Dipartimento di IngegneriaVia della Vasca Navale, 7900146 Roma, Italy

An Iterated Greedy

Metaheuristic for the Blocking

Job Shop Scheduling Problem

Marco Pranzo1, Dario Pacciarelli2

RT-DIA-208-2013 Novembre 2013

(1) Universita degli Studi Roma Tre,Via della Vasca Navale, 79

00146 Roma, Italy

(2) Universita degli Studi di SienaVia Roma 56

53100 Siena, Italy.

ABSTRACT

In this paper we consider a job shop scheduling problem with blocking (zero buffer) con-straints. Blocking constraints model the absence of buffers, whereas in the traditionaljob shop scheduling model buffers have infinite capacity. There are two known variantsof this problem, namely the blocking job shop scheduling (BJSS) with swap allowed andthe BJSS with no-swap. A swap is needed when there is a cycle of two or more jobs eachwaiting for the machine occupied by the next job in the cycle. Such a cycle representsa deadlock in no-swap BJSS, while with a swap all the jobs in the cycle move simulta-neously to their subsequent machine. This scheduling problem is receiving an increasinginterest in the recent literature, and we propose an Iterated Greedy (IG) algorithm tosolve both variants of the problem. The IG is a metaheuristic based on the repetition of adestruction phase, which removes part of the solution, and a construction phase, in whicha new solution is obtained by applying an underlying greedy algorithm starting fromthe partial solution. Comparison with recent published results shows that the iteratedgreedy outperforms other state-of-the-art algorithms on benchmark instances. Moreoverit is conceptually easy to implement and has a broad applicability to other constrainedscheduling problems.

Keywords: Job shop scheduling problem, blocking, swap, iterated greedy, alternativegraph.

2

1 Introduction

The job shop scheduling problem is one of the most studied problems in combinatorialoptimization. In this problem, a set of jobs must be processed on a set of machines andthe sequence of machines for each job is prescribed. The processing of a job on a machineis called an operation and cannot be interrupted. Each machine can host only one jobat a time, and whenever two jobs require the same machine, a processing order mustbe defined between the incompatible operations. The problem consists of scheduling alloperations on all machines to minimize the completion time of all jobs. The classicalversion of the problem assumes that jobs can be stored in a buffer of infinite capacitybetween consecutive operations. However, there are many practical situations in whichthe buffer capacity has to be taken into account [8, 26, 15, 22, 28, 5]. This is the case,for example, when there is no buffer to store work-in-progress [4] or it is not convenientto store jobs due to economic considerations [27]. The blocking constraint models thecomplete absence of storage capacity between consecutive machines [11]. In this setting,a machine is released by a job only when the subsequent machine is available for processingit. When the subsequent machine is unavailable the job remains on the current machine,thus blocking it, even after its operation is completed.

Two versions of the job shop scheduling problems with blocking can be distinguished:the blocking with swaps (BWS) and the blocking no-swap (BNS) versions. In order todefine a swap, let us introduce a deadlock as a cycle of two or more jobs each waiting forthe machine occupied by the next job in the cycle. A swap occurs whenever a deadlock issolved by simultaneously moving every operation in the cycle to the subsequent machine.If the swap is not allowed (BSN) a deadlock is infeasible. Mascis and Pacciarelli [18]show that the problem of finding a feasible extension of a partial solution is polynomiallysolvable for the BWS case and it is NP-complete for the BNS version.

Despite the practical relevance of the problem, only in the last years few algorithmiccontributions on job shop scheduling problems with blockings have been proposed in theliterature. One successful approach is based on the alternative graph formulation [18],which allows to model the BJSS problem in both variants, with and without swaps. In [18]a branch and bound algorithm is proposed to sove the problem exploiting the generaliza-tion to the blocking case of some properties holding for the traditional JSS problem withinfinite buffer capacity. In [29] a family of heuristics (AGH) extending the ones introducedin [18] has been proposed and tested. Based on the AGH heuristics, in [23] a rollout algo-rithm is proposed for solving four variants of the job shop scheduling problem including,among the others, the BWS and the BNS versions. In [16], a constructive algorithm withlocal search improvement is proposed for solving the blocking job shop scheduling problemwith parallel-machines and applied to solve a train scheduling problem. Computationalresults for the BWS case are also reported.

The literature on the blocking job shop scheduling problem includes several recentalgorithmic contributions specially tailored for the BWS version of the problem. Groflinand Klinkert [10] propose a tabu search algorithm in which the solution feasibility is recov-ered after every move. In [9], the neighborhood is enlarged to take into account machineflexibility and improved results are presented. A complete local search with memory isproposed by [37] which compares favourably with the greedy heuristic algorithms pre-sented in [18]. Recently, Oddi et al. [25] presented an iterative flattening search for theBWS problem obtaining state-of-the-art results.

3

The literature for the BNS case experienced a smaller number of contributions. Matiet al. [19, 20] study a flexible manufacturing system and model the production schedulingproblem as a BNS job shop scheduling problem in which each operation may require morethan a resource at a time to be executed. They develop a geometric approach to optimallysolve the BNS problem with two jobs. Mati and Xie [21] extend the previous works toinclude resource flexibility. They propose an insertion heuristic based on the geometricapproach for the problem with two jobs and incorporate this algorithm in a tabu searchframework to produce deadlock-free schedules for the problem with n jobs.

