Impact of permutation enforcement when minimizing total weighted tardiness in dynamic flowshops with...

Computers & Operations Research 34 (2007) 3055–3068www.elsevier.com/locate/cor

Impact of permutation enforcement when minimizing total weightedtardiness in dynamic flowshops with uncertain processing times

Rajesh Swaminathana,∗, Michele E. Pfundb, John W. Fowlerb,Scott J. Masona, Ahmet Kehab

aDepartment of Industrial Engineering, University of Arkansas, 4207 Bell Engineering Center, Fayetteville, AR 72701, USAbDepartment of Industrial Engineering, Arizona State University, P.O. Box 875906, Tempe, AZ 85287-5906, USA

Available online 10 January 2006

Abstract

This paper is motivated by the problem of meeting due dates in a flowshop production environment with jobs with differentweights and uncertain processing times. Enforcement of a permutation schedule to varying degrees for dynamic flowshops isinvestigated with the goal of minimizing total weighted tardiness (TWT). The approaches studied are categorized as follows:(1) pure permutation scheduling (2) shift-based scheduling (3) pure dispatching (which leads to non-permutation sequences). Asimulation-based experimental study was carried out to study the performance of the above methods with respect to minimizingTWT when new jobs arrive to the flowshop at every shift change. Results indicate significant gains in performance are possible whenthe permutation requirement is relaxed and shift-based scheduling is allowed. Shift-based scheduling yields competitive resultswith respect to the pure dispatching approach, even though dispatching has the advantage of a full relaxation of the permutationrequirement.� 2005 Published by Elsevier Ltd.

Keywords: Flowshop; Heuristic; Permutation; Manufacturing

1. Introduction

Often in manufacturing settings the jobs follow a fixed path and utilize specific resources in a certain order. Suchsettings are known as flowshops. The simplest examples typically have n stages with single machines at each stage(Fig. 1).

Scheduling is defined rather broadly as planning the utilization of various resources for periods of time at variousstages. Practical planning at the shop floor level requires finite capacity scheduling in contrast to infinite capacityscheduling which may be carried out at the company or aggregate level. It is common for manufacturers to either maketo order (MTO) or make to stock (MTS). At firms, which MTO, fulfilling customer orders is a priority and short-termproduction schedules are constructed accordingly. The due dates are generally firm for each customer order and theprimary goal is to manage resources efficiently to meet order due dates. In a MTS environment, order due dates areinternally specified, as they are based on current levels of finished goods. In this paper, total product processing timesare longer than short-term production planning periods. In such scenarios with long total processing times (such as

∗ Corresponding author.

0305-0548/$ - see front matter � 2005 Published by Elsevier Ltd.doi:10.1016/j.cor.2005.11.014

http://www.elsevier.com/locate/cor

3056 R. Swaminathan et al. / Computers & Operations Research 34 (2007) 3055–3068

M/c 1 M/c 2 M/c 3 M/c (n-1) M/c n

Fig. 1. Flowshop with single machines at each stage.

semiconductor manufacturing), technological constraints make the sequence in which jobs are performed pertinent [1].On the other hand, short cycle time environments pose a capacity balancing challenge.

Often jobs experience variance in their estimated processing times at the shop floor level. Seldom shop floor conditionsdo allow for completely smooth, controlled work flow and it is often difficult to identify and manage bottlenecks.Therefore, it becomes important to reduce the effect of uncertainty and use relevant performance metrics. Whileincreased capacity could help absorb the effects of variability, one may not have the luxury of that option. Clearly, ina finite capacity setting, job sequencing is important for realizing reliable, on-time delivery of the final product. Thispaper focuses on a flowshop environment where the machines are arranged in series.

Scheduling problems can be described by a triplet �|�|� [2]. The � field describes the machine environment, while the� field describes processing characteristics, restrictions, and constraints (such as release dates, batch, setup-dependentoperations). Finally, the � field represents the performance measure being considered. The flowshop problem with theobjective of minimizing total weighted tardiness (TWT) is denoted as Fm‖∑wiTj .

In this paper, new jobs arrive to the flowshop over time, thus the problem was studied over a rolling horizon. Theproducer is looking to maximize on-time delivery performance in an MTO environment with the jobs having differentlevels of importance; hence our performance measure of interest is TWT. Perez et al. [3] describe TWT as a measurethat incurs a penalty for each job that finishes processing after its promised delivery date. This penalty increases withthe magnitude of the tardiness, and therefore schedules that minimize the weighted (by job priority) sum of penaltiesprovide good on-time delivery performance, whereas higher than necessary values of TWT indicate that many importantjobs are not being delivered on time. TWT is the summation of the weighted tardiness over all jobs j = 1, 2, 3, . . . , n.It is denoted as wjTj , where Tj = max(0, Cj − dj ) and wj is the weight (priority) of job j . Further, Cj and dj referto the completion time and the due date of job j , respectively.

