An assignment-based lower bound for a class of two-machine flow shop problems

Computers & Operations Research 40 (2013) 1693–1699

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An assignment-based lower bound for a class of two-machine flow shop problems Mohamed Haouari, Mohamed Kharbeche n Department of Mechanical and Industrial Engineering, College of Engineering, Qatar University, Qatar

a r t i c l e i n f o

abstract

Available online 8 January 2013

We present an assignment-based lower bound that is valid for a wide class of two-machine flow shop problems with a regular additive performance criterion. We provide empirical evidence that this new P bound consistently outperforms state-of-the-art lower bounds in two important special cases: F2J C j P and F2J T j . Moreover, we illustrate its wide applicability and good performance on two additional P pffiffiffiffiffi P problems: F2J C j and F2J ln C j . & 2013 Elsevier Ltd. All rights reserved.

Keywords: Flow shop Linear assignment Total completion times Total tardiness

1. Introduction In this paper, we propose a lower bound that is valid for a class of two-machine flow shop problems with a regular additive performance criterion. More precisely, we investigate a generic two-machine flow shop problem which can be defined as follows. Each of n jobs from the set J ¼ f1,2, . . . ,ng has to be processed nonpreemptively on two machines M1 and M2 in that order. The processing time of job j A J on machine Mi is pij (i¼1,2). We assume that (i) at any time, each machine can process at most one job and each job can be processed on at most one machine, (ii) preemption is not allowed, (iii) all data are deterministic and integer, and (iv) all machines are ready from time zero onwards. Let S denote the set of feasible schedules of the n jobs. Clearly, each schedule s A S induces for each job j a well-defined completion time C j ðsÞ (or simply, Cj). For each job j, we are given a regular performance criterion f j ðÞ (i.e. a function that is nondecreasing in job completion time). The problem is as follows: Find a schedule sn A S satisfying: X jAJ

X f j ðC j ðsn ÞÞ ¼ min f j ðC j ðsÞÞ sAS

jAJ

Using the well-known three-field notation of Graham et al. [7], P this problem is denoted F2J f j ðC j Þ. Following Conway et al. [2], who show that there exists an optimal schedule that is a permutation schedule without any

n

Corresponding author. Tel.:þ 974 44034328, mobile: þ 974 33173507. E-mail addresses: [email protected] (M. Haouari), [email protected] (M. Kharbeche). URLS: http://faculty.qu.edu.qa/mhaouari (M. Haouari), http://faculty.qu.edu.qa/mkharbec (M. Kharbeche). 0305-0548/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cor.2013.01.001

unnecessary idle time, we shall restrict our attention to this class of schedules. Interestingly, it is easy to see that by appropriately setting the f j ðÞ’s several flow shop scheduling problems can be cast as P F2J f j ðC j Þ. For instance, by setting f j ðC j Þ ¼ C j for j A J, we obtain P the well-studied problem F2J C j [9]. Since this latter problem is P NP-hard [6], it follows that F2J f j ðC j Þ is NP-hard as well. Moreover, assume that each job j A J has a due date dj by which it should ideally be completed, and define the tardiness of job j as T j ¼ maxf0,C j dj g. Then, by setting f j ðC j Þ ¼ T j for j A J, the problem amounts to minimizing total tardiness in a two-machine flow shop [20]. At this point, it is worth emphasizing that the f j ðÞ’s might be nonlinear as well. For instance, we may consider the flow shop problem with discounted weighted sum of job completion times [15]. In this case, f j ðC j Þ ¼ wj ð1exprC j Þ=r for j A J where wj is the weight of job j and r 4 0 is a parameter. In addition, we may also consider the nonlinear objective that amounts to minimizing the quadratic sum of completion times (that is, P F2J C 2j ). It is noteworthy that even though this latter problem has been investigated by several authors (see [15,21,22]), no lower bound has ever been proposed so far. The main contribution of this paper is to present a general lower P bounding procedure for F2J f j ðC j Þ. This lower bound is based on solving an appropriate linear assignment problem. To provide evidence of the usefulness of the proposed approach, we shall use it to P derive lower bounds for the following special cases: (i) F2J C j , (ii) P P pffiffiffiffiffi P F2J T j , (iii) F2J C j , and (iv) F2J ln C j . Note that, this fourth special performance criterion is equivalent to minimizing the geometric mean of completion times (i.e. ðPnj¼ 1 C j Þ1=n ). The remainder of this paper is organized as follows. In Section 2, we present a description of the bounding procedure. In Sections 3–5,

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we provide the results of a comprehensive computational study of the proposed lower bound for the aforementioned four special cases. Finally, we provide some concluding remarks.