In this paper, we discuss and apply a conceptually very simple technique, known asIterated Greedy (IG), which aims at improving the performance of a greedy constructiveheuristic procedure in short computation times.

The paper is organized as follows. In Section 2 we formulate job shop schedulingproblems with blocking constraints using the alternative graph formulation. In Section 3we present our iterated greedy algorithm for solving the problem modeled by means ofthe alternative graph formulation. Computational results on a set of benchmark instancesare presented in Section 4, and finally some conclusions follow.

2 The job shop scheduling problem with blocking

constraints

In the job shop scheduling problem a set of jobs J must be processed on a set of machinesM , each processing at most one job at a time. The processing of a job on a machineis called an operation and cannot be interrupted. We let {o1, . . . , on} be the set of alloperations. The sequence of operations for each job is prescribed, while the sequence ofoperations for each machine has to be determined in such a way that the time needed tocomplete all operations, called the makespan, is minimum. More formally, the schedulingproblem consists in assigning a starting times ti to operation oi, i = 1, . . . , n, such that:(i) precedence constraints between consecutive operations of the same job are satisfied;(ii) each machine hosts at most one job at a time; and (iii) the makespan is minimized.

In the blocking job shop scheduling problem no intermediate storage is allowed betweentwo consecutive machines. Hence, once a job completes processing on machine Mh it eithermoves to the subsequent machine Mk (if it is available) or it remains on Mh, thus blockingit (if Mk is not available). The blocking job shop scheduling problem can be modeled byusing the alternative graph model of Mascis and Pacciarelli [18]. An alternative graphis a triple G = (N,F,A). N = {0, 1, . . . , n + 1} is a set of nodes, and each node i ∈ Nis associated to operation oi. Two additional nodes start 0 and finish n + 1 (dummyoperations o0 and on+1) model the beginning and the completion of the schedule, i.e., o0precedes all other operations while on+1 follows all other operations. F is a set of fixeddirected arcs and fij is the length of the fixed arc (i, j) ∈ F . A fixed arc (i, j) representsa precedence constraint on the starting time of operation oj, which is constrained to startat least fij time units after oi, i.e., tj ≥ ti +fij. Finally, A is a set of m pairs of alternativedirected arcs, and aij is the length of the alternative arc (i, j). If ((i, k), (h, j)) ∈ A,then arcs (i, k) and (h, j) form an alternative pair. Alternative pairs are used to modelprecedence relations between pairs of operations to be processed on the same machine.Let oi and oj be two operations to be processed on the same machine, and let oh and okbe the operations immediately following oi and oj in their respective jobs (see Figure 1).

4

The blocking constraint states that if oi is processed before oj, operation oj can start theprocessing only after the starting time of oh (i.e., when oi leaves the machine). Hence, werepresent this situation with the alternative arc (h, j). Conversely, arc (k, i) representsoperation oi preceding oj.

A selection S is a set of arcs obtained from A by choosing at most one arc fromeach pair, and G(S) indicates the graph (N,F ∪ S). The selection is feasible if thereare no positive length cycles in the resulting graph G(S). In fact, a positive length cyclerepresents an operation preceding itself, which is not feasible. A selected alternative arc(i, j) represents the precedence relation tj ≥ ti + aij.

When swaps are allowed (Figure 1 (a)) the length of all the alternative arcs is set tozero. In fact given an alternative pair ((i, k), (h, j)) ∈ A, oj can start immediately afterthe start of oh, i.e., ahj = 0. Note that, in this case, a cycle of alternative arcs has totallength zero, and it is therefore feasible. In the no-swap case a deadlock is infeasible. Sincea deadlock corresponds to a cycle of alternative arcs (see Figure 2), this infeasibility isachieved by setting the length of alternative arcs to a small positive value ε (Figure 1 (b)).This positive length corresponds to model the fact that an operation can start processingon a machine strictly after that the previous operations has left it.

J1

J2

(a)

0

0

pi

pj

J1

J2

(b)

pi

pj

ε ε

i

j k

h h

k j

i

Figure 1: (a) The alternative pair of blocking operations with swaps allowed. (b) Thealternative pair of blocking operations when swaps are not allowed.

J1

J2

(a)

0

0

(b)

ε

J1

M1 M2

J2

(c)

0

0

J1

J2

ε ε

ε

Figure 2: (a) A simple swap cycle between two operations (b) The alternative graphmodel with swaps allowed. (c) The alternative graph model when swaps are not allowed.

Given a feasible selection S, we denote the length of a longest path from i to j inG(S) by lS(i, j) (if there is no path from i to j we assume lS(i, j) = −∞). A solution Sis a selection of |A| alternative precedence relations, exactly one from each pair in A. Afeasible solution is optimal if the length lS(0, n + 1) of a longest path from 0 to n + 1 isminimum over all feasible solutions.

5

Finally, given a feasible selection S, we define LSij as the length of a critical (i.e.,

longest) path passing through the alternative arc (i, j):

LSij = lS(0, i) + aij + lS(j, n+ 1).