Permutation schedules in flowshops have identical job sequences on all machines. Permutation schedules are oftenappropriate in real world situations, such as factories with conveyors between machines for material transfer andassembly lines performing the final assembly of bulky products [4]. Permutation schedules allow for a planned,smooth, continuous flow of jobs. When there are more than two machines good non-permutation schedules are likelyto outperform their permutation counterparts. For the case of two machines with the makespan objective there exists apermutation schedule that is optimal [5]. The Fm‖∑wjTj scheduling problem is strongly NP-hard as it is a reductionfrom the 1‖∑wjTj problem, which is known to be strongly NP-hard [2]. The number of feasible non-permutationschedules where every job visits every machine is (n!)m. However, in the permutation case the search space is reducedto n! The problem of identifying good permutation schedules is itself difficult. When permutation scheduling is ineffect, “job passing” or job overtaking is not allowed.

The following are problem characteristics and assumptions in this paper.

• The flowshop has m stages with a single machine at each stage.• All jobs require processing on each machine.• The sequence of machine visits for each job is identical and unidirectional.• Some number of jobs (sampled from a uniform distribution) are released every shift. Machine breakdowns do not

occur. Jobs are released only at the beginning of every shift. Once a job begins its processing at a station it cannotbe preempted.

• Job processing times vary within some given percentage of an expected processing time. The information providedto the shift scheduler or dispatching rule is each job’s expected processing time. The actual job processing time ofeach operation is realized when the job is on the machine. Therefore, a priori, only partial knowledge about theprocessing times is known to the scheduler. Information about variability in job processing times is not used duringdispatching or schedule generation.

• Job weights and due dates are known a priori.• No setup cost or penalties are incurred when processing changes from one job type to another.

R. Swaminathan et al. / Computers & Operations Research 34 (2007) 3055–3068 3057

We examine three scheduling approaches in this paper, each of which enforces the permutation requirement toa varying degree. While the first, “pure permutation”, approach strictly enforces the permutation requirement, the“shift-based scheduling” approach schedules all jobs for an entire shift and enforces the resulting schedule during thesubsequent shift. However, under the shift-based schedule, jobs released in shift t may be overtaken by a job releasedin shift t + 1 at some point in time. The final approach, “pure dispatching,” represents the case where the flowshop’spermutation requirement is completely relaxed. Insights in understanding the inherent trade-offs between enforcingpermutation schedules and meeting customer due dates are examined. Initially, we expect better TWT performance asthe permutation requirement is relaxed. Without the permutation restriction being enforced, dispatching approachesshould be capable of producing job sequences that reduce the performance measure of interest.

2. Literature review

Wan [6] points out that scheduling methods may be classified by different dimensions. Scheduling problems may beclassified as static or dynamic, deterministic or stochastic, real time or offline. The typical goal in a static problem isto schedule a set of jobs optimally with a respect to a performance criterion. Dynamic problems feature a continuousarrival of new jobs. Deterministic scheduling problems assume fixed job processing times that are known a priori whilein stochastic problems one has limited knowledge about processing time characteristics along with random events suchas breakdowns. Additionally, scheduling may also be classified depending on whether the responsiveness of the methodis real time or offline. In real time methods, schedules may be changed after the occurrence of certain events (machinebreakdowns, new job arrivals, etc). Offline scheduling algorithms determine good schedules and execute them withoutany changes regardless of events that may occur after scheduling. Often the two approaches may be combined toproduce offline rescheduling where new schedules are constructed when pre-specified real time events occur. Previousresearch on flowshop scheduling has primarily been devoted to static, deterministic problems with offline scheduling.The majority of the vast literature on flowshop scheduling does not allow “job passing” due to the reduction in searchspace. The problem is identified in the literature as the permutation flowshop problem.

Researchers have focused on the makespan criterion for this problem for its relative simplicity and beneficial charac-teristics such as maximization of machine utilization. Makespan is defined as the completion time of the last job to leavethe system. A low value of makespan implies high utilization of the machines. Dudek et al. [7] cite a lack of applicationof flowshop algorithms in industry and state that real flowshop situations may be dynamic rather than static. They alsocriticize the selection of the makespan criterion due to its narrow applicability in industry. While makespan as a criterionis relevant with respect to machine utilization in a production line, it is a rather poor choice when individual orders needto be monitored. Ruiz and Moroto [8] present a comprehensive computational study on permutation flowshop heuristicsfor the makespan criterion. Their study included the well known Johnson’s algorithm [5] for flowshops, dispatchingrules, metaheuristics techniques such as tabu search, simulated annealing, genetic algorithms, iterated local searchesand other hybrid techniques. Their study compared the performance of various heuristics and mataheuristic techniqueson a large set of benchmark problem instances [9]. They concluded that the Nawaz–Enscore–Ham (NEH) heuristic[10] was the best heuristic for Taillard’s benchmarks. Tabu search, simulated annealing (SA) and genetic algorithms(GA) were shown to good metahueristics for the problem. They also determined the dispatching rules tested weresignificantly worse than even the RANDOM rule.

