nitt paper a fuzzy multi-objective based un-related parallel machine scheduling

Abstract--- In this paper, we have addressed a Multi Objective Problem (MOP) which covers-minimize the makespan, tardiness, load variation, flow time and secondary resources constraints for unrelated parallel machine scheduling problem with consideration of inherent uncertainty in processing times and due dates. To resolve the above computationally challenge problem in a reasonable time and to find a good approximation of Pareto frontier, we have proposed a Multi-Objective Evolutionary Algorithm (MOEA) based Fuzzy-Non-dominated Sorting Genetic Algorithm (FNSGA-II). Over randomly generated test problems, the performance of the proposed algorithm is validated and the results are analyzed for the benefit of the manufacturer. Finally, statistical analysis has been conducted and found that the proposed algorithm performs reasonably well in terms of quality, computational time, diversity and spacing metrics.

Keywords– Load Variation, MOEA, Makespan, Tardiness, Unrelated Parallel Machine

I. INTRODUCTION HE parallel machine scheduling problem (PMSP) is a very common production environment that can be found in

several manufacturing situations, in which the allocation of a set of jobs are assigned to a number of parallel machines in order to meet customer’s requirement. In general, from the literature it is found that PMSP is categorized as identical, uniform and unrelated parallel machine scheduling problem[1]. Among the above mentioned classifications, it is found that for unrelated PMSP (UPMSP) the processing time of each job depends on machines with different processing capabilities it is assigned to, where workstations are supposed to be non-identical. However, handling of UPMSP’s with real life cases is a challenge for researchers and Practitioners due

V.K.Manupati, Assistant professor, Department of Mechanical Engineering, KL University, Andhra Pradesh, India. E-mail: [email protected]

R. Sridharan, Professor, Department of mechanical Engineering, NIT Calicut, Kerala, India. E-mail: [email protected]

N. Arudhra, Student, Department of Mechanical Engineering, KL University, Andhra Pradesh, India. E-mail: [email protected]

P.Bharadwaj, Student, Department of Mechanical Engineering, KL University, Andhra Pradesh, India. E-mail:[email protected]

D.Trinath, Student, Department of Mechanical Engineering, KL University, Andhra Pradesh, India. E-mail:[email protected]

M.V.B.T.Santhi, Assistant professor, Department of Computer Science and Engineering, KL University, Andhra Pradesh, India. E-mail: [email protected]

to the fact that they are mostly NP-hard in nature and their special characteristics/requirements in practice [2].

The unrelated parallel machine system has been solved with many methods and techniques and it has been widely addressed in the literature from past few years. Leeet al. [6] addressed the UPMSP to minimize the objective function i.e., with dedicated machines and common deadlines kim et al.[4] and arnaout et al.[5] presented a simulated Annealing and ant colony algorithm optimization approach for the UPMSP with sequence dependent setup times.

In real world scheduling problems many factors are often uncertain. This research deals with scheduling of number of jobs on an unrelated parallel machine system with secondary resource constraints. Where, each job can only be processed if its required machine and other secondary resources are available

The remainder of this paper is organized as follows. In section 2, we give a detailed description of the problem with the basic assumptions and developed a mathematical model along with the constraints. Section 3 explains the mapping of NSGA-II, MPSO algorithms for solving the MOP. The experimentation with an illustrative example having different complex scenarios is illustrated in Section 4. In Section 5, the results and their discussions are detailed. The paper concludes with Section 6 suggesting the directions of the future work.

II. PROBLEM DESCRIPTION In this paper, we propose a multi-objective UPMSP with

non-zero ready times, job-sequence-and machine-dependent setup times, and the auxiliary resource constraints in a fuzzy based environment for improving the performance measures such as makespan, flow time, tardiness and machine load variation. Here we consider a set of jobs with j = {1,2.., n} of n that have to be processed on exactly one machine out of a set M= {1,2.., m} of m parallel machines. The jobs are processed on machines that are available continuously to process at most one job J at a time. When jobs are processing on the machines we have consider that pre-emption is not allowed and jobs are processing on machines k with their processing times, tardiness, makespan and load variation. In this problem, we consider a sequence dependent and machine dependent which deals with processing times, and setup times, when a machine switches its production from job i to job j on the same machine. We do not consider setup times before processing the first job on a machine. In this problem the occurrence of machine idle time is allowed. Idle time on a machine may be

A fuzzy Multi-Objective Based Un-Related Parallel Machines Scheduling Problem with Sequence and