LSij represents a rough lower bound of the longest path between 0 and n + 1 (i.e., of the

makespan) in any feasible solution obtained starting from the selection S ∪ {(i, j)}.We also make use of static implications. Two alternative pairs ((k, i), (h, j)) ((d, a), (c, b))

are statically implied if there is no feasible solution in which arcs (k, i) and (c, b) are se-lected or arcs (h, j) and (d, a) are selected. In other words, the selection of arc (d, a)implies the selection of arc (k, i), and the selection of arc (h, j) implies the selection of arc(c, b). Given an alternative arc (h, i), we define the set I(h, j) as the set of alternative arcsstatically implied by (h, j). Whenever an alternative arc pair (h, j) is selected/deselectedalso all arcs in I(h, i) are selected/deselected.

The following simple result allows to establish a correspondence between the selectionof arcs from different alternative pairs.

Proposition 1 (Implication) Given a selection S and two unselected alternative pairs((d, a), (c, b)) and ((h, j), (k, i)), if lS(a, h) ≥ 0 and lS(j, d) ≥ 0 then there is no feasiblesolution in which arcs (d, a) and (h, j) are both selected in S.

Proposition 1 can be applied in particular with the empty selection S = ∅, whenthe graph G(∅) = (N ,F) is composed of a set of chains of fixed arcs. Then, the pathslS(a, h) ≥ 0 and lS(j, d) ≥ 0 can be composed only by fixed arcs, i.e., the paths existonly if nodes a, h refer to a same job J1, and j, d refer to another job J2, sharing somemachines with J1. We call this implication static since it holds for any selection S. Theconditions of Proposition 1 hold for S = ∅ in the following two cases:

• J1 and J2 pass both through two consecutive machines M1 and M2 in the sameorder. In this case c ≡ i and d ≡ j (see Figure 3 (a)).

• J1 and J2 pass both through M1 and M2 consecutively, although in the oppositeorder (see Figure 3 (b)).

d

a c

b

h

j k

i J1

J2

a h c≡i J1

b k J2 d≡j

(a) (b)

Figure 3: (a) Jobs passing in the same order. (b) Jobs passing in opposite order.

6

Iterated GreedyInput:

Initial solution Sbegin

while stopping criterion is not fulfilled doS ′ = Destruction( S )S ′′ = Construction( S ′ )S = Acceptance Criterion( S, S ′′)

endend

Figure 4: The pseudocode of an iterated greedy algorithm.

Whenever one of the two above situations occurs, the selection of arc (d, a) impliesthe selection of arc (k, i), and the selection of arc (h, j) implies the selection of arc (c, b).Therefore I(d, a) = {(k, i)}, I(k, i) = {(d, a)}, I(h, j) = {(c, b)} and I(c, b) = {(h, j)}.

Static implications can be computed off-line very efficiently and have been successfullyapplied to train scheduling problems [4].

3 Iterated Greedy

The Iterated Greedy is a conceptually very simple approach that can be used to improvethe performance of a greedy heuristic, which is used as a black box within the algorithm.The basic idea is to iterate a Destruction-Construction cycle. A sketch of the algorithmis given in Figure 4. A Destruction-Construction cycle consists of three different phases,namely destruction (some components of the incumbent solution are removed), construc-tion (starting from the partial solution a new complete solution is built by the greedyheuristic) and acceptance criterion (the new solution is evaluated and possibly acceptedas a new incumbent). Finally a stopping criterion is adopted to terminate the search.

Given its simple structure, IG algorithm has been independently developed by differentresearch communities under different names, such as Ruin and Recreate [35], AdaptiveLarge Neighborhood Search [36] or Iterated Flattening Search [3]. IG algorithms havebeen successfully applied to the Set Partitioning Problem [17], to some variants of theVehicle Routing Problem [32, 31], to the Quadratic Assignement Problem [24], and to thepermutation flowshop scheduling in both versions without and with blocking [33, 34, 30].

One of the main advantage of the iterated greedy algorithm is that it does not relyon specific properties of the problem addressed, but all the problem specific knowledgeis embedded in the underlying heuristic solution procedure. This is particularly usefulwhen dealing with very general problems that suffer from a lack of specific properties tobe used in the design of efficient solution algorithms. Moreover, the framework is easy toimplement once a construction heuristic procedure is available. In the remaining part ofthis section we describe the three phases of our iterated greedy algorithm for the blockingjob shop problem with or without swaps.

7

3.1 Initial Solution

The first incumbent solution is obtained by simply ordering the jobs from 1 to |J | andthen, for each alternative pair involving jobs Ji and job Jj with i < j, selecting the arc inwhich Ji precedes Jj. Is it easy to show that the initial solution constructed in this wayis always feasible, since a cycle in G(∫) would require at least one arc from Jj to Ji, withi < j. The solution found by this initial heuristic has typically very poor quality but isguaranteed to be always feasible, both in the BWS and BNS cases. This is relevant since,starting from a partial selection, the problem of deciding whether a feasible solution existsis a NP -complete problem for the BNS case.

3.2 Destruction

In the Destruction phase a portion of the current selection S (representing the incumbentsolution) is deselected, and a new partial selection S ′ is obtained. More precisely, eachalternative pair is randomly deselected with probability p. The probability p is a param-eter of the Destruction procedure called perturbation strength and in our test we set it to0.8, i.e., in the destruction phase 80% of all alternative arcs are deselected. Recall that,since we use static implications, every time the algorithm deselects the alternative arc(i, j) ∈ ((i, j), (h, k)) all the implied alternative arcs in I(i, j) are also deselected. Thusthe actual fraction of removed arcs is at least p.