With regards to the flowshop problem for due date related objectives, the work by Gelders and Sambandam [11]involved constructive heuristics for the total tardiness problem. Some of the recent work on the tardiness probleminvolved the use of tabu search techniques by Kim [12] for the mean tardiness problem and later by Armentano andRonconi [13] for the total tardiness problem. Parthasarathy and Rajendran [14] developed two heuristics based onSA for the mean tardiness and the weighted mean tardiness problems. The two heuristics used different perturbationschemes namely the random insertion perturbation scheme (RIPS) and curtailed random insertion perturbation scheme(CRIPS). The proposed SA heuristics prevailed over the tabu search procedure by Kim [12] and the heuristic byGelders and Sambandam [11] in a computation evaluation study that used test problem instances generated by theauthors. Later, Hasija and Rajendran [15] used perturbation schemes and an improvement phase to improve upon theseed solution provided to the SA procedure for the total tardiness problem. A comparative study conducted on a set ofbenchmark problems selected from Taillard [9] revealed the superiority of their heuristic over the tabu search heuristicby Armentano and Ronconi [13] and SA heuristic by Parthasarathy and Rajendran [14].


Snomez and Baykasoglu [16] proposed a dynamic programming (DP) formulation that determined optimal solutionsfor the flowshop tardiness problem. The formulation produced permutation sequences with jobs having non-zero setuptimes at each machine and the setup times being sequence dependent. A zero setup time implementation of this DPperformed better than the heuristic methods available in Ref. [17]. The authors report that the computation time ofthe proposed DP formulation increases with problem size (main factor in determining problem size being number ofjobs) and this effect is more pronounced in the case of the DP when compared with other heuristic approaches for theproblem. This result is in line with a later study by Portougal and Scott [18] on the asymptotic convergence (AC) ofheuristics for flowshops with makespan and due date related objectives. Two important estimates when an objectivefunction is minimized are the upper bound (obtained through a heuristic procedure) and lower bound of all feasiblesolutions. When the relative difference between these two measures tends to zero as the number of jobs tends to infinitythe heuristic is said to be asymptotically convergent. This test for AC was proposed by Portougal [19]. With regards todue date related objectives Portougal and Scott [18] point out that the AC of a heuristic is guaranteed for objectives thatinvolve functions like maximum and mean but not true for functions like sum. They proved the AC of two heuristicsboth based on the first-come-first-served (FCFS) for the maximum tardiness and maximum weighted tardiness flowshopproblems.

3. Research methodology

This section describes each of the three flowshop scheduling approaches in greater detail. For each schedulingapproach, the various rules/methods for sequencing jobs within the flowshop are defined.

3.1. Pure permutation

Each of the pure permutation methods examined in this paper assigns job priorities when jobs are released into theflowshop, and they remain unchanged until the jobs exit the flowshop. The order of processing at the first machine isfollowed by all successive machines in the flowshop (i.e., the succeeding machines always follow a strict first in, firstout (FIFO) order). Jobs released in any of the previous shifts are not overtaken. The following methods were used todetermine the order of processing the jobs: FIFO, earliest due date (EDD), random (jobs are selected from a discreteuniform distribution DU [1, �1], where �1 is the number of jobs in the queue for the first machine), apparent tardinesscost (ATC), composite rank, ATC-grid, ATC-grid-NEH, and a GA. The following subsections describe these methodsin greater detail.

3.1.1. ATCThe ATC heuristic was originally proposed as a composite dispatching rule for single machines by Vepsalainen and

Morton [20]. The rule combines the weighted shortest processing time rule (WSPT) with the minimum slack rule (MS)in a single ranking index. The higher the values of the ATC index, the higher the priority of the job. The index isdetermined as follows:

IATC(t) = wj∑ipij

exp

(− max

(�j −∑

ipij − t, 0)

kp

). (1)

In (1) wj is the weight of job,∑

ipij is the sum of processing times of job j across all machines in the flowshop, �j isthe actual due date of job j (dj is not used since we use �j to depict internal job due dates), p is the average of totalprocessing times of all the jobs under consideration, t is the time at which the previous job is expected to finish andk is a scaling parameter. The following method is used to select k values:

k ={

4.5 + R, R < 0.5,

6 − 2R, R�0.5,(2)

R = (�max − �min)

Cmax, (3)


Cmax =∑

i

∑j

pij . (4)

In (2), (3) and (4) �max is the highest actual due date among the jobs in queue i, �min is the lowest actual due dateamong the jobs in queue i and Cmax is the estimated makespan.

3.1.2. Composite rankThis ranking scheme assigns higher priority to jobs with high weights, tight due dates and low sum of processing

times. Jobs are ranked based on: (1) decreasing order of weights; (2) increasing order of due dates; and (3) increasingorder of sum of processing times. The sum of all ranking positions of each job is determined. Jobs are ordered inincreasing order of the sum of all three ranks.

3.1.3. ATC-gridThe ATC heuristic is modified and adapted for use in a flowshop. The heuristic is based on ATC heuristic and the

ATC index for jobs is determined using (1). In this case, instead of determining the k value using (2)–(4), we performan iterative search within a fixed search space for k values that lead to good schedules. We use a grid search approachbased on the method proposed by Gadkari [21]. Values for k were selected from 0.4 to 14.0 with a step size of 0.4.The range was determined to ensure that frequently occurring values of k are not the extreme values in the grid. Thegrid search generates schedules with k values one at a time. The schedules are evaluated for performance on a static,deterministic flowshop (i.e., a forward deterministic simulation is performed). The schedule with the lowest TWT isselected.

