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Single machine batch scheduling problem with fuzzy batch size
Computers & Industrial Engineering 62 (2012) 688–692
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Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Single machine batch scheduling problem with fuzzy batch size Xuesong Li a,1, Hiroaki Ishii b,⇑, Teruo Masuda c a
Department of Mathematics, Harbin Institute of Technology, No. 92, West Da-Zhi Street, Harbin, Heilongjiang 150001, China Department of Mathematical Sciences, Kwansei Gakuin University, 2-1 Gakuen, Sanda, Hyogo 669-1337, Japan c Faculty of Business Administration, Tezukayama University, Tezukayama, Japan b
a r t i c l e
i n f o
Article history: Available online 24 December 2011 Keywords: Batch schedule Flexible upper bound of batch size Non-dominated schedule Efficient procedure Maximum completion time Flow time
a b s t r a c t In a batch scheduling problem, jobs are grouped (group is called batch) and scheduled in batches, and a setup time is incurred when starting a new batch. Processing times are assumed to be identical for all jobs. Setup times are assumed to be identical for all batches. Though all batch sizes cannot exceed a common upper bound, the upper bound is flexible and satisfaction degree with respect to the upper limit to be maximized is given. Also the other two objectives, i.e., the maximum completion time and the flowtime are to be minimized. Usually there exists no solution optimizing three objectives at a time. Therefore we define non-dominated solutions consisting of batch size, batch number and allocation of jobs to batches. First we propose an efficient algorithm for a sub-problem with fixed upper limit of batch size, fixed batch number based on a Lagrange relaxation procedure. Based on the properties of non-dominated solutions clarified in this paper, we propose an efficient algorithm to find some non-dominated solutions. Finally we summarize the results in this paper and discuss further research problems. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Modern technologies in flexible manufacturing lead to new types of scheduling problems including batch scheduling problems. In batch scheduling problems, jobs are grouped (each group is called batch) and scheduled in batches, and a setup time is incurred when starting a new batch. Jobs in the same batch have a same completion time, that is, the completion time of the batch is the completion time of the final job in the batch. There are two types of batching problems, denoted by p-batching problem and s-batching problem (Brucker, 2001). For p-batching problem (s-batch problem) the length of a batch is equal to the maximum (sum) of processing times of all jobs in the batch. The recent surveys of Potts and Kovalyov (2000) and Ng, Allahverdi, Cheng, and Kovalyov (2008) mention a class of problems in which the size of the batches is bounded. This constraint reflects many real life applications to a certain extent, where the number of items per batch cannot exceed a certain value due to limitations of the processor or of the production line. The classical single machine batch scheduling problem is to minimize the flow-time. Given the number of jobs, their processing times and setup times which are assumed to be identical, the optimal number of batches and their ⇑ Corresponding author. Tel./fax: +81 79 565 7129. E-mail addresses: [email protected] (X. Li), [email protected] (H. Ishii), [email protected] (T. Masuda). 1 Tel./fax: +86 451 85413857. 0360-8352/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.12.021
size are to be determined. Santos and Magazine (1985) first solved the relaxed version of the problem, i.e., the solution they obtained were non-integer batch sizes. Later algorithms for finding integer batch sizes were introduced by Naddef and Santos (1988), Coffman, Yannakakis, Magazine, and Santos (1990), Shallcross (1992). If n denote the number pffiffiffi of jobs, the computational time of these algorithms are OðnÞ, O n and OðlogðnÞÞ, respectively. Mosheiov, Oron, and Ritov (2005) introduced an alternative integer solution based on a simple rounding procedure of the non-integer batched obtained by Santos and Magazine (1985). Three years later, Mosheiov and Oron (2008a) extended the result obtained in Mosheiov et al., 2005. They introduced an efficient solution for both cases of an upper and a lower bound on the batch sizes. Besides, there are also many other researches on a batch problem, such as, Dobson, Karmarkar, and Rummel (1987), Hochbaum and Landy (1994), Cheng and Janiak (1995), Sung and Joo (1997), Cheng, Janiak, and Kovalyov (2001), Mosheiov and Oron (2008b), also see survey papers Potts and Van Wasenhove (1992). This paper treats one model of s-batch problem which is an extension of Mosheiov and Oron (2008a), that is, considers a single machine batch scheduling problem with the intent of introducing an important new problem and research direction in production and machine scheduling. Also motivated by an attempt to improve the resource usage and customer satisfaction which are important objectives in manufacturing. First in Section 2 formulates nonflexible version of our problem. This problem is bi-criteria and also extension of Mosheiov and Oron (2008a). Section 3 proposes an
X. Li et al. / Computers & Industrial Engineering 62 (2012) 688–692
efficient algorithm to find some non-dominated solutions after defining non-domination of solutions of the problem. For that purpose we derive some useful properties of non-dominated solution of the problem. Section 4 formulates our main problem and proposes a solution algorithm to find some non-dominated solutions after definition of non-domination with respect to our main problem. Section 4 also includes a toy example of our problem. Section 5 summarizes results in this paper and discusses further research problems. 2. Bicriteria single machine batch scheduling problem with fixed limit of batch size First we consider the following bi-criteria batch scheduling problem with a limited batch size: There exist n jobs with identical processing time. So without any loss of generality, we assume that identical processing time is 1. These jobs are grouped and scheduled in batches, and a setup time s is incurred when starting a new batch. Setup times are assumed to be identical for all batches. All batch sizes cannot exceed a common upper bound b. We consider two objectives, i.e., the maximum completion time and flow-time, which are to be minimized. That is, the problem is to find the optimal solution consisting of a batch number and allocation of jobs to batches minimizing the above two objectives under a common limited batch size. Let k be a batch number and nj ; j ¼ 1; 2; . . . ; k batch size of jth P batch. Then the completion time of jth batch is ji¼1 ni þ js. So the P Pj total flow time is f ¼ kj¼1 i¼1 ni þ js nj and maximum comple-
689
X 2 . If there exists no solution that dominates X, X is called nondominated solution of FP. Let t ¼ bs where de is the least integer not less than . Then we have the following property. Proposition 2. For X ¼ ðk; ðn1 ; n2 ; . . . ; nk ÞÞ with k > t, if b > ni P nj P 2 for some k P j > i; j i P t; then we can construct a solution X 0 ¼ ðk; ðn01 ; n02 ; . . . ; n0k ÞÞ better than (dominate X), where n01 ¼ n1 ; n02 ¼ n2 ; . . . ; n0i1 ¼ ni1 ; n0i ¼ ni þ 1; n0iþ1 ¼ niþ1 ; . . . ; n0j1 ¼ nj1 ; n0j ¼ nj 1; n0jþ1 ¼ njþ1 ; . . . ; n0k ¼ nk . Proof. From (2), the difference between total flow time of X and that of X 0 is
1 1 1 1 ðni þ isÞ2 þ ðnj þ jsÞ2 ðn0i þ isÞ2 ðn0j þ jsÞ2 2 2 2 2 1 1 1 1 2 2 ¼ ðni þ isÞ þ ðnj þ jsÞ ðni þ is þ 1Þ2 ðnj þ js 1Þ2 2 2 2 2 b ¼ ðni nj Þ þ ðj iÞs 1 P ðb 2Þ þ s 1 ¼ 1 s Since the batch number is same, this shows that X 0 dominates X. When nk ¼ 1, we have the following property:
h
Proposition 3. For X ¼ ðk; ðn1 ; n2 ; . . . nk1 ; 1ÞÞ with k > t, if b > ni P 1 for some k > i; k i P t; then we can construct a solution X 0 ¼ ðk 1; ðn01 ; n02 ; . . . ; n0k1 ÞÞ better than (dominate) X, where n01 ¼ n1 ; n02 ¼ n2 ; . . . ; n0i1 ¼ ni1 ; n0i ¼ ni þ 1; n0iþ1 ¼ niþ1 ; . . . ; n0k1 ¼ nk1 .
