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A Hybrid Bio-Inspired Algorithm for NP-Hard JobShop Scheduling Problem

Sundeep Vilasagarapu Ram MohanaReddy Guddeti

the date of receipt and acceptance should be inserted later

Abstract The job shop scheduling problem (JSSP) is known as a combi-natorial optimization problem that aims to find best sequence of operationswith optimal execution time called makespan. Research on optimization ofthe JSSP is one of the most significant and promising areas of optimization.Instead of the traditional optimization methods, this paper presents an investi-gation into the use of bio-inspired algorithms like Particle Swarm Optimization(PSO), Cat Swarm Optimization (CSO) and Ant colony optimization (ACO)to optimize the JSSP. And, by considering the merits and demerits of each bio-inspired technique mentioned above, we came up with two hybrid bio-inspiredalgorithms one is combination of PSO+ACO and CSO+ACO. Experimentalresults demonstrate that the hybrid bio-inspired techniques provides optimizedsolutions that the individual, and among the hybrid algorithms CSO+ACOoutperforms the PSO+ACO. Because of its inherently parallel nature, ACOis well-suited to GPU implementation. So to reduce the run time complexitywe have used GPU to implement the hybrid algorithms.

Keywords Scheduling Job Shop Particle Swarm Optimization CatSwarm Optimization Ant Colony Optimization GPU

1 Introduction

Best allocation of the resources to activities over time can be make throughgood scheduling which can greatly influence the production efficiency in man-ufacturing industries. JSSP is considered to be one of the well-known NP-hard

Sundeep VilasagarapuNational Institute of Technology Karnataka, Surathkal-575025, IndiaE-mail: [email protected]

Ram Mohana Reddy GuddetiNational Institute of Technology Karnataka, Surathkal-575025, IndiaE-mail: [email protected]

2 Sundeep Vilasagarapu, Ram Mohana Reddy Guddeti

combinatorial optimization problem, its difficulty to find out the optimal solu-tion among (n!)m combinations for n job and m machines, has always been thehot issue in industrial engineering. It is a very good essential problem which isto be solved in real-time applications like train scheduling, production units.Because even a small percentage of improvement in the efficiency of the overalloperation may bring significant financial return.

The following are some existing contributions for solving JSSP: Sadeh etal. [1] studied a version of the JSSP in which some operations have to be sched-uled within time windows. They developed an incipient look-back schemes thatavail the search procedure recover from so-called dead-end search states. PeterBrucker et al. [2] proposed a expeditious branch and bound algorithm for theJSSP which finds an optimal solution if the problem is of circumscribed size.Adams et al. [3] came up with SBP algorithm to solve JSSP. SBP has a verygood balance between computational intricacy and the quality of engenderedschedules. Paulli et al. [4] presented a tabu search (TS) algorithm by defininga neighborhood structure where there is no distinction between re-assigning orre-sequencing an operation. Jiang et al. [5] proposed a new Lagrangian relax-ation approach where operation precedence constraints are relaxed rather thanmachine capacity constraints. Mitsuo et al [20] developed a new method usinggenetic algorithm which is prodigiously time-consuming. Colorni et al. [6] in-troduced ant colony optimization for solving JSSP by a pheromone alterationstrategy which improves the basic ant system by utilizing the behaviour of ar-tificial ants to perform local search. Xia and Wu [7] proposed a particle swarmoptimisation algorithm for the multi-objective flexible JSSP, by combining lo-cal search (by self experience) and global search (by neighboring experience).

Table 1: Existing work and its limitationsAuthors Proposed Methodology Limitations

Peter Brucker et al. [2] Branch and bound algorithm Provides efficient solutionsin time only for small andmedium size probleminstances.

Adams et al. [3] Shifting bottleneck procedure Might obtain infeasiblesolutions. However, they didnot explain why the casemight exist.

Paulli et al. [4] Tabu search algorithm These approaches excel othersnot necessarily in solutionquality, but almost always incomputation time.

Mitsuo et al. [20] Genetic algorithm Works well when computationtime is of no concern.

Colorni et al. [6] Ant colony optimization the algorithm was far fromreaching state-of-the-artperformance.

Xia and Wu [7] Particle swarm optimization Infirmness local convergence& the slow convergence speed.

