Approximation Algorithm of Minimizing Makespan in Parallel Bounded Batch Scheduling

Transcription

1 Te 7t International Symposium on Operations Researc and Its Applications (ISORA 08) Lijiang Cina October Novemver 008 Copyrigt 008 ORSC & APORC pp Approximation Algoritm of Minimizing Makespan in Parallel Bounded Batc Sceduling Jianfeng Ren Yuzong Zang Sun Guo College of Operations Researc and Management Sciences Qufu Normal University Sangdong 7686 Cina Department of Matematics Qufu Normal University Sangdong 7686 Cina. Abstract We consider te problem of minimizing te makespan(c max ) on m identical parallel batc processing macines. Te batc processing macine can process up to B jobs simultaneously. Te jobs tat are processed togeter form a batc and all jobs in a batc start and complete at te same time. For a batc of jobs te processing time of te batc is equal to te largest processing time among te jobs in te batc. In tis paper we design a fully polynomial time approximation sceme (FPTAS) to solve te bounded identical parallel batc sceduling problem P m B < n C max wen te number of identical parallel batc processing macines m is constant. Keywords Approximation algoritm; Bounded batc sceduling; Makespan; FPTAS; Dynamic programming. Introduction Model: A batcing macine or batc processing macine is a macine tat can process up to B jobs simultaneously. Te jobs tat are processed togeter form a batc. Specifically we are interested in te so-called burn-in model in wic te processing time of a batc is defined to te maximum processing time of any job assigned to it. All jobs contained in te same batc start and complete at te same time since te completion time of a job is equal to te completion time of te batc to wic it belongs. Tis model is motivated by te problem of sceduling burn-in operations for large-scale integrated circuit(ic) cips manufacturing (see Lee [ for te detailed process). In tis paper we study te problem of sceduling n independent jobs on m parallel macines to minimize te makespan. Using te notation of Graam et al [ we denote tis problem as P m B < n C max. Karp [ sowed tat te problem P C max is NP-ard in te ordinary sense. Since it contains P C max as a special case te problem P m B < n C max is also NP-ard in te ordinary sense. Supported by te National Natural Science Foundation (Grant Number 06708) and te Province Natural Science Foundation of Sandong (Grant Number Y 005A0). Corresponding autor. qrjianfeng@6.com (J.F. Ren).

2 5 Te 7t International Symposium on Operations Researc and Its Applications Previous related work: Karp (97) sowed tat P C max is NP-ard in te ordinary sense and Sanin [ gave an FPTAS for it. For te problem of B C max Bartoldi (988) sowed tat it can be solved optimally by te algoritm of full batc longest processing time (FBLPT). As to minimize te makespan on parallel identical batc processing macines Lee et al. ([5) provided efficient algoritms under some assumptions. For te problem r j B C max Deng and zang [5 derived a PTAS algoritm. For te problem R B C max Zang (005) [7 presented a ( B + ε)-approximation algoritm. Our contributions: In tis paper we design a fully polynomial time approximation sceme (FPTAS) to solve te bounded identical parallel batc sceduling problem P m B < n C max wen te number of identical parallel batc processing macines m is constant. Problem Description Notation and Elementary Definitions Te sceduling model tat we analyze is as following. Tere are n independent jobs J J J n tat ave to be sceduled on m bounded identical parallel batc macines. Eac job J j ( j = n) as a non-negative processing time p j. All jobs are available for processing at time 0. Te goal is to sceduling te jobs witout preemption on m bounded identical parallel batc macines suc tat te makespan is minimized. Te set of real numbers is denoted by IR and te set of non-negative integers is denoted by IN; note tat 0 IN. Te base two logaritm of z denoted by logz and te natural logaritm by lnz. We recall te following well-known properties of binary relations on a set Z. Te relation is called reflexive if for any z Z: z z symmetric if for any zz Z: z z implies z z anti-symmetric if for any zz Z: z z and z z implies z = z transitive if for any zz z Z: z z and z z implies z z. A relation on z is called a partial order if it is reflexive anti-symmetric and transitive. A relation on Z is called a quasi-order if it is reflexive and transitive. A quasi-order on Z is called a quasi-linear order if any two elements of Z are comparable. For Z Z an element z Z is called a maximum in Z wit respect to if z z olds for all z Z. Te element z Z is called a maximal in Z wit respect to if te only z Z wit z z is z itself. Proposition [6 : For any binary relation on a set Z and for any finite subsetz of Z te following olds. (i) If is a partial order ten tere exists a maximal element in Z. (ii) If is a quasi-line order ten tere exists at least one maximum element in Z. Woeginger [6 sowed tat dynamic programming algoritm wit a special structure automatically lead to a fully polynomial time approximation sceme. Assume tat we ave an approximation algoritm tat always returns a near-optimal solution wose cost is at most a factor of ρ away from te optimal cost were ρ > is some real number: in minimization problems te near-optimal is at most a multiplicative factor of ρ above te optimum and in maximinization problems it is at most a factor of ρ below te optimum.