2. An assignment-based lower bound Let lkj denote a valid lower bound on the completion time of job j if scheduled on the kth position (j A J, k¼1,y,n) (a procedure for computing these bounds shall be described below). Also, define xkj as a binary variable that takes value 1 if job j is assigned to position k, and 0 otherwise (k,j ¼ 1, . . . ,n). Using these definitions, we consider the following linear assignment problem: n X n X

Z n ¼ Minimum

In the sequel, we assume with no loss of generality that J\fjg ¼ fi1 , . . . ,in1 g and that the jobs in J\fjg are indexed according to Johnson’s rule. That is, we have minfp1,ih ,p2,ih þ 1 g r minfp2,ih ,p1,ih þ 1 g

for h ¼ 1, . . . ,n2:

ð7Þ

Hence, an immediate consequence of Proposition 1 is that for computing lkj , (k¼2,y,n) we should find a subset Skj satisfying:

f j ðlkj Þxkj ,

ð1Þ

8j ¼ 1, . . . ,n,

ð2Þ

(C1): Skj DJ\fjg and 9Skj 9 ¼ k1, (C2): Skj should be sequenced according to Johnson’s rule. Hence, job ih A Skj is scheduled on position q (q ¼ 1, . . . ,k1) if and only if h A ½q,n1.

8k ¼ 1, . . . ,n,

ð3Þ

Let Dkj denote the family of subsets satisfying (C1) and (C2). Clearly, we have the following:

8k, j ¼ 1, . . . ,n:

ð4Þ

k¼1j¼1 n X

Subject to

Proof. Denote by C 1 ðSkj Þ and C 2 ðSkj Þ the completion times of Skj on M1 and M2, respectively. We have lkj ¼ maxfC 1 ðSkj Þ þ p1j ,C 2 ðSkj Þg þ p2j . We observe that C 1 ðSkj Þ is sequence-independent. Therefore minimizing the completion time of j amounts to minimizing C 2 ðSkj Þ. Clearly, this latter objective is achieved by sequencing the jobs using Johnson’s rule. &

xkj ¼ 1

k¼1 n X

xkj ¼ 1

j¼1

Proposition 2. xkj A f0,1g

Lemma 1. Zn is a valid lower bound for F2J

P

lkj ¼ min maxfC 1 ðSkj Þ þ p1j ,C 2 ðSkj Þg þ p2j : Skj A Dkj

f j ðC j Þ.

Proof. Let a permutation s ¼ ðs ð1Þ, . . . , s ðnÞÞ be an optimal assignment (i.e. xk, s ðkÞ ¼ 1 for k¼1,y,n is an optimal solution to (1)–(4)). P Let Z n ¼ nk ¼ 1 f s ðkÞ ðlk, s ðkÞ Þ denote the value of the corresponding objective function. Assume, by contradiction, that there exists a feasible schedule s ¼ ðsð1Þ, . . . , sðnÞÞ A S that satisfies: X f j ðC j ðsÞÞ o Z n : ð5Þ

ð8Þ

Now, we show that computing lkj (k¼2,y,n) is equivalent to finding shortest paths from one node to all other nodes in an acyclic bi-weighted graph G ¼ ðV,AÞ that is constructed as follows. The set of nodes is partitioned into n subsets V 0 ,V 1 , . . . ,V n1 where V0 is a singleton that contains a ‘start’ node 0, and Vq ðq ¼ 1, . . . ,n1Þ contains nodes ðq,ih Þ with h ¼ q, . . . ,n1. A node ðq,ih Þ A V q is associated with the decision of scheduling job ih at position q. The set of arcs is constructed in the following way:

