A Power Aware Solution for Multi-

machine Job Shop Scheduling

Congbo Li

The State Key Lab of Mechanical Transmission

Chongqing University, Chongqing, China

Email: congboli@cqu.edu.cn

Ying (Gina) Tang

Electrical and Computer Engineering

Rowan University, Glassboro, NJ 08028

Email: tang@rowan.edu

Introduction

Multi-machine Job Shop Scheduling

(MMJSS) Model and Optimization

Case Study

Conclusion

The energy consumption by the industrial sector has

almost doubled and the number continues to increase

that accounts for one-half of the world’s total energy

consumption

Introduction

Background

The rising severity of energy shortage together with

other environmental problems results in great efforts

towards energy saving and emission reduction

Manufacturing sector is responsible for 1/3 energy

consumption and 28% greenhouse gas in USA

Similarly, it is responsible for 59% energy consumption

in China – 20% higher than the international level

Introduction

Background (continued)

Machine tools are the main sources of energy

consumption

Existing methods in reducing energy consumption of

machine tools and improving their efficiency:

Change properties of machine tools

Choose better fit process parameters or opting out for a

lower energy consumption machine

However, the above methods require a large amount

of capital investments for hardware changes

Introduction

Solution

MMJSS is a well-studied optimization problem that

often minimize makespan, prodcution cost and other

factors of interest. And very few studies address the

problem with environmental consideration.

Develop an optimal scheduling procedure to achieve

the goal of energy reduction:

A MMJSS is established where both energy consumption

and makespan are considered

A modified multi-objective Genetic Algorithm (GA) is

applied to solve the optimization problem

MMJSS Model

Problem Formulation

There are a set of machines M = {Mj, j=1, 2, …, m} to

process a set of jobs J={Ji, i=1, 2, …, n}. Each job Ji

consists of a sequence of operations, which must be

accomplished according to a given manufacturing

process dependent order.

The route for each job in which the sequence of

operations are processed by the machines is known a

priori.

A schedule is a function S: Oij + {0} that for

each operation of the ith job at the jth machine, Oij ,

defines start time stij.

MMJSS Model

Problem Assumptions

Each job is nonpreemptive, requiring one and only

one machine at a time.

Each job does not visit the same machine twice.

Each machine can process at most one job at a time.

There are no precedence constraints among

operations of different jobs.

Setup time is included in processing time

Two Objectives

Energy consumption

MMJSS Model

ej is accumulated when the jth machine is on for its first

operation of the n jobs until it is off after its last operation

of the n jobs

ppj - The total power usage of the machine j when it is

processing jobs

ptj - The total processing time of the machine j itj - The total idle time of the machine j

ipj - The total power usage of the machine j when it is

idle

CTi - The completion time of the job i, Ji

m

j

jjjj

m

j

jitipptppeSE

11

)()(

Makespan of a schedule T(S) }{max)(i

i

CTST

MMJSS Model

The MMJSS Optimizing Total Energy and Makespan

))(),(min( STSE

ih

i

hjihij sttst )1(

kj

j

kikjijstxtst )1(

n

i

ijj tpt

1

n

i

ijijjij sttstit

1

)( '

Subject to:

Where: Oi’j is operated after Oij ;

is a very large integer constant.

is the precedence relationship parameter that equals

one if Ji is processed at Mh before Mj;

is the precedence relationship parameter that equals

one if Jk is processed at Mj before Ji

i

hj

j

kix

MMJSS Optimization

Multi-Objective Genetic Algorithm

Single-point

crossover

Vector evaluated

GA

Operation-based encoding scheme

Two JSS models, the proposed model and the one

considering makespan only, are applied to a 6X6

benchmark problem

The modified multi-objective GA is then applied to both

models 10 times and the best results are presented

Case Study

Correlation of Energy Consumption and Makespan

Is it normally the case that a shorter makespan yields

less energy consumption?

(m, t) (m, t) (m, t) (m, t) (m, t) (m, t)

Job 1: 3,1 1,3 2,6 4,7 6,3 5,6

Job 2: 2,8 3,5 5,10 6,10 1,10 4,4

Job 3: 3,5 4,4 6,8 1,9 2,1 5,7

Job 4: 2,5 1,5 3,5 4,3 5,8 6,9

Job 5: 3,9 2,3 5,5 6,4 1,3 4,1

Job 6: 2,3 4,3 6,9 1,10 5,4 3,1

Note: (m,t) means which machine processes the job and the

number of time units needed

Machine Number Idle power(KW) Processing power(KW)

1、3 3.1 12.5

2、4 0.7 3

3、6 1.8 8.8

Table 1: Benchmark problem Table 2: The idle and processing power of machines

Note: the energy data is adapted from D.N. Kordonowy, A

Power Assessment of Machining Tools. Bachelor of Science

Thesis in Mechanical Engineering, Massachusetts Institute of

Technology, Massachusetts, USA: MIT Press, 2002

Case Study

Correlation of Energy Consumption and Makespan (cont.)

