General Lower Bounds for the Total Completion Time in a Flowshop

Scheduling Problem

MaxPlus Approach

Nhat-Vinh Vo

, Pauline Fouillet

and Christophe Lent

Universit

e Franc¸ois-Rabelais de Tours, CNRS, LI EA 6300, OC ERL CNRS 6305, Tours, France

Universit

e Franc¸ois-Rabelais de Tours, Polytech, Tours, France

Keywords:

Scheduling, Flowshop, Total Completion Time, Lower Bound.

Abstract:

The ﬂowshop scheduling problem has been largely studied for 60 years. As a criterion, the total completion

time also receives a great amount of attention. Many studies have been carried out in the past but they are

limited in the number of machines or constraints. MaxPlus algebra is also applied to the scheduling theory but

the literature focuses on some concrete constraints. Therefore, this study presents a new method to tackle a

general permutation ﬂowshop problem, with additional constraints, to elaborate on lower bounds for the total

completion time. These lower bounds can take into account several constraints, like delays, blocking or setup

times, but they imply to solve a Traveling Salesman Problem. The theory is developed, based on a MaxPlus

modeling of ﬂowshop problems and experimental results are presented.

1 INTRODUCTION

The m-machine ﬂowshop scheduling problem has

been largely studied for 60 years. The makespan is

the most studied criterion, especially for permutation

ﬂowshop problems. However, the total completion

time criterion also receives a great amount of atten-

tion. It reﬂects ”the total manufacturing waiting time

experienced by all customers” (Emmons and Vairak-

tarakis, 2013). Even with only two machines, prob-

lem F

∑

is NP − hard in the strong sense and so

are problems with more machines. Therefore, results

that help to solve these problems are interesting.

Total completion time criterion has been stud-

ied largely. A branch-and-bound algorithm, incor-

porating a lower bound, dominance relation and an

upper bound is presented by Allahverdi and Al-

Anzi in (Allahverdi and Al-Anzi, 2006). That study

solves total completion time minimization problem

|perm,S

nsd

∑

where separate setup times are

taken into account. The number of visited nodes and

the percentage between this number and that of pos-

sible nodes are considered. This percentage shows us

that their lower bound is effective as the number of

visited nodes is quite small. Separate setup times are

also investigated by Su and Lee (Su and Lee, 2008) in

a two-machine ﬂowshop no-wait scheduling problem

with a single server in order to minimize total com-

pletion time. In another research, eleven heuristics

based on the Shortest Processing Time (SPT) rule are

implemented by Aydilek and Allahverdi (Aydilek and

Allahverdi, 2010). Their study is to minimize total

completion time of a two-machine ﬂowshop schedul-

ing problem, in which processing times are bounded.

A lower bound based on the ﬁrst machine of problem

∑

is presented as the sum of a previously ex-

isting lower bound and the optimum of an asymmet-

ric traveling salesman problem (ATSP) (Della Croce

et al., 1996). These aforementioned studies only deal

with limited number of machines and few constraints.

It is not easy to generalize them to any number of ma-

chines or any constraint.

In this study, the proposed approach is based on

MaxPlus algebra (see 2.1). It has been widely used

in control systems, especially in relation with Petri

Nets but rarely in the scheduling theory. Nevertheless,

some articles can be cited on project scheduling prob-

lems (Gifﬂer, 1963), on cyclic parallel machine prob-

lems (Hanen and Munier, 1995), on cyclic ﬂowshop

scheduling problems (Cohen et al., 1985; Gaubert,

1992) and on cyclic jobshop scheduling problems

(Gaubert and Maisresse, 1999). The MaxPlus al-

gebra is applied to modeling and scheduling ﬂow-

shop problems with minimal delays, setup and re-

moval times (Lent

e, 2001; Bouquard et al., 2006). It

is also applied to ﬂowshop problems with minimal-

382

Vo N., Fouillet P. and Lenté C..