jAJ

For each job j A J, we denote by s1 ðjÞ the position assigned to j in the sequence s. Since ls1 ðjÞ,j r C j ðsÞ, 8j ¼ 1, . . . ,n, and f j ðÞ is nondecreasing for j ¼ 1, . . . ,n, we have X X f j ðls1 ðjÞ,j Þ r f j ðC j ðsÞÞ: ð6Þ jAJ

jAJ

Since we get n X k¼1

Pn

j¼1

f j ðls1 ðjÞ,j Þ ¼

f sðkÞ ðlk, sðkÞ Þ o

n X

Pn

k¼1

f sðkÞ ðlk, sðkÞ Þ, then from (5) and (6)

f s ðkÞ ðlk, s ðkÞ Þ,

k¼1

which contradicts the fact that s is optimal. Thus, Zn is a valid lower bound. & Obviously, Lemma 1 remains correct even if there are more than two machines. However, we shall restrict our attention to the two-machine case and provide an effective procedure for computing the lkj ’s. A procedure for computing lkj for k¼2,y,n and j A J is the following (note that for k¼1, we trivially set l1j ¼ p1j þ p2j , 8j ¼ 1, . . . ,n). Denote by Skj D J\fjg the (unknown) subset of k  1 jobs that if they are scheduled before j, would yield the minimal possible completion time of j on M2. We set lkj equal to the makespan that results from sequencing Skj followed by j. Proposition 1. The jobs in Skj should be sequenced according to Johnson’s rule [10].

 For each node ð1,ih Þ A V 1 there is an arc ð0,ð1,ih ÞÞ.  For each node ðq,ih Þ A V q ðq ¼ 2, . . . ,n1, h ¼ q, . . . ,n1Þ there are hq þ1 arcs of the form ððq1,is Þ,ðq,ih ÞÞ for s ¼ q1, . . . , h1. An arc ððq1,is Þ,ðq,ih ÞÞ A A is associated with the decision of scheduling jobs is and ih at positions q1 and q, respectively. Each arc incident to node ðq,ih Þ ðq ¼ 1, . . . ,n1, h ¼ q, . . . ,n1Þ has an ordered pair ðp1,ih ,p2,ih Þ of weights associated to it. Fact 1. 9V9 ¼ 1 þðnðn1Þ=2Þ and 9A9 ¼ n1 þðnðn1Þðn2Þ=6Þ. Consider a path P ¼ ð0,ð1,j1 Þ,ð2,j2 Þ, . . . ,ðk1,jk1 ÞÞ in G between nodes 0 and ðk1,jk1 Þ. This path corresponds to sequencing jobs j1 ,j2 , . . . ,jk1 in that order. Fact 2. The paths in G between0 and a node in V k1 are in one-toone correspondence with the subsets Skj A Dkj . A two-dimensional label ðd1 ðPÞ, d2 ðPÞÞ is associated with P where d1 ðPÞ and d2 ðPÞ refer to the completion times on M1 and M2, respectively. A key observation is that computing lkj (according to Eq. (8)) requires finding a path P in G between 0 and ðk1,ih Þ ðh A ½k1,n1Þ and having a label ðd1 ðP Þ, d2 ðP ÞÞ such that maxfd1 ðP Þ þp1j , d2 ðP Þg is minimal. Definition 1. Let P 1 and P 2 denote two different paths in G between nodes 0 and ðq,jq Þ. P 1 dominates P 2 if and only if d1 ðP 1 Þ r d1 ðP 2 Þ and d2 ðP 1 Þ r d2 ðP 2 Þ. Definition 2. Given a set Q ðq,jq Þ of paths (or equivalently, labels) between nodes 0 and ðq,jq Þ, NDðQ ðq,jq ÞÞ is the set of paths that are obtained from Q ðq,jq Þ by dropping all the dominated paths.

M. Haouari, M. Kharbeche / Computers & Operations Research 40 (2013) 1693–1699

Definition 3. Given a path P in G between nodes 0 and ðq,jq Þ, P 0 is an extension of P if and only if P 0 is obtained by appending an arc ððq,jq Þ,ðqþ 1,jq þ 1 ÞÞ A A to P. Clearly, if P 0 is derived in this way then we have the following:

d1 ðP 0 Þ ¼ d1 ðPÞ þ p1,jq þ 1 d2 ðP 0 Þ ¼ maxfd1 ðPÞ þ p1,jq þ 1 , d2 ðPÞg þ p2,jq þ 1 Hence, lkj ðk ¼ 2, . . . ,n, j A JÞ can be computed using the following two-phase approach:

 Phase 1: Compute the set P kj of nondominated paths in G that connect node 0 to a node in V k1 .