A Hybrid Bio-Inspired Algorithm for NP-Hard Job Shop Scheduling Problem 3

In short, some of the existing work and its limitations are summarized inTable 1. Most studies indicate that a single technique can not solve this stub-born problem. Much work has been done on hybrid methods involving geneticalgorithm, simulated annealing, Tabu search, and Shifting bottleneck proce-dure techniques as hybrid methods can provide high-quality solution withinreasonable computing time. But using hybrid bio-inspired approach very lessresearch has been done. Li Li et al. [14] proposed a hybrid bio-inspired ap-proach using pso and aco for flexible job shop scheduling and tested for onlytwo problem sizes. Hence, we proposed a combination of bio-inspired algo-rithms using cso, pso and aco to find the nearest optimal schedules in mini-mum time.

The following are the key contributions of our work: To the best of our knowledge, we are the first to propose a hybrid combi-

nation of CSO and ACO to solve the JSSP. To the best of our knowledge, this is the first paper which parallelizes the

ant colony algorithm for solving the JSSP.The rest of this paper is organized as follows: Section 2 deals with the JobShop Scheduling Problem definition and representation; Section 3 focuses onthe bio-inspired algorithms adapted in this paper; Section 4 discusses proposedmethodology and Section 5 deals with Experimental Results and Analysis.Finally the Concluding remarks with future directions are given in Section 6.

2 Job Shop Scheduling Problem

2.1 Definition of JSSP

The classic JSSP is a machine scheduling and optimization problem which canbe defined as follows: Let us consider an n x m JSSP, in which n representsset of jobs J={J1, ..., Jn} and m represents set of machines M={M1, ...,Mm}.A predefined order is to be maintained to process each job through m ma-chines. This predefined order is known as operation precedence constraints.An operation can be defined as processing of a job on one machine uninter-rupted period of a given length. A set of operations can be represented asO={Oij |i [1, n], j [1,m]} where Oij represents processing of ith job on jthmachine.

2.2 Representation of JSSP

Special graph structure called disjunctive graph G = {V, Ca U Da} is usedto represent the job shop scheduling problems where V - vertices representingthe operations of jobs; Ca - conjunctive arcs represents the fixed precedencerelation among the operations; and Da - disjunctive arcs between every pairof operations that must be performed on the same machine.


Description for this disjunctive graph: Consider there are n jobs and, same-type m machines, we number the nodes from one to N, where N denotes thetotal no. of actual operations in disjunctive graph. Each arc (a, b) connectstwo actual nodes a and b, implying that operation a has to be processedimmediately before operation b. Each arc has the length of pa correspondingto the processing time of operation a.

Fig. 1: Graph for an instance n=3 and m=3 JSSP.

Figure 1 is modeled for n=3 and m=3 instance where each node representsan operation. Each tuple (j, m) p on every node implies the job number,machine number and time required for processing of an operation. For instance,[(1, 3) 2] at node 2 means operation 2 of job 1 has to be operated on machine3 for a processing time of 2. And lastly the length of the longest path from thedummy start node S to the dummy end node E is equivalent to the makespanof the schedule.

3 Bio-Inspired Algorithms

In the following sections we describe the bio-inspired algorithms PSO, CSOand ACO which we adapted for solving the JSSP.

3.1 Particle Swarm Optimization

It is an evolutionary computation technique which is proposed by Kennedyand Eberhart [21] in 1995, based on observations of the convivial comport-ment of birds in flocks or fish in schools, as well as on swarm theory. It hasbeen introduced to solve optimization problems like scheduling and routingproblems which impersonates the behavior of flying birds and the way theyexchange their information. It needs simple mathematical operators, and alsocomputationally competitive in terms of both time & memory requisites.


PSO Model: It performs search operating a population (called swarm) of in-dividuals (called particles) that are improved from iteration to iteration. Thecorrelation across the particles & swarm in PSO is akin to the correlationacross the chromosomes & population in a genetic algorithm. Particles movetowards either pbest or gbest position after every iteration where pbest is theindividual particle best position and gbest is the global best position amongswarm so far. The position and velocity of a particle designates its status. Thevelocity and position of a particle k with d-dimensions can be formulated asfollows [21]:

V kd = Iw Vkd + C1 r1() (Pkd Xkd) + C2 r2() (Pgd Xkd) (1)

X kd = Xkd + Vkd (2)

where Vkd is the particle k velocity, Pkd is the best position found by eachparticle so far, Pgd is the best position among swarm. The exploration andexploitation of a particle is controlled by inertia weight Iw, C1 and C2 are theconstants adopted to determine whether particles choose to move towards Pkdor Pgd. r1 and r2 are the two random functions return the values in range [0,1].

