Scheduling a single machine with precedence constraints

HANOI PEDAGOGICAL UNIVERSITY 2 DEPARTMENT OF MATHEMATICS——————–o0o——————— Nguyen Thanh Nga Scheduling a single machine with precedence constraints BACHELOR THESIS Major: Applied Mathemat

HANOI PEDAGOGICAL UNIVERSITY 2 DEPARTMENT OF MATHEMATICS

——————–o0o———————

Nguyen Thanh Nga

Scheduling a single machine with precedence constraints

BACHELOR THESIS

Major: Applied Mathematics

HANOI PEDAGOGICAL UNIVERSITY 2 DEPARTMENT OF MATHEMATICS

——————–o0o———————

Nguyen Thanh Nga

Scheduling a single machine with precedence constraints

BACHELOR THESIS Major: Applied Mathematics

Advisor:

Dr Le Xuan Thanh

I hereby confirm that this thesis is my original work completedunder the guidance of Dr Le Xuan Thanh The presented results areaccurate and correctly cited

Author

In particular, I would like to express my deep respect and gratitude

to Dr Le Xuan Thanh (Institute of Mathematics, Vietnam Academy

of Science and Technology) who has direct guidance and helps me

to complete this thesis Professionalism, seriousness and his rightorientations are important prerequisites for me to get the results inthis thesis

Due to limited time, capacity, and conditions, my thesis cannotavoid errors I look forward to receiving valuable comments fromreaders

Author

1.1 Thesis topic 6

1.2 Research methods 8

1.3 Results of the thesis 8

1.4 Structure of the thesis 8

2 Preliminaries 10 2.1 Some basic scheduling concepts 10

2.2 Mixed integer linear programming 12

3 Problem description and MIP formulations 14 3.1 Problem description 14

3.2 First formulation 16

3.3 Second formulation 18

3.4 Third formulation 20

3.5 Fourth formulation 21

4 Experiments 25 4.1 ZIMPL codes 25

4.2 Numerical results 30

Chapter 1

Introduction

1.1 Thesis topic

According to wikipedia, a schedule or a timetable, as a basic time

management tool, consists of a list of times at which possible tasks,events, or actions are intended to take place, or of a sequence of events

in chronological order in which such things are intended to take place.The process of creating a schedule - deciding how to order these tasksand how to commit resources between the variety of possible tasks -

is called scheduling.

Having good schedules makes our life more comfortable and organized In social services, industrial manufacturing, transporta-tion, etc., a good schedule results in higher productive and more com-petitive outcomes How to construct good schedules for real-life as well

well-as industrial activities is the main goal of scheduling problems Thus,one can say that scheduling problems arise naturally from practice andplay a vital role in operating systems In the past few decades, thepractical importance and complexity of the general scheduling prob-lems have motivated a significant volume of research in a wide variety

of scheduling environments

In (project) scheduling problems, a set of jobs (so-called activities,tasks) has to be processed on a set of machines in a given time hori-zon using limited resources (machines, processors, or staffs) Some

between the jobs) might be imposed in the problems Often there are

a finite (but usually very huge) set of possibilities for a feasible ule However, usually one has to optimize certain objective function.This means that we have to look for a best solution among a tremen-dous number of feasible solutions Furthermore, practical instances

sched-of scheduling problems might have very large size (up to hundreds

or even millions jobs and machines) That is the motivation for demic researchers to develop algorithms capable of solving large-sizescheduling problems in acceptable time

aca-The performance of the algorithms depends very much on the sign of the algorithms themselves and data structures For the sameproblems, there might be many solution approaches with differentlevels of efficiency Such a common approach is to use mixed integerprogramming (MIP) in modeling the problems This approach has

de-a two-fold de-advde-antde-age On one hde-and, it mde-akes use of de-avde-ailde-able MIPsolvers, since in the recent years the solvers became much more effi-cient On the other hand, it is easy to implement the models and thensolve the problems on computers