Notice that, in our model, k operations that must be processed on the same machinelead to k(k − 1)/2 alternative pairs. Clearly, only (k − 1) of them are necessary, whileall the others are redundant. Removing a redundant arc would not lead to a differentsolution, since a redundant arc is implied by the others and construction algorithm wouldselect it immediately. Hence, in order to reach a new solution, the destruction phase mustremove a large number of arcs, most of which are redundant.

3.3 Construction

Given a partial selection S, a constructive greedy procedure is applied to extend S ateach step with a new alternative arc (i, j), i.e., S = S ∪ {(i, j)} ∪ I(i, j), until a newfeasible solution S ′ is obtained or an infeasibility is detected. The latter case ariseswhen there is an alternative pair ((i, j), (h, k)) such that both G(S ∪{(i, j)}∪ I(i, j)) andG(S ∪ {(h, k)} ∪ I(h, k)) contain a positive cycle, and we say that S is infeasible even ifG(S ′) does not contain any cycle.

As constructive heuristic we use the AMCC (Avoid Maximum Current Completiontime) heuristic introduced in Mascis and Pacciarelli [18], a quite effective heuristic fordifferent versions of the job shop scheduling problem [12, 7, 4]. The AMCC heuristicchooses at each step the alternative pair ((i, j), (h, k)) ∈ A such that max{LS

ij, LShk} is

maximum over all unselected alternative pairs, and then selects (i, j)∪ I(i, j) if LSij < LS

hk

or (h, k) ∪ I(h, k) otherwise.If the selection of arc (i, j) would cause a positive length cycle in the graph, then the

arc (h, k) is selected. If also the selection of (h, k) would lead to a positive length cyclethe algorithm terminates and returns the (acyclic but infeasible) partial selection.

Observe that, as stand-alone heuristic, AMCC often fails in finding a feasible solutionfor large instances of both the BWS and BNS case. Hence, the use of a metaheuristic

8

AMCC HeuristicInput:

Alternative graph G = (N,F,A)Initial Selection S

beginwhile A 6= ∅ do

Let G = (N,F ∪ S)Choose pair ((i, j), (h, k)) ∈ A such that max{LS

ij, LShk} is maximum and LS

ij ≤ LShk.

Let S1 = S ∪ {(i, j)} ∪ I(i, j).if G(S1) is infeasible then let S2 = S ∪ {(h, k)} ∪ I(h, k).

if G(S2) is infeasible then STOP, failed in finding a feasible solution.else Update S = S2 and remove from A all pairs (h, k) and I(h, k) belong to.

else Update S = S1 and remove from A all pairs (i, j) and I(i, j) belong to.end

end.

Figure 5: Sketch of the AMCC algorithm

scheme is particularly useful to achieve the feasibility of the solutions, besides improvingtheir quality.

3.4 Acceptance Criterion

Once a candidate solution is built, the algorithm decides whether to accept it as the newincumbent or to discard it and retain the previous incumbent. Among the acceptancecriteria defined in the literature, in our experiments we assessed the following two:

• Random Walk (RW). When RW is adopted every feasible candidate solution gen-erated by the constructive phase is always accepted. If the construction phase doesnot terminate with a feasible solution then the IG maintains the previous incumbentsolution.

• Simulated Annealing like (SA). In the Simulated Annealing criterion [33, 32] anew feasible solution is always accepted if it improves the current best UB, i.e.,if Cmax(S ′) ≤ UB. If Cmax(S ′) > UB, then the candidate solution is accepted withprobability e[UB−Cmax(S′)]/T where T is a parameter called the temperature. In ourexperiments we set T = 0.5.

3.5 Stopping Criterion

Different stopping criteria can be devised, such as a time limit, a maximum numberof iterations and so on. In our tests we set a maximum time limit of computation of60 seconds for each run. To avoid stagnation and to improve the performance of thealgorithm we run the IG for 10 independent runs and retain the best solution found.

9

4 Computational Results

We tested the performance of our algorithm on the 58 benchmark instances for the jobshop scheduling problem by [6, 14, 2, 1], adapted to the BWS and BNS cases (for a totalof 116 instances). The optimum for these instances is known only for the 18 BWS and 18BNS instances with 10 jobs and 10 machines [18].

In a preliminary test phase, we tested 12 different parameters setting configurations,retaining the best solution found. The results obtained in this phase often improve thebest known solution, as illustrated in Sections 4.1 and 4.2. From this preliminary testphase we selected the two configurations exhibiting the best performance:

• Destruction: In the destruction phase we set perturbation strength equals to 0.8.

• Construction: The chosen constructive greedy heuristic is the AMCC algorithm.

• Acceptance Criterion: The two best configurations differ for the acceptance cri-terion, which is Random Walk (RW) in one configuration and Simulated Annealing(SA) in the other.

• Stopping Criterion: As stopping criterion we use a time limit of computationequal to 10 minutes. After every 60 seconds the current run is interrupted and anew independent run is started.

We refer to these algorithms as IG.RW and IG.SA, respectively. The algorithm is imple-mented in C++, uses a single thread and runs on an 3.0 GHz Core Duo Intel processorequipped with 2 Gb of RAM.