PSO algorithm:

1. Initialize a swarm of particles with random positions and velocities in theD-dimensional problem space.

2. For each particle, evaluate the desired optimization fitness function.3. Individual particles fitness value are with its pbest. If the current value is

better than pbest, then set the pbest value to be equal to the current value,and the pbest position equal to the current position in D dimensional space.

4. Compare the fitness evaluation value with the swarms best fitness valueobtained so far. If current value is better than gbest, then reset gbest to thecurrent particles fitness value.

5. Change the velocity and position of the particle according to Eqs. (1) and(2) respectively.

6. Loop to step (2) until a termination criterion is met, usually a sufficientlygood fitness or a specified number of generations.

3.2 Cat Swarm Optimization

Shu-Chuan et al. [15] noticed the behavior of cats and presented an evolution-ary computation technique called Cat Swarm Optimization (CSO). All catseven though have vigorous curiosity, mostly they are dormant. When cats arereposing they are generally awake with high vigilantness. This entire demeanorcan be represented in two methods:


Seeking mode (SM): This mode is the case where the cat is at rest, lookingnear by and exploring the next position to move using the four necessaryfactors in this sub-model which are described in the below:

SMP Seeking memory pool is for initializing the no. of cats in searchmode.

SRD Seeking range of the selected dimension is the maximum differencebetween the new and old values in the dimensions selected.

CDC Counts of dimension to change is the size culled to carry themutation

SPC self-position considering is a boolean value indicating if a cat canbe selected for the tracing mode or not.

The seeking mode working mechanism can be illustrated in the below:

1. put j copies of the present position of catk, with j = SMP. If the value ofSPC is true, let j=(SMP-1), and retain the cat as one of the candidates.

2. Generate a random value of SRD.3. Calculate the fitness values (FS) of all candidate points.4. If the FS is not equal, calculate the probability of each candidate by Eq

(3), else the default probability value of each candidate is 1.

Pi =|FSi FSb|

FSmax FSmin (3)

5. perform mutation and replace the current position.

Tracing mode (TM): The TM represents the hunting mode of the cat. TheTM process is described as follows:

1. Update the velocities of each catk according to the Eq (3).

V k = Vk + r1 c1 (Xbest Xk) (4)2. Check if the velocities are of highest order or not.3. Update the position of Catk according to Eq (4).

X k = Xk + V k (5)

where Xbest is the position of the cat who has the best fitness value; Vk is thecurrent velocity and Xk is the actual position of catk. c1 is a constant and r1is a random value in the range of [0,1].

CSO Algorithm

1. Create N cats in the process.2. Randomly sprinkle the cats into the M-dimensional solution space and

randomly select values, which are in-range of the maximum velocity, to thevelocities of each cat. Then haphazardly pick number of cats and set theminto tracing mode according to MR, and the others set into seeking mode.


3. Evaluate the fitness value of each cat by applying the positions of cats intothe fitness function, which represents the criteria of our goal, and keep thebest cat into memory. Note that we only need to remember the position ofthe best cat (Xbest) due to it represents the best solution so far.

4. Move the cats according to their flags, if catk is in seeking mode, apply thecat to the seeking mode process, otherwise apply it to the tracing modeprocess. The process steps are presented above.

5. Re-pick number of cats and set them into tracing mode according to MR,then set the other cats into seeking mode.

6. Check the termination condition, if satisfied, terminate the program, andotherwise repeat step3 to step5.

3.3 Ant Colony Optimization

Through inspecting the behavior of ants, Marco Dorigo in 1992 presented aevolutionary computation technique called Ant Colony Optimization (ACO)to search for an optimal path in a graph. The original idea has since diversifiedto solve a wider class of numerical problems, and as a result, several problemshave emerged, drawing on various aspects of the behavior of ants.

