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

Performance measures Makespan

Setup Cost

Earliness and tardiness cost

Personnel cost