4.1 Performance analysis on Blocking with Swap

In this section, we compare our results with the best known solutions available in theliterature. As a benchmark for the BWS case, we consider two tabu search algorithms[10, 9], a rollout algorithm [23], a complete local search with memory [37], simple heuristictechniques reported in [29, 16] and the recent algorithms presented in Oddi et al. [25]. Thebest performing algorithms described in these papers are the GPB tabu search algorithm[9], the IFS algorithm [25] and the CP-OPT constraint programming solver of IBM Ilog[25] which implements the approach described in [13].

In Table 1 we compare the results found by our IG against the best known solutionsfrom the literature. The first two columns report the instance name and size (number ofjobs × number of machines). The reference to the paper and the best known solution foreach instance are shown in Column 3 and 4, respectively. Finally, in Column 5 we showthe best result found by all the tested configurations of IG, highlighted in bold when someconfiguration of the IG improves over the best known solution.

The overall CPU time required for each instance by the 12 configurations of IG toobtain the results in Table 1 is 7200 seconds on a 3.0 GHz processor. The CPU timesrequired by the other best performing algorithms are as follows. GK’09 performs 5 in-dependent runs of 1800 seconds each on a 2.8GHz processor, for a total of 9000 seconds.IFS’12 requires two runs of 1800 seconds each. Since 16 configurations of IFS’12 are pre-sented in [25] and Table 1 shows the best result, the total computation time required byIFS’12 for one instance is 16 hours of computation. CP-OPT’12 performs a single run

10

of 1800 seconds. Both the IFS’12 and CP-OPT’12 run on a AMD Phenom II X4 Quad3.5GHz.

The 12 configurations of IG presented in this paper are able to attain or to improvethe best solution known in the literature in 44 over 58 benchmark instances. Over the40 instances for which a proven optimum is not known, IG finds a new best known in 19cases. Among the other 18 instances for which the optimum is given in [18], IG achievesthe optimum in 17 cases.

11

Instance

|J|×|M|

Referen

ceBestknown

BestIG

Instance

|J|×|M|

Referen

ceBestknown

BestIG

abz5

(10×

10)

BB

-[18]

1468

1468

abz9

(20x15)

GK

-[10]

1166

1111

abz6

(10×

10)

BB

-[18]

1145

1145

ft06

(6x6)

GK

-[10]

63

63

abz7

(20x15)

GK

-[10]

1224

1106

ft10

(10×

10)

BB

-[18]

1068

1068

abz8

(20x15)

GK

-[10]

1226

1099

ft20

(20×

5)

GK

-[10]

2047

1434

la01

(10×

5)

CP-O

PT

-IF

S-[25]

793

793

la21

(15×

10)

IFS-[25]

1521

1490

la02

(10×

5)

IFS-[25]

793

793

la22

(15×

10)

CP-O

PT

-[25]

1379

1339

la03

(10×

5)

IFS-[25]

715

715

la23

(15×

10)

CP-O

PT

-[25]

1497

1445

la04

(10×

5)

IFS-[25]

743

743

la24

(15×

10)

IFS-[25]

1498

1434

la05

(10×

5)

CP-O

PT

-IF

S-[25]

664

664

la25

(15×

10)

IFS-[25]

1424

1392

la06

(15×

5)

IFS-[25]

1064

1102

la26

(20×

10)

CP-O

PT

-[25]

2035

1989

la07

(15×

5)

IFS-[25]

1038

1020

la27

(20×

10)

IFS-[25]

2104

2017

la08

(15×

5)

IFS-[25]

1062

1071

la28

(20×

10)

IFS-[25]

2027

2039

la09

(15×

5)

IFS-[25]

1185

1162

la29

(20×

10)

CP-O

PT

-[25]

1898

1846

la10

(15×

5)

IFS-[25]

1110

1134

la30

(20×

10)

IFS-[25]

2095

2049

la11

(20×

5)

IFS-[25]

1466

1498

la31

(30×

10)

CP-O

PT

-[25]

2921

3018

la12

(20×

5)

IFS-[25]

1272

1271

la32

(30×

10)

CP-O

PT

-[25]

3237

3338

la13

(20×

5)

IFS-[25]

1465

1482

la33

(30×

10)

CP-O

PT

-[25]

2844

2909

la14

(20×

5)

CP-O

PT

-[25]

1506

1513

la34

(30×

10)

CP-O

PT

-[25]

2848

3016

la15

(20×

5)

IFS-[25]

1517

1517

la35

(30×

10)

CP-O

PT

-[25]

2923

3133

la16

(10×

10)

BB

-[18]

1060

1060

la36

(15×

15)

IFS-[25]

1793

1755

la17

(10×

10)

BB

-[18]

929

929

la37

(15×

15)

CP-O

PT

-[25]

1952

1870

la18

(10×

10)

BB

-[18]

1025

1025

la38

(15×

15)

IFS-[25]

1708

1728

la19

(10×

10)

BB

-[18]

1043

1043

la39

(15×

15)

IFS-[25]

1783

1731

la20

(10×

10)

BB

-[18]

1060

1060

la40

(15×

15)

IFS-[25]

1777

1743

orb

01

(10×

10)

BB

-[18]

1175

1175

orb

06

(10×

10)

BB

-[18]

1199

1199

orb

02

(10×

10)

BB

-[18]

1041

1041

orb

07

(10×

10)

BB

-[18]

483

492

orb

03

(10×

10)

BB

-[18]

1160

1160

orb

08

(10×

10)

BB

-[18]