Ant colonies exhibit very interesting behaviors: one ant has limited capa-bilities, but the behavior of a whole ant colony is highly structured. They arecapable of finding the shortest path from their nest to a food source, withoutusing visual cues but by exploiting pheromone information. While walking,ants can deposit some pheromone on the path. The probability that the antscoming later choose the path which is proportional to the amount of pheromoneon the path, previously deposited by other ants.

The basic model of the ACS is that: m ants are initially placed at a startpoint with n decision points to the destination. Each ant builds a tour byrepeatedly applying a stochastic greedy rule, which is called the state transitionrule.

S ={argmaxqEr [(p, q)][(p, q) ], if e < e0 (exploitation)S, otherwise

(6)

(p, q) represents an edge between point p and q, and (p, q) stands for thepheromone on edge (p, q). (p, q) is the desirability of edge (p, q), which isusually defined as the inverse of the length of edge (p, q). e is a random numberuniformly distributed in [0, 1], e0 is a user-defined parameter with (0e01), is the parameter controlling the relative importance of the desirability. Eris the set of edges available at decision point r. S is a random variable selectedaccording to the probability distribution given below.

P (p, s) =

[(r, s)] [(r, s) ]

qEr[(r, s)] [(r, s) ], if e < e0(exploitation)

0, otherwise(7)


The selection strategy used above is also called roulette wheel selection sinceits mechanism is a simulation of the operation of a roulette wheel.

A certain amount of pheromone is dropped when an ant goes by. It is a con-tinuous process, but we can regard it as a discrete release by some rules. Herewe introduce two kinds of pheromone update strategies, called local updatingrule and the global updating rule.

Local updating rule: While constructing its tour, an ant will modify the amountof pheromone on the passed edges by applying the local updating rule.

(p, s) = (1 ) (p, s) + 0 (8)where is the coefficient representing pheromone evaporation (0


The next step is to compute the fitness of each particle for its new positionthrough Eqs. (1) and (2). The absolute value of Vkd and Xkd may be great.Thus, the particle may overshoot the problem space. Therefore, the Vkd islimited to the range of [-n, n] and Xkd is limited to a range of [1, n]. Thecomputational result of Eq. 2 will generate repetitive code (job number) inevery segment, i.e. one job is processed on the same machine repeatedly. Thisbreaches the constraint conditions in JSSP. We call the computation resultsthat breach constraints illegal solutions. Illegal solutions can be converted tolegal solutions by modification.

The process of modifying solutions is as follows:1. Inspect each particle on each machine and note the job numbers which are

repetitive and absent.2. According to the increment order of their process, sort the absent job num-

bers on each machine of a particle.3. Then for each particle replace those absent job numbers for repetitive ones

on each machine from low to high dimensions.From the above steps we obtain the legal solution and as ACO has the

best positive feedback system for meteoric discovery of best solutions. Weintegrated the ACO module to the PSO which is shown in Figure 2. Thecomplete procedure of the ACO Module is described below:

ACO Module:1. Initialize all the parameters.2. calculate the number of subsets according to the jobs number.3. Initialize the positions with the optimum solutions obtained from the above

module.4. Construct the first scheduling according to transition possibilities and state

transition rule. Compute object function for the first scheduling;5. Carry out local search. Adjust the initial scheduling obtained by step 4

according to local search method. Compute object functions. Save the bestone and carry out local updating according to it;

6. Carry out global updating according to the best scheduling obtained fromthe very beginning to now;

7. According to the maximal generation defined in the algorithm, repeat step4 to step 6 until the terminated condition is met. Make the best schedulingobtained as the optimum scheduling of this subset;

8. Obtain the optimum scheduling of this subset. Repeat step 3 to step 7 untilthe number of maximal subsets is met;

9. Compare the optimum scheduling obtained in step 8 and make the bestone as the global optimum scheduling.

4.2 JSSP Using CSO and ACO

CSO Module: For n-jobs and m-machines, the solution is presented by a se-quence of n x m operations. In order to apply CSO to the JSSP, a generic


Fig. 2: Combined PSO and ACO Architecture.

solution of the problem should be encoded. Each solution is represented by asequence of all operations. The operations are numbered from 1 to n*m. Everyprocess is represented by a pair moi and toi (z [1,(n*m)] ), where moi andtoi represents the machine on which the process oi will be executed, and theprocessing time of operation ok.