Machine Dependent Setup Times V.K.Manupati, R. Sridharan, N. Arudhra, P. Bharadwaj, D.Trinath and M.V.B.T. Santhi

T

International Conference on Advances in Design and Manufacturing (ICAD&M'14) 1275

ISBN 978-93-84743-12-3 © 2014 Bonfring

required to complete a job on its due date, avoiding earliness. However, to meet the requirements from real-world production, we have considered more than two objectives corresponding to constraints for Multi objective problem. The above mentioned problem makes several assumptions that are worth highlighting. 2.1 Assumptions

1. Each job requires an operation that can be done on all machines,

2. Job’s setup times are sequence and machine dependent,

3. Jobs may have different arrival (ready) times, 4. Assignment of a job to a machine is permitted if the

required secondary resource(s) (e.g., tool, die) is (are) available,

5. Processing times of jobs are machine dependent, 6. Preemption and machine breakdown are not allowed, 7. Processing times and due dates of jobs due to possible

fluctuations in real world are subject toepistemic uncertainty (i.e. lack of knowledge in estimating these parameters precisely).

The indices, parameters and variables used to formulate the problem mathematically are described below.

1 0

1

0

1

sin

Decision varibles :if job i precedes job j on machine kotherwise

if job i is assigned to machine kotherwise

if proces g of job i is finished before

Xijk

Y jk

Zij

=

=

= sin 0

proces g of job j startsotherwise

2.2 Mathematical Model By allocating the above mentioned assumptions and

notations, the problem is being formulated as the following fuzzy mixed–integer non–linear programming (FMINLP) model.


ISBN 978-93-84743-12-3 © 2014 Bonfring

Objectives :

N= w c (1)i i

i=1N

= w (c -c )(D) (2)i i+1 i

MinZ1

Min Z2

M

i=1

= w (cin -s )i ii

Z3 i

∑

∑

N (3)

=1M k= (D)(c -c ) (4)max max

k=1

Subjected to constraints :

1 ; 1 1

MinZ4

N m

Xijk ji ki j

∑

∑

= ∀∑ ∑= =≠

;

(5)

(6)

1, ,

,1

;

1

1

NX Yijk jk j k

ii j

NX Yijk ik

ji N

j

= ∀∑=≠

≤∑=

= …

≠

(7)

, , (8)

, ) (9)

,

(1 ) Max{c } ; ,

(1 ) (c , ) ; (1

1;

c A Xijk s s pj N i ijk j jkM

c A z Max s p Yj N ij i j jk jkk

i j k

i

z zij ji

jÎS i jr

i jÎ

+ − ≥ + +

+ − ≥ + ∑=

+ =

≠

( )

(10),

, (11),

;

; 1

S i jr

i jÎS i jr

T C di

MX zijk ij

i

k

i i

≠

≠≤∑=

≥ ∀─

{ } ( )

(12)

0; 0; , 0,1 ; , , . 13 , ,C T I X Y Z Î i j ki i ijk ik ij≥ ≥

The above mentioned objectives, i.e. minimization of total weighted makespan(Z1), total weighted flow time(Z2)total weighted tardiness(Z3), machine load variation(Z4) are given by Eqs. (1) – (4), respectively. Eq (5)assures that each job is only assigned to one position on a machine. Eq(6) suggests that if job jis allotted to machine k, then it is adopted by another job indicating dummy job 0. Constraint (7)specify that at most one job can immediately adopt the previously allotted job i on machine k. Constraint (8)computes the completion time of job when it is processed instantly after job i on machine . Constraint (9 and 10)ensures that if job I and j needs same tool, one must be finished earlier before starting the other. Constraint (11)gives the relation between Xijk and Zij.Eq(12)computes tardiness of job i.Eventually Eq(13)points the non-negativity and integrality constraints.

III. MULTI-OBJECTIVE BASED EVOLUTIONARY ALGORITHM

Most of the UPMSPs are NP-hard in nature and due to its characteristic it is necessary to solve more than two objectives simultaneously to find out the optimal solutions that are very close to reality. In general, it is very difficult to solve the multiple objectives simultaneously thus we adopted multi-objective based evolutionary algorithms for solving the above mentioned complex problem.[7] In this paper, we apply a NSGA-II which includes parito ranking and crowding system mechanism for selection of the individuals

3.1.Nondominated sorting Genetic Algorithm Goldberg was the first who suggested the concept of non-

dominated sorting genetic algorithm (NSGA) and Srinivas and Deb were the first to implement it. Where NSGA differs from well known simple genetic algorithm is only in the way the selection operator works. Although NSGA has been used to solve a variety of MOPs, its main drawbacks include: (i)high computational complexity of non-dominated sorting, (ii) lack of elitism, and (iii) the need for specifying the tunable parameter. With the advent of NSGA-II, the above mentioned drawbacks have been resolved.

