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Introduction INSTRUCTOR: DR. OSWALDO AGUIRRE
COURSE: MFG 5321
SEMESTER: FALL 2015
Modeling/ Analysis and Mfg Process
Planning & Scheduling
Planning and scheduling are decision-making processes that are used in many manufacturing and service industries
These techniques play an important role in areas such as manufacturing, transportation, information, communication, etc.
The optimization of these tasks relay on mathematical techniques and heuristics methods to allocate limited resources.
An Automobile Assembly Line
Different car models
Different features
Bottlenecks Paint shop
MAXIMIZE THE CAR PRODUCTION
Scheduling a Sport Tournament
Games have to be scheduled over a fixed number of rounds
Each team should have a schedule that alternates between games at home and games away
If a city has two teams both player can play at the same time
Scheduling has to take in consideration transportation between cities
Schedule has to maximize tv ratings
MAXIMIZE TOTAL PROFIT
Scheduling Nurses in a Hospital
Every hospital has a staffing requirements that change from day to day
The number of worker needed is different each day
Different shifts involve different cost
All nurses work a specific number of shits per week
Consecutive shifts can not be assigned to the same worker
MINIMIZE COST
Scheduling Problems
A System installation Project: The project involves different stages such as manufacturing, testing, etc. Complete the entire process in the shortest time
Planning and scheduling in a supply chain: In a supply chain system a collaboration between customer and manufacturer is required The overall goal is to minimize the total cost including production costs,
transportation costs and inventory holding costs
Routing and scheduling Airplanes: Based on the demand, the airline can estimate the profit of assessing a specific type of aircraft to a flight. The objective is the optimization of the utilization of scare resources.
Challenges in scheduling Airline Crew Scheduling: State-of-the-art o Gopalakrishnan, B. & Johnson, E.
Solving Challenging Real-World Scheduling Problems o Kyngas, J (Dissertation)
The Top ten job scheduling Challenges and how to solve them o Norman Martin
Is the scheduling a solved problem? o Smith, F
Air line Scheduling During the past years there has been an increase in sizes in all areas of
airline planning ◦ Airlines expansion
◦ Increase in people using the service
Air planning consists of fleet scheduling and crew scheduling ◦ Crew scheduling is considering a challenging problem in air planning
◦ Crew cost represent one of the largest cost factors for airlines
◦ Effective assignments of crews to flight is a very important aspect of air planning
◦ U.S. Airlines schedule more than 2500 flights per day over 150 cities.
◦ Due to the scale of the problem is impractical to preform it manually.
◦ The crew scheduling problem is relatively easy to model mathematically and interpret as an optimization problem
◦ Solving the model efficiently is a very challenging task
Air line Scheduling Crew scheduling
◦ Is the process of assigning crews to aerate an airline system
Crew pairing ◦ Sequence of flights that begin and end at a crew base such that in a sequence the arrival city of
a flight coincides with the departure city of the next flight
The crew scheduling problem is a challenging problem to solve due to the following reasons ◦ The number of parings is extremely large (over 100 million)
◦ Many rules and regulations have to be satisfied
◦ Crew cost depend on complex crew pay guaranties and are highly nonlinear
The purpose of the airline crew paring problem is to generate a set of minimal cost crew parings covering all flights legs
Air line Scheduling Solution methods
◦ Trip pairing for airline crew scheduling
◦ Linear programming algorithms
◦ Volume algorithms
◦ Integer programming algorithms
◦ Approximation algorithm
Nurse scheduling problem A NP- Hard problem
NP-Hard o Non deterministic polynomial-time hard
o Refers to the time to solve a problem in computational complexity theory
What is a NP-Hard Problem ?
What is a NP problem?
What is a P problem?
P,NP, NP-Hard P problem Yes or no problem that can be solved easily
Problem solved in polynomial time
Problem can be solved quickly
2*3 =6
123456789101112*121110987654321= 1.4952x10^28
NP Problem Yes or not problem solved in non deterministic polynomial time
Problem easily checkable if you get an the yes answer
?*?= 1.4952x10^28
NP Problems Why are NP Problems difficult?
◦ Np Problems involve searching
◦ Searching is a time consuming process
It is possible to find a solution without searching
No-One has prove that is not possible
Find a needle in a haystack
P,NP, NP-Hard P = NP or P≠NP A $1,000,000 question
NP- Hard problem A problem at least as difficult as NP Problem
A very hard to solve problem
A problem that can not be solve in realistic time
Many real world scheduling Problem are NP, NP-Complete, or NP-hard problems
Scheduling Models
Manufacturing models Manufacturing systems
Deterministic models
Machine job models
Service Models Provide services
Resources are not fixed
Denying a customer a service is a more common practice than not delivering a product
Manufacturing Models
Job Machine model Single machine model
Parallel machine model
n jobs
n jobs
m machines
Job Machine Models
Flow Shop Model Jobs have to undergo multiple operations on a number of different
machines
The routes for all jobs are identical
n jobs
m machines
Job Machine Models
Job Shop Model In a multi- operation shops, jobs have different routes
Not all jobs has to visit all the machines
Job #1
Job #2
Machine #1
Machine #2
Machine #3
Machine Models Parameters
Processing time (pij): Time of job j has to spend in machine i
Release date(rj): It is the time that the job arrives at the system
Due date (dj): it is the due date for a job
Weight (wj): It is a priority factor reflecting the importance of job j
Starting time(Sij): Time when job j starts its processing on machine i
Completion time(Cij): It is the time when job j is completed on machine i
Example #1
Assume sequence J2 -> J1-> J3
Calculate completion times
Job Pj rj dj wj
1 12 2 15 20
2 10 8 20 10
3 5 4 18 30
Example #1
Sequence J2 -> J1-> J3
Calculate completion times
Job Pj rj dj wj Cj Sj
1 12 2 15 20 30 18
2 10 8 20 10 18 8
3 5 4 18 30 35 30
J2
0 8 18
10
J1
30
12
J3
35
5
Performance Measures
Makespan: The total time to process all the jobs in all the machines
Flow Time: the sum of all completion times
Lateness: amount of time above the due time of specific job
Maximum Lateness
𝐶𝑚𝑎𝑥 = max(𝐶1, … , 𝐶𝑛)
𝐿𝑖 = (𝐶𝑗 − 𝑑𝑗)
𝐿𝑚𝑎𝑥 = max(𝐿1, … , 𝐿𝑛)
𝐶𝑗
𝑛
𝑗=1
Performance Measures
Tardy jobs: Number of hobs that were completed after their due dates
Tardiness: lateness of a tardy job
Average tardiness
Weighted tardiness
𝑇𝑖 = max(𝐶𝑗 − 𝑑𝑗 , 0)
𝑇𝑗
𝑛
𝑛
𝑗=1
𝑤𝑗𝑇𝑗
𝑛
𝑗=1
Example Given J3-> J2>J1
Evaluate: a) Makespan
b) Flow Time
c) # of tady jobs
d) Total tardiness
e) Weighted tardiness
Job Pj rj dj wj
1 12 2 15 20
2 10 10 20 10
3 5 4 18 30
𝑤𝑗𝑇𝑗
𝑛
𝑗=1
Example Given J3 J2 J1
Evaluate: a) Makespan =32
b) Flow Time =61
c) # of tady jobs = 1
d) Total tardiness =17
e) Weighted tardiness =340
Job Pj rj dj wj
1 12 2 15 20
2 10 10 20 10
3 5 4 18 30