The matrix information shown below has 5 rows and (n*m) columns. It isdeveloped to represent all the information about each operation:

O1 O2 O3 O4 O5 O6 O7 O8 OiJo1 Jo2 Jo3 Jo4 Jo5 Jo6 Jo7 Jo8 JoiSo1 So2 So3 So4 So5 So6 So7 So8 SoiMo1 Mo2 Mo3 Mo4 Mo5 Mo6 Mo7 Mo8 MoiTo1 To2 To3 To4 To5 To6 To7 To8 Toi

where Oi: The number of operations in schedule (i [1,(n*m)]).

Joi : The job of operation oi.Soi : The sequence of operation oi in corresponding job.Moi : The machine name where the operation oi will be processed.Toi : The processing time of operation oi.


1. Initialize the position of the cats with a random solution and its flag byMR.

2. To correct the random solution consider a new as the new solution andold as the current solution. The process of correction is described asfollow:

(a) Scan the vector solution old.(b) The first available operation is added to task queue of new and deleted

from old.(c) Repeat (a) and (b) until old vector is cleared. Finally we will obtain

a valid solution.3. Now calculate the makespan (schedule length) of the valid schedule solution

and update xbest.4. Changing the position of each cat with respect to its mode indicated by its

flag5. Updates the value of the flag of each cat according to MR.6. Check for the termination condition. If yes, exit else repeat the steps 3, 4

and 5.

To integrate the above module with the ACO, the optimum solutions ob-tained from the CSO module are given as input and repeat the steps describedin the above ACO module.

4.3 Parallel Framework of ACO

The structure of sequential ACO algorithm is highly suitable for paralleliza-tion. The behavior of a single ant during the iteration is totally independentof the behavior of all other ants during that iteration which provides scope fortask parallelism. This behavior of the ants was the motivation to develop analgorithmic strategy for parallelization.

Stefan [22] proposed a simple parallelization strategy for the ACO. Wemodified it according to our problem and made the ants to compute the sched-ules in parallel. This would result in a master-slaves paradigm as shown inFigure 5. The master initializes the information about the problem (the initialpheromone trail 0 and the data set n x m matrix), and captures the globalknowledge (the pheromone matrix, the best found solution) acquired duringthe search process represented by the workers. Each worker handles an antprocess. At each iteration, the master distributes the pheromone matrix to allslaves. Each slave receives the pheromone matrix, constructs a feasible sched-ule and sends the result to the master. After receiving all the solutions, themaster updates the trail levels and checks for the best schedule found so far,and then the process is iterated.


Fig. 3: Combined CSO and ACO Architecture.

Fig. 4: Master-Slave paradigm for parallel ant colony algorithm


5 Experimental Results and Analysis

All the algorithms have been coded in MATLAB 2011a and tested on the HPdesktop with Intel I7 3.40 GHz Processor, 8 GB RAM and NVIDIA NVS 300GPU having 2 compute units and 512 MB RAM. The proposed experimentis to obtain the optimum solutions for the JSSP using hybrid bio-inspiredalgorithms.

To explain the effectiveness and performance of the proposed algorithms inthis paper, various kinds of benchmark job-shop instances with different sizeshave been selected to compute. All instances are downloaded from the OR-Library (http://people.brunel.ac.uk/mastjjb/jeb/orlib/files/jobshop1.txt).Table 2 contain the results of bio-inspired algorithms tested on forty Lawrenceinstances LA01-LA40 and table 3 constain the other different benchmark in-stances Abz5-Abz9,Orb1-Orb10,FT06,FT10,FT20. The columns in the table,problem is for instance name, n for no. of jobs and m for no. of machines,BKS for best known solution so far, the error percentage is obtained by Eq(11), combination of pso-aco and cso-aco are the best solutions obtained byapplying these hybrid algorithms to each instance on average 15 times.

Error% = (BestBKS)BKS

100 (11)

Setting parameters: In PSO Swarm size of 20, the maximum number of itera-tive generations is set to 400, inertia weight of 0.9 and acceleration constantsC1, C2 each equal to 2.0. In CSO, SMP 5, CDC 8, C1 2.05, r1 [0,1], inertia weight of 0.8. In ACO, 2, 0.1, 0.1 and themaximum number of iterative generations is set to 800.

