Minimizing Total Weighted Completion Time on Uniform Machines with Unbounded Batch

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1 The Eighth Internationa Symposium on Operations Research and Its Appications (ISORA 09) Zhangiaie, China, September 20 22, 2009 Copyright 2009 ORSC & APORC, pp Minimizing Tota Weighted Competion Time on Uniform Machines with Unbounded Batch Cuixia Miao, 2, Yu-Zhong Zhang Jianfeng Ren Institute of Operations Research, Qufu Norma University, Rizhao, Shandong, , China 2 Schoo of Mathematica Sciences, Qufu Norma University, Qufu, Shandong, 27365, China Abstract In this paper, we consider the scheduing probem of minimizing tota weighted competion time on uniform machines with unbounded batch. Each of the machine M ( =,,m) has a speed s and can process up to B( n) obs simutaneousy as a batch. The processing time of a batch denoted by P(B) is given by the processing time of the ongest ob in it, and the running time of the batch on machine M is P(B) s. We present some usefu properties of the optima schedue, and then design an O(n m+2 ) time backward dynamic programming agorithm for the probem. Keywords Uniform machines with unbounded batch; batch-spt; dynamic programming agorithm; poynomiay sovabe Introduction The batching schedue is motivated by burn-in operations in semiconductor manufacturing. Lee et a. ([]) provided a background description, Webster and Baker ([2]) presented an overview of agorithms and compexity resuts for scheduing batch processing machines. This processing system has been extensivey studied in the ast decade ([3], [9], [2]). For the batching schedue, there are two distinct modes: the bounded mode, in which the bound B for each batch size is effective, i.e., B < n, where n is the number of obs, the unbounded mode, in which there is effectivey no imit on the size of batch, i.e., B n or B( ). Probems of bounded mode arise in the manufacture of integrated circuits ([]), the critica fina stage in the production of circuits is the burn-in operation, in which chips are oaded on boards which are then paced in an oven and exposed to high temperatures, each chip has a pre-specified minimum burn-in time and the burn-in oven has a imited capacity; scheduing probems of unbounded mode arise for instance in situations where compositions need to be hardened in kins and the kin is sufficienty arge that it does not restrict batch sizes ([4]). Zhang ([9]) gave a survey on both of the two modes. In this paper, we ony address the unbounded mode and the obective is to minimizing the tota weighted competion time. This paper was supported by the Nationa Natura Science Foundation (06708), the PH.D Funds of Shandong Province (2007BS004) and the Funds of Qufu Norma University (XJ074). Corresponding author. E-mai address: miaocuixia@26.com

2 Minimizing Tota Weighted Competion Time on Uniform Machines 403 Previous reated resuts: Under the off-ine setting, for the probem of minimizing tota weighted competion time on a singe machine with unbounded batch, when a the obs are reeased at the same time, Brucker ([4]) presented a dynamic programming agorithm in poynomia time; for genera reease times, Deng and Zhang ([5]) proved that the probem is NP-hard, and they further showed that severa important specia cases of the probem can be soved in poynomia time, and for the genera case, Li et a. ([6]) gave a poynomia time approximation scheme (PTAS), and they ([7]) aso presented a poynomia time approximation scheme (PTAS) for probem of minimizing tota weighted competion time on identica parae unbounded batch machines. Under the on-ine setting and on a singe unbounded batch machine, Chen ([8]) provided an on-ine agorithm with 0 3 competitive. ([4], [0]) give the motive of this paper. Our contributions: We address the scheduing probem of minimizing tota weighted competion time on uniform machines with unbounded batch for the first time. When obs are reeased at the same time, we present some usefu properties of the optima schedue, and design an O(n m+2 ) time backward dynamic programming agorithm in this paper. 2 Assumptions, Notations and Preiminaries We are given a set of n independent obs J = {J,,J n }. Each ob J ( =,,n) requires processing time p which specifies the minimum time needed to process the ob, and a weight w which is a measure of its importance, a the obs are reeased at time zero. There is a set of m uniform machines M = {M,,M m }. Each machine M ( =,,m) has a speed s, and can process up to B( n) obs simutaneousy as a batch. The obs that are processed together form a batch, we aso denote it by B, and the processing time of the batch denoted by p(b) is given by the processing time of the ongest ob in the batch, i.e., p(b) = max{p J B}, and the running time of the batch on machine M is P(B) s. A obs contained in the same batch start and compete at the same time. Once processing of a batch is initiated, it can not be interrupted and other obs cannot be introduced into the batch unti processing is competed. The obective is to schedue the obs on the set of m uniform machines with unbounded batch so as to minimize w C, where C denotes the competion time of ob J. Using the 3 fied notation of Graham et a. ([]), we denote our probem as: Q m B( ) w C, and Q m B( ) C for w = ( =,,n). Definition We say that a sequence is in batch-spt if for any two batches B x and B y in the sequence, where B x is processed before B y, there is no pair of obs J i, J such that J i B y, J B x and p i < p. 3 Dynamic Programming Agorithm for Q m B( ) w C In this section, for the probem Q m B( ) w C, we present some properties of the optima schedue, and design a backward dynamic programming agorithm in poynomia time for it. Firsty, we assume that the obs are numbered in SPT order so that p p n.