tion time
C max ¼ n þ ks
P where kj¼1 nj ¼ n; 1 6 nj 6 b; j ¼ 1; 2; . . . ; k; integer: Further from simple calculation, it holds that
f ¼
j k X X j¼1
i¼1
ð1Þ
1 1 1 1 1 ðni þ isÞ2 þ ð1 þ ksÞ2 kðk þ 1Þð2k þ 1Þs2 ðn0i þ isÞ2 þ 2 2 12 2 12 ðk 1Þkð2k 1Þs2
!
k k X 1 1X ni þ js nj ¼ n2 þ n2 þ s ini 2 2 i¼1 i i¼1
k 1 1X 1 2 s kðk þ 1Þð2k þ 1Þ ¼ n2 þ ðni þ isÞ2 2 2 i¼1 12
ð2Þ
Under the above setting, we formulate the first problem (denoted by FP) as follows:
FP :
Proof. From (2), the difference between total flow time of X and that of X0 is
Min f
1 1 1 1 2 ðni þ isÞ2 þ ð1 þ ksÞ2 ðni þ is þ 1Þ2 k s2 2 2 2 2 b ¼ ðk iÞs ni P s ðb 1Þ ¼ 1 s ¼
Since the batch number of X 0 is less than that of X, this shows that X 0 dominates X. h
Min C max
P subject to kj¼1 nj ¼ n; 1 6 nj 6 b; j ¼ 1; 2; . . . ; k; integer . Note that in FP, k; nj ; j ¼ 1; 2; . . . ; k are decision variables. FP is a bi-criteria problem and so usually there exists no solution minimizing both objectives f and C max at a time. Therefore after definition of non-domination below, we seek some non-dominated solutions of FP. Also note that minimizing C max is equivalent to minimizing k and for fixed k minimizing f is equivalent to that of Pk 2 i¼1 ðni þ isÞ . As is easily shown from (2), we have the following property: Proposition 1. For fixed k, a schedule minimizing f satisfies nonincreasing order of batch size n1 P n2 P P nk . Hereafter we consider a solution satisfying proposition 1. 2.1. Non-dominated solution of FP 1
For two solutions X 1 ¼ ðk1 ; N ¼ ðn1j ; j ¼ 1;2;...;k1 ÞÞ; X 2 ¼ ðk2 ; N 2 ¼ P 2 1 2 s k1 ðk1 þ 1Þ ð nj ; j ¼ 1; 2; . . . ; k2 ÞÞ; if k1 6 k2 , 12 ki¼1 ðn1i þ isÞ2 12 Pk 2 1 2 1 2 ð2k1 þ 1Þ 6 2 i¼1 ðni þ isÞ 12 s k2 ðk2 þ 1Þð2k2 þ 1Þ and at least one inequality holds without equality, we call that X 1 dominates
Repeatedly applying Proposition 1–3, finally we expect batch sizes of batch 1 to batch (k–t) may be upper bound b and only other batches have batch size less than b. Next we show this is true when n b 1
þ 2s þ 2 > k > t, where bc is the greatest integer not greater b
b than . Note that nb þ 2s þ 12 is the optimal batch number in a single machine batch scheduling problem with bounded batch size and total flow time to be minimized due to Mosheiov and Oron (2008a). Note that in Mosheiov and Oron (2008a), batch number is also decision variable and not restricted. Further in case that the upper limit of batch size is also not restricted, the optimal batch number and a batch size of each batch are l m k ¼ ð14 þ 2n Þ1=2 12 and an integer batch size produced after mods si; i ¼ 1; 2; . . . ; k to an ifying each relaxed solution ni ¼ nk þ sðkþ1Þ 2 integer by Algorithm Rounding due to Mosheiov et al. (2005). 2.2. Algorithm rounding P Step 1: Calculate Di ¼ ni bni c; i ¼ 1; 2; . . . ; k; a ¼ ki¼1 Di . Step 2: Let A denote the set of a batches with the largest values. Round up all batches in the set A. Round down all the remaining batches.
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Mosheiov and Oron (2008a) have considered bounded batch size version of Mosheiov et al. (2005), and obtained an optimal batch number and batch size of each batch from the following Lagrange relaxation method. That is, following problem (denoted by FPC) is considered as a relaxed one without integer condition of batch size. FPC: Minimize f
subject to
k X
nj ¼ n; 0 6 nj 6 b; j ¼ 1; 2; . . . ; k;
j¼1
where k; n1 ; n2 ; . . . ; nk are decision variables. Note that f is a quadratic function and so FPC is a convex programming except that k is also decision variable. Therefore an optimal solution is a feasible solution of the following Kuhn Tucker condition:
" ! # k k X X @ f k ni n ki ðb ni Þ 6 0; @nj i¼1 i¼1 " ! # k k X X @ f k ni n ki ðb ni Þ ¼ 0; nj @nj i¼1 i¼1
j ¼ 1; 2; . . . ; k;
ð3Þ
j ¼ 1; 2; . . . ; k; ð4Þ
k X
ðk lÞ2 ðk lÞ 6
2ðn lbÞ ; s
ð14Þ
ðk lÞ2 ðk lÞ 6
2ðkb nÞ : s
ð15Þ
The solution of the above two quadratic inequalities is
b b k ¼ nb þ 2s þ 12 ; l ¼ nb 2s 12 in case that k is a decision variable. Integer batch sizes are obtained by Algorithm Rounding. These batch sizes denoted by n1 ; n2 ; . . . ; nk and k constitute an optimal solution when batch number is not restricted. In our case that batch number
b k is fixed as a parameter, only k 6 k ð¼ nb þ 2s þ 12 Þ should be consid ered since solution with k over k cannot become non-dominated solution of FP as is easily shown. Further above solution X ¼ ðk ; ðn1 ; n2 ; . . . ; nk ÞÞ is a non-dominated solution of FP. There fore, we assume that k 6 k 1 and obtain a non-dominated solu tion with each batch number k 6 k 1 for FP. In our case, k is fixed and corresponding relaxed solution is obtained by the same procedure as the process to find X ¼ ðk ; ðn1 ; n2 ; . . . ; nk ÞÞ. Only difference is that we seek an optimal l satisfying (14) and (15). Proposition 4. For fixed k such max nb ; t ; l ¼ k t satisfies (14).