3.1.1. Initial population generation

Fig.1: Initial Population Generation. We randomly generate the initial population for a given

population size. According to the characteristics of the proposed problem, encoding schema and its representation is designed. An example of the encoding scheme for the problem is illustrated in Fig 1. Here, we have considered the number of jobs(1 to 6) as chromosome which is having length seven, where jobs should process on machines (1 to 4) in such a way that the secondary resources ( 1 to 3 ) must available. In the second set of the chromosome representation the machines and their secondary resources and the encoding scheme of this layer is shown in Fig 2b. In this layer, zeroes indicate, no job assign on the machine. One more advantage of indicating zeroes is for maintaining the chromosome length as uniform.

3.1.2. Evolutionary operators

Cross over Crossover chooses genes from parent chromosomes and

produce a new offspring. Crossover is done by swapping portions between two parent chromosomes. It is based on basic parameter called crossover probability. Specific crossover built for a particular problem can improve performance of the genetic algorithm.

Mutation


ISBN 978-93-84743-12-3 © 2014 Bonfring

Mutation randomly used for alternating the values of particular genes to prevent diminishing all solutions in population into a local optimum of solved problem.

Non dominated sorting Non dominating sorting is done by choosing the best

pareto front which is spread throughout the space by tournament selection based on Pareto domination.

Fig.2: Schematic Procedure for Processing NSGA-II.

OP –Offspring population PP - Parent population Fig 2, describes the schematic algorithm throughout the

processing of NSGA-II (Deb et al. 2000)

crowding distance Crowding distance can be calculated for all chromosomes

of same pareto front, then individual having large crowding distance is selected to obtain uniform distribution.

Stopping rule If the maximum generation is reached then stop.

IV. EXPERIMENTATION Here, Card (g) describes the cardinality of test sample, and

α, β represents the control of job arrival and control of tightness respectively.

Table 1: Test Sample (TS) Form Machine Shop

Table 1 represents a raw data of random test sample (TS)

taken from a cable and manufacturing company that is incorporated as unrelated parallel machine which have sequence dependent and machine dependent setup times .

V. RESULTS AND DISCUSSION

Figure 3: Pareto Optimal Front for Sets Considering Makespan vs. Flow Time.

Figure 4: Pareto Optimal Front for Sets Considering Makespan vs. Tardiness.

Figure 5: Pareto Optimal Front for Sets Considering Makespan vs. Machine Load Variation.

Pareto optimal front is obtained using the evolutionary operators based on concerned performance measures(Fig:3, Fig:4, Fig: 5) such as minimization of makespan, flowtime, tardiness, and machine load variation, for randomly selected test samples. The make span is the total completion time of jobs, flow time is the time taken by the required job to complete, tardiness is lateness or slow processing i.e., more than due date, machine load variation is the time to setup or calibration of machine during processing of jobs on the machines. Results show better solutions and achieved Pareto optimal solutions

VI. CONCLUSION AND FUTURE WORK This paper proposes a fuzzy multi objective UPMSP with

non zero ready times, sequence and machine dependent setup times. Moreover, we have considered secondary resource

0

200

400

45.542.940.931.123.120.916.614.2 7.3

flow

time

Makespan

Makespan vs FlowtimeSet 1Set 2

0.00

100.00

200.00

45.5

42.9

40.9

32.3

31.9

23.8

20.9

16.6

14.2 7.3

Tard

ines

s

Makespan

Makespan vs Tardiness

Set 3

0.00

200.00

25.8

33.9

18.7

45.9 9.3

24.9

46.8

48.3

13.4

35.6

Mac

hine

Loa

d va

riatio

n

Makespan

Makespan vs Machine load variation Set 1

Set 2Set 3


ISBN 978-93-84743-12-3 © 2014 Bonfring

constraints to simultaneously solving different performance measures to obtain pareto optimal solutions. Here, we have considered different performance measures as minimization of makespan, total weighted tardiness, total weighted flow time, machine load variation. To solve the above mentioned complex NP hard problem it is necessary to adopt a multi-objective based evolutionary algorithm i.e., NSGA-II. With different problem sets by varying its cardinality starting time and due date the performance of the algorithm is examined. Experimental results show that the proposed approach is performing better and obtained optimal solutions in the future work, one can compare with different multi-objective evolutionary algorithms and also on large data sets that can bring the experimentation close to reality.

