Sequencing Models

OPERATIONS RESEARCHOPERATIONS RESEARCH

Midterm ( Jan‐June‐2012)( )

Implication of Operations Research

Linear Programming

Transportation Models

Assignment Models

Traveling Salesman Problem

10:45 am to 12:15 pm 03-30-12

Dr. Anannya Deb Roy MFM, NIFT,Kolkata

Dr. Anannya Deb RoyMFM ,NIFT ,Kolkata

O ROPERATIONS RESEARCH

End term ( Jan‐June‐2012)

Inventory Management

Sequencing Models

Q i ThQueuing Theory

Project Management

10:45 am to 12:15 pm 03-30-12

Dr. Anannya Deb Roy MFM, NIFT, Kolkata

SEQUENCING MODELSSEQUENCING MODELS

Determination of an optimal order or sequenceDetermination of an optimal order or sequence of performing a series of jobs by a no. of facilitiesfacilities

F iliti d i ifi dFacilities are arranged in a specific order

10:45 am to 12:15 pm 03-30-12

OBJECTIVES

l l d ( k )Minimize total elapsed time (markspan)

Minimizing mean flow time (mean time in the job shop)Minimizing mean flow time (mean time in the job shop)

Minimizing idle time of the machinesMinimizing idle time of the machines

Minimizing total tardiness (+ve Lateness)

10:45 am to 12:15 pm 03-30-12

OBJECTIVES

f d bMinimize no. of Tardy Jobs

Minimizing in Process Inventory CostMinimizing in Process Inventory Cost

Minimizing Idle time of the MachinesMinimizing Idle time of the Machines

Minimizing the Cost of Being Late

10:45 am to 12:15 pm 03-30-12

DIFFERENT CASES OF SEQUENCING

n Jobs and 1 Machine An Jobs and 1 Machine A

n Jobs and 2 Machines A and B (all jobs processed in the ordern Jobs and 2 Machines A and B (all jobs processed in the order say AB)

n Jobs and 3 Machines A, B and C

b d h2 jobs and m Machines

n Jobs and mMachinesn Jobs and m Machines 6

10:45 am to 12:15 pm 03-30-12

ASSUMPTION IN THE SEQUENCING PROBLEMSQ

Only 1 Operations is Carried out on a Machines at a Particular Time

Each operation Once Started Must be Completed

An Operation Must be Completed before its Succeeding Operation Can Startp

Only 1 Machine of Each Type is Available

10:45 am to 12:15 pm 03-30-12

ASSUMPTION IN THE SEQUENCING PROBLEMS (CONTINUE)ASSUMPTION IN THE SEQUENCING PROBLEMS (CONTINUE)

A job is processed as soon as possible but only in the orderspecified

Processing time is independent of the order of performing theoperationsoperations

The transportation time from one machine to another ispnegligible

Jobs are completely known and ready for processing

The cost of in‐process inventory for each job is same andnegligibly smallnegligibly small 8

10:45 am to 12:15 pm 03-30-12

PROCESSING OF N JOBS THROUGH 1 MACHINE

Shortest Processing Time (SPT) Rule

Weighted Shortest Processing Time (WSPT) Rule

Earliest Due Date (EDD) Rule

Slack Time Remaining (STR) Rule

First Come First Served (FCFS) Rule

10:45 am to 12:15 pm 03-30-12

S P T (SPT) RSHORTEST PROCESSING TIME (SPT) RULE

Sequencing the jobs in a way that the job with least processingi i i k d fi f ll d b h i h lltime is picked up first followed by the one with next smallestprocessing time

Objectives:Objectives:

Minimizing mean waiting time

Minimizing mean flow time

Minimizing mean lateness

Minimizing the mean no. of jobs waiting as in‐processiinventory

10:45 am to 12:15 pm 03-30-12

WEIGHTED SHORTEST PROCESSING TIME(WSPT) RULE

If the importance of the jobs varies, a weight wi is assigned to each job

Larger value indicates greater importance

Processing time (ti) is divided by the weight (wi)