that
k 1PkP
Proof. Note that
nj ¼ n;
ð5Þ
k6k 1¼
j¼1
b nj P 0;
j ¼ 1; 2; . . . ; k;
kj ðb nj Þ ¼ 0;
j ¼ 1; 2; . . . ; k;
k; kj ; nj P 0;
n b 1 n b 1 16 þ : þ þ b 2s 2 b 2s 2
ð6Þ
Therefore,
ð7Þ
l¼kt ¼k
ð8Þ
Left hand side of (14):
b n b 1 b n b 1 6 þ ¼ ; s b 2s 2 s b 2s 2
If k is an optimal batch number (in our case k is fixed) and so nj > 0; j ¼ 1; 2; . . . ; k, it holds that
2 1 1 ðk lÞ2 ðk lÞ ¼ t2 t ¼ t 2 4 2 2 b 1 1 b b ¼ þ ; 6 þ1 s 2 4 s s
ðnj þ js k þ kj Þ ¼ 0; j ¼ 1; . . . ; k;
and the right hand side of (14):
From (4), it holds that
nj ðnj þ js k þ kj Þ ¼ 0; j ¼ 1; 2; . . . ; k;
ð9Þ
ð10Þ
Due to proposition 1, we assume that the first l batches are of size b, i.e., ni ¼ b; i ¼ 1; 2; . . . ; l. It follows from (7) that k‘þ1 ¼ k‘þ2 ¼ ¼ kk ¼ 0: Clearly 0 6 l 6 k, where l ¼ 0 reflects the case of non-binding capacity bound due to Mosheiov et al. (2005) and in any case k P nb . Therefore, the last k l equations in (10) become,
n‘þ1 þ ðl þ 1Þs k ¼ 0; n‘þ2 þ ðl þ 2Þs k ¼ 0; .. .
ð11Þ
nk þ ks k ¼ 0: Solving these equations, the size of last batch is
n lb k l 1 s; nk ¼ nk1 s ¼ nk ¼ kl 2 ¼ nlþ1 ðk l 1Þs:
nlþ1
ð17Þ
ð18Þ
2 2 n nb 2sb 12 b 2ðn lbÞ b b ¼ P þ ; s s s s Since t 12 > 0, from (18) and (19), proposition 4 holds.
ð19Þ h
Let denote r ¼ k ‘ . Then from adding (14) and (15), it holds that r 6 bs þ 1 .But from proposition 2 and 3, further reduce r 6 bs results. Now we solve the following set of two quadratic inequalities with respect to r.
2b 2ðn kbÞ r 6 0; r2 1 þ s s r2 r
2ðkb nÞ 6 0: s
ð20Þ
ð21Þ
Then it holds that
ð12Þ
Especially,
n lb k l 1 þ s: ¼ kl 2
ð16Þ
ð13Þ
Since nk P 0; nlþ1 6 nl ¼ b and nlþ1 þ s > b by a simple calculation, we obtain following two inequalities:
80 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 9 2 < 1 b 1 b kb nA = @ ;1 6r þ þ max 2 : 2 s ; 2 s s 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 2 <b 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2ðkb nÞ 1 b 1 b kb n = ; þ ; þ ; þ þ þ 2 6 min : s 2 4 s 2 s 2 s s ; r : integer;
ð22Þ
X. Li et al. / Computers & Industrial Engineering 62 (2012) 688–692
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 is a positive number since þ bs þ bs 2 kbn 2 s k 2 ½max nb ; t ; k 1. Note that possible number of r is at most b 1: We construct a solution (n1, n2, . . . , nk) for corresponding l s to r satisfying above (22) (that is, ‘ = k r) and choose one giving the smallest f. Note that rounding algorithm as above can be used in order to make a integer solution of batch size (validity to produce integer batch size giving the smallest f is the very same as Mosheiov and Oron (2008a) since Mosheiov and Oron (2008a) does not use an explicit form of k in order to show validity). For each integer k in the interval ½max nb ; t ; k 1 (denoted as ½kL ; kU ; kL ¼ n max b ; t and kU ¼ k 1) we seek a non-dominated solution of FP by the above procedure. These non-dominated solutions are dewhere
1 2
1
2
mðbÞ
noted by ðk ðbÞ; N 1 ðbÞÞ; ðk ðbÞ; N 2 ðbÞÞ; . . . ; ðk ðbÞ; N mðbÞ ðbÞÞ, where mðbÞ is the number of non-dominated solutions found in 1
2
mðbÞ
FP with upper bound of batch size b, k ðbÞ < k ðbÞ < < k i
and N ðbÞ ¼
ðni1 ðbÞ;
ni2 ðbÞ; . . . ; niki ðbÞ ðbÞÞ
sizes corresponding i = 1, 2, . . . ,m(b).
to
ith
ðbÞ
the vector consisting of batch
non-dominated
solution
found,
3. Main problem with three objectives Now we consider the main problem FCB with flexible bound of batch size. For that purpose, we introduce a satisficing level with respect to b by a membership function lB ðbÞ; b ¼ bL ; . . . ; j; . . . ; bU satisfying lB ðbL Þ P lB ðbL þ 1Þ P P lB ðbU Þ. This reflects the situation that the pallet size containing produced goods is flexible, that is, if the pallet becomes full, then it is moved to another processing or quality checking or shipping. This case means produced goods in a same pallet constitute a batch and using bigger pallet costs more, i.e., reduces a satisfaction. Another situation is packing in a box after producing some amount containable in the box and there are different sizes of boxes. Hereafter we denote lB ðjÞ with lj ; j ¼ bL ; . . . ; bU simply. Then we consider the following three criteria batch scheduling problem (denoted by FCB).