3.1.4. ATC-grid-NEHThe NEH heuristic, which was originally proposed for the flowshop problems with the makespan objective was

adapted for the TWT objective. The modified version of the NEH heuristic considers the job priorities obtained withthe ATC-grid approach instead of the sum of the jobs processing times on all machines. The method for prioritizing n

jobs is described as follows:

1. Obtain job priorities using the ATC-grid approach and sort them in descending order of job priorities.2. Pick the two jobs from the first and second position of the list of Step 1, and find the best sequence for these two

jobs by calculating TWT for the two possible sequences using a forward deterministic simulation. Do not changethe relative positions of these two jobs with respect to each other in the remaining steps of the algorithm. Set i = 3.

3. Pick the job in the ith position of the list generated in Step 1 and find the best sequence by placing it at all possiblei positions in the partial sequence found in the previous step, without changing the relative positions to each otherof the already assigned jobs.

4. If i = n, STOP, otherwise set i = i + 1 and go to Step 3.

3.1.5. Genetic algorithmGenetic algorithms (GA) have been used by several authors [22,23] for flowshop scheduling problems. Authors

have generally concentrated on the makespan criterion for static instances of flowshop problems. In our GA, the jobsequences are themselves viewed as chromosomes. For example in a job sequence {32514}, job 3 gets processedfirst, and then job 2 and so on. The initial population consists of randomly chosen chromosomes along with solutionsobtained from the ATC-grid and the ATC-grid-NEH algorithms. Six identical copies of each solution are included inthe initial population for the first generation. The fitness value assigned to chromosome i is determined by first runninga deterministic forward simulation using expected job processing times and the actual job due dates for all the jobsin the flowshop. Then the final fitness value is calculated as fi = ri/

∑ri where ri is the reciprocal of the TWT of

the chromosome i and∑

ri is the sum of the reciprocals of all chromosomes in the population. The best availablesolution is always retained in the population for subsequent generations.

A “roulette wheel mechanism” [24] is used to select chromosomes for the succeeding generation. The processrandomly selects chromosomes based on their respective fitness values. Given P is the population size, one wouldexpect to have Pf i copies of the chromosome i in the pool. Once P chromosomes are in the pool, the crossover and


mutation operators are applied. We use a “uniform order-based crossover” [25] for the crossover operation. Givenparent 1 and parent 2, the following steps dictate the creation of child 1:

1. Generate a random bit string (0’s and 1’s) that is the same length as the parents.2. Fill in the positions on child 1 by copying them from parent 1 wherever the bit string contains a “1”. This leaves

gaps in child 1 wherever the bit string contained a “0”.3. Make a list of elements from parent 1 associated with a “0” in the bit string.4. Permute these elements so that they appear in the same order they appear on parent 2.5. Fill these permuted elements in the gaps on child 1 in the order generated in Step 4.

For example, if parents 1 and 2 were 234516 and 452361 and the generated bit string were 011001 then child 1 wouldbe 534261. A similar process is carried out to obtain child 2 by reversing the roles of parents 1 and 2. The crossoveroperation is carried out until P children are obtained. The crossover operator is not applied on the same chromosomemore than once. The mutation operator is applied with a probability pmu on all chromosomes in the population. Whenapplied the operator randomly picks two positions in the chromosomes and reverses the order of the genes betweenthese two points.

In our problem, which is examined over a rolling horizon, the GA is required to solve a new problem instance withthe new arrivals each shift and a different number of jobs are present in the system at different points in time. Thus,a problem instance encountered in the 50th shift might have different characteristics and be more difficult to analyzethan one in the 25th shift due to more total jobs arriving in that shift for the pure permutation case or due to more jobsbeing present in the system in the shift-based scheduling approach.

In our experimentation (see Section 4) we consider two different start rate cases. The high start rate cases have a highernumber of jobs arriving each shift and take longer to solve. Therefore, tuning experiments to determine appropriateGA parameters were carried out separately for low and high start rate cases. The GA parameters chosen for low andhigh start rates are as follows:

• Crossover probability = 1.0 for both low and high start rate cases.• Mutation probability = 0.1 for both low and high start rate cases.• Population size (P ) = 30 for low start rate case, 40 for high start rate case.• Generations = 1500 for low start rate case 2000 for high start rate case.

3.2. Shift-based scheduling

It is generally convenient to implement a permutation sequence on the shop floor. However, when new jobs arriveevery shift, some of these jobs may have higher priority than jobs that are already in the system. Thus, one could benefitby making a new permutation schedule for the shift. This allows for new, high priority jobs to be scheduled ahead ofolder, lower priority jobs.

In this section, we describe our approach towards scheduling jobs in the flowshop. Four methods for generating jobsequences based on the priority of the jobs in the flowshop were implemented: composite rank, ATC-grid, ATC-grid-NEH, and a GA. The GA is expected to have long computation times but comes with the expectation of better qualitysolutions. The composite rank, ATC-grid and the ATC-grid-NEH methods, on the other hand, are fast algorithms thatare expected to provide reasonably good solutions. Previously considered FIFO, EDD, RANDOM and ATC were notincluded in the shift-based scheduling approach as they did not perform competitively in the pure permutation approach.

