.54 6.58 10.9

Scheduling of unrelated parallel machines 929

7. Conclusions

In this research, we address the scheduling of the bottle-neck (drilling) operation of a PWB line. The drilling linehas unrelated parallel machines, and we are interestedin multiple criteria. Due to setup reduction techniques,setups are negligible in the system. Therefore, we con-structed an IP model with a special structure – unimod-ularity which ensured that the corresponding linearprogramming optimal solution always satisfies the in-tegrality constraints. However, the objective of the IPmodel – sum of weighted completion time, is in mostsituations not consistent with other objectives of thescheduling system, like makespan and balanced utiliza-tion of machines. To alleviate this problem, we proposedthe introduction of a Lagrangian Relaxation Heuristicthat helps the realization of other objectives by absorbingappropriate constraints into the objective function. Anumber of simulations were conducted. Compared withthe network model, the LRH method provides shortermakespans, smaller mean flow times, and a lower numberof overtime lots and total overtime. The results of theLRH method showed obvious advantages over themodified FIFO method, which is similar to the humanscheduling procedure. By introducing the LRH method,the plant expects to improve productivity and to balancetheir workload. The computational efficiency is well-within the plant’s expectation.We observed that although the LRH method always

guarantees feasible solutions for each lot and each ma-chine by scheduling feasibility constraints (Equations(9)–(11)), the performance of the scheduling method isaffected by the decision of the production mix. For ex-ample, if releasing many similar lots from upstream,which can only be processed on fewer machines with along processing time on all the possible machines, itwould keep these machines extremely busy, while leavingothers idle. Also it would result in longer makespan,larger tardiness, and worse balance of utilization. Re-leasing these kinds of lots in a proper proportion in eachrelease is a very important issue to keep a smooth andhigh-performance operation of the scheduling system. Weare continuing the research on integration of lot sizingand scheduling on unrelated parallel machines.

Acknowledgments

The authors would like to thank the two anonymousreferees for their helpful comments and suggestions andDr. Takahashi Sekiguchi and Mr. Go Yoshikawa fortheir contribution to the project. John W. Fowler ispartially supported by NSF DMI-9713750 and SRC-97-FJ-492.

References

Beasley, J.E. (1995) Lagrangian relaxation, in Modern HeuristicTechniques for Combinatorial Problems, Reeves, C.R. (ed.), Mc-Graw-Hill, London.

Chen, T.C.E. and Sin, C.C.S. (1990) A state-of-the-art review of par-allel-machine scheduling research. European Journal of Opera-tional Research, 47, 271–292.

Hariri, A.M.A. and Potts, C.N. (1991) Heuristics for scheduling un-related parallel machines. Computers and Operations Research,18(3), 323–331.

Hitomi, K. (1992) Manufacturing Systems Engineering, (in Japanese),Kyouri, Tokyo, Japan.

Horowitz, E. and Sahni, S. (1976) Exact and approximate algorithmsfor scheduling nonidentical processors. Journal of the Associationfor Computing Machinery, 24, 317–327.

Ibarra, O.H. and Kim, C.E. (1977) Heuristic algorithms for schedulingindependent tasks on nonidentical processors. Journal of the As-sociation for Computing Machinery, 24, 280–289.

Karp, R.M. (1972) Reducibility among combinatorial problems,in Complexity of Computer Computations, Miller, R.E.and Thatcher, J.W. (eds.), Plenum Press, New York, pp. 85–103.

Kolen, A.W.J. and Kroon, L.G. (1991) On the computational com-plexity of (maximum) class scheduling. European Journal of Op-erational Research, 54, 23–38.

Lawler, E.G., Lenstra, J.K., Rinnooy Kan, A.H.G. and Shmoys, D.B.(1993) Sequencing and scheduling: algorithms and complexity,in Handbooks in OR & MS, Vol. 4, Graves, S.C., RinnooyKan, A.H.G. and Zipkin, P.H. (eds.), Elsevier, Amsterdam,Ch. 9.

Lenstra, J.K., Shmoys, D.B. and Tardos, E. (1990) Approximationalgorithms for scheduling unrelated parallel machines. Mathe-matical Programming, 46(9), 259–271.

Martello, S., Soumis, F. and Toth, P. (1997) Exact and approximationalgorithms for makespan minimization on unrelated parallelmachines. Discrete Applied Mathematics, 75, 169–188.

Murty, K.G. (1992) Network Programming, Prentice Hall, EnglewoodCliffs, NJ.