FCB : Maximize
lB ðbÞ
Minimize C max Minimize f subject to
k P
nj ¼ n; 1 6 nj 6 b; j ¼ 1; 2; . . . ; k; integer
j¼1
b ¼ bL or bL þ 1 or or bU Usually there exists no solution that optimizes three objectives at a time and so we seek some non-dominated solutions given as follows: 3.1. Non-dominated solution of FCB For two solutions X 1 ¼ðb1 ; k1 ;N 1 ¼ðn1j ; j¼1;2;...;k1 ÞÞ; X 2 ¼ ðb2 ; k2 ; P N ¼ ðn2j ; j¼1;2;...;k2 ÞÞ; if lB ðb1 Þ6 lB ðb2 Þ, k1 6k2 ; 12 ki¼1 ðn1i þisÞ2 P 1 2 1 2 s k1 ðk1 þ1Þð2k1 þ1Þ 6 12 ki¼1 ðn2i þisÞ2 12 s k2 ðk2 þ1Þð2k2 þ1Þ and 12 at least one inequality holds without equality, we call that X 1 dominates X 2 . If there exists no solution that dominates X, X is called non-dominated solution of FCB. Then we have the following algorithm for seeking some nondominated solutions where FP(b) denotes FP with fixed upper bound of batch size b, NB(b) a set of some non-dominated solution of FP(b) found in the solution procedure of Section 2 and NB a tentative set of non-dominated solutions currently produced. Further we assume that r ¼ k ‘–0 since k ¼ ‘ corresponds to k ¼ nb and solution is unique, that is, all batch size equals to b. 2
691
3.2. Solution procedure for FCB Step 1: Set NB ¼ /; b ¼ bL : Go to Step 2. Step 2: Solve FP(b) and find NB(b) (={X1(b), X2(b), . . . Xm(b)(b)}) j where X j ðbÞ ¼ ðb; k ðbÞ; N j ðbÞÞ; j ¼ 1; 2; . . . ; mðbÞ. Set NB NB[ NBðbÞ and delete some elements of NB dominated by other solutions of NB from NB (Set NB as that after this deletion). Step 3: Set b b þ 1: If b ¼ bU þ 1, then terminate. Current NB is a set of non-dominated solutions. Otherwise return to Step 2. 3.3. An example Assume n ¼ 31; s ¼ 2; bL ¼ 6; bU ¼ 8; lB ð6Þ ¼ 1; lB ð7Þ ¼ lB ð8Þ ¼ 0:7. So we only consider the case b = 6, 8. First consider FP(6), that is, the batch size b = 6 31
6 1 Then t ¼ 62 ¼ 3 and 31 k ¼ 6 þ 22 þ 2 ¼ 7, k ¼ 7. From k 1 ¼ 6 P k P max 6 ; 3 , we obtain k ¼ 6 and r ¼ 2 using (22). By ‘ ¼ k r ¼ 4, from (12) and (13) we have n1 ¼ n2 ¼ n3 ¼ n4 ¼ 6; n5 ¼ 4 12 ; n6 ¼ 2 12. Using Algorithm Rounding, a ¼ D5 þ D6 ¼ ð4 12 4Þ þ ð2 12 2Þ ¼ 1, the solution is X 1 ð6Þ 1 ¼ ð6; k ð6Þ; N 1 ð6ÞÞ ¼ ð6; 6; ð6; 6; 6; 6; 5; 2ÞÞ. The corresponding flow time is 6 ð2 þ 6Þ þ 6 ð2 2 þ12Þ þ 6 ð2 3 þ 18Þ þ 6 ð2 4 þ 24Þ þ 5 ð2 5 þ 29Þ þ 2 ð2 6 þ 31Þ ¼ 761 and C max ¼ n þ ks ¼ 31 þ 6 2 ¼ 43: NBð6Þ ¼ fX 1 ð6Þg Secondly consider FP(8), that is, the batch size b = 8. Similarly to the bound b = 6, from t ¼ 82 ¼ 4 and 31 8 k ¼ 8 þ 22 þ 12 ¼ 6, k 1 ¼ 5 P k P max 31 ; 4 ¼ 4, that is, 8 k ¼ 4; 5. Case k = 4 From (22), we obtain r = 1 and ‘ ¼ k r ¼ 4 1 ¼ 3. From (12) and (13) we have n1 ¼ n2 ¼ n3 ¼ 8; n4 ¼ 7. That is, the solution is 3 X 2 ð8Þ ¼ ð8; k ð8Þ; N 3 ð8ÞÞ ¼ ð8; 4; ð8; 8; 8; 7ÞÞ. Corresponding flow time is 767 and C max ¼ n þ ks ¼ 39. Case k = 5 From (22), we obtain r1 ¼ 2; r 2 ¼ 3. By ‘ ¼ k r, we obtain ‘1 ¼ 3; ‘2 ¼ 2. Subcase (a). k ¼ 5 and ‘1 ¼ 3. From (12) and (13) we have n1 ¼ n2 ¼ n3 ¼ 8; n4 ¼ 4 12 ; n5 ¼ 2 12. In order to obtain integer batch sizes, we apply Algorithm Rounding:a = 1 and the solution is 3 X 3 ð8Þ ¼ ð8; k ð8Þ; N 3 ð8ÞÞ ¼ ð8; 5; ð8; 8; 8; 5; 2ÞÞ and corresponding flow time is 747 and C max ¼ n þ ks ¼ 41. Subcase (b). k ¼ 5 and ‘2 ¼ 2. From (12) and (13) we obtain n1 ¼ n2 ¼ 8; n3 ¼ 7; n4 ¼ 5; n5 ¼ 3. That is, the solution is 4 X 4 ð8Þ ¼ ð8; k ð8Þ; N 4 ð8ÞÞ ¼ ð8; 5; ð8; 8; 7; 5; 3ÞÞ. Corresponding flow time is 746 and C max ¼ n þ ks ¼ 41. X 3 ð8Þ is dominated by X 4 ð8Þ. NBð8Þ ¼ fX 2 ð8Þ; X 4 ð8Þg. Above all, NB(the set of non-dominated solutions) ¼ NBð6Þ[ NBð8Þ ¼ fX 1 ð6Þ; X 2 ð8Þ; X 4 ð8Þg . 4. Conclusion We have considered a single machine batch scheduling problem with three objectives and proposed a solution procedure to find some non-dominated solutions. Though we suspect the batch number with batch size less than b, that is, k satisfying (14) and (15) may be unique, to show this conjecture is hard. Therefore in a current situation, time complexity of our solution procedure is not polynomial and in this meaning, our algorithm is not efficient. Therefore we should prove this conjecture. Another urgent thing to do is to utilize
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proposition 2 and 3 in our solution procedure. When b is changed, we need not solve FP(b) from scratch since the batch number needed may be non-increasing when b increases. Overall we refine the solution procedure. Moreover big b is not realistic since satisficing level may become low drastically due to many reasons and so we restrict possible upper limit of batch size to some range of moderate size. Anyway, we should consider the more realistic models. References Brucker, P. (2001). Scheduling algorithms. Springer-Verlag. Cheng, T. C. E., & Janiak, A. (1995). Single machine batch scheduling with deadline and resource dependent processing times. Operations Research Letters, 17, 243–249. Cheng, T. C. E., Janiak, A., & Kovalyov, M. Y. (2001). Single machine batch scheduling with resource dependent setup and processing times. European Journal of Operational Research, 135, 177–183. Coffman, E. G., Jr., Yannakakis, M., Magazine, M. J., & Santos, C. (1990). Batch sizing and job sequencing on a single machine. Annals of Operations Research, 26, 135–147. Dobson, G., Karmarkar, U. S., & Rummel, J. L. (1987). Batching to minimize flow times on one machine. Management Science, 33, 784–799. Hochbaum, D. S., & Landy, D. (1994). Scheduling with batching: Minimizing the weighted number of tardy jobs. Operations Research Letters, 16, 79–86.