995

995

orb

04

(10×

10)

BB

-[18]

1146

1146

orb

09

(10×

10)

BB

-[18]

1039

1039

orb

05

(10×

10)

BB

-[18]

995

995

orb

10

(10×

10)

BB

-[18]

1146

1146

Tab

le1:

Com

par

ison

sw

ith

the

bes

tknow

nso

luti

ons

for

BW

S

12

Table 2 compares the results of the two best configurations of IG with the threebest performing algorithms from the literature: GPB’11, IFS’12 and CP-OPT’12. Werestrict the comparison to the 40 BWS instances of the Lawrence benchmark, the onlyinstances for which a published result is available for all the three benchmark algorithms.Under IFS’12 we report the results of the best performing configuration among the 16IFS variants presented in [25], namely the one using the slack-based procedure and withγ = 0.5.

In Table 2, columns from 1 to 4 show the instance name, size, the value of thebest known solution from Table 1 and the paper in which the solution has been pre-sented, respectively. Columns 5, 6 and 7 report the results obtained by GBP’11, IFS’12and CP-OPT’12, respectively. Finally, Columns 8 and 9 report the results achieved bytwo Iterated Greedy configurations IG.RW and IG:SA. For each algorithm we present,in the last row of Table 2, the relative percentage error over the best known solutions(error = 100 ∗ (found − best)/best) from the literature. The computation times of thealgorithms are as follows: GPB’11 uses 5 independent runs of 1800 seconds each on a3.0GHz processor. IFS’12 requires two runs of 1800 seconds each on a 3.5GHz processor.CP-OPT’12 performs a single run of 1800 seconds on a 3.5GHz processor. Finally, thecomputation time of the IG.RW and IG.SA is 600 seconds for both algorithms, obtainedwith 10 independent runs of 60 seconds each on a 3.0 GHz processor.

From Table 2, it turns out that both IG.RW and IG.SA outperform the other algo-rithms in terms of solution quality and computation time. In fact, both the computationtime of IG.RW and its average distance from the best known is one third of the best per-forming algorithm from the literature (CP-OPT’12), i.e., 600 against 1800 seconds and1.25% against 3.84%, respectively. Moreover, the two IGs are able to improve the bestknown solution in 11 out of 40 benchmark instances (the values are shown in bold in thetable).

4.2 Performace analysis on Blocking No Swap

In this section we focus on the BNS case. As in the previous section, we first compare theoverall performance of the 12 configurations of IG with the current state-of-the-art. Wenext compare the two best performing configurations against the best algorithm for JSPwith blocking and no swaps. To the best of our knowledge, besides the exact approach in[18] referred to as BB, the only available heuristic results on JSP in the BNS case are atabu search algorithm by Mati and Xie [21] referred to as MX’11, a rollout algorithm [23]and simple heuristic techniques [29]. Among these approaches the tabu search of Matiand Xie (MX’11) is the best performing algorithm. Thus we omit the other approachesin the comparison.

13

Instance |J | × |M | Best Known Reference GPB CP-OPT IFS IG.RW IG.SAla01 (10× 5) 793 CP-OPT - IFS [25] 820 793 793 793 793la02 (10× 5) 793 IFS - [25] 817 815 793 793 793la03 (10× 5) 715 IFS - [25] 740 790 740 715 715la04 (10× 5) 743 IFS - [25] 764 784 776 743 743la05 (10× 5) 664 CP-OPT - IFS - [25] 666 664 664 664 664la06 (15× 5) 1064 IFS - [25] 1180 1131 1112 1102 1120la07 (15× 5) 1038 IFS - [25] 1084 1106 1081 1062 1040la08 (15× 5) 1062 IFS - [25] 1125 1129 1135 1089 1093la09 (15× 5) 1185 IFS - [25] 1223 1267 1257 1192 1192la10 (15× 5) 1110 IFS - [25] 1203 1168 1158 1140 1151la11 (20× 5) 1466 IFS - [25] 1584 1520 1501 1550 1529la12 (20× 5) 1272 IFS - [25] 1391 1308 1321 1342 1332la13 (20× 5) 1465 IFS - [25] 1541 1528 1471 1531 1523la14 (20× 5) 1506 CP-OPT - [25] 1620 1506 1567 1538 1545la15 (20× 5) 1517 IFS - [25] 1630 1571 1547 1593 1575la16 (10× 10) 1060 BB - [18] 1142 1150 1086 1060 1060la17 (10× 10) 929 BB - [18] 977 996 1000 930 930la18 (10× 10) 1025 BB - [18] 1078 1135 1120 1040 1061la19 (10× 10) 1043 BB - [18] 1093 1108 1077 1043 1068la20 (10× 10) 1060 BB - [18] 1154 1119 1166 1080 1080la21 (15× 10) 1521 IFS - [25] 1545 1579 1521 1514 1490la22 (15× 10) 1379 CP-OPT - [25] 1458 1379 1490 1368 1390la23 (15× 10) 1497 CP-OPT - [25] 1570 1497 1538 1445 1473la24 (15× 10) 1498 IFS - [25] 1546 1523 1498 1434 1498la25 (15× 10) 1424 IFS - [25] 1499 1561 1424 1422 1473la26 (20× 10) 2035 CP-OPT - [25] 2125 2035 2179 2013 2012la27 (20× 10) 2104 IFS - [25] 2175 2155 2172 2044 2017la28 (20× 10) 2027 IFS - [25] 2071 2062 2132 2039 2155la29 (20× 10) 1898 CP-OPT - [25] 1990 1898 1963 1928 1860la30 (20× 10) 2095 IFS - [25] 2097 2147 2125 2137 2085la31 (30× 10) 2921 CP-OPT - [25] 3137 2921 3771 3095 3115la32 (30× 10) 3237 CP-OPT - [25] 3316 3237 3852 3415 3420la33 (30× 10) 2844 CP-OPT - [25] 3061 2844 3741 2970 2958la34 (30× 10) 2848 CP-OPT - [25] 3146 2848 3796 3016 3057la35 (30× 10) 2923 CP-OPT - [25] 3171 2923 3818 3193 3169la36 (15× 15) 1793 IFS - [25] 1919 1952 1891 1755 1764la37 (15× 15) 1952 CP-OPT - [25] 2029 1952 1983 1870 1884la38 (15× 15) 1708 IFS - [25] 1828 1880 1708 1728 1720la39 (15× 15) 1783 IFS - [25] 1882 1813 1848 1731 1752la40 (15× 15) 1777 IFS - [25] 1925 1928 1831 1743 1771

