Scheduling of Jobs and Maintenance Activities on Parallel Machines

Transcription

1 Scheduling of Job and Maintenance Activitie on Parallel Machine Chung-Yee Lee* Department of Indutrial Engineering Texa A&M Univerity College Station, TX Zhi-Long Chen** Department of Sytem Engineering Univerity of Pennylvania Philadelphia, PA September 1998 (Accepted by Naval Reearch Logitic) *Supported in part by NSF Grant DMI **Supported in part by the Univerity of Pennylvania Reearch Foundation

2 ABSTRACT Mot machine cheduling model aume that the machine are available all the time. However, in many realitic ituation, machine need to be maintained and hence may become unavailable during a certain period. In thi paper, we tudy the problem of proceing a et of ob on parallel machine in which each machine mut be maintained once in the planning horizon. Our obective i to chedule ob and maintenance activitie o that total weighted completion time of ob i minimized. Two cae are tudied in thi paper. In the firt cae, there i a ufficient amount of reource and hence different machine can be maintained imultaneouly if neceary. In the econd cae, only one machine can be maintained at any given time. In thi paper, we firt how that, even when all the ob have the ame weight, both cae of the problem are NP-hard. We then propoe branch and bound algorithm baed on the column generation approach for olving both cae of the problem. The algorithm are capable of olving optimally medium ized problem within a reaonable computational time. 1

3 1. Introduction The maority of machine cheduling literature aume that the machine are available for proceing ob all the time in the planning horizon. However, in many manufacturing ituation, machine are maintained periodically to prevent malfunction from happening. During a maintenance period, a machine i not available for proceing any ob. Therefore, a more realitic cheduling model hould take into account aociated machine maintenance activitie. The chedule of maintenance activitie can be determined either before the chedule of ob, or ointly with the chedule of ob. In the firt cae, maintenance period are already known and fixed at the time when ob are to be cheduled. The problem of cheduling ob with thi type of maintenance reduce to the problem often referred in the literature a cheduling with machine availability contraint becaue during maintenance period the machine i not available for proceing ob. Schmidt (1984, 1988) tudie a parallel machine problem where each machine ha different availability interval. Adiri et al. (1989) and Lee and Liman (1992) tudy a ingle machine problem with a machine availability contraint. Lee (1991) tudie a parallelmachine problem where machine are available tarting from different time. Lee and Liman (1993) conider a two-parallel-machine problem where one machine ha an availability contraint. Moheiov (1994) tudie the ame problem by auming that each machine i available in an interval. Lee (1996, 1997) conider cheduling problem with availability contraint under different machine configuration and performance meaure. Note that all thee reult aume that the machine unavailability i known and fixed in advance. In our experience with indutry application, epecially in emiconductor manufacturing, it i often oberved that a machine i idle, yet ob are waiting and the machine i not broken down but waiting for maintenance peronnel to come do preventive maintenance. Thi i due to lack of coordination between production planning peronnel and maintenance peronnel. Obviouly, a careful coordination between maintenance and ob proceing will turn out a better overall chedule. Hence the cheduling problem to be tudied in thi paper i to determine imultaneouly when to 2