Mosheiov, G., & Oron, Daniel. (2008a). A single machine batch scheduling problem with bounded batch size. European Journal of Operational Research, 187, 1069–1079. Mosheiov, G., & Oron, Daniel. (2008b). Open-shop batch scheduling with identical jobs. European Journal of Operational Research, 187, 1282–1292. Mosheiov, G., Oron, Daniel., & Ritov, Yaacov. (2005). Minimizing flow-time on a single machine with integer batch sizes. Operations Research Letters, 33, 497–501. Naddef, D., & Santos, C. (1988). One-pass batching algorithms for the one-machine problem. Discrete Applied Mathematics, 21, 133–145. Ng, C. T., Allahverdi, A., Cheng, T. C. E., & Kovalyov, M. Y. (2008). A survey of scheduling problems with setup times or costs. European Journal of Operational Research, 187, 958–1032. Potts, C. N., & Kovalyov, M. Y. (2000). Scheduling with Batching: A review. European Journal of Operational Research, 120, 228–249. Potts, C. N., & Van Wasenhove, L. N. (1992). Interacting scheduling with batching and lot sizing: A review of algorithm and complexity. Journal of Operational Research Society, 43, 395–406. Santos, C., & Magazine, M. (1985). Batching in single operation manufacturing systems. Operations Research Letters, 4, 338–343. Shallcross, D. F. (1992). A polynomial algorithm for a one machine batching problem. Operations Research Letters, 11, 213–218. Sung, C. S., & Joo, U. G. (1997). Batching to minimize weighted mean flow time on a single machine with batch size restrictions. Computers and Industrial Engineering, 32, 333–340.
Contents lists available at SciVerse ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Single machine batch scheduling problem with fuzzy batch size Xuesong Li a,1, Hiroaki Ishii b,⇑, Teruo Masuda c a
Department of Mathematics, Harbin Institute of Technology, No. 92, West Da-Zhi Street, Harbin, Heilongjiang 150001, China Department of Mathematical Sciences, Kwansei Gakuin University, 2-1 Gakuen, Sanda, Hyogo 669-1337, Japan c Faculty of Business Administration, Tezukayama University, Tezukayama, Japan b
a r t i c l e
i n f o
Article history: Available online 24 December 2011 Keywords: Batch schedule Flexible upper bound of batch size Non-dominated schedule Efficient procedure Maximum completion time Flow time
a b s t r a c t In a batch scheduling problem, jobs are grouped (group is called batch) and scheduled in batches, and a setup time is incurred when starting a new batch. Processing times are assumed to be identical for all jobs. Setup times are assumed to be identical for all batches. Though all batch sizes cannot exceed a common upper bound, the upper bound is flexible and satisfaction degree with respect to the upper limit to be maximized is given. Also the other two objectives, i.e., the maximum completion time and the flowtime are to be minimized. Usually there exists no solution optimizing three objectives at a time. Therefore we define non-dominated solutions consisting of batch size, batch number and allocation of jobs to batches. First we propose an efficient algorithm for a sub-problem with fixed upper limit of batch size, fixed batch number based on a Lagrange relaxation procedure. Based on the properties of non-dominated solutions clarified in this paper, we propose an efficient algorithm to find some non-dominated solutions. Finally we summarize the results in this paper and discuss further research problems. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Modern technologies in flexible manufacturing lead to new types of scheduling problems including batch scheduling problems. In batch scheduling problems, jobs are grouped (each group is called batch) and scheduled in batches, and a setup time is incurred when starting a new batch. Jobs in the same batch have a same completion time, that is, the completion time of the batch is the completion time of the final job in the batch. There are two types of batching problems, denoted by p-batching problem and s-batching problem (Brucker, 2001). For p-batching problem (s-batch problem) the length of a batch is equal to the maximum (sum) of processing times of all jobs in the batch. The recent surveys of Potts and Kovalyov (2000) and Ng, Allahverdi, Cheng, and Kovalyov (2008) mention a class of problems in which the size of the batches is bounded. This constraint reflects many real life applications to a certain extent, where the number of items per batch cannot exceed a certain value due to limitations of the processor or of the production line. The classical single machine batch scheduling problem is to minimize the flow-time. Given the number of jobs, their processing times and setup times which are assumed to be identical, the optimal number of batches and their ⇑ Corresponding author. Tel./fax: +81 79 565 7129. E-mail addresses: [email protected] (X. Li), [email protected] (H. Ishii), [email protected] (T. Masuda). 1 Tel./fax: +86 451 85413857. 0360-8352/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.12.021
size are to be determined. Santos and Magazine (1985) first solved the relaxed version of the problem, i.e., the solution they obtained were non-integer batch sizes. Later algorithms for finding integer batch sizes were introduced by Naddef and Santos (1988), Coffman, Yannakakis, Magazine, and Santos (1990), Shallcross (1992). If n denote the number pffiffiffi of jobs, the computational time of these algorithms are OðnÞ, O n and OðlogðnÞÞ, respectively. Mosheiov, Oron, and Ritov (2005) introduced an alternative integer solution based on a simple rounding procedure of the non-integer batched obtained by Santos and Magazine (1985). Three years later, Mosheiov and Oron (2008a) extended the result obtained in Mosheiov et al., 2005. They introduced an efficient solution for both cases of an upper and a lower bound on the batch sizes. Besides, there are also many other researches on a batch problem, such as, Dobson, Karmarkar, and Rummel (1987), Hochbaum and Landy (1994), Cheng and Janiak (1995), Sung and Joo (1997), Cheng, Janiak, and Kovalyov (2001), Mosheiov and Oron (2008b), also see survey papers Potts and Van Wasenhove (1992). This paper treats one model of s-batch problem which is an extension of Mosheiov and Oron (2008a), that is, considers a single machine batch scheduling problem with the intent of introducing an important new problem and research direction in production and machine scheduling. Also motivated by an attempt to improve the resource usage and customer satisfaction which are important objectives in manufacturing. First in Section 2 formulates nonflexible version of our problem. This problem is bi-criteria and also extension of Mosheiov and Oron (2008a). Section 3 proposes an
X. Li et al. / Computers & Industrial Engineering 62 (2012) 688–692
efficient algorithm to find some non-dominated solutions after defining non-domination of solutions of the problem. For that purpose we derive some useful properties of non-dominated solution of the problem. Section 4 formulates our main problem and proposes a solution algorithm to find some non-dominated solutions after definition of non-domination with respect to our main problem. Section 4 also includes a toy example of our problem. Section 5 summarizes results in this paper and discusses further research problems. 2. Bicriteria single machine batch scheduling problem with fixed limit of batch size First we consider the following bi-criteria batch scheduling problem with a limited batch size: There exist n jobs with identical processing time. So without any loss of generality, we assume that identical processing time is 1. These jobs are grouped and scheduled in batches, and a setup time s is incurred when starting a new batch. Setup times are assumed to be identical for all batches. All batch sizes cannot exceed a common upper bound b. We consider two objectives, i.e., the maximum completion time and flow-time, which are to be minimized. That is, the problem is to find the optimal solution consisting of a batch number and allocation of jobs to batches minimizing the above two objectives under a common limited batch size. Let k be a batch number and nj ; j ¼ 1; 2; . . . ; k batch size of jth P batch. Then the completion time of jth batch is ji¼1 ni þ js. So the P Pj total flow time is f ¼ kj¼1 i¼1 ni þ js nj and maximum comple-
689
X 2 . If there exists no solution that dominates X, X is called nondominated solution of FP. Let t ¼ bs where de is the least integer not less than . Then we have the following property. Proposition 2. For X ¼ ðk; ðn1 ; n2 ; . . . ; nk ÞÞ with k > t, if b > ni P nj P 2 for some k P j > i; j i P t; then we can construct a solution X 0 ¼ ðk; ðn01 ; n02 ; . . . ; n0k ÞÞ better than (dominate X), where n01 ¼ n1 ; n02 ¼ n2 ; . . . ; n0i1 ¼ ni1 ; n0i ¼ ni þ 1; n0iþ1 ¼ niþ1 ; . . . ; n0j1 ¼ nj1 ; n0j ¼ nj 1; n0jþ1 ¼ njþ1 ; . . . ; n0k ¼ nk . Proof. From (2), the difference between total flow time of X and that of X 0 is
1 1 1 1 ðni þ isÞ2 þ ðnj þ jsÞ2 ðn0i þ isÞ2 ðn0j þ jsÞ2 2 2 2 2 1 1 1 1 2 2 ¼ ðni þ isÞ þ ðnj þ jsÞ ðni þ is þ 1Þ2 ðnj þ js 1Þ2 2 2 2 2 b ¼ ðni nj Þ þ ðj iÞs 1 P ðb 2Þ þ s 1 ¼ 1 s Since the batch number is same, this shows that X 0 dominates X. When nk ¼ 1, we have the following property:
h
Proposition 3. For X ¼ ðk; ðn1 ; n2 ; . . . nk1 ; 1ÞÞ with k > t, if b > ni P 1 for some k > i; k i P t; then we can construct a solution X 0 ¼ ðk 1; ðn01 ; n02 ; . . . ; n0k1 ÞÞ better than (dominate) X, where n01 ¼ n1 ; n02 ¼ n2 ; . . . ; n0i1 ¼ ni1 ; n0i ¼ ni þ 1; n0iþ1 ¼ niþ1 ; . . . ; n0k1 ¼ nk1 .