3 Approximation Algoritm of Minimizing Makespan 55 Suc an approximation algoritm is called a ρ-approximation algoritm. A family of ( + ε)-approximation algoritms over all ε > 0 wit polynomial running time is called a polynomial time approximation sceme (PTAS). If te time complexity of a PTAS is also polynomially bounded in ( ε ) ten it is called a fully polynomial time approximation sceme (FPTAS). An FPTAS is te strongest possible polynomial time approximation result tat we can derive for an NP-ard problem unless P=NP. Woeginger et al. considered a GENEric optimization problem ( for sort GENE) and provided a uniform approac to design te fully polynomial time approximation sceme for it. A DP-simple optimization problem GENE is called DP-benevolent iff tere exist a partial order dom a quasi-wine order qua and a degree-vector D suc tat its dynamic programming formulation DP fulfills te Conditions C.-C.. Te lemma [6 in section denotes tat te DP-benevolent problems are easy to approximate. Te Dynamic Programming As we can firstly scedule all jobs wose processing times are zero and let all nonzero processing times be integers by enlarge te same times and keep te same structure of optimal scedule we propose tat all p j ( j = n) are non-negative integers. We renumber te jobs suc tat p p p n. lemma Tere exists an optimal scedule in wic all macines process te jobs increasing order of index. Moreover an optimal scedule will not contain any macine idle time. A straigtforward job intercange argument can proof te lemma. Now let α = and β = m. For k = n define te input vector X k = [p k. A state S = [s () s() s() s() S k encodes a partial scedule for te first jobs J J J k : te coordinate s (l) measures te total processing time on te l-t macine in te partial scedule and s (l) measures te processing time of te last batc on te l-t macine in te partial scedule. Te two additional coordinates s (l) and s (l) respectively stores te least and te largest index of te last batc on te l-t macine in te partial scedule. Te set F consists of m functions F(l) F(l) l = m. [p ks () s() s() [s () s() s() s() s() s(l ) s (m) [p ks () s() s() [s () s() s() s() s (l ) s (l ) s (l ) = s (l) s() s(l ) s (m) s (l ) s (l ) s (l ) = F(l) s(l) s(l) k s(l+) s (l+) s (l+) s (l+) s (l) + p k p k kks (l+) s (l+) s (l+) s (l+) Intuitively speaking te function scedule te job J k on te l-t macine and put it into te last batc of te partial scedule S = [s () s() s() s() S k for te jobs J J J k. i.e Adds job J k to te l-t macine so tat it does not start te last batc.

4 56 Te 7t International Symposium on Operations Researc and Its Applications Te function scedule te job J k to te l-t macine end of te partial scedule S = [s () s() s() s() s m S k for te jobs J J J k. i.e Adds job J k so tat it starts te last batc. Te functions and in H correspond to and respectively. [p ks () s() s() s() = k s (l) + B [p ks () s() s() s() 0 l = m. Now te iterative computation in Line 5 of DP for all functions in F reads If k s (l) + B 0 ten add s(l ) [s () s() s() s (m) If 0 0 ten add [s () s() s() s() s() s (l ) s (l ) s (l ) s (m) l = m. Finally set G[s () s() s() s (l+) s (l) s(l ) s (l ) s (l ) s (l ) s (l) s(l) s(l) ks(l+) s (l+) s (l+) s (l+) + p k p k kks (l+) s (l ) s (l ) s (l ) s (l) s(l) s(l) s(l) s(l+) s() s(l ) s (l) s (l+) s (l+) s (l+) s (l+) s (l+) s(l+) and ini- s() s(l ) s (m) s (m) s (m) s (m) =max[s () tialize te space S 0 = {[00 0}. Next we proof te problem P m B < n C max is benevolent. Let te degree vector D = [ Note tat te coordinates according to te state variable s (l) (l = m) are and all oter coordinates are 0. Let S = [s () s() s() S = [s () s() s () s () s () Te dominance relation is defined: S dom S s (l) and Te quasi-linear order is defined: S k S k for = ; l = m S qua S m for l = m l= l= Teorem For any > for any F F for any SS IN m te following olds: (i) If S is [D -close to S and if S qua S ten (a) F(XS) qua F(XS ) olds and F(XS) is [D -close to F(XS ) or (b) F(XS) dom F(XS ). (ii) If S dom S ten F(XS) dom F(XS ). Were X = [p k. k = n Proof. (i) Consider a real number > two vectors S = [s () s() s() s() s(m) s (m) s (m) s (m) S = [s () s () s () s () s (m) tat fulfill S is [D -close to S and S qua S. From S is [D -close to S we get tat s (l) s (l) s (l) and s (l) for l = m; = () As S qua S so m. for l = m () l= l=