Jobs present at a given time in the flowshop can be classified into one of the following three types.

• Jobs that have just been released. These jobs are present in the queue of the first machine.• Jobs released in previous shifts currently in queue at various stages of processing present in the flowshop.• Jobs that are currently being processed on a machine in the flowshop.

The scheduler is supplied the following information about all the jobs present at the flowshop: (1) Expected processingtimes of the jobs on all the machines; (2) Due dates of the jobs; (3) Job weights; and (4) Current job position in theflowshop. The permutation sequence is fed into a subroutine that constructs schedules for each workstation in the


flowshop while considering current shop status at the time of scheduling. Scheduling is performed periodically attime interval T , which represents the duration of a shift. Using the permutation sequence, separate schedules for themachines are built taking into consideration the various technological constraints for the flowshop.

3.3. Pure dispatching

The permutation requirement in the flowshop is completely relaxed in order to assess the “cost” of enforcingpermutation schedules. The following dispatching rules are applied on individual machines: FIFO, EDD, random, andATC. These approaches are disadvantaged due to the lack of global information pertaining to the flowshops and arehence myopic in nature. However, the complete relaxation of the permutation requirement may offset the informationdisadvantage.

Since the methods described in this approach dispatch at each machine, the job processing times and the internal(operational) due dates at the given machine are used. The external or actual job due dates are not appropriate here(except at the last machine) as the total job processing time across all machines is not used. The calculation of internaljob due dates is described below.

�j =(�j −∑m

i=1pij − rj)

m, (5)

�mj = �j , (6)

�ij = �i+1

j − pi+1,j − �j ∀ i ∈ {(m − 1), (m − 2), . . . , 1}. (7)

In (5), (6) and (7), �j is the actual or external job due date, �j is the average operation slack for job j ,∑m

i=1 pij is thesum of processing times of job j on all machines, m is the number of machines in the flowshop, �i

j is the internal duedate of jobj for machine i, pi+1,j is the processing time of job j on machine i + 1, and rj is the job release date. Weassume the slack for a job is evenly spread through all its operations.

In our EDD approach, �ij is used to represent each job’s due date. When employing ATC, we again substitute �i

j

in place of the conventional dj term. Also, we set the value of p as the average processing time of all the jobs in thequeue at each machine. Finally, the � terms in (3) are replaced by corresponding �i terms and Cmax =∑n

j=1 pij whendetermining the scaling parameter k under pure dispatching.

3.4. System considerations and experimental conditions

We used simulation to model the rolling horizon case for flowshops and to test the algorithms under the fourapproaches. The simulation model was built in SLAM II [26] and executed on a SUN Ultra-60 server. The experimentalruns were conducted with shift lengths of 720 time units. The simulation was run for 120 shifts, with data collectedfrom the first 20 shifts discarded. A benchmark test suite selected from those available in the OR-library [27] was usedin testing: instance car5. The data set provides ten job processing time sets for a six machine flowshop. During jobrelease, one of the ten job processing time sets from the data set was randomly picked and assigned to the job. Theassigned processing time set represents the expected processing time of that job on individual machines. Each job isassigned a weight sampled from a discrete uniform distribution DU [1,20].

The computation times for solving a single problem instance under the different methods vary greatly. Simpledispatching rules such FIFO, EDD, RANDOM and the ATC rule used in the pure dispatching approach solve a singlereplication of 120 shifts in about 1 s. A complete replication involves that uses the ATC-Grid method consumes between80 and 90 s per replication. The ATC-grid-NEH rule that uses the ATC-grid solution and conducts a search based on theNEH rule solves a replication in about 150 s on average. The GA, which is the most computationally intensive method,takes on average 1100–1600 s on average for low and high start rate cases, respectively.


4. Experimentation and results

Three important, practical factors characterize problem instances in our experiments: processing time variation, jobdue dates, and flowshop start rate. The actual processing time of a job j at machine i is sampled from a symmetricaltriangular distribution with the expected job processing time of that job as the mean and the upper and lower limitcontrolled by the extent of variability.

pij = TRIAG(E[pij ] − XE[pij ], E[pij ], E[pij ] + XE[pij ]). (8)

In (8), pij is the actual processing time of job j on machine i, E[pij ] is the expected processing time of job j , and Xis the variability factor, which was varied over three levels (0%, 30%, 60%). The external (actual) job due dates are setbased on the approach of Parthasarathy and Rajendran [28]:

�j = TRIAG[Z − 0.3, Z, Z + 0.3], (9)

�j = T +[(

�j ∗∑

i

pij

)+∑

i

pij

]. (10)

In (9) and (10), �j is the actual (or external) job due date, �j is due date tightness, T is the current time,∑

i pij is sumof processing times of the job j on all machines, and Z is the due date tightness factor. Finally, the two flowshop startrate levels of interest were devised to target specific bottleneck tool utilizations/loading. The low-level start rate drivesbottleneck tool utilization to 75–80%, while this utilization increases to 90–95% under the high start rate scenario. Thus,based on these bottleneck utilization targets, the number of jobs released every shift was sampled from a triangulardistribution with parameters (8,11,14) for the low start rate case and a triangular distribution with parameters (10,13,16)for the high start rate case.