General Lower Bounds for the Total Completion Time in a Flowshop Scheduling Problem - MaxPlus Approach.

DOI: 10.5220/0004833403820389

In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 382-389

ISBN: 978-989-758-017-8

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

maximal delays for two-machines (Bouquard and

Lent

e, 2006) or for any number of machines (Augusto

et al., 2006). In these studies, jobs are associated

to MaxPlus square matrices and lower bounds, up-

per bounds and/or dominance conditions are derived

by applying transformations to those matrices. More-

over, this approach is used effectively to model ﬂow-

shop problems with minimal-maximal delays, setup

and removal times and to highlight a central problem

(Vo and Lent

e, 2013).

The objective of this study is to address a general

permutation ﬂowshop problem in terms of constraints

taken into account. We elaborate on lower bounds for

the total completion time that are based on the reso-

lution of two sub-problems: one problem similar to

the one machine total completion time minimization

problem and the other similar to a traveling salesman

problem. These lower bounds are incorporated in a

branch-and-bound procedure to be tested.

The next section presents the background of the

study: MaxPlus algebra and ﬂowshop scheduling

problem. We recall in section 3 how MaxPlus alge-

bra can be used to model a general ﬂowshop prob-

lem. The lower bound construction is then explained

in section 4. Finally, a branch-and-bound algo-

rithm is explained and some tests concerning prob-

lem F3|perm; S

nsd

∑

and problem F

|perm|

∑

are presented as experimental results.

2 CONTEXT AND DEFINITIONS

2.1 MaxPlus Algebra

MaxPlus algebra is brieﬂy described as follows; a

more detailed presentation can be found in (Gunawar-

dena, 1998).

In MaxPlus algebra, the maximum is denoted by

⊕ and the addition by ⊗. The former, ⊕, is idempo-

tent, commutative, associative and has a neutral ele-

ment (−∞) denoted by 0. The latter, ⊗, is associa-

tive, distributive on ⊕ and has a neutral element (0)

denoted by 1. The null element, 0, is an absorbing

element for ⊗. These properties can be summarized

by stating that R

max

= (R ∪{−∞},⊕,⊗) is a dioid. It

is important to note that in MaxPlus algebra in partic-

ular, and in dioids in general, the ﬁrst operator ⊕ can

not be simpliﬁed, that is a ⊕ b = a ⊕ c 6⇒ b = c. Fur-

thermore, in R

max

, the second operator ⊗ is commu-

tative, and except 0, every element is invertible: the

inverse of x is denoted by x

−1

or 1/x. For simplicity,

we denote the ordinary subtractions by x/y instead of

x ⊗ y

−1

and by xy the product x ⊗ y.

It is possible to extend these two operators to m ×

m matrices of elements of R

max

. Let A and B be two

matrices of size m × m, operators ⊕ and ⊗ are deﬁned

∀(i, j) ∈ {1, ...,m}

,[A ⊕B]

i j

= [A]

i j

⊕ [B]

i j

∀(i, j) ∈ {1, ...,m}

,[A ⊗B]

i j

k=1

[A]

⊗ [B]

k j

where [.]

i j

is the element at the i

row and j

col-

umn of the corresponding matrix. It is not difﬁcult to

show that the set of m × m matrices in R

max

is a dioid.

However, ⊗ is not commutative and not every matrix

is invertible.

The two following lemmas can be derived from

the previous deﬁnitions. They will be useful for the

development of the lower bound.