4 perform each maintenance activity and when to proce each ob o that a certain performance meaure i optimized. Little reearch ha been done in the literature for the model of ointly cheduling maintenance activitie and ob. Qi et al (1997) conider a problem where multiple maintenance activitie need to be cheduled ointly with ob on a ingle machine. Grave and Lee (1998) clarify the complexity of variou ingle machine problem with thi model. Lee and Leon (1997) tudy a ingle machine problem with thi model where proceing time of ob may change after a maintenance activity. Our experience with indutry alo how that preventive maintenance i uually performed within every precribed number of period. In ome planning horizon we may not have maintenance activity. However, after certain planning horizon, ome period have elaped, and one maintenance activity ha to be cheduled for each machine within the current planning horizon. In thi paper we tudy the problem in which the maintenance for each machine mut be completed within T period. More pecifically, we tudy the problem of ointly cheduling a et of n ob, N={1, 2,..., n}, and maintenance activitie on m identical parallel machine. Each machine mut be maintained exactly once with a contant maintenance length t during the given time interval [, T], where T t. Each ob ha a proceing time p and weight w. Our goal i to find a oint chedule of ob and maintenance activitie to minimize total weighted completion time of ob Σ N w C, where C i the completion time of ob in a chedule. We aume that the ob proceing i nonreumable, i.e. a ob mut be reproceed fully after the maintenance if it proceing i interrupted by the maintenance activity on a machine. We conider two cae of maintenance. In the firt cae, call it independent cae, there i a ufficient amount of maintenance reource uch that any number of machine can be maintained imultaneouly if neceary. In the econd cae, call it dependent cae, the amount of maintenance reource i limited o that only one machine can be maintained at any given time and hence the maintenance activity on one machine cannot be overlapped with that on another machine. In thi cae, we aume that T mt for otherwie there i imply no feaible chedule. Following the three-field notation of 3

5 Pinedo (1995), we denote the two correponding problem a, Pm ind w C and Pm dep w C, in the independent and dependent cae of maintenance, repectively, when the number of machine i fixed a m. The remainder of thi paper i organized a follow. In the next ection, we give ome baic propertie aociated with thee problem and analyze their computational complexity. More pecifically, we how that Pm ind C and Pm dep C, problem with equal weight, are NP-hard even though the correponding claical problem, the one without maintenance i polynomially olvable. Thi mean that our problem are more difficult than the claical counterpart without maintenance. Note that the claical total weighted completion time problem without maintenance, Pm w C, can be olved by a dynamic programming approach (Rothkopf 1966) with the time complexity O(nmP m-1 ), where P i the total proceing time of all the ob. However, it can be expected that an algorithm with uch high complexity can only olve problem with a mall ize. Belouadah and Pott (1995) ue a Lagrangian relaxation baed branch and bound algorithm to olve the problem Pm w C with up to 2 ob and 5 machine or 3 ob and 4 machine. By a column generation approach, van den Akker (1995) and Chen and Powell (1996) are able to olve thi ame problem with a much larger ize (1 ob and 2 machine). Chen and Powell (1998, 1999), demontrate that the column generation approach i very effective for olving many other parallel-machine cheduling problem. It can be hown that we can modify the dynamic programming approach by Rothkopf (1966) to olve our problem Pm ind w C and Pm dep w C. However, the time complexity will be even higher than O(nmP m-1 ). Clearly, uch an approach can only olve our problem with a very mall ize. Hence, in Section 3, we propoe optimal algorithm for the problem Pm ind w C and Pm dep w C by uing the column generation approach. Our computational reult in Section 4 demontrate that the column generation baed algorithm are capable of olving problem of medium ize to optimality within a reaonable computational time. Finally, we conclude the paper in Section 5. 4

6 2. Optimality Propertie and Complexity Analyi We firt give ome baic propertie of the problem Pm ind w C and Pm dep w C in Section 2.1. Then we analyze the complexity of Pm ind C and Pm dep C in Section Optimality Propertie In thi ection, we introduce ome propertie aociated with optimal chedule of the problem Pm ind w C and Pm dep w C. Thee propertie will be ued later to retrict the earch pace when we earch for optimal chedule for thee problem. For eae of preentation, we ay a et of ob are in their WSPT order if thee ob are equenced in the non-increaing order of the ratio w /p. Property 1: (1) For both problem Pm ind w C and Pm dep w C, there exit an optimal chedule uch that on each machine ob cheduled before the maintenance are in their WSPT order, and thoe cheduled after the maintenance are in their WSPT order a well. (2) For the problem Pm ind w C, there exit an optimal chedule uch that there i no idle time on each machine, and the maintenance tart at time zero or at the completion time of ome ob. Proof: By adacent pairwie interchange rule. We omit the detail of the proof. QED Note that for the problem Pm dep w C, in an optimal chedule, there may exit idle time between the completion time of a ob and the tarting time of the maintenance. Property 2: There exit an optimal chedule for the problem Pm ind w C uch that on each machine, (i) the lat ob cheduled before the maintenance i completed no later than the time R 1 = [P+(m-1)t-p max ]/m + p max (1) (ii) the lat ob cheduled after the maintenance i completed no later than the time R 2 = [P-p max ]/m + t + p max (2) 5

