0. Elements of Sequencing and Scheduling by Kenneth R. Baker
Introduction
Chapter 1
Elements of Sequencing and Scheduling by Kenneth R. Baker
Byung-Hyun Ha

1. Outline Schedule and scheduling Mathematical models
Categorization and description
Solution to scheduling problem
Algorithms and time complexity

2. Introduction to Sequencing and Scheduling
Schedule
A tangible plan or document
a bus schedule, a class schedule
a. When things will happen; a plan for the timing of certain activities
when will be the dinner served, when the laundry will be done
b. In which sequence things will happen
the North bus departs right after cleaning and maintenance are finished
Scheduling
Process of generating schedule
Given tasks to be carried out, resources available to perform the tasks
Determining the detailed timing of the tasks within resource capacity
preparing a dinner, doing the laundry, problem in industry

3. Introduction to Sequencing and Scheduling
Decision-making hierarchy
Scheduling follows more basic decisions, some earlier
dinner preparation  menu items, recipes, ...
planning decisions in industry
demand, design, technology for products
A production planning and control hierarchy for pull system (next slide)
Assuming planning decisions have been made already, and we are given the following information
Tasks that are well defined and completely known
Resources to perform these tasks are specified and available

4. Capacity/facility planning Sequencing & scheduling
Marketing
parameters
Forecasting
Production/process parameters
Capacity/facility planning
Workforce planning
Labor
policies
Capacity plan
Personnel plan
Aggregate planning
Aggregate plan
Strategy
WIP/quota setting
Customer demand
Master production schedule
Demand management
WIP position
Sequencing & scheduling
Tactics
Control
Real-time simulation
Work schedule
Work forecast
Shop floor control
Production tracking
A production planning and control hierarchy for pull system (source: Hopp & Spearman, Factory Physics, 2000)

5. Introduction to Sequencing and Scheduling
Defining scheduling problem
Resource: type, amount
Task: resource requirement, duration, available time, due date, technological constraints ( precedence restriction)
Finding solution
Formal problem-solving approaches are required because of complexity
Formal models
For understanding problem and finding solution
Graphical, algebraic, and simulation models
Gantt chart
Visualizing problem, measuring performance, and comparing schedules
Resource 1 1 2 4 3
Resource 2 2 1 4 3
Resource 3 3 2 1 4
time

6. Scheduling Theory Mathematical models Objective function
Primary concerns in this course
Development of useful models
Leading to solution techniques and practical insights
Interface between theory and practice
Quantitative approach
Capturing problem structure in mathematical form
Beginning with a description of resources and tasks with translation of decision-making goals into explicit objective function
Objective function
Ideally, it should consist of all costs that depend on scheduling decisions
Not practical (difficult to identify, isolate, and fix)
Practical and prevalent goals
Turnaround -- time required to complete a task
Timeliness -- conformance of a task’s completion to a given deadline
Throughput -- amount of work completed during a fixed period of time

7. Scheduling Theory Categorization of scheduling models
By number of machines
one or several
By capacity of machine
available in unit amount or in parallel
By job availability
static -- available jobs does not change over time, or
dynamic -- new jobs appear over time
Although dynamic models, which is less tractable than static models, would appear to be more important for practical application, static models often capture the essence of dynamic systems, and the analysis of static problems frequently uncovers valuable insights and sound heuristic principles that are useful in dynamic situation.
By certainty
deterministic or stochastic

8. Scheduling Theory
Description of scheduling problem (source: Pinedo, 2008)
Using triple  |  | 
 -- machine environment
 -- processing characteristics and constraints
 -- objective to be minimized
Machine environment
Single machine -- 1, identical machines in parallel -- Pm, flow shop -- Fm, job shop -- Jm, ...
Processing characteristics and constraints
Release dates -- rj, preemptions -- prmp, precedence constraints -- prec, sequence dependent setup times -- sjk, batch processing -- batch(b), ...
pj = p, dj = d, ...
Objective to be minimized
Makespan -- Cmax, maximum lateness -- Lmax, total weighted completion time -- wjCj, weighted number of tardy jobs -- wjUj, ...
Examples
1 | rj, prmp | wjCj, 1 | sjk | Cmax, Fm | pij = pj | wjCj, Jm || Cmax

9. Scheduling Theory Solution to scheduling problem
Answering the following decisions
Which resources will be allocated to perform each task? (allocation decision)
When will each task be performed? (sequencing decision)
by feasible resolution of the following common constraints
Limits on capacity of machines
Technological restriction on order in which some jobs can be performed
...
Solution methodologies
Mathematical programming models
Combinatorial procedures
Simulation techniques
Heuristic solution approaches

10. Scheduling Theory Algorithms and time complexity Time complexity
Computing effort required by a solution algorithm
The number of computations required by an algorithm to solve a problem of “size n (amount of information needed to specify problem)”
Size of sorting problem -- number of entries
Size of travelling salesman problem (TSP) -- number of cities, roughly
Described by order-of-magnitude notation, O()
A function f(n) is O(g(n)) whenever there exists a constant c such that |f(n)|  c|g(n)| for all values of n  0
Insertion sort -- O(n2), quick sort -- O(nlogn)
All enumeration algorithm: knapsack problem -- O(2n), TSP -- O(n!)
Polynomial time algorithm
An algorithm with complexity such as O(1), O(n2), O(n8), ...
Exponential time algorithm
An algorithm with complexity such as O(2n), O(3n), ...

11.  105 sec.  1 day, 41017 sec.  age of the universe
Scheduling Theory
Algorithms and time complexity (cont’d)
Running time example
Comparing algorithms with complexities of n, nlogn, n2, n8, 2n, 3n
Assuming that our computer executes 106 computations per second
Assuming that our computer has become 103 times faster than above
 105 sec.  1 day, 41017 sec.  age of the universe

12. With computer 100 times faster With computer 1000 times faster
Scheduling Theory
Algorithms and time complexity (cont’d)
Running time example (cont’d)
Size of largest problem instance solvable in 1 hour (source: Garey and Johnson, 1969)
Time complexity
With present computer
With computer 100 times faster
With computer 1000 times faster n n2 n3 n5 2n 3n N1 N2 N3 N4 N5 N6
100 N1
10 N2
4.64 N3
2.5 N4 N N
1000 N1
31.6 N2
10 N3
3.98 N4 N N

13. Scheduling Theory Algorithms and time complexity (cont’d)
NP-complete problems
A class of problems which includes many well-known and difficult combinatorial problems
Knapsack problem, TSP, graph coloring problem, ...
All the problems are equivalent in the sense that if one of them can be solved by polynomial algorithm, then so can the others
No one has found polynomial algorithm, nor proved no algorithm exists
Garey and Johnson, 1969
Proof of NP-completeness of a problem
By reducing a well-known NP-complete problem to the problem
Application of NP-complete theory
If we are faced with the need to solve large versions of an NP-complete problem, we might be better off to use a so-called heuristic solution procedure, rather than pursuing optimal solution