Experiment No. The proposed model

Makespan (Day) Energy consumption (109J)

1 57 49.3128

2 59 48.744

3 58 48.7512

4 59 48.8484

5 55 49.3416

6 59 49.1004

7 60 48.7224

8 59 48.78

9 58 48.7512

10 59 49.1688

Average 58.3 48.95208

Experiment No. The baseline model

Makespan (Day) Energy consumption (109J)

1 55 50.0184

2 55 50.04

3 55 49.518

4 57 50.7204

5 55 50.4504

6 55 49.7304

7 55 50.0112

8 57 50.7528

9 55 49.7052

10 57 49.7412

Average 55.6 50.08212

Schedules with the same makespan may vary

considerably in energy consumption

It is not necessarily true that a shorter makespan yields

less energy use

The average value of energy consumption of the proposed

model is 2.25 percent lower than that of the baseline

model

Case Study

Correlation of Energy Consumption and Makespan (cont.)

(a) (b)

For the baseline model as shown in (a), when the best and

mean values of makespan are decreasingly convergent over

iterations, those of energy consumption are fluctuating with

no pattern to seek.

The proposed model takes into consideration of both energy

consumption and makespan in MMJSS, the mean and best

values of both energy consumption and makespan are

convergent over iterations as shown in Fig. (b)

Case Study

Performance of Modified GA

The modified multi-objective GA is applied to several

benchmark problems with different complexity from Fisher

and Thompson (1963) [1] and Lawrence (1984) [3]

The results are then compared to the ones using 2ST-GA [2]

and 2ST-PSO [4]

1. H. Fisher, and G.L. Thompson, “Probabilistic learning combinations of local job shop scheduling

rules”. in Industrial scheduling. New Jersey, Prentice-Hall, pp. 225-251

2. V. Kachitvichyanukul, and S. Sitthitham, “A two-stage genetic algorithm for multi-objective job shop

scheduling problems,” Journal of Intelligent Manufacturing, vol. 22, no. 3, pp. 355-365, 2011

3. S. Lawrence, “Supplement to resource constrained project scheduling: An experimental

investigation of heuristic scheduling techniques,” in Technical report, GSIA, Carnegie Mellon

University. 1984

4. T. Pratchayaborirak, and V. Kachitvichyanukul, “A two-stage PSO algorithm for job shop scheduling

problem,” International Journal of Management Science and Engineering Management, vol. 6, no. 2,

pp. 84-93, 2011

Case Study

Performance of Modified GA (cont.)

The modified multi-objective GA improves on the

performance of the other two methods for the small-scale

benchmark problems

When the problem size increases, 2ST-PSO beats the

others, but the modified multi-objective GA still can deliver

fairly good results with way much less time in comparison to

2ST-GA (average 89% time reduction).

Problem Size nXm BKS Modified multi-objective GA 2ST-GA 2ST-PSO

Mean Time (s) Mean Time (s) Mean Time(s)

FT06 6X6 55 55.7 7.36 55 833 55 9

LA01 10X5 666 666 9.6 666 1273 666 15

LA06 15X5 926 926 17.2 926 2480 926 34

LA11 20X5 1222 1222 27.2 1222 4254 1222 59

LA16 10X10 945 993.8 105.5 955.3 567.9 945 31

LA26

20X10 1218 1304.8 658 1288.6 17345 1266 111

Note: FT problems are designed by Fisher and Thompson (1963) and LA problems are designed by Lawrence (1984).

Contribution:

A power-aware multi-machine job shop scheduling

model with the objectives of minimizing both energy

consumption and makespan is proposed.

The relationship of energy consumption and

makesan of a schedule is thoroughly analyzed, from

which a MMJSS solution with relatively better energy

consumption and makespan is ensured.

The simulation experiments concluded that a shorter

makespan does not necessarily yield less energy

consumption

Conclusion

Future Work:

Comprehensive energy models with respect to processing

loads and speeds are worth of investigation for more

complex industrial cases

More sophisticated power-aware scheduling models are

needed for the practical flexible job shop scheduling

problem

The extension of this work for large-scale real industrial

MMJSS problems to investigate the intrinsic links between

scheduling and energy consumption is necessary

Conclusion