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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
1
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
2
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
3
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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)
4
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
5
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
7
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
9
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
11
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
12
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
13
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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.
14
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
15
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
16
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
10:45 am to 12:15 pm 03-30-12
TABLE 1:TABLE 1:Job Time for
Turning(minutes)Time for Threading(minutes)
1 3 8
2 12 102 12 10
3 5 9
4 2 6
5 9 35 9 3
6 11 117
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
10:45 am to 12:15 pm 03-30-12
FINAL SEQUENCE:
4 1 3 2 5 6
18
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
19
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
20
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
21
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
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Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
23
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
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Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
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Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
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Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
27
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
28
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
Job 1
Sequence A B C D E
Time(hrs.) 1 2 3 5 1
Job 2
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
29
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
Job 1
Sequence A B C D E
Time(hrs.) 6 8 4 12 4
Job 2
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
30
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
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
l i d d l b h h j btotal time needed to complete both the job
Job 1
Sequence C A E F D BSequence C A E F D B
Time(hrs.) 2 3 4 5 6 1
Job 2
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
31
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
REFERENCE
Gupta, Prem Kumar and D.S. Hira, (2010),Gupta, Prem Kumar and D.S. Hira, (2010), “Operations Research”, S. Chand, New Delhi, India
32
Dr. Anannya Deb Roy MFM, NIFT, Kolkata
10:45 am to 12:15 pm 03-30-12
Thank YouMFM, NIFT, Kolkata
01:15 pm to 4:15 amMFMM03-20-12
33
DR. ANANNYA DEB ROY
10:45 am to 12:15 pm 03-30-12
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