tion time
C max ¼ n þ ks
P where kj¼1 nj ¼ n; 1 6 nj 6 b; j ¼ 1; 2; . . . ; k; integer: Further from simple calculation, it holds that
f ¼
j k X X j¼1
i¼1
ð1Þ
1 1 1 1 1 ðni þ isÞ2 þ ð1 þ ksÞ2 kðk þ 1Þð2k þ 1Þs2 ðn0i þ isÞ2 þ 2 2 12 2 12 ðk 1Þkð2k 1Þs2
!
k k X 1 1X ni þ js nj ¼ n2 þ n2 þ s ini 2 2 i¼1 i i¼1
k 1 1X 1 2 s kðk þ 1Þð2k þ 1Þ ¼ n2 þ ðni þ isÞ2 2 2 i¼1 12
ð2Þ
Under the above setting, we formulate the first problem (denoted by FP) as follows:
FP :
Proof. From (2), the difference between total flow time of X and that of X0 is
Min f
1 1 1 1 2 ðni þ isÞ2 þ ð1 þ ksÞ2 ðni þ is þ 1Þ2 k s2 2 2 2 2 b ¼ ðk iÞs ni P s ðb 1Þ ¼ 1 s ¼
Since the batch number of X 0 is less than that of X, this shows that X 0 dominates X. h
Min C max
P subject to kj¼1 nj ¼ n; 1 6 nj 6 b; j ¼ 1; 2; . . . ; k; integer . Note that in FP, k; nj ; j ¼ 1; 2; . . . ; k are decision variables. FP is a bi-criteria problem and so usually there exists no solution minimizing both objectives f and C max at a time. Therefore after definition of non-domination below, we seek some non-dominated solutions of FP. Also note that minimizing C max is equivalent to minimizing k and for fixed k minimizing f is equivalent to that of Pk 2 i¼1 ðni þ isÞ . As is easily shown from (2), we have the following property: Proposition 1. For fixed k, a schedule minimizing f satisfies nonincreasing order of batch size n1 P n2 P P nk . Hereafter we consider a solution satisfying proposition 1. 2.1. Non-dominated solution of FP 1
For two solutions X 1 ¼ ðk1 ; N ¼ ðn1j ; j ¼ 1;2;...;k1 ÞÞ; X 2 ¼ ðk2 ; N 2 ¼ P 2 1 2 s k1 ðk1 þ 1Þ ð nj ; j ¼ 1; 2; . . . ; k2 ÞÞ; if k1 6 k2 , 12 ki¼1 ðn1i þ isÞ2 12 Pk 2 1 2 1 2 ð2k1 þ 1Þ 6 2 i¼1 ðni þ isÞ 12 s k2 ðk2 þ 1Þð2k2 þ 1Þ and at least one inequality holds without equality, we call that X 1 dominates
Repeatedly applying Proposition 1–3, finally we expect batch sizes of batch 1 to batch (k–t) may be upper bound b and only other batches have batch size less than b. Next we show this is true when n b 1
þ 2s þ 2 > k > t, where bc is the greatest integer not greater b
b than . Note that nb þ 2s þ 12 is the optimal batch number in a single machine batch scheduling problem with bounded batch size and total flow time to be minimized due to Mosheiov and Oron (2008a). Note that in Mosheiov and Oron (2008a), batch number is also decision variable and not restricted. Further in case that the upper limit of batch size is also not restricted, the optimal batch number and a batch size of each batch are l m k ¼ ð14 þ 2n Þ1=2 12 and an integer batch size produced after mods si; i ¼ 1; 2; . . . ; k to an ifying each relaxed solution ni ¼ nk þ sðkþ1Þ 2 integer by Algorithm Rounding due to Mosheiov et al. (2005). 2.2. Algorithm rounding P Step 1: Calculate Di ¼ ni bni c; i ¼ 1; 2; . . . ; k; a ¼ ki¼1 Di . Step 2: Let A denote the set of a batches with the largest values. Round up all batches in the set A. Round down all the remaining batches.
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X. Li et al. / Computers & Industrial Engineering 62 (2012) 688–692
Mosheiov and Oron (2008a) have considered bounded batch size version of Mosheiov et al. (2005), and obtained an optimal batch number and batch size of each batch from the following Lagrange relaxation method. That is, following problem (denoted by FPC) is considered as a relaxed one without integer condition of batch size. FPC: Minimize f
subject to
k X
nj ¼ n; 0 6 nj 6 b; j ¼ 1; 2; . . . ; k;
j¼1
where k; n1 ; n2 ; . . . ; nk are decision variables. Note that f is a quadratic function and so FPC is a convex programming except that k is also decision variable. Therefore an optimal solution is a feasible solution of the following Kuhn Tucker condition:
" ! # k k X X @ f k ni n ki ðb ni Þ 6 0; @nj i¼1 i¼1 " ! # k k X X @ f k ni n ki ðb ni Þ ¼ 0; nj @nj i¼1 i¼1
j ¼ 1; 2; . . . ; k;
ð3Þ
j ¼ 1; 2; . . . ; k; ð4Þ
k X
ðk lÞ2 ðk lÞ 6
2ðn lbÞ ; s
ð14Þ
ðk lÞ2 ðk lÞ 6
2ðkb nÞ : s
ð15Þ
The solution of the above two quadratic inequalities is
b b k ¼ nb þ 2s þ 12 ; l ¼ nb 2s 12 in case that k is a decision variable. Integer batch sizes are obtained by Algorithm Rounding. These batch sizes denoted by n1 ; n2 ; . . . ; nk and k constitute an optimal solution when batch number is not restricted. In our case that batch number
b k is fixed as a parameter, only k 6 k ð¼ nb þ 2s þ 12 Þ should be consid ered since solution with k over k cannot become non-dominated solution of FP as is easily shown. Further above solution X ¼ ðk ; ðn1 ; n2 ; . . . ; nk ÞÞ is a non-dominated solution of FP. There fore, we assume that k 6 k 1 and obtain a non-dominated solu tion with each batch number k 6 k 1 for FP. In our case, k is fixed and corresponding relaxed solution is obtained by the same procedure as the process to find X ¼ ðk ; ðn1 ; n2 ; . . . ; nk ÞÞ. Only difference is that we seek an optimal l satisfying (14) and (15). Proposition 4. For fixed k such max nb ; t ; l ¼ k t satisfies (14).