Lemma 1. ∀ j ∈ {1,. . . , m} :

[A ⊗B]

1 j

k=1

[A]

⊗ [B]

k j

≥ [A]

⊗ [B]

1 j

(1)

Lemma 2. ∀`, j ∈ {1,. . . , m} :

[A ⊗B]

1 j

≥ [A]

⊗ [B]

` j

(2)

[A ⊗B]

` j

≥ [A]

⊗ [B]

` j

(3)

2.2 Flowshop Scheduling Problem

Since the paper of Johnson (Johnson, 1954), ﬂow-

shop problems have been studied largely (Em-

mons and Vairaktarakis, 2013). Basically, a ﬂow-

shop scheduling problem consists of a set of n-

jobs J = {J

,..., J

} and another set of m-machines

,..., M

}. Each job must go through all ma-

chines in the same predeﬁned order, let us say from

to M

and each machine can load only one job at

a time (Brucker, 2006). If all jobs must be executed

in the same order over all machines, the problem is

called a permutation ﬂowshop problem. In this case,

there exists an ordered list of jobs (or a sequence) σ

that is identically scheduled on all machines. We limit

our current study to permutation ﬂowshop problems.

Each job J

is composed of m operations O

(1 ≤ k ≤

m): one per machine. An operation is at least de-

scribed by a processing time p

: the processing time

of job J

on machine M

(or equivalently, the process-

ing time of the k

operation of job J

). The comple-

tion time of job J

on machine M

) and the com-

pletion time of job J

) are related by C

= C

Over the years, several additional constraints have

been taken into consideration (Emmons and Vairak-

tarakis, 2013). Some of them can be modeled us-

ing MaxPlus algebra (Vo and Lent

e, 2013). One of

GeneralLowerBoundsfortheTotalCompletionTimeinaFlowshopSchedulingProblem-MaxPlusApproach

383

the most common constraints is the permutation con-

straint (perm) which has just been mentioned above.

A constraint of no − wait appears in problems where

there is no delay allowed between two successive op-

erations of a job. On the contrary, constraints of

min − delay, max − delay, min − max delay indicate

a ﬂowshop problem with delays between two suc-

cessive operations of a job. Depending on the case,

these delays may have to meet a lower bound, an up-

per bound or both. It may also exist separate non-

sequence dependent setup times (S

nsd

) and/or removal

times (R

nsd

) before and after each operation. Finally,

some authors have considered blocking constraints,

due to the non-existence of intermediate storage be-

tween consecutive machines or to speciﬁc interactions

between machines. These constraints are referred to

as RSb, RCb and RCb

∗

in (Trabelsi et al., 2012).

The most studied criterion is the makespan, or the

maximal completion time (C

max

). It is deﬁned by the

completion time of the last operation scheduled on the

last machine (M

). In this article we focus on the total

completion time (

∑

) which is the sum of the com-

pletion times of the different jobs in a given schedule.

3 MaxPlus MODELING OF

FLOWSHOP SCHEDULING

PROBLEMS

Our problem can be noted F

|perm β|γ using

notations proposed by Graham et al. (Graham

et al., 1979). It is a m machine permuta-

tion ﬂowshop problem with a set of constraints

β that is a subset of {min − max delay, no −

wait, S

nsd

, R

nsd

, RSb,RCb,RCb

∗

}. Criterion γ can

be whatever we desire since it does not interfere in

the modeling process. The total completion time cri-

terion is investigated in the following of this article.

Basically, the modeling process follows this

scheme:

• Given the k

operation O

of a job J

, four dates

are considered: date δ

of availability of machine

(before execution of operation O

), starting

time St

of operation O

, its completion time

and date of liberation D

of machine M

(af-

ter execution of operation O

), that is the date

when job J

leaves machine M

to be placed in

a stock or on the following machine. In most

ﬂowshop problems, dates C

and D

are equal;

however, they can be different in case of block-

ing constraints or removal times. Date of liber-

ation of the last machine (D

) is equal to the

completion time C

of the job, except if there ex-

ist removal times. In this case D

is equal to

plus the removal time of operation O

. Fig-

ure 1 shows an example of ﬂowshop problem

|perm,min − max delay, S

nsd

∑

where

triangles illustrate setup and removal times and

rectangles illustrate processing times.

• Formulate the system (S) of inequalities that link

these different variables.

• Calculate the smallest (D

) (1 ≤ k ≤ m),(1 ≤ i ≤

n) solutions of the system (S).