 Phase 2: Set lkj ¼ minP A P fmaxfd1 ðPÞ þ p1j , d2 ðPÞgg þp2j . kj

Phase 1 could be achieved by using a label-setting algorithm for generating nondominated paths that is similar to the so-called generalized Dijkstra algorithm for the constrained shortest path problem [5]. This algorithm is the following:

Algorithm. A generalized Dijkstra algorithm for computing lkj ðk ¼ 2, . . . ,nÞ. Step 1: Initialization: Q ð1,ih Þ ¼ fðp1,ih ,p1,ih þ p2,ih Þg for h ¼ 1, . . . ,n1 Step 2: Computation of nondominated paths For q ¼ 2, . . . ,n1 For h ¼ q, . . . ,n1 Step 2.1. Set Q ðq,ih Þ ¼ Set of paths obtained by appending arcs ððq1,is Þ,ðq,ih ÞÞ to paths Q ðq1,is Þ ðs ¼ q1, . . . ,h1Þ, Step 2.2. Set Q ðq,ih Þ’NDðQ ðq,ih ÞÞ End (For h) End (For q) Step 3: Computation of lkj , k¼2,y,n For k¼2,y,n Set lkj ¼ minh ¼ k1,...,n1 fminP A Q ðk1,ih Þ fmaxfd1 ðPÞ þ p1j , d2 ðPÞg þp2j gg End (For k)

1695

~ We have removed from G. ~ ¼ ðn1Þ þ 9A9

n2 n2 X 1X ðqn þ 1Þ9V q 99V q þ 1 9 ¼ O n q3 2q¼1 q¼1

! ¼ Oðnn4 Þ

Clearly, a path in G~ that connects node (0,0,0) to node ðq,ih ,tÞ represents a sequence of q jobs such that job ih is scheduled at position q and the completion time on M1 is t. The weight of this path (hereafter referred to by gðq,ih ,tÞÞ corresponds to the minimum makespan of this latter sequence. This makespan can be computed using the following recursions: gð0,0,0Þ ¼ 0 gðq,ih ,tÞ ¼ minfmaxft,gðq1,il ,tp1,ih Þgg þp2,ih il o ih

for ðq,ih ,tÞ A V~ \fð0,0,0Þg:

ð9Þ

Clearly, a path in G~ that connects node (0,0,0) to node ðq,ih ,tÞ corresponds to a path P in G that connects 0 to node ðq,ih Þ and whose associated label is ðd1 ðPÞ, d2 ðPÞÞ ¼ ðt,gðq,ih ,tÞÞ. Finally, the value of lkj is given by

lkj ¼

min

ðk1,ih ,tÞ A V~

fmaxft þ p1j ,gðk1,ih ,tÞgg þ p2j for k ¼ 2, . . . ,n:

Recursions (9) show that the computation of gðq,ih ,tÞ for ~ time (or equivalently, Oðnn4 Þ time). Thus, ðq,ih ,tÞ A V~ requires Oð9A9Þ for each j A V the computation of lkj requires Oðnn4 Þ time. & Example 1. Consider the following 5-job instance [4].

p1j p2j

1

2

3

4

5

1 5

3 2

4 7

4 8

8 1

If the jobs are indexed according to Johnson’s rule, we obtain:

p1j p2j

1

2

3

4

5

1 5

4 7

4 8

3 2

8 1

Lemma 2. The computation of the lkj ’s can be achieved in pseudopolynomial time.