that
k 1PkP
Proof. Note that
nj ¼ n;
ð5Þ
k6k 1¼
j¼1
b nj P 0;
j ¼ 1; 2; . . . ; k;
kj ðb nj Þ ¼ 0;
j ¼ 1; 2; . . . ; k;
k; kj ; nj P 0;
n b 1 n b 1 16 þ : þ þ b 2s 2 b 2s 2
ð6Þ
Therefore,
ð7Þ
l¼kt ¼k
ð8Þ
Left hand side of (14):
b n b 1 b n b 1 6 þ ¼ ; s b 2s 2 s b 2s 2
If k is an optimal batch number (in our case k is fixed) and so nj > 0; j ¼ 1; 2; . . . ; k, it holds that
2 1 1 ðk lÞ2 ðk lÞ ¼ t2 t ¼ t 2 4 2 2 b 1 1 b b ¼ þ ; 6 þ1 s 2 4 s s
ðnj þ js k þ kj Þ ¼ 0; j ¼ 1; . . . ; k;
and the right hand side of (14):
From (4), it holds that
nj ðnj þ js k þ kj Þ ¼ 0; j ¼ 1; 2; . . . ; k;
ð9Þ
ð10Þ
Due to proposition 1, we assume that the first l batches are of size b, i.e., ni ¼ b; i ¼ 1; 2; . . . ; l. It follows from (7) that k‘þ1 ¼ k‘þ2 ¼ ¼ kk ¼ 0: Clearly 0 6 l 6 k, where l ¼ 0 reflects the case of non-binding capacity bound due to Mosheiov et al. (2005) and in any case k P nb . Therefore, the last k l equations in (10) become,
n‘þ1 þ ðl þ 1Þs k ¼ 0; n‘þ2 þ ðl þ 2Þs k ¼ 0; .. .
ð11Þ
nk þ ks k ¼ 0: Solving these equations, the size of last batch is
n lb k l 1 s; nk ¼ nk1 s ¼ nk ¼ kl 2 ¼ nlþ1 ðk l 1Þs:
nlþ1
ð17Þ
ð18Þ
2 2 n nb 2sb 12 b 2ðn lbÞ b b ¼ P þ ; s s s s Since t 12 > 0, from (18) and (19), proposition 4 holds.
ð19Þ h
Let denote r ¼ k ‘ . Then from adding (14) and (15), it holds that r 6 bs þ 1 .But from proposition 2 and 3, further reduce r 6 bs results. Now we solve the following set of two quadratic inequalities with respect to r.
2b 2ðn kbÞ r 6 0; r2 1 þ s s r2 r
2ðkb nÞ 6 0: s
ð20Þ
ð21Þ
Then it holds that
ð12Þ
Especially,
n lb k l 1 þ s: ¼ kl 2
ð16Þ
ð13Þ
Since nk P 0; nlþ1 6 nl ¼ b and nlþ1 þ s > b by a simple calculation, we obtain following two inequalities:
80 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 9 2 < 1 b 1 b kb nA = @ ;1 6r þ þ max 2 : 2 s ; 2 s s 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 2 <b 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2ðkb nÞ 1 b 1 b kb n = ; þ ; þ ; þ þ þ 2 6 min : s 2 4 s 2 s 2 s s ; r : integer;
ð22Þ
X. Li et al. / Computers & Industrial Engineering 62 (2012) 688–692
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 is a positive number since þ bs þ bs 2 kbn 2 s k 2 ½max nb ; t ; k 1. Note that possible number of r is at most b 1: We construct a solution (n1, n2, . . . , nk) for corresponding l s to r satisfying above (22) (that is, ‘ = k r) and choose one giving the smallest f. Note that rounding algorithm as above can be used in order to make a integer solution of batch size (validity to produce integer batch size giving the smallest f is the very same as Mosheiov and Oron (2008a) since Mosheiov and Oron (2008a) does not use an explicit form of k in order to show validity). For each integer k in the interval ½max nb ; t ; k 1 (denoted as ½kL ; kU ; kL ¼ n max b ; t and kU ¼ k 1) we seek a non-dominated solution of FP by the above procedure. These non-dominated solutions are dewhere
1 2
1
2
mðbÞ
noted by ðk ðbÞ; N 1 ðbÞÞ; ðk ðbÞ; N 2 ðbÞÞ; . . . ; ðk ðbÞ; N mðbÞ ðbÞÞ, where mðbÞ is the number of non-dominated solutions found in 1
2
mðbÞ
FP with upper bound of batch size b, k ðbÞ < k ðbÞ < < k i
and N ðbÞ ¼
ðni1 ðbÞ;
ni2 ðbÞ; . . . ; niki ðbÞ ðbÞÞ
sizes corresponding i = 1, 2, . . . ,m(b).
to
ith
ðbÞ
the vector consisting of batch
non-dominated
solution
found,
3. Main problem with three objectives Now we consider the main problem FCB with flexible bound of batch size. For that purpose, we introduce a satisficing level with respect to b by a membership function lB ðbÞ; b ¼ bL ; . . . ; j; . . . ; bU satisfying lB ðbL Þ P lB ðbL þ 1Þ P P lB ðbU Þ. This reflects the situation that the pallet size containing produced goods is flexible, that is, if the pallet becomes full, then it is moved to another processing or quality checking or shipping. This case means produced goods in a same pallet constitute a batch and using bigger pallet costs more, i.e., reduces a satisfaction. Another situation is packing in a box after producing some amount containable in the box and there are different sizes of boxes. Hereafter we denote lB ðjÞ with lj ; j ¼ bL ; . . . ; bU simply. Then we consider the following three criteria batch scheduling problem (denoted by FCB).