In this thesis, we would like to highlight the fact that there mighthave different MIP formulations for representing a scheduling prob-lem, and having a good formulation is crucial in solving numericalinstances of such problem For that we focus on studying the schedul-ing problem on a single machine with precedence constraints betweenthe jobs and minimizing the total flow time More precisely, we con-sider the situation in which one has to schedule a set of jobs on a singlemachine Each job has its processing time and is processed withoutpreemption Between the jobs there may be arbitrary precedence con-straints that impose which jobs have to be completed before startinganother job The objective is to find a schedule that minimizes thetotal flow time, i.e the total value of completion times of all jobs.The scheduling problem studied in this thesis is one of the mostbasic scheduling problems For a comprehensive study on schedulingproblems, we refer to [2] It is also shown in [2] that the problemunder our consideration is NP-hard, therefore MIP-based approaches

like to highlight “In integer programming, formulating a good model

is of crucial importance to solving the model” has already quoted in[4]

1.2 Research methods

We use MIP-based approaches for solving our studied schedulingproblem due to the mentioned advantages of the methods That is, weformulate the problem as different mixed integer programming formu-lations We will also test the obtained models on different instancesthat are randomly generated, with the help of some modeling languageand integer programming solver

1.3 Results of the thesis

We describe the scheduling problem on a single machine withprecedence constraints between the jobs, no job preemption, and mini-mizing the total flow time We present four different MIP formulationsfor the problem Based on experimental results on some tested in-stances, we find out the best performance formulation for numericallysolving the scheduling problem

1.4 Structure of the thesis

Apart from this chapter, the thesis consists of four more chapters

• Chapter 2: Preliminaries In this chapter we present some

pre-liminaries on the scheduling problem of our consideration and ashort introduction to mixed integer programming

• Chapter 3: Problem description and MIP formulations This

chapter first describes the scheduling problem studied in thisthesis, and then presents several MIP formulations for the prob-

• Chapter 4: Experiments This chapter describes how do we

generate tested instances for numerical experimenting the mulations mentioned in Chapter 3 Then we present how toimplement the formulations on computers After that, we ana-lyze the performance of the formulations based on our obtainednumerical results

for-• Chapter 5: Conclusions We close the thesis by some

conclu-sions

Chapter 2

Preliminaries

In this chapter, we introduce some basic concepts related to ourconsidered scheduling problem Then we present a short introduction

to mixed integer programs

2.1 Some basic scheduling concepts

Resources for a scheduling problem include jobs and machines

Here jobs mean any activities, tasks, or works that should be done and completed for some specific purposes Machines can be persons

who are responsible for doing the jobs, tools for processing the jobs,

or spaces in which the jobs are done For example, a student needs

to review some subjects before the final exam of a semester In thiscontext, the student can be considered as a machine, and reviewingeach subject can be thought as a job

In general scheduling problems, we are given n jobs j = 1, , n which have to be processed on m machines M k (k = 1, , m) Often

we assume that each machine can process at most one job at any time,and each job cannot be processed on more than one machine simul-taneously In some models a job can be interrupted when processing

on a machine and resumed later either on the same machine or onanother one In such contexts we say that the jobs can be preempted,

or equivalently, job preemption is allowed However, in this thesis we

consider a scheduling problem on a single machine (i.e m = 1) with

no job preemption (i.e., once a job is started, it must continue until

its completion on the machine)

Associated data for a job j may include:

• processing time p j which is the amount of time units needed tocomplete the job,

• release date r j which is the point of time before that the jobcannot start,

• deadline ˜d j which is the point of time before that the job must

be finished,

• due date d j which is the point of time before that the job should

be completed

Between jobs there may be precedence constraints imposing which

jobs must be finished before doing certain jobs The notation i → j

means that the job i must be completed before the job j can be started.

To visualize precedence constraints one often use an acyclic directedgraph in which the nodes represent the jobs and the arcs represent theprecedence relations

Example 1 The graph in Figure 2.1 visualize an instance of

job 1 must be completed before the starting time of job 3, and similar

A schedule on single machine without job preemption can be

com-pletely described by the starting times S j of all jobs j, or equivalently their completion times C j = S j + p j Such schedules are often visual-ized by Gantt charts

Example 2 Figure 2.2 shows a feasible schedule for a problem

processing order of the jobs is 1, 3, 2, 4, the completion times of the

C4 = p1 + p3 + p2 + p4 = 11.