For the combination of cso and aco maximum no of instances x% optimalresults the error percentage is 0% for all these instances. Using combination ofcso and aco for most of the instances LA01-LA12, it reaches the BKS in lessthan 60 seconds. And for instances LA31-LA35, it reaches the BKS within1800 seconds for all instances. On the other side pso and aco combinationgot optimal results for y% and it takes on average less than 90 seconds forthe instances LA01-LA15 and less than 210 seconds for the instances 2600seconds.

With the support of ACO for inherent parallelism, we implemented themodule of ACO in parallel using MATLAB. There is no much difference in theoptimal results further compared to the serial hybrid algorithm but obtainedthese results in less time. From the graph shown in Figure X, we can easilyconclude that serial implementation of hybrid algorithms are having greatestrunning time compared to parallel implementation . This is because GPUsare more suitable for running data-parallel workloads than CPUs, and theACO approach utilizes parallelization. And the speed up factor of cso-acocombination is 2.19 and pso-aco combination is 1.76.


Table 2: Results obtained for hybrid algorithms using Lawrence instancesProblem N M BKS Using PSO and ACO Using CSO and ACO

LA01 10 5 666 666 666LA02 10 5 655 655 655LA03 10 5 597 597 597LA04 10 5 590 590 590LA05 10 5 593 593 593LA06 15 5 926 926 926LA07 15 5 890 890 890LA08 15 5 863 863 863LA09 15 5 951 951 951LA10 15 5 958 958 958LA11 20 5 1222 1222 1222LA12 20 5 1039 1039 1039LA13 20 5 1150 1150 1150LA14 20 5 1292 1292 1292LA15 20 5 1207 1207 1207LA16 10 10 945 945 945LA17 10 10 784 784 784LA18 10 10 848 848 848LA19 10 10 842 842 842LA20 10 10 902 902 902LA21 15 10 1046 1047 1053LA22 15 10 927 927 927LA23 15 10 1032 1032 1032LA24 15 10 935 938 938LA25 15 10 977 977 977LA26 20 10 1218 1218 1218LA27 20 10 1235 1236 1235LA28 20 10 1216 1216 1216LA29 20 10 1157 1164 1157LA30 20 10 1355 1355 1355LA31 30 10 1784 1784 1784LA32 30 10 1850 1850 1850LA33 30 10 1719 1719 1719LA34 30 10 1721 1721 1721LA35 30 10 1888 1888 1888LA36 15 15 1268 1269 1268LA37 15 15 1397 1401 1397LA38 15 15 1196 1208 1173LA39 15 15 1233 1240 1233LA40 15 15 1222 1226 1228

6 Conclusion and Future Work

To improve the performance for large scale data, bio-inspired algorithms likeparticle swarm optimization (PSO), cat swarm optimization (cso), ant colonyoptimization (ACO) techniques are used. The main aim is to develop a hybridbio-inspired algorithm that gives the best optimum solution. Experimentalresults shows that combination CSO-ACO outperforms PSO-ACO with lesspercentage of error for benchmark data sets. In addition to that, the parallelcombination have a speed up factor of 2.19 for cso-aco and 1.76 for pso-aco


Table 3: Results obtained for hybrid algorithms using Abz,Orb,FT instancesProblem N M BKS Using PSO and ACO Using CSO and ACO

Abz5 10 10 1234 1234 1234Abz6 10 10 943 943 943Abz7 20 15 654 654 654Abz8 20 15 634 634 634Abz9 20 15 656 656 721Orb1 10 10 1059 1059 1059Orb2 10 10 888 888 888Orb3 10 10 1005 1005 1005Orb4 10 10 1005 1005 1005Orb5 10 10 888 889 888Orb6 10 10 1010 1013 1010Orb7 10 10 397 401 397Orb8 10 10 899 899 899Orb9 10 10 934 934 934

Orb10 10 10 944 944 944Ft06 6 6 55 55 55Ft10 10 10 930 930 930Ft20 20 5 1165 1175 1165

Fig. 5: Results of serial and parallel hybrid algorithms.

combinations which shows that running time of parallel hybrid algorithms ismuch lesser than their serial implementation.

Acknowledgements The authors are grateful to Liu et al. [1] for their proposed workwhich provided scope and the foundation for our work.


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