Proof. We show that the generalized Dijkstra algorithm amounts to ~ that is derived from computing paths in an expanded graph G~ ¼ ðV~ , AÞ ¼ maxj A J fp1j g, pmin ¼ minj A J fp1j g, and n ¼ pmax G as follows. Let pmax 1 1 1 pmin 1 . Given that the processing times are integers, we replace each node ðq,ih Þ A V q ðq ¼ 1, . . . ,n1Þ by qn þ 1 nodes ðq,ih ,tÞ (t ¼ qpmin 1 , . . . ,qpmax 1 Þ. Node ðq,ih ,tÞ represents the state where job ih is scheduled at the qth position and finishes processing on M1 at time t. We have 9V~ 9 ¼ 1þ

n1 X

ðqn þ1Þ9V q 9 ¼ O n

q¼1

n1 X

! q2

¼ Oðnn3 Þ:

q¼1

Furthermore, the arcs in G~ are constructed as follows: if there exists an arc in A that links (q,i Þ to ðq þ 1,i Þ then we create an arc in A~ that l

h

links ðq,il ,tÞ to ðqþ 1,ih ,t þ p1,ih Þ. Recall that there are 129V q 99V q þ 1 9 arcs in G that connect Vq to V q þ 1 ðq ¼ 1, . . . ,n2Þ. Therefore there are

n þ 1Þ9V q 99V q þ 1 9 arcs of the form ððq,il ,tÞ,ðqþ 1,ih ,t þp1,ih ÞÞ.

1 2ðq

Finally, we create a ‘‘start’’ node (0,0,0) that is connected to nodes (1,ih ,p1,ih Þ (ih A V 1 Þ. Obviously, nodes having no incoming arcs can be

In this example, we shall compute lk1 for k ¼ 1, . . . ,5. Clearly, l11 ¼ 6. Next, the figures describe how lk1 can be computed for k ¼ 2, . . . ,5. V 1 ¼ fð1,2Þ,ð1,3Þ,ð1,4Þ,ð1,5Þg (see Fig. 1). In this figure, we added on the top of each node the two-dimensional label of each nondominated path. l11 ¼ minfmaxf4þ 1,11g þ 5,maxf4þ 1,12g þ 5, maxf3þ 1,5g þ 5,maxf8 þ1,9g þ5g ¼ 10. From Fig. 2, we see that V 2 ¼ fð2,3Þ,ð2,4Þ,ð2,5Þg. We have l31 ¼ minfmaxf8 þ1,19g þ5,maxf7 þ1,13g þ5,maxf11 þ 1, 12g þ 5g ¼ 17. We observe that the path ð0,0Þ-ð1,3Þ-ð2,4Þ having a label (7,14) is dominated by the path ð0,0Þ-ð1,2Þ-ð2,4Þ having a label (7,13). From Fig. 3, we see that V 3 ¼ fð3,4Þ,ð3,5Þg. We have l41 ¼ min fmaxf11 þ 1,21g þ 5, maxf15 þ 1,16g þ 5g ¼ 21. The path ð0,0Þð1,2Þ-ð2,3Þ-ð3,5Þ is dominated by the path ð0,0Þ-ð1,2Þð2,4Þ-ð3,5Þ. From Fig. 4, we have V 4 ¼ fð4,5Þg and l51 ¼ maxf19 þ1,22g þ 5 ¼ 27. 3. Minimizing total completion times P Problem F2J C j has been intensely investigated since the 1960s when Ignall and Schrage [9] published their seminal paper. In

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M. Haouari, M. Kharbeche / Computers & Operations Research 40 (2013) 1693–1699

Fig. 4. Computation of l51 .

Hoogeveen et al. [8] and is based upon the following integer linear P programming formulation of F2J C j . Decision variables:

 ykj : binary variable that takes value 1 if job j A J is assigned to position k ðk ¼ 1, . . . ,nÞ, and 0 otherwise.

 Ik : continuous variable that represents the idle time on M2 Fig. 1. Computation of l21 .

before the start of the processing kth job (k ¼ 1, . . . ,nÞ on that machine. Using these definitions, F2J

Minimize

n X n X

P

C j can be formulated as follows:

ðnkþ 1Þp2j ykj þ

k¼1j¼1

Subject to

n X

n X

ðnkþ 1ÞIk ,

ð10Þ

k¼1

ykj ¼ 1

8k ¼ 1, . . . ,n,

ð11Þ

j¼1

n X

ykj ¼ 1

8j ¼ 1, . . . ,n,

ð12Þ

k¼1 k1 X n X

p2j yij 

i¼1j¼1

Fig. 2. Computation of l31 .