Figure 2.2: Gantt chart of a schedule

Usually a schedule is made in order to achieve some optimality

criterion The most common objective is to minimize the makespan

C max = max{C j , j = 1, , n } The makespan let us know the latest

point of time that all jobs are finished Another common objective is

to minimize the total flow time∑n

i=1 (C j −r j), or equivalently, ∑n

i=1 C j,

since the values of parameters r j are given constants The flow time of

a job j is the total time it spends in the system and equals C j −r j, i.e.the difference between its completion time and release time Thereforethe total flow time let us know how long all the jobs spent in thesystem In this thesis, we focus on studying the objective of minimizethe total flow time

2.2 Mixed integer linear programming

Many scheduling problems can be formulated as integer linear grams A linear program (abbreviated by LP) is a system of the form

where a ij , c j , b i ∈ R are parameters with given values, and x j are

de-cision variables whose values are needed to be decided Here, n is the number of variables and m is the number of constraints An LP

of the form (2.1) − (2.3) with the additional restriction x j ∈ Z (i.e.,

all variables are required to have integer values) is called an integer

linear program (abbreviated by ILP or shortly IP) A mixed teger linear program (abbreviated by MIP) contains both integer

in-and continuous variables

The sum in (2.1) is called objective function whose value is needed

to be optimized, while the system (2.2) and (2.3) are called constraints Any value of decision vector x which satisfies the constraints is called

a feasible solution A feasible solution x ∗ which attains the minimum

value of objective function (2.1) is called an optimal solution.

Having an instance of MIPs, we can implement it on computers byusing some mathematical modeling language Our choice of modelingtool is ZIMPL [3] To numerically solve the instance, one can usevarious available MIP solvers Our choice of solvers is SCIP [1] which

is known as the best non-commercial solver for solving MIPs

Chapter 3

Problem description

and MIP formulations

In this chapter, we first formally describe the problem of our sideration, then present four different (M-)IP formulations for thisproblem

con-3.1 Problem description

We consider the scheduling problem with the following features

• We are given n jobs j = 1, , n that have to be processed on a

single machine

• For each job j its processing time p j ∈ Z+is known exactly (that

means, job j needs p j > 0 time units to process on the machine).

• Every job j has realease date r j = 0, i.e., all the jobs can start

from beginning of the time horizon t = 0.

• No due date or deadline is given for the jobs, i.e., each job can

be completed whenever we want

• Job preemption is not allowed, i.e., once each job starts, it must

• Between the jobs there may be arbitrary precedence constraintsthat define which job must be completed before the start ofanother job.

• The objective is to minimize the total flow time of all jobs.Following the 3-field notation proposed in [2], this problem can beshortly denoted by

Here the number 1 in the first field means that there is a single

ma-chine The second field consists of the keyword prec which means that

there are precedence constraints between jobs The last field consists

of the objective function of total flow time which needs to be mized

mini-Example 3 Consider an instance of the scheduling problem with n =

5 jobs of processing times p1 = 5, p2 = 3, p3 = 7, p4 = 4, p5 = 5 and

schedules (a) and (b) are shown in the left part of Figure 3.1 We can process job 3 before job 2 or vice versa without violating the given precedence constraints However, the schedule (c) in the right part of Figure 3.1 is not feasible since it violates the precedence constraint

schedules For schedule (a) its total flow time is 5+8+15+19+24 = 71 For schedule (b) its total flow time is 5 + 12 + 15 + 19 + 24 = 75.

1 3 2 4 5

5 15 19 24

t

3 7 12 17 0

Since a schedule on a single machine without job preemption can

be completely described by the completion times C j of all jobs j, in this formulation we use C j with j = 1, , n as variables Using these

variables, it is easy to see that the total flow time is ∑n

j=1 C j, andtherefore the objective function is

Since all jobs start after the time origin 0 and each job j processes in

p j time units, we must have

For each job precedence relation i → j, since job j starts after the

completion time C i of job i and processes in p j time units, we musthave the following constraint on its completion time:

C j − C i ≥ p j ∀i, j = 1, , n, i → j.

Since the jobs are processed in a single machine, for the other pairs

before job j, or vice versa To determine which job is scheduled before another in the schedule, we use binary variables y ij with i, j = 1, , n and i ̸= j, in which

j must be scheduled before job i This adapts to the latter constraint,

which now becomes

To summarize, we obtain the following IP formulation for problem

the scheduling problem (P ).

Since there is a single machine and job preemption is not allowed,

it is obvious that an optimal schedule must have no gap (free time)between jobs Thus the total time for processing all the jobs is fixedand computed by

For the second formulation, we need the time horizon T and introduce

nT binary variables x jt ∈ {0, 1} with j = 1, , n and t = 1, , T , in