Job sequenced in order of increasing ti/wi

10:45 am to 12:15 pm 03-30-12

EARLIEST DUE DATE (EDD) RULE

Job are sequenced in the order of non‐decreasing due dates

Obj iObjective:

Minimize maximum job lateness

Minimize maximum job tardiness

10:45 am to 12:15 pm 03-30-12

PROCESSING N JOBS THROUGH 2 MACHINES

Only 2 machines are involved

Each job is processed in the order of ABEach job is processed in the order of AB

Set‐up cost of the machines are independent of thesequence

No job is required more urgently than other

10:45 am to 12:15 pm 03-30-12

PROCEDUREStep 1: Examine the columns of processing times onmachines A and B and find the smallest value

i.e. Min(Ai, Bi)

Step 2: if this value falls in column A, schedule this job firston machine A. If this value falls in column B, schedulehi j b l hi Athis job last on machine A.

If there are equal minimum values one in each column,schedule the one in first column first on machine A; andthe one in the second column, last on machine A.

10:45 am to 12:15 pm 03-30-12

PROCEDURE (CONTINUE)

f b h l l h f l ( ) l h hIf both equal values in the first column(A), select the one with lowest entry in column B first.

If both equal values in the second column(B), select the one with lowest entry in column A first.

Step 3:

h b d d h l hCross out the job assigned and continue the process, placing the job next to the first or next to the last till all the jobs are scheduled

10:45 am to 12:15 pm 03-30-12

PROBLEM:1

A machine operator has to perform 2 operations,turning and threading, on a no. of different jobs. Theg g, jtime required to perform these operations (in minutes)for each job is known. Determine the order in which thejobs should be processed in order to minimize the totaljobs should be processed in order to minimize the totaltime required to turn out all the jobs

10:45 am to 12:15 pm 03-30-12

TABLE 1:TABLE 1:Job Time for

Turning(minutes)Time for Threading(minutes)

2 12 102 12 10

5 9 35 9 3

6 11 117

10:45 am to 12:15 pm 03-30-12

FINAL SEQUENCE:

4 1 3 2 5 6

10:45 am to 12:15 pm 03-30-12

PROBLEM 2:There are 7 jobs, each of which has to go through themachines A and B in the order AB Processing times in hrsmachines A and B in the order AB. Processing times in hrsare given as:

Job 1 2 3 4 5 6 7

Machine A 3 12 15 6 10 11 9

Machine B 8 10 10 6 12 1 3Machine B 8 10 10 6 12 1 3

Determine a sequence of these jobs that will minimize theDetermine a sequence of these jobs that will minimize thetotal elapsed time T. Also find T and idle time for eachmachines A and B

10:45 am to 12:15 pm 03-30-12

PROCESSING N JOBS THROUGH 3 MACHINES

Only 3 machines A, B and C are involved

Each job is processed in the order of ABCEach job is processed in the order of ABC

No passing of jobs is permitted

Set‐up cost of the machines are independent of the p psequence

No job is required more urgently than other

10:45 am to 12:15 pm 03-30-12

NO GENERAL SOLUTION IS AVAILABLE AT PRESENT

Methods of the n jobs and 2 machines can be applied if either one or both of the following conditions hold good

1. The minimum time on machine A is >= maximum time on Machine Bmaximum time on Machine B

2. The minimum time on machine C >= maximum time on machine B

10:45 am to 12:15 pm 03-30-12

PROCEDURE

The problems of n jobs and 3 machines isp jreplaced by n jobs and 2 machines

GTwo machines denoted by G and H where

Gi = Ai +Bi and Hi= Bi+CiGi = Ai +Bi and Hi= Bi+Ci

10:45 am to 12:15 pm 03-30-12

PROBLEM 3:PROBLEM 3:

A machine operator has to perform 3 operations:t i th di d k li b fturning, threading and knurling on a number ofdifferent jobs. The time required to perform theseoperations for each job is known. Determine the orderp jin which the jobs should be processed in order tominimize the total time required to turn out all the jobs.Also find the idle time for the total operationsAlso find the idle time for the total operations

10:45 am to 12:15 pm 03-30-12

TABLE : 3

J b Ti f Ti f Ti fJob Time for turning (m)