3 404 The 8th Internationa Symposium on Operations Research and Its Appications 3. Properties of Optima Schedue Theorem For the probem Q m B( ) w C, there exists an optima schedue satisfying the foowing properties: (i) In each batch of the schedue, the indices of its obs are consecutive in SPT order, i.e., one batch B i = {J ni,j ni +,,J ni+ } in the schedue. (ii) On each machine, the batches are sequenced in batch-spt, i.e., if the schedue on machine M ( =,,m) is π = {B,,B n }, then P(B ) < < P(B n ), where B ( =,,n ) is the -th batch and n is the number of batches assigned to M in the schedue. Proof. We consider any optima schedue π = {π,,π m } on the number of m uniform machines, where π = {B,,B n } is the subschedue on machine M ( =,,m) in π. (i) Suppose that there are two different batches B x and B 2 y, and there are three obs J, J +, J +2 with p p + p +2 and J, J +2 beong to B x, J + beongs to B 2 y. We distinguish between two cases: Case. = 2 =, that is, B x and B 2 y are on the same machine. Case.. x < y We can get a new subschedue π = {B,,B x {J + },,B y {J + },,B n } on M by moving ob J + from B y to B x, and we get a new schedue π = {π,, π,, π m }. Since P P + P +2, we have that p(b x {J + }) = p(b x), p(b y {J + }) p(b y). Accordingy, the competion time of ob J + decreases from C(B y) to C(B x), whie the competion times of the other obs do not increase, i.e., C + (π ) < C + (π), and C i (π ) C i (π), for i =,,n, i = +, obviousy, w + C + (π ) < w + C + (π), and w i C i (π ) w i C i (π), for i =,,n, i = +. This contradicts the optima schedue π. Case.2. x > y We can get a new schedue π = {B,,B y {J },,B x {J },,B n } on M by moving ob J from B x to B y, and we get a new schedue π = {π,,π,,π m }. Since p p + p +2, we have that p(b y {J }) = p(b y), p(b x {J }) = p(b x). Consequenty, C (π ) < C (π), and C i (π ) = C i (π), for i =,,n, i =. A contradiction. Case 2. = 2, that is, B x and B 2 y are on different machines, w..o.g et < 2 m. We distinguish three cases: Case 2.. C(B x ) < C(B 2 y ) We get π = {B,,B x {J + },,B n }, π 2 = {B 2,,B 2 y {J + },,B 2 n2 } by moving ob J + from B 2 y to B x, and we get a new schedue π = {π,,π,,π 2,,π m }. Since p p + p +2, we have that p(b x {J + }) = p(b x ), p(b 2 y {J + }) p(b 2 y ).

4 Minimizing Tota Weighted Competion Time on Uniform Machines 405 Consequenty, the competion time of ob J + decrease from C(B 2 y ) to C(B x ), whie the competion times of other obs do not increase, i.e., C + (π ) <C + (π), and C i (π ) C i (π), for i =,,n, i = +. A contradiction. Case 2.2. C(B x ) > C(B 2 y ) We get π = {B,,B x {J },,B n }, π 2 = {B 2,,B 2 y {J },,B 2 n2 } by moving ob J from B x to B 2 y, and we get a new schedue π = {π,,π,,π 2,,π m }. Since p p + p +2, we have that p(b x {J }) = p(b x ), p(b 2 y {J }) = p(b 2 y ). Consequenty, C (π ) < C (π), and C i (π ) = C i (π), for i =,,n, i =. A contradiction. Case 2.3. C(B x ) = C(B 2 y ) We get π = {B,,B x {J },,B n }, π 2 = {B 2,,B 2 y {J },,B 2 n2 } by moving ob J from B x to B 2 y, and we get a new schedue π = {π,,π,,π 2,,π m }. Since p p + p +2, we have that p(b x {J }) = p(b x ), p(b 2 y {J }) = p(b 2 y ). Consequenty, the competion times of a obs are unchanged, then the obective vaue is unchanged. A finite number of repetitions of this procedure yieds an optima schedue of the required form. Now, the concusion of (i) hods. (ii) For the subschedue π = {B,,B x,,b y,,b n } on machine M ( =,, m) in π, to prove P(B ) < < P(B x) < < P(B y) < < P(B n ), assume the opposite, w..o.g suppose that P(B x) P(B y), from (i), we know that a the processing time of obs in batch B x are not smaer than those of obs in B y. If we move a the obs in batch B y to batch B x, then the obective function vaue is stricty decreased, which contradicting the optima schedue π. So, the concusion of (ii) hods. This competes the proof of Theorem. 3.2 Agorithm and Exampe Based on Theorem, we present an O(n m+2 ) time backward dynamic programming agorithm in this subsection. Let F ( J,..., J m ) be the minimum tota weighted competion time in SPT order schedue on the number of m uniform machines with unbounded batch, and each machine M ( =,,m) contains the ob subset J among {J,J +,,J n }. Where m = J {J,J +,,J n }, and m = J = n +. Where J denotes the cardinaity of set J, i.e., J is the tota number of obs on machine M among {J,J +,,J n }. Here, we must use the cardinaity J to denote the number of obs instead of n, which is a natura number, because, when we cacuate the obective in the iteration, we woud consider the weights of obs in set J. =