error (%) 0.00 5.48 3.84 6.69 1.25 1.62

Table 2: Comparisons with the best performing algorithms for BWS on Lawrence in-stances.

14

Instance

|J|×|M|

Referen

ceBestknown

BestIG

Instance

|J|×|M|

Referen

ceBestknown

BestIG

abz5

(10×

10)

BB

-[18]

1641

1641

abz6

(10×

10)

BB

-[18]

1249

1249

ft10

(10×

10)

BB

-[18]

1158

1158

la01

(10×

5)

MX

-[21]

881

881

la21

(15×

10)

MX

-[21]

1700

1627

la02

(10×

5)

MX

-[21]

900

900

la22

(15×

10)

MX

-[21]

1519

1426

la03

(10×

5)

MX

-[21]

808

808

la23

(15×

10)

MX

-[21]

1731

1574

la04

(10×

5)

MX

-[21]

862

859

la24

(15×

10)

MX

-[21]

1633

1502

la05

(10×

5)

MX

-[21]

742

732

la25

(15×

10)

MX

-[21]

1655

1533

la06

(15×

5)

MX

-[21]

1243

1225

la26

(20×

10)

MX

-[21]

2310

2146

la07

(15×

5)

MX

-[21]

1194

1133

la27

(20×

10)

MX

-[21]

2362

2191

la08

(15×

5)

MX

-[21]

1216

1219

la28

(20×

10)

MX

-[21]

2291

2245

la09

(15×

5)

MX

-[21]

1349

1311

la29

(20×

10)

MX

-[21]

2184

2030

la10

(15×

5)

MX

-[21]

1270

1237

la30

(20×

10)

MX

-[21]

2312

2242

la11

(20×

5)

MX

-[21]

1725

1641

la31

(30×

10)

MX

-[21]

3403

3219

la12

(20×

5)

MX

-[21]

1479

1465

la32

(30×

10)

MX

-[21]

3701

3567

la13

(20×

5)

MX

-[21]

1683

1627

la33

(30×

10)

MX

-[21]

3363

3201

la14

(20×

5)

MX

-[21]

1745

1686

la34

(30×

10)

MX

-[21]

3380

3202

la15

(20×

5)

MX

-[21]

1749

1680

la35

(30×

10)

MX

-[21]

3373

3494

la16

(10×

10)

BB

-[18]

1148

1148

la36

(15×

15)

MX

-[21]

2026

1835

la17

(10×

10)

BB

-[18]

968

968

la37

(15×

15)

MX

-[21]

2177

1931

la18

(10×

10)

BB

-[18]

1077

1077

la38

(15×

15)

MX

-[21]

1968

1813

la19

(10×

10)

BB

-[18]

1102

1102

la39

(15×

15)

MX

-[21]

1992

1811

la20

(10×

10)

BB

-[18]

1118

1118

la40

(15×

15)

MX

-[21]

1958

1815

orb

01

(10×

10)

BB

-[18]

1256

1256

orb

06

(10×

10)

BB

-[18]

1266

1275

orb

02

(10×

10)

BB

-[18]

1144

1149

orb

07

(10×

10)

BB

-[18]

527

527

orb

03

(10×

10)

BB

-[18]

1311

1311

orb

08

(10×

10)

BB

-[18]

1139

1139

orb

04

(10×

10)

BB

-[18]

1246

1246

orb

09

(10×

10)

BB

-[18]

1130

1130

orb

05

(10×

10)

BB

-[18]

1203

1203

orb

10

(10×

10)

BB

-[18]

1367

1367

Tab

le3:

Com

par

ison

sw

ith

the

bes

tknow

nso

luti

ons

for

BN

S

15

In Table 3 we show the current state-of-the-art performance on BNS instances ofJSP. The instance name and its size are reported in the first two columns. The bestknown solution from the literature is shown in Column 3, while Column 4 shows thepaper reporting first the best solution. Finally, in Column 5 (Best IG) we report the bestsolution found by the 12 IG configurations. In bold are highlighted the new best values.The proposed IG configurations are able to attain or to improve the best known solutionin all but 3 instances. For the 40 instances for which a proven optimum is not known,IG finds a new best known solution in 29 cases. Moreover, for the 18 known optima,IG attains the optimum in 16 cases. Regarding the CPU times, the algorithm MX’11described in [21] requires a shorter CPU time since the results are obtained after 2000seconds on a 2.8GHz processor, while the 12 IG configurations require in total 2 hours ona 3GHz processor.