Time for threading (m)

Time for knurling (m)

1 3 8 131 3 8 13

2 12 6 14

3 5 4 93 5 4 9

4 2 6 12

5 9 3 8

6 11 1 13

10:45 am to 12:15 pm 03-30-12

TABLE : 4

J b Gi Ti f (t i Hi Ti fJob Gi =Time for (turning + threading ) in mnts

Hi = Time for (threading = knurling) in mnts

1 3 + 8 11 8 + 13 211 3 + 8 =11 8 + 13 =21

2 12 +6 =18 6 + 14 =20

3 5 +4 = 9 4 + 9 = 133 5 +4 = 9 4 + 9 = 13

4 2 +6 = 8 6 + 12 = 18

5 9 +3 =12 3 + 8 = 115 9 +3 =12 3 + 8 = 11

6 11 +1 =12 1 + 13 = 14

10:45 am to 12:15 pm 03-30-12

PROBLEM 4:PROBLEM 4:There are 5 jobs each of which is to be processed through3 machines A, B and C in the order ABC. Processing timein hrs are:in hrs are:

Job Time in A ( i t )

Time in B ( i t )

Time in C ( i t )(minutes) (minutes) (minutes)

1 3 4 7

2 8 5 92 8 5 9

3 7 1 5

4 5 2 6

5 4 3 10

10:45 am to 12:15 pm 03-30-12

PROBLEM 4 (CONTINUES):PROBLEM 4 (CONTINUES):Find the sequence that minimize the total elapsed timerequired to complete the following tasks. Each task is

d i t f th hi A B d C iprocessed in any two of the machines A, B and C in anyorder

Machine 1 2 3 4 5 6 7

A 12 6 5 3 5 7 6A 12 6 5 3 5 7 6

B 7 8 9 8 7 8 3

C 3 4 11 5 2 8 4C 3 4 11 5 2 8 4

10:45 am to 12:15 pm 03-30-12

PROCESSING 2 JOBS THROUGH M MACHINES

Only m machines A, B ,C …..k are involved

Only 2 jobs A and B are to be performedOnly 2 jobs A and B are to be performed

The actual or expected processing times are known

Each machines can work only 1 job at a timey j

Storage space for in process inventory is available

10:45 am to 12:15 pm 03-30-12

PROBLEM:5

Using graphical method, determine the optimal sequence neededto process jobs 1 and 2 on 5 machines, A, B, C, D and E. For eachmachine find the job which should be done first. Also calculate the

l i d d l b h h j btotal time needed to complete both the job

Sequence A B C D E

Time(hrs.) 1 2 3 5 1

Sequence C A D E B

Ti (h ) 3 4 2 1 5Time(hrs.) 3 4 2 1 5

10:45 am to 12:15 pm 03-30-12

PROBLEM:5

Using graphical method, determine the optimal sequence neededto process jobs 1 and 2 on 5 machines, A, B, C, D and E. For eachmachine find the job which should be done first. Also calculate the

Sequence A B C D E

Time(hrs.) 6 8 4 12 4

Sequence B C A D E

Ti (h ) 10 8 6 4 12Time(hrs.) 10 8 6 4 12

10:45 am to 12:15 pm 03-30-12

PROBLEM:5

Using graphical method, determine the optimal sequence neededto process jobs 1 and 2 on 6 machines, A, B, C, D ,E and F. For eachmachine find the job which should be done first. Also calculate the

Sequence C A E F D BSequence C A E F D B

Time(hrs.) 2 3 4 5 6 1

Sequence B A E F C D

Time(hrs.) 3 2 5 3 2 3

10:45 am to 12:15 pm 03-30-12

REFERENCE

Gupta, Prem Kumar and D.S. Hira, (2010),Gupta, Prem Kumar and D.S. Hira, (2010), “Operations Research”, S. Chand, New Delhi, India

10:45 am to 12:15 pm 03-30-12

Thank YouMFM, NIFT, Kolkata

01:15 pm to 4:15 amMFMM03-20-12

DR. ANANNYA DEB ROY

10:45 am to 12:15 pm 03-30-12

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