Ozdamar, L. and Bozyel, M.A. (1998) Simultaneous lot sizing andloading of product families on parallel facilities of differentclasses. International Journal of Production Research, 36(5), 1305–1324.

Ozdamar, L. and Sevket, _II.B. (1998) Hybrid heuristics for the capaci-tated lot sizing and loading problem with setup times an overtimedecisions. European Journal of Operational Research, 110(5), 525–547.

Palekar, U.S., Rama, N. and Toaffe, K. (1991) Duality based relax-ations for makespan minimization for unrelated parallel ma-chines. TIMS/ORSA Bulletin, 31(MC2.2), 21.

Piersma, N. and Van Dijk, W. (1996) A local search heuristics forunrelated parallel machine scheduling with efficient neighborhoodsearch. Mathematical Computer Modelling, 24(9), 11–19.

Pinedo, M. (1995) Scheduling Theory, Algorithms, and Systems, Pren-tice Hall, Englewood Cliffs, NJ.

Potts, C.N (1985) Analysis of a linear programming heuristic forscheduling unrelated parallel machines. Discrete Applied Mathe-matics, 10, 155–164.

Schrijver, A. (1986) Theory of Linear and Integer Programming, Wiley,Chichester, UK.

Suresh, V. and Chaudhuri, D. (1996) Bicriteria scheduling problem forunrelated parallel machines. Computers and Industrial Engineer-ing, 30(1), 77–82.

Van de Velde, S.L. (1993) Duality based algorithms for schedulingunrelated parallel machines. ORSA Journal on Computing, 5, 192–205.

930 Yu et al.

Biographies

Lian Yu is affiliated with the Industrial Engineering and ComputerScience & Engineering Departments at Arizona State University. Pre-viously, she was on the faculty of the Department of Management andInformation Systems Engineering of Nagaoka University of Technol-ogy since 1999. She received her Ph.D. in Electrical and ComputerEngineering from Yokohama National University in 1999. Her mainareas of research are production planning and scheduling, fuzzymodeling in production management, and software engineering.

Heloisa Martins Shih received her Doctor of Engineering degree fromYokohama National University (Department of Electrical and Com-puter Engineering) in 1991. She was an Assistant Professor at theDepartment of Industrial Engineering of Universidade de Sao Paulofrom 1983 to 1993. In September 1993 she joined the faculty of theDepartment of Industrial Engineering and Engineering Managementof Hong Kong University of Science and Technology. Her main re-search areas are applications of Petri nets, operations research andfuzzy inference to production management problems.

Michele Pfund is a Ph.D. student in the Department of IndustrialEngineering at Arizona State University, specializing in the area ofoperations research. She received her masters degree from PurdueUniversity and her undergraduate degree from Case Western ReserveUniversity. She serves as the editor for the INFORMS student journalOR/MS Tomorrow.

W. Matthew Carlyle is an Assistant Professor of Industrial Engi-neering at Arizona State University. He received his Ph.D. in Op-

erations Research from Stanford University in 1997 and his B.S. inInformation and Computer Science from Georgia Tech in 1992. Hisresearch is focused on effective models and solution procedures forcombinatorial optimization problems in manufacturing and produc-tion systems. Applications of this research have included large-scalemodeling and analysis of printed circuit-card assembly systems, un-derground mining operations, and semiconductor manufacturingoperations. Dr. Carlyle is co-director of the Operations Researchand Production Systems Laboratory, and an associate of the Mod-eling and Analysis of Semiconductor Manufacturing Laboratory, atASU.

John W. Fowler is an Associate Professor in the Industrial EngineeringDepartment at Arizona State University. Prior to his current position,he was a Senior Member of Technical Staff in the Modeling, CAD, andStatistical Methods Division of SEMATECH. He received his Ph.D. inIndustrial Engineering from Texas A&M University and spent the last1.5 years of his doctoral studies as an intern at Advanced Micro De-vices. His research interests include modeling, analysis, and control ofsemiconductor manufacturing systems. Dr. Fowler is the co-director ofthe Modeling and Analysis of Semiconductor Manufacturing Labo-ratory at ASU. The lab has had research contracts with NSF, SRC,SEMATECH, Infineon Technologies, Intel, Motorola, ST Microelec-tronics, and Tefen, Ltd. He is also an Associate Editor of IEEETransactions on Electronics Packaging Manufacturing and on the Edi-torial Board for IIE Transactions on Scheduling and Logistics. He is amember of ASEE, IIE, IEEE, INFORMS, POMS, and SCS.

Contributed by the Scheduling Department

Scheduling of unrelated parallel machines 931