FCB : Maximize
lB ðbÞ
Minimize C max Minimize f subject to
k P
nj ¼ n; 1 6 nj 6 b; j ¼ 1; 2; . . . ; k; integer
j¼1
b ¼ bL or bL þ 1 or or bU Usually there exists no solution that optimizes three objectives at a time and so we seek some non-dominated solutions given as follows: 3.1. Non-dominated solution of FCB For two solutions X 1 ¼ðb1 ; k1 ;N 1 ¼ðn1j ; j¼1;2;...;k1 ÞÞ; X 2 ¼ ðb2 ; k2 ; P N ¼ ðn2j ; j¼1;2;...;k2 ÞÞ; if lB ðb1 Þ6 lB ðb2 Þ, k1 6k2 ; 12 ki¼1 ðn1i þisÞ2 P 1 2 1 2 s k1 ðk1 þ1Þð2k1 þ1Þ 6 12 ki¼1 ðn2i þisÞ2 12 s k2 ðk2 þ1Þð2k2 þ1Þ and 12 at least one inequality holds without equality, we call that X 1 dominates X 2 . If there exists no solution that dominates X, X is called non-dominated solution of FCB. Then we have the following algorithm for seeking some nondominated solutions where FP(b) denotes FP with fixed upper bound of batch size b, NB(b) a set of some non-dominated solution of FP(b) found in the solution procedure of Section 2 and NB a tentative set of non-dominated solutions currently produced. Further we assume that r ¼ k ‘–0 since k ¼ ‘ corresponds to k ¼ nb and solution is unique, that is, all batch size equals to b. 2
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3.2. Solution procedure for FCB Step 1: Set NB ¼ /; b ¼ bL : Go to Step 2. Step 2: Solve FP(b) and find NB(b) (={X1(b), X2(b), . . . Xm(b)(b)}) j where X j ðbÞ ¼ ðb; k ðbÞ; N j ðbÞÞ; j ¼ 1; 2; . . . ; mðbÞ. Set NB NB[ NBðbÞ and delete some elements of NB dominated by other solutions of NB from NB (Set NB as that after this deletion). Step 3: Set b b þ 1: If b ¼ bU þ 1, then terminate. Current NB is a set of non-dominated solutions. Otherwise return to Step 2. 3.3. An example Assume n ¼ 31; s ¼ 2; bL ¼ 6; bU ¼ 8; lB ð6Þ ¼ 1; lB ð7Þ ¼ lB ð8Þ ¼ 0:7. So we only consider the case b = 6, 8. First consider FP(6), that is, the batch size b = 6 31
6 1 Then t ¼ 62 ¼ 3 and 31 k ¼ 6 þ 22 þ 2 ¼ 7, k ¼ 7. From k 1 ¼ 6 P k P max 6 ; 3 , we obtain k ¼ 6 and r ¼ 2 using (22). By ‘ ¼ k r ¼ 4, from (12) and (13) we have n1 ¼ n2 ¼ n3 ¼ n4 ¼ 6; n5 ¼ 4 12 ; n6 ¼ 2 12. Using Algorithm Rounding, a ¼ D5 þ D6 ¼ ð4 12 4Þ þ ð2 12 2Þ ¼ 1, the solution is X 1 ð6Þ 1 ¼ ð6; k ð6Þ; N 1 ð6ÞÞ ¼ ð6; 6; ð6; 6; 6; 6; 5; 2ÞÞ. The corresponding flow time is 6 ð2 þ 6Þ þ 6 ð2 2 þ12Þ þ 6 ð2 3 þ 18Þ þ 6 ð2 4 þ 24Þ þ 5 ð2 5 þ 29Þ þ 2 ð2 6 þ 31Þ ¼ 761 and C max ¼ n þ ks ¼ 31 þ 6 2 ¼ 43: NBð6Þ ¼ fX 1 ð6Þg Secondly consider FP(8), that is, the batch size b = 8. Similarly to the bound b = 6, from t ¼ 82 ¼ 4 and 31 8 k ¼ 8 þ 22 þ 12 ¼ 6, k 1 ¼ 5 P k P max 31 ; 4 ¼ 4, that is, 8 k ¼ 4; 5. Case k = 4 From (22), we obtain r = 1 and ‘ ¼ k r ¼ 4 1 ¼ 3. From (12) and (13) we have n1 ¼ n2 ¼ n3 ¼ 8; n4 ¼ 7. That is, the solution is 3 X 2 ð8Þ ¼ ð8; k ð8Þ; N 3 ð8ÞÞ ¼ ð8; 4; ð8; 8; 8; 7ÞÞ. Corresponding flow time is 767 and C max ¼ n þ ks ¼ 39. Case k = 5 From (22), we obtain r1 ¼ 2; r 2 ¼ 3. By ‘ ¼ k r, we obtain ‘1 ¼ 3; ‘2 ¼ 2. Subcase (a). k ¼ 5 and ‘1 ¼ 3. From (12) and (13) we have n1 ¼ n2 ¼ n3 ¼ 8; n4 ¼ 4 12 ; n5 ¼ 2 12. In order to obtain integer batch sizes, we apply Algorithm Rounding:a = 1 and the solution is 3 X 3 ð8Þ ¼ ð8; k ð8Þ; N 3 ð8ÞÞ ¼ ð8; 5; ð8; 8; 8; 5; 2ÞÞ and corresponding flow time is 747 and C max ¼ n þ ks ¼ 41. Subcase (b). k ¼ 5 and ‘2 ¼ 2. From (12) and (13) we obtain n1 ¼ n2 ¼ 8; n3 ¼ 7; n4 ¼ 5; n5 ¼ 3. That is, the solution is 4 X 4 ð8Þ ¼ ð8; k ð8Þ; N 4 ð8ÞÞ ¼ ð8; 5; ð8; 8; 7; 5; 3ÞÞ. Corresponding flow time is 746 and C max ¼ n þ ks ¼ 41. X 3 ð8Þ is dominated by X 4 ð8Þ. NBð8Þ ¼ fX 2 ð8Þ; X 4 ð8Þg. Above all, NB(the set of non-dominated solutions) ¼ NBð6Þ[ NBð8Þ ¼ fX 1 ð6Þ; X 2 ð8Þ; X 4 ð8Þg . 4. Conclusion We have considered a single machine batch scheduling problem with three objectives and proposed a solution procedure to find some non-dominated solutions. Though we suspect the batch number with batch size less than b, that is, k satisfying (14) and (15) may be unique, to show this conjecture is hard. Therefore in a current situation, time complexity of our solution procedure is not polynomial and in this meaning, our algorithm is not efficient. Therefore we should prove this conjecture. Another urgent thing to do is to utilize
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proposition 2 and 3 in our solution procedure. When b is changed, we need not solve FP(b) from scratch since the batch number needed may be non-increasing when b increases. Overall we refine the solution procedure. Moreover big b is not realistic since satisficing level may become low drastically due to many reasons and so we restrict possible upper limit of batch size to some range of moderate size. Anyway, we should consider the more realistic models. References Brucker, P. (2001). Scheduling algorithms. Springer-Verlag. Cheng, T. C. E., & Janiak, A. (1995). Single machine batch scheduling with deadline and resource dependent processing times. Operations Research Letters, 17, 243–249. Cheng, T. C. E., Janiak, A., & Kovalyov, M. Y. (2001). Single machine batch scheduling with resource dependent setup and processing times. European Journal of Operational Research, 135, 177–183. Coffman, E. G., Jr., Yannakakis, M., Magazine, M. J., & Santos, C. (1990). Batch sizing and job sequencing on a single machine. Annals of Operations Research, 26, 135–147. Dobson, G., Karmarkar, U. S., & Rummel, J. L. (1987). Batching to minimize flow times on one machine. Management Science, 33, 784–799. Hochbaum, D. S., & Landy, D. (1994). Scheduling with batching: Minimizing the weighted number of tardy jobs. Operations Research Letters, 16, 79–86.
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