International Journal of Current Research and Review
ISSN: 2231-2196 (Print)ISSN: 0975-5241 (Online)
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IJCRR - 9(6), March, 2017

Pages: 15-21

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A Heuristic Approach to Minimize Utilization Time in N * 2 Specially Structured Flow Shop Scheduling Problem Including Setup Time, Transportation Time and Jobs in a String of Disjoint Job Blocks

Author: Deepak Gupta, Shashi Bala, Pradeep Bishnoi

Category: General Sciences

Abstract:The present paper is an attempt to obtain a sequence of jobs through heuristic method to optimize the utilization time of machines for specially structured n-job and 2-machine flow shop scheduling problem. Also, the jobs are to be processed in a string of disjoint job blocks having sequence independent setup times separated from processing times each associated with their respective probabilities including transportation time. In flow shop scheduling minimization of total elapsed time may not always result in minimization of utilization time of machines. To minimize the utilization time an algorithm is proposed and a numerical problem is solved to authenticate the algorithm.

Keywords: Elapsed time, Expected processing time, Expected setup time, Jobs in a String

Full Text:

INTRODUCTION

Scheduling models deals with determination of an optimal sequence to provide service to customers or to perform a set of jobs in order to minimize the total elapsed time or some other suitable performance measure. In today’s manufacturing and distribution systems, scheduling have significant role to meet customer requirements as quickly as possible while maximizing the profits. In flow shop scheduling problem n-jobs are processed on m-machines and the processing order i.e. the order in which various machines are required for completing the job is given. The common objectives in flow shop scheduling problems are to minimize some performance measures such as make span, mean flow time, mean tardiness, mean setup time, number of tardy jobs and mean number of setups. Johnson(1) developed an algorithm for two stage production schedule for minimizing the make span. Palmer DS(2) developed a heuristic algorithm for sequencing jobs to minimize the total elapsed time. Gupta JND(3) studied specially structured flow shop scheduling problem to obtain an optimal sequence of jobs. Gupta D, Sharma S and Bala S(4) investigated specially structured two stage flow shop scheduling problem under rental situation.

Corwin BD et al(5) studied two machine flow shop scheduling problems with sequence dependent setup time. Setup time includes the time to prepare the machines, obtaining, adjusting and returning tools for an operation, cleaning up the machines, setting the necessary jigs and fixtures and inspecting and positioning the process material. Setup time has an important part as reduction in setup time leads to increase in output, profitability and customer satisfaction in an organization. The setup times in scheduling problems can be classified into two categories viz. sequence-independent setup times and sequence-dependent setup times. Sequence-independent setup time depends solely on the current job to be processed regardless of previously processed job. Sequence dependent setup time depend both on the current job and previously processed job.

In most manufacturing and distribution systems, semi finished jobs are moved from one processing facility to another for further processing and finished jobs are delivered to customers or store houses by suitable modes. In job sequencing the time required in moving a job from one machine to another machine during the processing of jobs is known as transportation time. Maggu and Dass(6) considered a two machine flow shop problem including transportation time of jobs from first machine to the second machine. Langston(7) gave heuristic solution to minimize make span for a k-station flow shop problem where each station has a number of machines that can be used to process jobs and there is only one transporter with a capacity to transport one job with transportation times dependent on the physical locations of the starting and destination machines. Chung et al(8) studied machine scheduling problems with explicit transportation considerations. They considered scheduling models for transporting a semi-finished job from one machine to another for further processing and for delivering finished jobs to the customers or warehouses. Gupta D, Sharma S and Bala S(9) applied heuristic algorithm to minimize the utilization time and rental cost of machines for two stage specially structured flow shop scheduling problem involving transportation time. 

Maggu and Das(10) established the basic concept of equivalent job for job block in scheduling theory. The string of disjoint job blocks consist of two disjoint job blocks such that in one job block the order of jobs is fixed and in second job block the order of jobs is arbitrary. Anup and Maggu(11) gave an optimal schedule for n × 2 flow shop problem with job blocks of jobs in which first job in each job block being the same. Heydari(12) studied flow shop scheduling problem with processing of jobs in a string of disjoint job blocks. Singh TP, Kumar V and Gupta D(13) studied n × 2 flow-shop scheduling problem in which processing time, set up time each associated with probabilities along with jobs in a string of disjoint job blocks. Gupta D, Sharma S and Gulati N(14) studied n×3 flow-shop scheduling problem in which processing time, set up time each associated with probabilities along with jobs in a string of disjoint job blocks. Gupta et al(15) considered specially structured two stage flow shop scheduling problem having jobs in a string of disjoint job blocks.

In this paper we investigate n-job and two machines specially structured flow shop scheduling problem with sequence independent setup times separated from processing times each associated with probabilities including transportation time and jobs to be processed as string of disjoint job blocks. The aim of the study is to obtain a sequence of jobs that optimizes the utilization time of machines. An algorithm is proposed to solve the problem and is validated with the help of a numerical example.                   

PRACTICAL SITUATION

The service centres and industrial units must utilize their resources in an optimal manner to increase their profits and productivity. For optimal utilization of available resources there must be a proper scheduling system and this makes scheduling a highly important aspect of industrial establishments and service sectors. Specially structured two machine flow shop scheduling problem has been considered as there are many realistic situations where the processing times of jobs on the two machines are related in specific manner. In many practical situations such as chemical, food processing, pharmaceutical, metal processing, printing etc. setup time is required while shifting from one operation to another.