k X n X

p1j yij þ

i¼1j¼1

ykj A f0,1g,Ik Z0

k X

Ii Z 0

8k ¼ 1, . . . ,n,

ð13Þ

i¼1

8k,j ¼ 1, . . . ,n,

ð14Þ

where the objective (10) is to minimize the sum of completion times. Constraint (11) ensures that each position is covered by exactly one job. Constraint (12) requires that each job is assigned to exactly one position, and (13) states the logical restrictions on Pk1 Pn the idle times (note that we used the convention i¼1 j¼1 p2j yij ¼ 0 for k¼1). We refer to [8] for a detailed discussion of this model. Clearly, solving the LP relaxation of model (10)–(14) yields a valid lower bound. Proposition 3. k X n X

Fig. 3. Computation of l41 .

i¼1j¼1

p2j yij þ

k X j¼1

Ij Z

n X

lkj ykj , k ¼ 1, . . . ,n

ð15Þ

j¼1

is a valid inequality for model (10)–(14). particular, lower bounds and exact approaches have been investigated in numerous papers (see [1,3,4,8]). However, according to Hoogeveen et al. [8], a major motivation for the study of the P F2J C j is related to its notorious intractability which has been attributed to the lack of relatively strong lower bounds. In this section, we compare the performance of the proposed assignmentbased lower bound with the tightest polynomial-time lower bound that has been published so far. This bound has been proposed by

Proof. It suffices to observe that the LHS corresponds to the completion time on M2 of the job that is assigned to position k, and that, using the definition of lkj , the RHS corresponds to a lower bound on the completion time of the job that is assigned to this position. & We carried out a computational study to assess the quality of the proposed lower bound. In this section, we use as a lower bound measure of performance the average percentage relative deviation (APRD) with respect to an upper bound that is obtained

M. Haouari, M. Kharbeche / Computers & Operations Research 40 (2013) 1693–1699

Table 1 P Performance of the lower bounds for F2J C j . n

LBHNV

þ LBAP

LBAP

APRD

TIME

APRD

TIME

APRD

TIME

10 20 30 40 50 60

1.63 1.21 1.06 0.83 0.64 0.56

– – 0.01 0.02 0.04 0.06

1.15 0.87 0.80 0.63 0.51 0.45

– – 0.04 0.29 1.33 4.01

0.58 0.54 0.60 0.54 0.41 0.37

– 0.02 0.08 0.37 1.47 4.32

Average

0.99

0.02

0.74

0.94

0.52

1.04

(–) means that the mean CPU time is less than 0.001 s.

P by solving F2J f j ðC j Þ using a genetic algorithm [12]. All the results presented in this paper were obtained on a Pentium IV 3.3 GHz PC with 3 Gbytes RAM and all the LPs were solved using CPLEX 12.1. Furthermore, the assignment problems were solved using Jonker and P Volgenant’s algorithm [11]. We have randomly generated F2J C j instances with up to 60 jobs, where the processing times were drawn from the uniform distribution [1,100]. For each problem size, 20 instances were randomly generated. All of the test problems solved in this paper are publicly available at http://faculty.qu.edu.qa/mkharbec/Thesis.aspx. In Table 1, we report the performance of the following lower bounds:

 LBHNV : The lower bound obtained by solving the LP relaxation of model (10)–(14).

 LBAP : The proposed assignment-based lower bound. þ  LBAP : The LP bound obtained by including (15) in model (10)–(14). For each lower bound and each problem size, we report the average percentage relative deviation (APRD) as well as the average CPU time (in seconds). We see from Table 1 that LBAP outperforms LBHNV and yields an average percentage deviation of 0.74% whereas the gap yielded by the best bound from the literature is 0.99%. More precisely, we found that LBAP strictly dominates LBHNV for 92% of the instances (at this point, it is worth mentioning that LBHNV largely dominates all the polynomial lower bounds that were proposed so far). Interestingly, we see that the proposed valid inequality consistently improved the quality of the LP relaxation. Indeed, we þ observe that LBAP exhibits a good performance and yields a remarkable average percentage deviation of 0.52%. Nevertheless, we see that both LBAP and LBAP þ require much longer CPU times than LBHNV does. In particular, we observe that a significant CPU time is required for n Z50.