In Table 4 we compare the performance obtained by the two best configurations ofIG, namely IG.RW and IG.SA, against that of MX’11 over the 40 BNS instances of theLawrence benchmark. Columns 1 and 2 show the instance name and size, respectively.The best known solutions are shown in Column 3. Except for the 5 instances with 10jobs and 10 machines for which a proven optimum is known [18], the best known isgiven by MX’11. In columns 4, 5 and 6 we report the results for MX’11, IG.RW andIG.SA. For each algorithm, the last row of Table 4 report the relative percentage error(error = 100 ∗ (found − best)/best) over the best known solution from the literature.A negative value corresponds to a performance improving on average the best knownsolutions. The computation times of the algorithms are as follows: MX’11 performs 10independent runs of 200 seconds on a 2.8GHz processor, for a total of 2000 seconds, whilethe IG is the best over 10 independent runs of 60 seconds on a 3.0 GHz processor, for atotal of 600 seconds.

The results show that IG clearly outperforms MX’11 both in terms of computationtime and quality of the solutions. In particular, the two IG algorithms improve the bestknown solutions 28 times out of the 35 instances for which the proven optimum is notknown. The best average results are attained by the IG.SA (-2.25% on average).

5 Conclusions

In this paper we developed an Iterated Greedy metaheuristic to solve the job shop schedul-ing problem with blocking constraints, in the two variants with and without swaps. Themain advantages of the iterated greedy algorithm are the ease of implementation and theindependence of the framework from specific properties of the problem addressed.

Even if the proposed approach is conceptually very simple and easy to implement, theperformance obtained clearly outperforms all the previously published algorithms fromthe literature. A new best known solution has been found for 20 BWS instances and for29 BNS instances. It is also important to notice that the Iterated Greedy has a broadapplicability since it can be easily applied to any complex scheduling problem modeledby means of the alternative graph formulation.

Future research directions should address the study of local search based approachesfor the blocking job shop scheduling problem and the adaptation of the IG algorithms toreal world applications with more complex constraints.

16

Instance |J | × |M | Best Known Reference MX IG.RW IG.SAla01 (10× 5) 881 MX - [21] 881 919 881la02 (10× 5) 900 MX - [21] 900 906 900la03 (10× 5) 808 MX - [21] 808 808 808la04 (10× 5) 862 MX - [21] 862 859 859la05 (10× 5) 742 MX - [21] 742 732 732la06 (15× 5) 1243 MX - [21] 1243 1247 1257la07 (15× 5) 1194 MX - [21] 1194 1173 1143la08 (15× 5) 1216 MX - [21] 1216 1224 1213la09 (15× 5) 1349 MX - [21] 1349 1311 1341la10 (15× 5) 1270 MX - [21] 1270 1237 1277la11 (20× 5) 1725 MX - [21] 1725 1683 1684la12 (20× 5) 1479 MX - [21] 1479 1525 1524la13 (20× 5) 1683 MX - [21] 1683 1695 1642la14 (20× 5) 1745 MX - [21] 1745 1701 1686la15 (20× 5) 1749 MX - [21] 1749 1728 1682la16 (10× 10) 1148 BB - [18] 1205 1148 1186la17 (10× 10) 968 BB - [18] 1020 968 979la18 (10× 10) 1077 BB - [18] 1156 1077 1099la19 (10× 10) 1102 BB - [18] 1191 1124 1124la20 (10× 10) 1118 BB - [18] 1204 1164 1164la21 (15× 10) 1700 MX - [21] 1700 1653 1627la22 (15× 10) 1519 MX - [21] 1519 1435 1435la23 (15× 10) 1731 MX - [21] 1731 1574 1628la24 (15× 10) 1633 MX - [21] 1633 1584 1530la25 (15× 10) 1655 MX - [21] 1655 1558 1569la26 (20× 10) 2310 MX - [21] 2310 2159 2164la27 (20× 10) 2362 MX - [21] 2362 2211 2191la28 (20× 10) 2291 MX - [21] 2291 2280 2319la29 (20× 10) 2184 MX - [21] 2184 2074 2054la30 (20× 10) 2312 MX - [21] 2312 2263 2342la31 (30× 10) 3403 MX - [21] 3403 3482 3405la32 (30× 10) 3701 MX - [21] 3701 3649 3576la33 (30× 10) 3363 MX - [21] 3363 3326 3255la34 (30× 10) 3380 MX - [21] 3380 3306 3352la35 (30× 10) 3373 MX - [21] 3373 3582 3522la36 (15× 15) 2026 MX - [21] 2026 1835 1894la37 (15× 15) 2177 MX - [21] 2177 1974 1931la38 (15× 15) 1968 MX - [21] 1968 1813 1831la39 (15× 15) 1992 MX - [21] 1992 1820 1811la40 (15× 15) 1958 MX - [21] 1958 1818 1815

error (%) 0.00 0.84 -2.11 -2.25

Table 4: Comparisons with the best performing algorithms for BNS on Lawrence in-stances.

17

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