The idea of job block has practical importance to deal with ordering of jobs so as to ensure priority in service to the preferred customers and/or jobs and hence maximize profits. Scheduling models with jobs in a string of disjoint job blocks are necessary in cases where certain orderings of jobs are prescribed either by technological constraints or by externally imposed policy(10).

During the processing of jobs in many production and distribution units, semi-finished tasks are transferred from one machine to another through various modes such as automated guided vehicles and conveyors, and finished jobs are delivered to consumers or storehouses by vehicles such as trains or trucks. Machine scheduling models that take into account the job transportation time are indeed more realistic than those scheduling models that do not take into consideration these parameters.

NOTATIONS

The following notations have been used throughout the paper:

σ :   Sequence of n- jobs obtained by applying Johnson’s algorithm.

σ k:  Sequence of jobs obtained by applying the proposed algorithm, k = 1, 2, 3, ------.

Mj:  Machine j, j= 1, 2.

aij:   Processing time of ith job on machine Mj.

sij:   Set up time of ith job on machine Mj.

pij:   Probability associated to the processing time aij.

qij:   Probability associated to the set up time sij.

Aij:  Expected processing time of ith job on machine Mj.

Sij:   Expected set up time of ith job on machine Mj.

Ti,12 : Transportation time of ith job from first machine to second machine.

tij (σ k):  Completion time of ith job of sequence σ k on machine Mj.       

T (σ k):   Total elapsed time for jobs 1, 2, --------, n for sequence σ k.

Uj (σ k):  Utilization time for which machine Mj is required for sequence σ k.

Aij (σ k):  Expected processing time of ith job on machine Mj for sequence σ k.

α :  Fix order job block.        

β :  Job block with arbitrary order.

βk : Job block with jobs in an optimal order obtained by applying the proposed algorithm,

       k = 1, 2, 3, ------.

S:  String of job blocks α and β i.e. S = (α, β)

S?: Optimal string of job blocks α and βk.

ASSUMPTIONS

The assumptions for the proposed algorithm are stated below:

  1. Jobs are independent to each other and are processed thorough two machines M1 and M2 in order M1 M2.
  2. Pre-emption is not allowed. Once a job is started on a machine the process on that machine cannot be stopped unless the job is completed.
  3. Processing times must satisfy the structural conditions mini{Gi}  ≥ maxi{Hi}  or maxi{Gi}  ≤ mini{Hi} .
  4. Each job has two operations and each job is processed through each of the machine once and only once.
  5. The independency of processing times of jobs on the schedule is maintained.
  6. Only one machine of each type is available.
  7. Jobs i1, i2, ---------------, ih are to be processed as a job block (i1, i2, ----------------, ih) showing priority of job i1 over i2 etc. in that order in case of a fixed order job block.                                                                                                                                                

    DEFINITION

    Completion time of ith job on machine Mj is given by,

                  tij = max (ti−1, j + S i−1, j,  ti, j−1 + ti,  12 ) + Aij ; j ≥ 2,

    where Aij = Expected processing time of ith job on machine Mj and Sij = Expected set up time of ith job on machine Mj.

    THEOREM

    If Ai1  Aj2 for each i and j, then K1, K2, ----------, Kn is a monotonically increasing sequence, where Kn = i=1nAi1  -  i=1n-1Ai2 .

    Proof: Let Kn = i=1nAi1  -  i=1n-1Ai2

                                                                                                     

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Awards, Research and Publication incentive Schemes by IJCRR

Best Article Award: 

One article from every issue is selected for the ‘Best Article Award’. Authors of selected ‘Best Article’ are rewarded with a certificate. IJCRR Editorial Board members select one ‘Best Article’ from the published issue based on originality, novelty, social usefulness of the work. The corresponding author of selected ‘Best Article Award’ is communicated and information of award is displayed on IJCRR’s website. Drop a mail to editor@ijcrr.com for more details.

Women Researcher Award:

This award is instituted to encourage women researchers to publish her work in IJCRR. Women researcher, who intends to publish her research work in IJCRR as the first author is eligible to apply for this award. Editorial Board members decide on the selection of women researchers based on the originality, novelty, and social contribution of the research work. The corresponding author of the selected manuscript is communicated and information is displayed on IJCRR’s website. Under this award selected women, the author is eligible for publication incentives. Drop a mail to editor@ijcrr.com for more details.

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‘Emerging Researcher Award’ is instituted to encourage student researchers to publish their work in IJCRR. Student researchers, who intend to publish their research or review work in IJCRR as the first author are eligible to apply for this award. Editorial Board members decide on the selection of student researchers for the said award based on originality, novelty, and social applicability of the research work. Under this award selected student researcher is eligible for publication incentives. Drop a mail to editor@ijcrr.com for more details.


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List of Awardees

A Study by Ese Anibor et al. "Evaluation of Temporomandibular Joint Disorders Among Delta State University Students in Abraka, Nigeria" from Vol 13 issue 16 received Emerging Researcher Award


A Study by Alkhansa Mahmoud et al. entitled "mRNA Expression of Somatostatin Receptors (1-5) in MCF7 and MDA-MB231 Breast Cancer Cells" from Vol 13 issue 06 received Emerging Researcher Award


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