4. Minimizing total tardiness As a second illustrative application of the proposed assignmentbased lower bound, we consider the problem of sequencing n jobs in a two-machine flow shop with the objective of minimizing total P tardiness (F2J T j ). This problem has received a lot of attention in the scheduling literature and most of these efforts have been devoted to the development of lower bounds and solution algorithms (see [20,13,14,18,17,19], to quote just a few). Not surprisP ingly, F299 T j is NP-hard [16,14]. Actually, even the solution of a small-sized 20-job problem constitutes a serious challenge for state-of-the-art exact approaches [19]. Clearly, the development P of an improved lower bound for F2J T j is of crucial importance with regard to the design of effective solution algorithms. In this

1697

section, we compare the performance of a customized LBAP with the best lower bound from the literature. This bound has been proposed by Pan and Fan [18] and will be referred to as LBPF. For the sake of brevity, the description of LBPF is omitted. In our experiments, we used the procedure that is described P in Pan et al. [17] to randomly generate F2J T j instances. The processing times were drawn from the discrete uniform distribution [1,10]. The due dates were generated as a function of the due date range (R) and the tardiness factor (t). Denote P by T ¼ minj A J p1j þ j A J p2j a lower bound on the minimum makespan, then the due dates were randomly drawn from the discrete uniform distribution ½Tð1tR=2Þ,Tð1t þ R=2Þ. In our experiments, we set t A f0:25,0:5,0:75g, R A f0:25,0:5,0:75,1:0g, and n A f10,20,30,40,50,60g. For each problem combination, 10 instances were generated. Hence, we obtained 720 instances. In Table 2, we report the results of an analysis of the performance of LBAP and LBPF. In this table, each entry in the column entitled ‘‘AAD’’ indicates the average absolute deviation from an upper bound obtained using the genetic algorithm [12]. It is worth noting that we found that the mean CPU time is not dependent on parameters t and R. Therefore, only one single value of the mean time is reported for each value of n. In Table 2, we see that LBAP exhibits a very good overall performance since it consistently outperforms LBPF. Hence, we found that the ratio of the sum (over all instances) of the absolute deviations yielded by LBAP to the sum of those yielded by LBPF is 0.67. Moreover, a detailed pairwise comparison reveals that LBAP is strictly smaller than LBPF for 559 instances out of 720 (78%) and that both bounds yield the same deviations for the remaining instances. However, we observe that LBPF is extremely fast and that LBAP requires relatively longer CPU times. Remark 1. We performed further experiments and we observed that the CPU times increased by an average factor of 4.6 when the processing times are drawn from the interval [1,100]. 5. Results on F2J

P pffiffiffiffiffi P C j and F2J ln C j

To provide further evidence of the applicability of the proposed lower bound, we assessed its performance on two ‘‘nonstandard’’ two-machine flow shop problems having nonlinear performance P pffiffiffiffiffi P criteria: F2J C j and F2J ln C j . To the best of our knowledge, no lower bound has ever been proposed for any of these problems. We carried out the computational experiments using the same P generation procedure that was employed for F2J C j . Here again, the APRDs were computed with respect to upper bounds provided by the genetic algorithm [12]. The results are displayed in Table 3. We see that, LBAP exhibits a good performance. Indeed, for both problems the APRDs are consistently less than 0.71%. Remark 2. We performed additional tests on instances having processing times drawn from the uniform distribution [1,10] (instead of [1,100]). We found that in this case, LBAP exhibits a better performance and requires shorter CPU times as well. P P pffiffiffiffiffi Indeed, we observed that, for problems F2J C j , F2J C j , and P F2J ln C j , the mean percentage deviations are 0.51%, 0.44%, and þ 0.40%, respectively. Moreover, the enhanced LP bound LBAP yielded a mean percentage deviation of 0.30%. 6. Conclusion In this paper, we presented an assignment-based lower bound for a large class of two-machine flow shop problems with a regular additive performance criterion. This class includes as important special cases two notoriously difficult scheduling P P problems: F2J C j and F2J T j . We presented the results of

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M. Haouari, M. Kharbeche / Computers & Operations Research 40 (2013) 1693–1699