5 406 The 8th Internationa Symposium on Operations Research and Its Appications Processing the first batch in the schedue starts at time zero on machine M ( =,,m). Whenever a new batch is added to the beginning of this schedue and assigned to M, there is a corresponding deay to the processing of a those batches on that machine. Suppose that a batch {J,J +,,J k }, which has processing time p k, is inserted at the start of the schedue and assigned to M. For obs {J k,,j n }, the tota weighted competion time of some of them which schedued on M increases by p k Ji J k w i, s where J k is the ob set of the schedue among {J k,,j n } on M, whie the tota weighted competion time of {J,J +,,J k } is p k k i= w i s. As J k {J,,J k } = J, thus, the overa increase in the tota weighted competion time is p k Ji J k w i + p k k i= w p i k Ji J w i =. s s s A forma description of backward Dynamic Programming Agorithm is given beow. ALGORITHM DP(Dynamic Programming) Step. Re-index a obs according to the SPT order so that p p n. Step 2. Let F n+ ( ϕ,..., ϕ ) = 0. If ( ; J,..., J m ) = (n + ; ϕ,..., ϕ ), then F ( J,..., J m ) =. Let = n. Step 3. For each tupe ( J,..., J m ) such that J {0,,,n + }, =,,m, and m = J = n +, m = J = {J,,J n }. Compute the foowing F ( J,..., J m ) = min <k n+ {F k ( J,...,( J {J,J +,,J k } ),..., J m ) p k Ji J w i + : m}. s If =, go to step 4, otherwise, et =, repeat step 3. Step 4. Define F = min{f ( J,..., J m ) : J m {0,,,n}, =,,m,and J = n, m J = J}, = which is the optima vaue, then to find the optima schedue by backtracking. Remarks:. The time compexity of ALGORITHM DP is O(n m+2 ), and our probem is poynomiay sovabe if the number of machines m is constant. 2. In step 3, if F ( J,..., J m ) = F k ( J,...,( J {J,J +,,J k } ),..., J m ) + = p k Ji J w i, s

6 Minimizing Tota Weighted Competion Time on Uniform Machines 407 then batch {J,J +,,J k } is inserted from the start on machine M. Exampe: Consider the two-machine probem with the foowing dates. J = {J,J 2,J 3 } with processing times p = 2, p 2 = 4, p 3 = 6 and weights w = 3,w 2 =,w 3 = 5. M = {M,M 2 }, and the speeds are s = and s 2 = 2. The order of obs is aready in SPT order. If we use ALGORITHM DP, we can get the schedue: J is assigned to M, J 2 and J 3 are in one batch assigned to M 2, and the obective function vaue is F = w C = 24, which is the optima vaue. 4 The Specia Case of w = for =,...,n In this section, we discuss the specia case of w = for =,,n, i.e., the probem Q m B( ) C. Theorem is vaid for the specia case. In section 3, if we repace the cardinaity J by a natura number n for =,,m and =,,n, we can get the simiar dynamic programming agorithm for the probem Q m B( ) C. We ony recount it in sketch: Let F ( J,..., J m ) = F (n,...,n m). Suppose that a batch {J,J +,,J k } which has processing time p k is inserted at the start of the schedue and assigned to M. The overa increase in the tota competion time is The iterative formua is (k + n k )p k s = n p k s. F (n,...,n m) = min <k n+ {F k (n,...,n (k ),...,n m) + n p k s : m}. Exampe: Consider the two-machine probem with the foowing dates. J = {J,J 2,J 3,J 4 } with processing times p =, p 2 = 2, p 3 = 4, p 4 = 6. M = {M,M 2 }, and the speeds are s = and s 2 = 2. The schedue is that J 2 is assigned to M, J as a batch, is assigned to M 2 from the start, J 2 and J 3 are in one batch assigned to M 2 after J. The optima obective function vaue is F = min{f (4,0),F (0,4),F (2,2),F (,3),F (3,)} = min{20,0,0,9.5,3} = Concusion In this paper, we present a backward dynamic programming agorithm for Q m B( ) w C, and it is poynomiay sovabe if the number of machine m is constant. An interesting probem for further research is Q m r,b( ) w C. References [] C.-Y. Lee, R. Uzsoy and L.A. Martin-Vega, Efficient agorithms for scheduing semiconductor burn-in operations, Operations Research, 40, 992,

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