Figure 1: Example of ﬂowshop problem: F

|perm,min −

max delay,S

nsd

∑

Whatever the set of constraints β is, these calcula-

tions lead to a MaxPlus linear relation between dates

of liberation D

and dates of availability δ

(Lent

2011), (Vo and Lent

e, 2013). More precisely, we can

state the following proposition:

Proposition 1 (Matrix associated to a job). Let

(resp.

) be the line vector of the m dates δ

(resp.

): it exists a m × m MaxPlus matrix T

computed

from data of job J

such that

δ ⊗ T

(4)

Matrix T

is called the associated matrix of job J

: it

entirely characterizes job J

Various elements of matrix T

will be denoted t

in other words, t

= [T

]

. This matrix sums up the

job data (processing times, setup times, delays and so

on) and the ﬂowshop constraints.







... t

... ... ... . . .

... t







(5)

These results can be generalized to a sequence of

jobs (Lent

e, 2011), (Bouquard et al., 2006).

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Deﬁnition 1 (Matrix associated to a sequence). Let

σ be a sequence of ν jobs: its associated matrix is

matrix T

deﬁned by

i=1

σ(i)

(6)

Proposition 2. If

δ is the vector of dates of availabil-

ity of machines and

the vector of dates of libera-

tion of machines, after the execution of sequence σ,

we have the relation

δ ⊗ T

(7)

4 PROPOSED LOWER BOUNDS

This section presents lower bounds for problem

Fm|perm, β|

∑

, with β ⊂ {min − max delay, no −

wait, S

nsd

, R

nsd

, RSb,RCb, RCb

∗

}. To develop the

calculations, we assume that C

= D

(1 ≤ i ≤ n). It

is true unless there exists removal times: this particu-

lar case will be discussed at the end of this section.

4.1 The ﬁrst lower bound

We ﬁrst present a lower bound of the completion time

of the k

job in a sequence before elaborating on a

lower bound for the total completion time.

4.1.1 Lower bound of completion time of a job

Proposition 3. Let σ a sequence of jobs and

δ the

line vector of dates of availability of the machines (

= (δ

,δ

,..., δ

)). The completion time of the job in

position in the sequence veriﬁes relation:

i f k = 1 : C

σ(1)

≥ δ



σ(1)



i f k = 2 : C

σ(2)

≥ δ



σ(1)

σ(2)



i f k > 2 : C

σ(k)

≥ δ

k−2

j=1

σ( j)

)



σ(k−1)

σ(k)



Proof. Let τ be the sub-sequence composed of the

ﬁrst k jobs of sequence σ. Proposition 2 and deﬁni-

tion 1 result in:

δ ⊗ T

δ ⊗

i=1

τ(i)

δ ⊗

i=1

σ(i)

(8)

Moreover

σ(k)

and by assumption C

σ(k)

σ(k)m

, which is the last element of vector

σ(k)

, we

have,

σ(k)

δ ⊗

i=1

σ(i)

(9)

If k = 1, the application of lemma 1 results in:

σ(1)

≥



σ(1)



= δ



σ(1)



(10)

If k ≥ 2, by iteratively applying this lemma into equa-

tion (9), we obtain:

σ(k)

≥

⊗

k−2

j=1



σ( j)





σ(k−1)

σ(k)



(11)

Inequality (11) can be rewritten as

σ(k)

≥ δ

k−2

j=1

σ( j)

)



σ(k−1)

σ(k)



(12)

4.1.2 Lower bound of the Total Completion

Time

Deﬁnition 2 (Lower Bound LB

V FL

.).

Given a sequence σ of n-jobs, we deﬁne:

(σ) =

n−1

j=1

σ( j)

)

n− j

(σ) =



σ(1)



j=2



σ( j−1)

σ( j)



σ( j−1)

Proposition 4.