The GA serves the purpose of validating the quality of solutions provided by constructive heuristics (ATC-grid, ATC-grid-NEH). The GA is seeded with the ATC-grid and ATC-grid-NEH solutions. Thus, the GA will provide solutionsthat are no worse than the solutions provided by the constructive heuristics (at the very least). We conducted a set oftest runs on static problem instances in order verify the quality of the solutions provided by the GA. These resultspointed to the superiority of the GA with respect to the static instances. These results do not necessarily translate tobetter performance in the rolling horizon case and the superiority of the GA over the other heuristics in the rollinghorizon case is by no means guaranteed. A full factorial design of experiments and analysis of variance (ANOVA) wasconducted. The average TWT values from five replications of each algorithm/scheduling approach for the probleminstances were used as the response. The model’s adequacy to the data was checked in the residual plots and theyappeared to be randomly and normally distributed.

4.1. Pure permutation approach results

The ratio of TWT values obtained by applying each method and those obtained by applying the composite rankrule as the base case were studied. The average results for these ratios are presented in Table 1. The results arepresented by problem instance characteristics to observe possible variations in the performance of algorithms fordifferent combinations of the inputs factors. FIFO, RANDOM and the composite rank rule did not perform competitively.FIFO and RANDOM are the only rules that yield performance ratios higher than one (i.e., worse than the CompositeRank rule which serves as the benchmark).

The performance of EDD as compared to the composite rank method suggests that the due date of the job and itsslack take precedence over job weights. The composite rank method, which assigns higher priority to high weight jobs,is handicapped by the fact that some of these jobs have high slack. Also, the method gives equal importance to job duedates, weights and processing and is not nearly as sophisticated as the more complex set of rules that belong to theATC family. EDD also gives better on average performance than ATC. However, ATC produces superior results in thetight due date cases where a majority of jobs have little slack.

Results from a Fisher’s-LSD procedure [29] indicate that there is no statistical difference between the overall meansobtained from the GA, ATC-grid-NEH and ATC-grid in this approach at 95% level of confidence. Fig. 2 shows a boxplot of performance ratios of different algorithms with respect to composite rank. As can be seen from the box plot


Table 1Average performance for the pure permutation approach

FIFO EDD RANDOM ATC Composite rank ATC-grid ATC-grid-NEH Genetic algorithm

Variability (%) 0 1.0477 0.8734 1.0813 0.9059 1.0000 0.5422 0.5488 0.531730 1.0837 0.8643 1.0820 0.9616 1.0000 0.5576 0.5703 0.552660 1.0762 0.8774 1.1217 0.9240 1.0000 0.5900 0.6134 0.5966

Due dates Tight 0.9909 0.9050 1.0038 0.7532 1.0000 0.7572 0.7240 0.7236Medium 1.0136 0.7784 1.0361 0.8917 1.0000 0.5638 0.5501 0.5376Loose 1.2032 0.9318 1.2451 1.1466 1.0000 0.3688 0.4584 0.4197

Start rate Low 1.0340 0.7043 1.0399 0.7368 1.0000 0.3558 0.3759 0.3478High 1.1045 1.0391 1.1501 1.1242 1.0000 0.7707 0.7791 0.7728

Overall 1.0692 0.8717 1.0950 0.9305 1.0000 0.5633 0.5775 0.5603

Algorithm

Per

f Rat

io w

rt C

om

po

site

Ran

k

RANDOMATC-Grid-NEHATC-GridGAFIFOEDDATC

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Boxplot of Perf Ratio wrt Composite Rank by Algorithm

Fig. 2. Box plot for the pure permutation approach.

best case–worst case ranges of the ATC-grid-NEH heuristic and the GA in most cases are completely contained insidethat of the ATC-grid. Generally, the results obtained from the ATC-grid, ATC-grid-NEH and the GA are very close inall the cases with GA being the best among the three on average.

The ATC-grid-NEH method, which comes with the expectation of better quality solutions, does not perform onaverage as well as the ATC-grid method in this approach. The ATC-grid-NEH relies upon the accuracy of the NEHmethodology to make improvements. These improvements are dependent on the information made available. In thisapproach only the new jobs are prioritized. The information on the job locations and remaining work on old jobs


Table 2Average performance for the shift-based scheduling approach

ATC-grid ATC-grid-NEH Genetic algorithm Composite rank

Variability (%) 0 0.3943 0.3512 0.3228 1.000030 0.4058 0.3621 0.3379 1.000060 0.4324 0.3963 0.3723 1.0000

Due dates Tight 0.6694 0.5708 0.5742 1.0000Medium 0.3943 0.3347 0.3215 1.0000Loose 0.1689 0.2041 0.1372 1.0000

Start rate Low 0.3348 0.3294 0.2885 1.0000High 0.4869 0.4103 0.4001 1.0000

Overall 0.4109 0.3699 0.3443 1.0000

(downstream) is not made available. The impact of these old jobs on the NEH sequence sometimes produces counter-productive results.