7 where P = N p and p max = max{p N}. Proof: (i) Conider it partial chedule on any machine k for a given optimal chedule. Let ob be the lat ob cheduled before the maintenance on machine k and let be the tarting time of ob. At time, any other machine i (i k) mut be buy (either proceing ob or doing maintenance). For otherwie, we hould have aigned ob to that machine. Hence by Property 1, we mut have m P - p + (m-1)t Thi implie that the completion time of will be C = + p [P + (m-1)t p ]/m + p R 1. (ii) A imilar proof can be ued to prove the relation (2). QED. Property 3: For the problem Pm dep w C, there i an optimal chedule uch that on each machine, (i) the lat ob cheduled before the maintenance i completed no later than R 3 = [P+(m-1)t+(m-2)p max ]/m + p max, (3) (ii) the lat ob cheduled after the maintenance i completed no later than R 4 = [P+(m-1)p max ]/m + t + p max (4) Proof: Firt note that in an optimal chedule if ome ob are cheduled after the maintenance on a machine, then on that machine the idle time immediately before the maintenance mut be le than p max. For otherwie, we can move a ob cheduled after the maintenance to the poition right before the maintenance to improve the chedule. (i) Let be the lat ob cheduled before the maintenance on a machine k in an optimal chedule and let be the tarting time of ob. At time, any other machine i (i k) mut not have idle time more than p. For otherwie, we hould have aigned ob to that machine. Hence we have m (P - p ) + (m-1)t + (m-1)p (P - p ) + (m-1)t + (m-1)p max Thi implie that the completion time of will be C [P + (m-1)t p +(m-1)p max ]/m + p R 3. (ii) The relation (4) can be proved imilarly. QED. 6

8 2.2. NP-hardne of Pm ind C and Pm dep C Note that the claical parallel-machine problem Pm C can be olved eaily by cheduling ob in the nondecreaing order of their proceing time. In thi ection, we how, however, that the problem Pm ind C and Pm dep C are NP-hard. Hence the problem with general weight Pm ind w C and Pm dep w C are NP-hard a well. The following Partition Problem, a well-known NP-hard problem, will be ued later in the NP-hardne proof. Partition Problem (PP): Given poitive integer a 1, a 2,..., a h, and A uch that 2A, the problem ak if there exit a ubet S H={1, 2,, h} uch that a A. Theorem 1: The problem Pm dep C i NP-hard. S = h a i i= 1 Proof: We prove it by a reduction from the Partition Problem. Given an intance of PP, we contruct the following intance for the problem Pm dep C : m=2 machine, n=h+1 ob, N=H {h+1}={1, 2,, h+1} proceing time: p =a, for H; and p h+1 =B, where B=2hA 2 maintenance length t = B maintenance for each machine mut be finihed before or at T=A + 2B obective function threhold y = 2B In the following we how that there i a chedule for the contructed intance of the problem Pm dep C with the total completion time no more than y if and only if there i a partition S H for the PP intance uch that a S = A. = Given a partition S H for the PP intance, we contruct a chedule a follow: Machine 1: chedule all the ob in et S, followed by the maintenance Machine 2: chedule all the ob in et H\S, then ob h+1, followed by the maintenance It i eay to check that thi chedule i feaible becaue the maintenance on the firt machine tart at time A and that on the econd machine tart at time A+B. The completion time of each ob in H i no more than A, and that of ob h+1 i A + B, o the total completion time i no more than ha + A + B < y. 7