∀σ sequence :

i=1

σ(i)

≥ δ

⊗ A

(σ) ⊗B

(σ)

Proof. Considering proposition 3, we have:

i=1

σ(i)

≥ δ



σ(1)



⊗

i=2

i−2

j=1

σ( j)

)



σ(i−1)

σ(i)



(13)

Rearranging the factors on the right side of inequality

(13):

i=1

σ(i)

≥ (δ

)

⊗

n−2

i=1



σ(i)



n−i−1

⊗



σ(1)



i=2



σ(i−1)

σ(i)



(14)

GeneralLowerBoundsfortheTotalCompletionTimeinaFlowshopSchedulingProblem-MaxPlusApproach

385

and then multiplying the inequality (14) by

n−1

i=1

n−1

i=1

we complete the proof.

At this point, we can obtain a lower bound of the

Total Completion Time by computing the optimal val-

ues of factors A

and B

. The two following proposi-

tions explain how to do.

Proposition 5 (Minimisation of A

Let σ

SPT

the sequence obtained by sorting jobs in

non-decreasing order of the coefﬁcient t

. This se-

quence minimizes criterion A

Proof.

(σ) =

n−1

j=1

σ( j)

)

n− j

j=1

σ( j)

)

n− j+1

j=1

σ( j)

)

(15)

The second factor is a constant, so we have to mini-

mize

j=1

σ( j)

)

n− j+1

. It is similar to the total comple-

tion time criterion in a one-machine problem (1||

∑

)

where processing times are the t

s. This criterion is

minimized by using Smith’s rule (Smith, 1956).

Proposition 6 (Minimisation of B

Let us consider an Asymmetric Traveling Salesman

Problem (ATSP) deﬁned by the following distances

between n + 1 towns, numbered from 0 to n:











∀i ∈ {1,. . . , n}: d(0, i) = [T

]

∀i ∈ {1,. . . , n}: d(i, 0) = 1 (= 0)

∀(i, j) ∈ {1, ...,n}

: d(i, j) =

]

(16)

Let sequence σ

AT SP

be an optimal cycle of this ATSP:

(σ

AT SP

) is the optimal value of criterion B

Proof. With these notations, B

(σ) can be rewritten

as the length of a cycle:

(σ) = d(0,σ(1))

n−1

i=1

d(σ(i),σ(i + 1))

d(σ(n),0)

(17)

All these results lead to the next proposition.

Proposition 7 (Lower Bound LB

V FL

Let LB

V FL

= (δ

)

⊗ A

(σ

SPT

) ⊗ B

(σ

AT SP

): LB

V FL

is a lower bound of the Total Completion Time.

In usual notations, this lower bound is deﬁned by:

V FL

= nδ

+ A

(σ

SPT

) +B

(σ

AT SP

) (18)

It is needed to solve a traveling salesman prob-

lem to compute this lower bound; however, the proce-

dures for solving that problem are rather effective on

medium size instances.

This lower bound is similar to the one presented

by Della Croce et al. (Della Croce et al., 1996) for

two machines, but it works with m machines and more

constraints.

4.1.3 Existence of Removal Times

If there are removal times, the date of liberation of

machine M

by job J

) is equal to the sum of

completion time C

of job J

and removal time of the

last operation of O

of J

. Thus, the total sum of

(1 ≤ i ≤ n) is equal to the total completion time

plus a constant term which is equal to the sum of re-

moval times of all last operations. Therefore, to ob-

tain a lower bound of the total completion time we

only have to subtract this constant from LB

V FL

4.2 Additional Similar Lower Bounds

A similar approach to the construction of the ﬁrst

lower bound can be developed to achieve the `

lower

bound (∀` ∈ {1,...,m}). Using iteratively lemma 2,

we obtain:

σ(1)

≥ δ



σ(1)



σ(2)

≥ δ



σ(1)

σ(2)



∀i > 2 : C

σ(i)

≥ δ

i−2

j=1

σ( j)



σ(i−1)

σ(i)



(19)

Deﬁning A

(σ) and B

(σ):

(σ) =

n−1

j=1

σ( j)

)

n− j

(20)

(σ) =



σ(1)



j=2



σ( j−1)

σ( j)



σ( j−1)

(21)

we have

i=1

σ(i)

≥ δ

(σ)B

(σ) (22)

Similarly to propositions 5 and 6, we can ﬁnd

SPT

to minimize A

(σ) and σ

AT SP

to minimize

(σ). The `

lower bound of the total completion

time of the initial ﬂowshop problem is then:

V FL

= (δ

)

(σ

SPT

(σ

AT SP

) (23)

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386

Table 1: The performance of branch-and-bound procedure for different k values.