The GA, which is seeded with the ATC-grid-NEH and the ATC-grid solutions, tries to improve on these solutions andis expected to be superior across the board. However, the ATC-grid method performs better than the GA in some caseswith loose job due dates and high variability. It is worth noting that with loose due dates the base ATC rule reduces to theminimum slack rule. Loose due date settings can lead to situations where there are many solutions with zero tardiness(i.e.) there are multiple optimal solutions. All the algorithms function with the knowledge of the expected processingtimes. The GA evaluates chromosomes by performing a forward deterministic simulation. The chromosomes are ofthe same quality when viewed under purely deterministic conditions with no variation in processing times. Because ofrandom processing times in many of these cases the GA does not differentiate between nearly tardy cases and tardycases by ignoring job slack unlike a constructive heuristic like ATC-grid. Thus, it is not unlikely that the GA willschedule a job so it is scheduled to complete just before its due date. When both loose due dates and high variabilityare in play with the high probability of the realized processing time of the job (or those before it) being higher than theexpected value, the job may end up tardy. Therefore, the GA that is expected to be superior is not always superior.

4.2. Shift-based scheduling approach results

The ratio of TWT values obtained by applying each method and those obtained by applying the composite rank ruleas the base case were studied. The average results for these ratios are presented in Table 2. The performance of theATC-grid-NEH is a little over 4% better than ATC-grid in this approach. The ATC-grid approach performs better thanthe ATC-grid-NEH approach in loose due date settings. Again, in this approach the GA is seeded with the ATC-gridand ATC-grid-NEH solutions. The GA outperforms the ATC-grid on average and in the best case–worst case scenarios.The GA successfully beats the ATC-grid solution in these cases since all the old jobs are scheduled again with the newjobs in this approach.

This provides a better opportunity to make improvements. The GA does a better job at exploiting this than any con-structive heuristic would. ATC-grid-NEH produces the most consistent performance with the minimum best case–worstcase range width. The GA, however, produces the best average results in all cases. However, results from the Fisher’s-LSD procedure [29] indicate that there is no statistical difference between the overall means obtained from the GA,ATC-grid-NEH and ATC-grid in this approach at a 95% level of confidence. Fig. 3 shows a box plot of performanceratios of different algorithms with respect to composite rank.

4.3. Pure dispatching approach results

The ratio of TWT values obtained by applying each method and those obtained by applying the ATC dispatchingrule as the base case were studied. The average results for these ratios are presented in Table 3. As indicated in thetable, the ATC rule far outperforms the EDD, RANDOM and FIFO rules in all of the problem instances.


Algorithm

Per

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rt C

om

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site

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ATC-Grid-NEHATC-GridGA

0.7

0.6

0.5

0.4

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0.1

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Boxplot of Perf Ratio wrt Composite Rank by Algorithm

Fig. 3. Box plot for shift-based scheduling approach.

Table 3Average performance for the pure dispatching approach

ATC dispatching EDD RANDOM FIFO

Variability (%) 0 1.0000 5.0932 4.4673 4.678730 1.0000 4.8089 4.2200 4.737460 1.0000 4.5774 3.9685 4.4648

Due dates Tight 1.0000 2.0075 1.9637 1.9569Medium 1.0000 4.0397 3.8139 3.8827Loose 1.0000 8.4323 6.8782 8.0414

Start Rate Low 1.0000 4.7409 4.1864 4.5787High 1.0000 4.9121 4.2508 4.6754

Overall 1.0000 4.8265 4.2186 4.6270

4.4. Cost of the permutation requirement

The ATC-grid-NEH rule as implemented in the pure permutation and shift-based scheduling approaches wereselected for evaluating the effect of the permutation requirement. The ATC dispatching rule was selected from thepure dispatching approaches that completely relax the permutation requirement. The ratio of TWT values obtained byapplying the algorithms under various approaches and those obtained by applying the ATC-grid-NEH under the purepermutation approach were determined. The average results for these ratios are presented in Table 4.

The pure permutation approach as expected is outperformed by the other approaches. The shift-based schedulingand the pure dispatching approaches exhibited a 32% and 38% improvement over the pure permutation approach onaverage, respectively. The shift-based scheduling approach performs competitively with respect to the pure dispatchingapproach. The algorithms from the pure permutation, shift-based scheduling and pure dispatching approaches arestatistically indistinguishable at the 95% confidence level as per results from the Fisher’s-LSD test [29]. However, wenote that pure dispatching seems to work better than shift-based scheduling when the variability in processing timesis high, when the due dates are loose, and when the start rate is high. Any increase in variability causes a deviationfrom the schedule and shift-based scheduling does not have a mechanism to counter the resulting effects. On the otherhand, the absence of a schedule allows pure dispatching to select other jobs based on current status. Any effort tominimize tardiness in the tight due date cases results in a consequent increase in tardiness of other jobs. Jobs weights


Table 4Average performance with different approaches

Pure permutation Shift based Pure dispatching

ATC-grid-NEH ATC-grid-NEH ATC

Variability (%) 0 1.0000 0.6867 0.669330 1.0000 0.6803 0.617660 1.0000 0.6752 0.5491

