Introduction

Chapter 1

Elements of Sequencing and Schedulingby Kenneth R. Baker

Byung-Hyun Ha

2

Outline

Schedule and scheduling

Mathematical models

Categorization and description

Solution to scheduling problem

Algorithms and time complexity

3

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

4

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

5

Forecasting

Capacity/facilityplanning

Workforceplanning

Capacityplan

Personnelplan

Aggregate planning

Aggregate plan

WIP/quota setting

Sequencing & scheduling

Production tracking

Laborpolicies

Marketingparameters

Production/processparameters

Demand managementMaster production schedule

Work schedule

Customer demand

WIP position

Real-time simulation

Strategy

Tactics

Control

Shop floor controlWork forecast

A production planning and control hierarchy for pull system (source: Hopp & Spearman, Factory Physics, 2000)

6

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

1

2

3 2

1

2

4

1

3

4 3

4

Resource 1

Resource 2

Resource 3

time

7

Scheduling Theory

Mathematical models 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

8

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

9

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, seq

uence 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

10

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 ...

11

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 proble

m 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!) http://en.wikipedia.org/wiki/Computational_complexity_theory#Big_O_n

otation• 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), ...

12

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, 41017 sec. age of the universe

00.0010.0020.0030.0040.0050.0060.0070.0080.009

0 20 40 60 80 100

0E+001E+052E+053E+054E+055E+056E+057E+058E+059E+05

0 20 40 60 80 100

01E+172E+173E+174E+175E+176E+177E+178E+179E+17

0 20 40 60 80 100

00.0010.0020.0030.0040.0050.0060.0070.0080.009

0 20 40 60 80 100

0E+001E+052E+053E+054E+055E+056E+057E+058E+059E+05

0 20 40 60 80 100

01E+172E+173E+174E+175E+176E+177E+178E+179E+17

0 20 40 60 80 100

13

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

N5 + 6.64

N6 + 4.19

1000 N1

31.6 N2

10 N3

3.98 N4

N5 + 9.97

N6 + 6.29

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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 http://en.wikipedia.org/wiki/Np-complete

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 probl

em, we might be better off to use a so-called heuristic solution procedure, rather than pursuing optimal solution