Jobs PV N

(k = 0.3) PVN

V FL

(k = 0.3) PVN

(k = 0.5) PVN

V FL

(k = 0.5) PVN

(k = 0.8) PVN

V FL

(k = 0.8)

7 5.12 × 10

−1

6.86 × 10

−1

4.39 × 10

−1

7.01 × 10

−1

5.12 × 10

−1

10.36 × 10

−1

8 2.20 × 10

−1

2.60 × 10

−1

2.74 × 10

−1

2.66 × 10

−1

2.92 × 10

−1

3.08 × 10

−1

9 5.11 × 10

−2

7.77 × 10

−2

6.46 × 10

−2

6.82 × 10

−2

6.05 × 10

−2

10.03 × 10

−2

10 3.56 × 10

−2

1.63 × 10

−2

2.79 × 10

−2

1.94 × 10

−2

2.40 × 10

−2

2.64 × 10

−2

11 1.63 × 10

−2

0.49 × 10

−2

1.28 × 10

−2

0.58 × 10

−2

1.43 × 10

−2

0.74 × 10

−2

12 3.08 × 10

−3

0.71 × 10

−3

2.74 × 10

−3

1.12 × 10

−3

3.12 × 10

−3

1.44 × 10

−3

13 6.31 × 10

−4

1.87 × 10

−4

6.30 × 10

−4

3.48 × 10

−4

4.71 × 10

−4

3.87 × 10

−4

14 5.02 × 10

−5

4.79 × 10

−5

1.22 × 10

−4

0.52 × 10

−4

1.37 × 10

−4

1.30 × 10

−4

15 2.38 × 10

−5

1.74 × 10

−5

2.19 × 10

−5

1.48 × 10

−5

2.46 × 10

−5

2.13 × 10

−5

16 1.20 × 10

−5

0.18 × 10

−5

1.21 × 10

−5

0.17 × 10

−5

1.24 × 10

−5

0.50 × 10

−5

17 5.70 × 10

−6

0.34 × 10

−6

5.30 × 10

−6

1.03 × 10

−6

5.60 × 10

−6

1.33 × 10

−6

18 5.40 × 10

−7

5.52 × 10

−7

5.00 × 10

−7

4.75 × 10

−7

5.20 × 10

−7

1.62 × 10

−7

20 - 7.04 × 10

−9

- 0.67 × 10

−9

- 3.78 × 10

−9

5 BRANCH-AND-BOUND

ALGORITHM

In order to validate the lower bounds we proposed, we

have incorporated them in a branch-and-bound proce-

dure. A branch-and-bound procedure is an enumer-

ation method that builds dynamically a search tree.

Lower bounds or other criteria like dominance re-

lations are used to cut some useless branches. We

have used the separation scheme introduced by Ignall

and Schrage (Ignall and Schrage, 1965): a partial se-

quence is progressively built as we go deeper in the

search tree. A node corresponds to a partial sequence

and a set of free jobs. The separation of a node con-

sists in adding a free job at the end of the sequence.

A node has as many children as its free jobs. The

branching strategy is Depth-First-Search (DFS). An

upper bound is computed at the root node and updated

at each node. For this purpose, we have used heuris-

tic PR4(15) presented by Pan and Ruiz (Pan and Ruiz,

2013).

The branch-and-bound procedure is detailed in

Algorithm 1 and numerical results are presented in

section 6. In this algorithm, L is the list of nodes that

have not yet been separated and LC the list of child

nodes built after separation of a node.