Due dates Tight 1.0000 0.7799 0.7406Medium 1.0000 0.6564 0.5855Loose 1.0000 0.6058 0.5099

Start rate Low 1.0000 0.8790 0.8591High 1.0000 0.4825 0.3649

Overall 1.0000 0.6807 0.6120

Approach

Per

f Rat

io w

rt A

TC

-Gri

d-N

EH

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ach

1

ATC-Grid-NEH -Shift based SchedulingATC Dispatching

1.0

0.9

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0.7

0.6

0.5

0.4

0.3

0.2

Boxplot of Perf Ratio wrt ATC-Grid-NEH- PurePermutation by Approach

Fig. 4. Box plot for evaluation of effect of permutation requirements.

become very decisive in these cases although a lack of dispersion in job weight values can further constrict the choice.This situation does not arise in loose due date cases. Pure dispatching has the option of prioritizing low slack jobswithout contributing to the tardiness of other jobs. The effect is pronounced when high variability is also present in thesystem. The low start rate case represents the easier of the two start rate cases. Shift-based scheduling does a relativelygood job in the low start rate case. The lack of potential for improvement explains the imperceptible improvementmade by pure dispatching over shift-based scheduling. The high start rate case presents a stiff capacity managementchallenge to shift-based scheduling especially when it is also confronted with high variability. The restriction placed onshift-based scheduling to stick to a schedule for the shift leads to its relatively poor performance when compared withpure dispatching. Fig. 4 shows a box plot of performance ratios of different algorithms with respect to ATC-grid-NEHalgorithm in the pure permutation approach.

5. Conclusions and future research

This paper examined the inherent trade-offs between due date performance and level of enforcement of permutationsequences in flowshops. Several heuristic algorithms for the flowshop problem with due date related objectives were


developed. The first method, ATC-grid provides solutions of reasonable solution quality and is not computationallyintensive. This ATC-grid approach is a significant modification of the ATC rule for single machines. This was furtherimproved upon by conducting a search based on the NEH rule for permutation flowshops with the ATC-grid solution. Agenetic algorithm was developed for flowshops for the due date related objective (TWT). The GA also helped validatethe quality of solutions provided by other algorithms. We focused on developing shift-based scheduling methods thatdo not yield pure permutation sequences but also uphold the integrity of the permutation sequence within a shift. Weobserved a 32% improvement on average when the permutation was enforced on shift basis. The shift-based schedulingapproach only trails the pure dispatching by about 6% on average. The method proves to be effective while at the sametime maintaining permutation sequences within shifts.

Results indicate that it is possible to maintain a permutation sequence for an extended period of time without asignificant drop in performance. These results must be viewed from the standpoint of being portable and applicable tomore complex manufacturing systems like hybrid flowshops and cellular manufacturing cells that handle the productionof part families with varied tooling, sequencing and setup requirements. Production environments like job shops havetraditionally been employed because of their flexibility often sacrificing the efficiency offered by flowshop environments.The implementation of efficient material handling systems can be a costly and arduous venture. The idea of segregatingparts into families such that all the parts in a family have similar manufacturing requirements can be appealing.Individual manufacturing cells can then be setup to resemble flowshops. These manufacturing cells would resemblepure flowshops except for the existence multiple part families and can be scheduled on an individual basis, thus breakingthe overall problem into smaller manageable problems. Permutation sequences can be enforced on a shift basis due toease of implementation and schedules can be recreated on a global level across all cells at the end of the each shiftowing to the large number of part families and interchangeable resource requirements. Andres et al. [30] and Francaet al. [31] have addressed problems dealing with group technology in hybrid flowshops and flowshop manufacturingcells with a focus on throughput related objectives. This research will help researchers and practitioners in attackingthese problems with the goal of meeting customer due dates.

Several areas for further research have been identified. The use of a statistical framework like Bayesian Inference tomodel processing time uncertainty and incorporate it in the decision making process is worth exploring. Commencingwith an assumed likely prior distribution of underlying data, a likelihood function of observed data may be constructed.The likelihood function and the prior distribution may then be used to arrive at the posterior distribution using theBayesian Paradigm. Processing time values supplied to the scheduler may be sampled from this posterior distribution,thus factoring in the inherent variability in the process instead of using the expected value of theoretical distributions.

Using appropriate penalties when breaking permutation requirements and incorporating it in the objective mighthelp us obtain quantifiable estimates on the effect of the enforcement on due date performance. The apparent tardinesscost heuristic for setups (ATCS) proposed by Lee et al. [32] can be adapted easily to study the case of minimizingweighted tardiness when faced with non-zero setup times. The inclusion of machine breakdowns and urgent job ordersare two important extensions to testing the robustness of approaches described in this paper. Incorporating informationon upstream jobs while making decisions downstream could also be fruitful. In future research the case of jobs withmissing operations may be considered. The non-permutation-scheduling (NPS) algorithm [33] proposed to handle suchcases may be used to exploit this feature and achieve performance gain. Finally, the research may also be extended toflexible and reentrant flowshops to further investigate more practical, real world manufacturing settings.

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