5 Approximation Algoritm of Minimizing Makespan 57 From () we ave m and m + p k + p k () l= l= l= l= () yields tat (XS) qua (XS ) and (XS) qua (XS ) for l = m. From S is [D -close to S and () we ave m l= s (l) m l= m l= From S is [D -close to S and () we ave ( m l= s (l) + p k) m l= s (l) and s (l) = s (l) for l = m; = () ( + p k) s (l) = for l = m; = l= (5) () (5) imply tat (XS) is [D -close to F(l) (XS ) and (XS) is [D -close to (XS ) (for l = m). Hence for functions and Teorem(i) old. (ii) As S dom S we ave s (l) and for l = m; = (6) + p k s (l) + p k and for l = m; = (7) Ten (6) (7) respectively yields tat (XS) dom (XS ) and (XS) dom (XS ) for l = m. Teorem For any > for any H H for any SS IN m te following olds: (i) If S is [D -close to S and S qua S ten H(XS ) H(XS). (ii) If S dom S ten H(XS ) H(XS). Proof. (i) By S is [D -close to S and S qua S applying te definition of te quasiorder relation we ave (XS) = H(l) (XS ) and (XS) = H(l) (XS ) for l = m. (ii) By te definition of te dominance relation we can easily get (XS) = H(l) (XS ) and (XS) = H(l) (XS ) for l = m. Teorem Let g = ten for any > and for any SS IN m te following olds: (i) If S is [D -close to S and if S qua S ten G(S ) g G(S) = G(S). (ii) If S dom S ten G(S ) G(S). Proof. (i) From S is [D -close to S and S qua S we get s (l) s (l) s (l) for l = m. So max l m {s(l) } max l m {s (l) } max l m {s(l) } i.e G(S) G(S ) G(S). Of course olds G(S ) G(S). (ii) From S dom S and (6) we ave s (l) for l = m. Ten olds G(S ) G(S) Teorem (i) Every F F can be evaluated in polynomial time. Every H H can be evaluated in polynomial time. Te function G can be evaluated in polynomial time. Te relation qua can be decided in polynomial time. (ii) Te cardinality of F is polynomially bounded in n and logx. (iii) For every instance I of P m B < n C max te state space S 0 can be computed in time tat is polynomially bounded in n and logx. As a consequence also te cardinality of te

6 58 Te 7t International Symposium on Operations Researc and Its Applications state space S 0 is polynomially bounded in n and logx. (iv) For an instance I of P m B < n C max and for a coordinate l ( l m) let V l (I) denote te set of te l-t components of all vectors in all state spaces S k ( k n). Ten te following olds for every instance I. For all coordinate l ( l m) te natural logaritm of every value in V l (I) is bounded by a polynomial π (nlogx) in n and logx. Moreover for coordinate l wit d l = 0 te cardinality of V l (I) is bounded by a polynomial π (nlogx) in n and logx. Proof. (i) (ii) (iii) are straigtforward. For (iv) note tat te coordinates are 0 only take te n sums of job processing or te index elements n ence (iv) is also olds. Based on te above dynamic programming we gave following algoritm (named MTDP). Algoritm MTDP. Step 0 Delete all jobs wit zero processing times and cange all oter processing times into integers by multipling te same parameter. We denote te new instance I. For I go to Step Step Initialize T 0 := S 0 Step For k = to n do Step Let U k := φ Step For every T T k and every F F do Step 5 If H F (X k T ) 0 ten add F(X k T ) to U k Step 6 Endfor Step 7 Compute a trimmed copy T k of U k Step 8 Endfor Step 9 Output min {G(S) : S S n } Step 0 Scedule te jobs of instance I according to I and insert sufficient zero batces scedule jobs (deleted in Step 0) wit zero processing times in te aead of partial sceduling. Teorem- sows tat te problem P m B < n C max is DP-benevolent. Applying lemma we can get te following lemma5. Teorem5 Algoritm MTDP is an FPTAS for problem P m B < n C max. Proof. From Lemma we get tat MTDP is an FPTAS for problem P m B < n C max. Note: F =m T k ( + ( gn ε )π (nlogx) + + π (nlogx) m and te running time of deciding te relation qua on T k is O( m(m+) ). So te total running time of te algoritm MTDP is O[( nm(m+) ) ( + ( gn ε )π (nlogx) + + π (nlogx) m were x = n p j. j= References [ C Y. Lee R. Uzsoy and L. A. Martin Vega Efficient algoritms for sceduling semiconductor burn-in operations Operation Researc 99 0: [ R. L. Graam Lawler J. K. Lenstra and A. H. G. Rinnooy Kan Optimization and approximation in deterministic sequencing and sceduling: a survey Annals of Discrete Matematics 979 5:-5. [ R.M. Karp Reducibility among combinatorial problems In R.E. Miller and J.W. Tatcer editors Complexity of Computer Computations

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