1. Some Topics in OR

2. (1) Linear programming (LP)
-Transportation problem( cost, time)
-Assignment problem, travelling salesman problem and knapsack problem
-Integer LP
(2) Non Linear programming (NLP)
(3) Dynamic programming (DP)
(4) Game Theory
(5) Project Scheduling by PERT and CPM
(6) Inventory Model
(7) Queuing Theory
(8) Simulation

3. (9) Machine Scheduling Problem
(10) Reliability Theory
(11) Genetic Algorithms and Local Search
(12) Multi Criteria Problems
(13) Control Problems
* The Future of OR
* Math. Model of problem in OR
* Methods of Solutions in OR
* Difficulties in OR
- P_ Type Problem
- NP_ Hard Problem
- Open Problem

4. Scheduling Problems

5. General Idea of the Problem. Allocation of Resources
General Idea of the Problem * Allocation of Resources * Allocation of Time Slots * Constraints * Optimization

6. Definition of the Problem. J = { J1 , … , Jn }. M = { M1 , … Mm}
Definition of the Problem * J = { J1 , … , Jn } * M = { M1 , … Mm} * Schedule – Mapping of jobs to machines and processing times * The Schedule is subject to feasibility constraints and optimization objectives

7. Schedule Constraints Each machine can only process one job at a time
Schedule Constraints Each machine can only process one job at a time . * * Each job can only be processed by one machine at any time * Once a machine has started processing a job , it will continue running on that job until the job is finished.

8. Classification of Problems. Single – machine problems
Classification of Problems * Single – machine problems * Multi – machine problems * Single – stage problems * Multi – stage problems

9. Other Concepts. Processing time pi. Release dates ri. Due dates di
Other Concepts * Processing time pi * Release dates ri * Due dates di * Weights wi * Setup times tij * Precedence constraints

10. Classification of Single Stage Multi – Machine Problems
Classification of Single Stage Multi – Machine Problems * Parallel Machine Problems - Identical parallel machine problems - Uniform parallel machine problems - Unrelated parallel machine problems

11. Classification of Multi - Stage Multi – Machine Problems
Classification of Multi - Stage Multi – Machine Problems * Flow Shop Problems * Open Shop Problems * Job Shop Problems * Group Shop Environment

12. Definitions Completion Time Ci is earliest time at which Ji is completely processed. Lateness Li : = Ci - di Tardiness Ti : = max { Ci – di , 0 } Earliness Ei : = max { di – Ci , 0 }

13. Objective Functions
Two types of objective functions are most common: *bottleneck objective functions max {fj(Cj) | j= 1, ... , n}, and *sum objective functions S fj(Cj) = f1(C1) + f2(C2) fn(Cn) .

14. Objective Functions Cmax and Lmax symbolize the bottleneck
objective functions with fj(Cj) = Cj
(makespan) and fj(Cj) = Cj - dj (maximum
lateness), respectively.
Common sum objective functions are:
*S Cj (mean flow-time) and S wj Cj (weighted flow- time)

15. Classification of Scheduling Problems
Classes of scheduling problems can be specified in terms of the three-field classification a | b | g where *a specifies the machine environment, *b specifies the job characteristics, and *g describes the objective function(s).

16. Machine Environment
To describe the machine environment the following symbols are used:
*1 single machine *P parallel identical machines *Q uniform machines *R unrelated machines
*MPM multipurpose machines *J job-shop *F flow-shop *O open-shop
The above symbols are used if the number of machines is part of the input. If the number of machines is fixed to m we write Pm, Qm, Rm, MPMm, Jm, Fm, Om.

17. Job Characteristics
*pmtn preemption *rj release times *dj deadlines *pj = 1 or pj = p or pj  {1,2} restricted processing times *prec arbitrary precedence constraints
*intree (outtree) intree (or outtree) precedences *chains chain precedences *series-parallel a series-parallel precedence graph

18. Objective Functions * Maximum Completion Time ( makespan ) Cmax := max { C1 , … , Cn } * Sum of the ( weighted ) completion times Wi * Ci ∑ * Total Weighted Tardiness Ti *Wi ∑

19. Objective Functions
*S Uj (number of late jobs) and S wj Uj (weighted number of late jobs) where Uj = 1 if Cj > dj and Uj = 0 otherwise. *S Tj (sum of tardiness) and S wj Tj (weighted sum of tardiness) where the tardiness of job j is given by Tj = max { 0, Cj - dj }.

20. Examples
*1 | prec; pj = 1 | S wj Cj *P2 | | Cmax *P | pj = 1; rj | S wj Uj *R2 | chains; pmtn | Cmax *J3 | n = 3 | Cmax *F | pij = 1; outtree; rj | S Cj *Om | | pj = 1 | S Tj

21. Complexity Theory
Polynomial algorithms Classes P and NP NP- complete and NP- hard problems

22. How to Live with NP - hard Scheduling Problems?
Small sized problems can be solved by
*Mixed integer linear programming
*Dynamic programming
*Branch and bound methods
To solve problems of larger size one has to apply
*Approximation algorithms
*Heuristics

23. Machine Scheduling Example
3 machines and 7 jobs
job times are [6, 2, 3, 5, 10, 7, 14]
possible schedule 6 13 A
Example schedule is constructed by scheduling the jobs in the order they appear in the given job list (left to right); each job is scheduled on the machine on which it will complete earliest. 2 7 21 B 3 13 C
time >

24. Machine Scheduling Example6 13 A 2 7 21 B 3 13 C
time >
Finish time = 21
Objective: Find schedules with minimum finish time.

25. LPT Schedules Longest Processing Time first.
Jobs are scheduled in the order
14, 10, 7, 6, 5, 3, 2
Each job is scheduled on the machine on which it finishes earliest.

26. LPT Schedule [14, 10, 7, 6, 5, 3, 2] Finish time is 16! 14 16 A 10 15B 7 13 16 C
Finish time is 16!

27. LPT Schedule
*LPT rule does not guarantee minimum finish time schedules.
*(LPT Finish Time)/(Minimum Finish Time) <= 4/3 - 1/(3m) where m is number of machines.
*Usually LPT finish time is much closer to minimum finish time.
*Minimum finish time scheduling is NP- hard.

28. Other Types of Scheduling Problems
There are other classes of scheduling problems.
Some of them are .
*Due-date scheduling *Batching problems *Multiprocessor task scheduling *Cyclic scheduling *Scheduling with controllable data
*Shop problems with buffers *Inverse scheduling *No-idle time scheduling *Multi-criteria scheduling *Scheduling with no- available constraints

29. Other Types of Scheduling Problems
Scheduling problems are also discussed in
connection with other areas:
*Scheduling and transportation
*Scheduling and game theory
*Scheduling and location problems
*Scheduling and supply chains

30. Applications
*Production scheduling *Robotic cell scheduling *Computer processor scheduling *Timetabling *Personnel scheduling *Railway scheduling *Air traffic control *etc.