6 EXPERIMENTAL RESULTS

There are few studies on exact resolution of ﬂow-

shop scheduling problems with criterion of total com-

pletion times. We decided to compare our branch-

and-bound procedure to the one developed by Al-

lahverdi and Al-Anzi (Allahverdi and Al-Anzi, 2006)

for problem F3|perm; S

nsd

∑

. According to the

approach proposed by Allahverdi and Al-Anzi, the

processing and setup times values were randomly

generated respectively from the uniform distribu-

tion on the interval [1,100] and on the interval

[0,100k]. We considered problems of n-jobs (n =

Algorithm 1: Branch-and-Bound.

1: procedure BRANCH-AND-BOUND

2: BestUB ← ∞

3: Generate Root tree NRoot // an empty node

4: Compute LB(NRoot) and UB(NRoot)

5: Add NRoot to list L

6: while L is not empty do

7: N ← top(L) // move the ﬁrst node of list L

8: BestUB ← min{BestUB,U B(N)}

9: if LB(N) < BestUB then

10: Generate children list LC of N

11: for each NChild in LC do

12: Compute LB(NChild)

13: Compute UB(NChild)

14: end for

15: Sort LC

16: // in non-increasing order of LB(NChild)

17: for each NChild in LC do

18: if LB(NChild) < BestUB then

19: top(L) ← NChild

20: else

21: Delete NChild

22: end if

23: end for

24: else

25: Delete N

26: end if

27: end while

28: end procedure

7,8,9,10,11, 12, 13, 14, 15, 16, 17, 18, 20). A class of

thirty instances was generated for each number of jobs

and each k value. The k value for each data set was

assigned to 0.3, 0.5 and 0.8. It was assumed that

all machines were available from the time zero (δ

0, 1 ≤ k ≤ m). To compute lower bounds LB

V FL

we used the ATSP solving procedure developed by

G. Carpaneto, M. Dell’amico and P. Toth (Carpaneto

et al., 1995). The used machine is based on an Intel

Duocore 2.6GHz 4GB RAM.

In their study, Allahverdi and Al-Anzi did not in-

GeneralLowerBoundsfortheTotalCompletionTimeinaFlowshopSchedulingProblem-MaxPlusApproach

387

dicate computation times, they prefer computing the

percentage of visited nodes to solve an instance rela-

tively to the total number of nodes of the whole search

tree. Therefore, we did the same in order to perform

a comparison. We have reported in table 1 the mean

percentage of visited nodes over the thirty instances

of each class (n,k) of problems for our branch-and-

bound (columns PV N

V FL

(k = 0.3), PV N

V FL

(k = 0.5)

and PV N

V FL

(k = 0.8)) and for Allahverdi and Al-

Anzi’s branch-and-bound (columns PV N

(k = 0.3),

PV N

(k = 0.5), PV N

(k = 0.8)).

Furthermore, we have indicated in table 2 the

mean computation times (in second) of our branch-

and-bound procedure for each class of instances. For

instances with strictly less than 18 jobs, we used the

three lower bounds LB

V FL

,LB

V FL

and LB

V FL

in the

branch-and-bound, while for instances of 18 jobs we

use only LB

V FL

and LB

V FL

and for instances of 20

jobs, we use LB

V FL

and LB

V FL

In the other hand, we developed another version

of this branch-and-bound procedure using only a very

simple lower bound SLB. Lower bound SLB of a node

is equal to the total completion time of its correspond-

ing partial sequence. We limited computation times

to 1500 (3000 and 9000, respectively) seconds in case

of n ∈ {7,8, 9, 10, 11, 12, 13, 14,15} (n ∈ {16,17} and

n ∈ {18,20}, respectively). The mean computation

times of that branch-and-bound have been also re-

ported in table 2. When it appears something like ”¿

1500”, it means that the branch-and-bound has never

found the optimal solution within time limit of 1500

seconds over the thirty instances of the class. This

version allowed us to evaluate the effectiveness of our

proposed lower bounds.