REFERENCES [1] T. Cheng, C. Sin, A state-of-the-art review of parallel-machine

schedulingresearch, European Journal of Operational Research 47 (1990) 271–292.

[2] S.A. TORABI, N.Sahebjamnia, S.A.Mansouri,M. Aramon bajestani, “A particle swarm optimization for fuzzy multi-objective unrelated parallel machines scheduling problem”,Applied Soft Computing, 13(2013)4750-4762.

[3] J.-Y. Bang, Y.-D. Kim, Scheduling algorithms for a semiconductor probing facil-ity, Computers and Operations Research 38 (2011) 666–673.

[4] J.-P. Arnaout, G. Rabadi, R. Musa, A two-stage ant colony optimizationalgorithm to minimize the makespan on unrelated parallel machines withsequence-dependent setup times, Journal of Intelligent Manufacturing 21(2010) 693–701.

[5] F.A. Cappadonna,A. Costa,S. Fichera, “Make span minimization of unrelated parallel machines with limited human resources”,Procedia CIRP 12(2013)450-455.

[6] Lee, Cheng-Hsiung, Ching-Jong Liao, and Chien-Wen Chao. "Unrelated parallel machine scheduling with dedicated machines and common deadline." Computers & Industrial Engineering (2014).

[7] Deb, Kalyanmoy. "Multi-objective optimization." Search methodologies. Springer US, 2014. 403-449.

Dr. Vijay Kumar Manupati received the B-Tech degree in Department of mechanical engineering from Siddhardha Engineering College, and M-Tech in Department of Mechanical engineering with Industrial Engineering and Management as a specialization from National Institute of Technology Calicut. He received his PhD in Department of Industrial and Systems Engineering from Indian Institute of Technology Kharagpur. His current research interests belong include

Intelligent manufacturing systems, agent/multi-agent/mobile-agent systems for distributed control, simulation, integration of process planning and scheduling in manufacturing. He has published papers inInternational Journal of Production Research, Computers and Industrial Engineering, and International Journal of Advanced Manufacturing Technology.

Dr.R.Sridhran completed his B. E. Honours in Mechanical engineering from madras University, M-Tech in Department of Industrial Engineering and Operation Research specialization from IIT Kharagpur, and Ph. D. in department of Industrial Engineering from IIT Bombay. He had published many publications and technical reports in reputed journals. His areas of research are Operation Management, Supply Chain

Management, Decision Modelling, Simulation Modelling and Analysis

Arudhra Nerella pursuing his Bachelor degree in Mechanical Engineering from KLUniversity (Vijayawada, Andhra Pradesh, India). He worked on several projects like “Controlling the Defects in Centrifugal Casting”, “Developed a Flexible Manufacturing System using Cellular Manufacturing Techniques”. He is a Member of American Society of

Mechanical Engineers. His research interest includes manufacturing, production, and Supply chain management

TRINATH DESIBOYINA, was born on June 12th 1992 and perusing B .Tech in mechanical engineering at KLUniversity(2011-2015).He has done internship at GENTING LANCO Power(India) Private Limited, Kondapalli,and done an automobile oriented project on ‘Aero car’s, an environmental oriented project on ‘Solar cell’, and a technical project on ‘Conditional monitoring on rotating machines using vibrational

analysis’ these were done as a part of curriculum

BHARADWAJ POPURI: was born on July 30th 1993 and pursuing B.Tech in mechanical engineering at KLUniversity (2011-2015),he has done internship on Quality control charts for variables and attributes at “Model Dairy Private Limited”, and done his project’s on ‘ wire electric discharge machining’ and ‘lean manufacturing in processing industry’ as a part of curriculum.

M.V.B.T.Santhi received the M-Tech degree in Computer Science and Engineering from AN University, in 2010 & B-Tech degree in Computer Science and Engineering from Jawaharlal Nehru Technology University, Hyderabad, in 2003 .She is currently working as an Assistant Professor in Department of Computer Science & Engineering, Koneru Lakshmaiah University, Vaddeswaram, Guntur

Dist.A.P


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