Table 2 P Performance of the different lower bounds for F2J T j . n

10

t

0.25

0.50

0.75

R

LBPF

0.25 0.50 0.75 1.00 0.25

0.25

Sum 20

0.25

0.50

0.75

0.25 0.50 0.75 1.00 0.25

0.25

Sum 30

0.25

0.50

0.75

0.25 0.50 0.75 1.00 0.25

0.25

Sum

LBAP AAD

n

AAD

TIME

7.9 12.0 14.2 18.6 20.5 0.50 0.75 1.00 26.7 0.50 0.75 1.00

–

296.8

0.1

144.2

1.1

14.5 11.8 7.9 28.8 59.5 0.50 0.75 1.00 111.7 0.50 0.75 1.00

–

11.7 9.1 6.1 26.6 42.4 72.8 84.2 159.9 45.6 152.4 173.0 171.7

0.01

1048.2

0.1

618.9

1.2

29.0 23.0 17.7 4.6 101.1 0.50 0.75 1.00 200.5 0.50 0.75 1.00

–

22.2 17.7 15.7 4.1 68.9 147.2 182.7 254.2 96.1 262.7 292.3 364.5

0.02

1879.5

0.1

1260.4

2.4

4.5 6.4 10.1 14.8 12.5 25.6 39.0 38.8 5.8 28.7 34.4 30.4

TIME –

LBPF AAD

40

41.0 36.4 2.9 1.0 201.0 12.9 22.6 18.7 341.2 7.9 14.8 13.2 3185.9

50

81.2 55.9 34.9 12.6 316.8 54.3 70.5 119.4 496.0 49.7 83.8 99.7 5046.0

60

70.2 21.2 0.0 0.0 338.8 124.8 162.9 228.9 782.4 116.5 163.4 239.2 6080.4

LBAP TIME –

AAD 28.4 30.6 1.6 1.0 133.5 268.0 249.9 378.6 145.5 519.5 576.2 570.2

0.1

2129.3

–

62.8 44.3 30.0 12.6 233.1 370.4 479.3 763.1 221.0 763.4 786.6 885.8

TIME 0.13

215.8 218.0 325.6 248.5 347.7 433.1 15.6 0.46

299.2 437.8 704.7 372.0 466.3 647.4

0.1

3531.2

–

51.9 14.3 0.0 0.0 232.3 322.4 459.5 542.2 351.9 1163.7 1268.6 1111.4

1.58

4071.2

189.6

0.1

55.2

249.8 431.1 512.1 589.9 759.4 878.5

(–) means that the mean CPU time is less than 0.001 s.

Table 3 P P pffiffiffiffiffi C j and F2J ln C j . Performance of the lower bounds for F2J n

F2J

P pffiffiffiffiffi Cj

F2J

P

ln C j

APRD

TIME

APRD

TIME

10 20 30 40 50 60

0.71 0.68 0.68 0.57 0.45 0.49

– 0.01 0.05 0.30 1.31 4.01

0.31 0.37 0.37 0.34 0.24 0.27

– 0.01 0.05 0.30 1.31 4.00

Average

0.60

0.94

0.32

0.94

computational experiments that demonstrate that the proposed bound exhibits a better quality than the state-of-the-art lower P P bounds for F2J C j and F2J T j for 92% and 78% of the instances, P respectively. Also, we showed that, for problem F2J C j , a good LP-based bound can be obtained by appending a suitable valid inequality to the previously proposed formulation. Furthermore, we provided illustrative applications of the proposed bound on

P pffiffiffiffiffi P C j and F2J ln C j . We found that for these problems F2J latter problems, the proposed lower bound exhibits a good performance. To the best of our knowledge, this is the first lower bound that exhibits a good performance on several flow shop problems with different optimization criteria. Nevertheless, we emphasize that we do not claim that LBAP would necessarily P exhibit a good performance on any F2J f j ðC j Þ special case. For instance, preliminary experiments carried out on the special case P F2J C 2j show that the mean percentage deviations are around 3%. Hence, future investigation is required to better identify additional special cases for which the proposed lower bound proves effective. Furthermore, to assess the practical usefulness of the assignment-based lower bound, we propose to embed it within a branch-and-bound algorithm.

Acknowledgments The authors thank two anonymous referees for a number of careful remarks and insightful comments that have helped to improve the presentation in this paper.

M. Haouari, M. Kharbeche / Computers & Operations Research 40 (2013) 1693–1699

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