Table 2: The mean computation time for each class.

Jobs k=0.3 k=0.5 k=0.8 LB

0.048 0.049 0.058 LB

V FL

,LB

V FL

,LB

V FL

0.030 0.039 0.041 SLB

0.100 0.098 0.176 LB

V FL

,LB

V FL

,LB

V FL

0.099 0.102 0.121 SLB

0.212 0.165 0.193 LB

V FL

,LB

V FL

,LB

V FL

0.683 0.602 0.709 SLB

0.291 0.285 0.353 LB

V FL

,LB

V FL

,LB

V FL

6.654 5.629 6.748 SLB

0.730 0.838 0.932 LB

V FL

,LB

V FL

,LB

V FL

73.674 66.924 74.569 SLB

1.313 1.779 2.33 LB

V FL

,LB

V FL

,LB

V FL

906.030 1173.178 905.417 SLB

4.589 7.593 7.995 LB

V FL

,LB

V FL

,LB

V FL

¿ 1500 ¿ 1500 ¿ 1500 SLB

16.317 15.306 34.569 LB

V FL

,LB

V FL

,LB

V FL

¿ 1500 ¿ 1500 ¿ 1500 SLB

68.559 50.609 68.467 LB

V FL

,LB

V FL

,LB

V FL

¿ 1500 ¿ 1500 ¿ 1500 SLB

152.996 155.756 404.959 LB

V FL

,LB

V FL

,LB

V FL

¿ 3000 ¿ 3000 ¿ 3000 SLB

419.347 1478.93 1894.091 LB

V FL

,LB

V FL

,LB

V FL

¿ 3000 ¿ 3000 ¿ 3000 SLB

7984.230 7638.810 2694.640 LB

V FL

,LB

V FL

¿ 9000 ¿ 9000 ¿ 9000 SLB

2850.320 346.510 1640.350 LB

V FL

,LB

V FL

¿ 9000 ¿ 9000 ¿ 9000 SLB

Table 1 shows that lower bounds LB

V FL

, LB

V FL

and LB

V FL

are useful in reducing the number of vis-

ited nodes. They are really effective when compared

to the performance in the study of Allahverdi and Al-

Anzi (Allahverdi and Al-Anzi, 2006) for instances

with more than nine jobs.

Table 2 shows that with a small number of jobs, it

takes a very short time to achieve the optimum. As the

number of jobs is increasing, the version using SLB

proves that an effective lower bound like LB

V FL

very important to achieve the optimum within a time

limit. In other words, LB

V FL

s are effective to elimi-

nate unworthy branches. However, as the number of

jobs is large, we need to have also a strategy in or-

der to shorten the computation time. This strategy is

under investigation.

7 CONCLUSIONS

We proposed a MaxPlus approach to tackle a m-

machine ﬂowshop problem with several additional

constraints. The MaxPlus approach enables the trans-

formation of a general ﬂowshop problem into a matrix

problem. Then some computations over these matri-

ces allow us to highlight new lower bounds for the

total completion time criterion, based on the resolu-

tion of a one-machine problem and an asymmetric

traveling salesman problem. Despite the necessity of

solving an NP-hard problem, experimental results and

comparison to a previously published research have

shown the effectiveness of these lower bounds.

Our further research will aim at improving these

lower bounds LB

V FL

as well as improving a branch-

and-bound algorithm. In some cases as the number of

jobs is large, a strategy in order to improve the qual-

ity of lower bounds and to shorten the computation

time of the whole branch-and-bound algorithm will be

also studied. Moreover, more speciﬁc constraints for

a ﬂowshop problem can be studied such as no − wait,

min − max delay, S

nsd

, limited stocks between

machines or blocking constraints by modifying only

matrix T

associated to job J

. The study can be also

extended to the weighted total completion time crite-

rion

∑

i=1

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