9 Given a chedule with the total completion time no more than y, firt we can ee that no ob hould be cheduled after the maintenance on any machine becaue of the following reaon. Suppoe that there i ome ob cheduled after the maintenance on ome machine. There are two cae. If thi ob i from H, then it completion time i more than B. Since ob h+1 mut have a completion time no le than B, thi mean that the total completion time of all the ob i more than 2B=y. On the other hand, if the ob cheduled after the maintenance i ob h+1, then it completion time will be at leat 2B, which implie that the total completion time of all the ob mut be more than 2B=y. Furthermore, we can ee that in the given chedule no ob from H can be cheduled after ob h+1. Thi can be hown uing the ame argument a above. Without lo of generality, we aume that ob h+1 i proceed on machine 2. Thu the given chedule ha the following tructure: Machine 1: chedule a ubet of ob of H, denoted a S, followed by the maintenance Machine 2: chedule the ret of ob in H, H\S, then ob h+1, followed by the maintenance. Now uppoe that p S > A, then the maintenance on the firt machine will occupy the time interval [r, q] with r>a and q>a+b, which implie that the maintenance on the econd machine will not be feaible. On the other hand, if p S < A, then H \ S p > A, which alo implie that the maintenance on the econd machine will not be feaible. Therefore, p S = A, and hence there i a olution to the PP intance. QED. Problem. The following theorem can be proved imilarly by a reduction from the Partition Theorem 2: The problem Pm ind C i NP-hard. 3. Branch and Bound Algorithm for Pm ind w C and Pm dep w C In thi ection, we develop branch and bound algorithm baed on column generation for olving optimally thee problem. Our algorithm are baed on a imilar 8

10 approach propoed recently by Chen and Powell (1996) for olving other parallel machine cheduling problem. A we will how in Section 4, our algorithm can efficiently olve problem of medium ize. Column generation method ha been ued in the literature to olve large-cale combinatorial optimization problem, including vehicle routing (Derocher et al., 1992), cutting tock (Vance, et al., 1994), capacitated lot izing (Cattrye et al., 1993), and air crew cheduling (Lavoie et al., 1988). Recently, it ha been uccefully applied to olving large-cale cheduling problem, ee for example, van den Akker et al.(1995), and Chen and Powell (1996, 1998, 1999). Chan and Simchi- Levi (1998) provide error bound analyi of uing column generation for olving parallelmachine cheduling problem. Following the column generation approach, we will firt formulate in Section 3.1 the problem Pm ind w C and Pm dep w C a et partitioning type formulation. In thee formulation, there are an exponential number of column, each of which repreent a partial chedule on a ingle machine. Then branch and bound algorithm can be deigned to olve the et partitioning type formulation. In the branch and bound tree, each node i a linear relaxation of a et partitioning type problem. Thi linear relaxation problem i olved in Section 3.2 following the idea of the tandard column generation procedure (ee, e.g. Ladon 197) for large-cale linear programming problem. In thi procedure, neceary column are generated iteratively by olving ome ingle-machine cheduling problem uing peudo-polynomial dynamic programming algorithm. The olution of a branch and bound node i ued to quantify ome value aociated with individual ob and pair of ob. Branching procedure (decribed in Section 3.3) are baed on thee value Set Partitioning Type Formulation Clearly, to find an optimal chedule for the problem Pm ind w C, we only need to conider chedule atifying Propertie 1 and 2. Similarly, for the problem Pm dep w C, we only need to conider chedule atifying Propertie 1 and 3. Define a feaible partial chedule on a machine a a oint chedule of a ubet of ob and a maintenance activity where the maintenance i completed before or at the time 9

11 T. Let S k be the et of all feaible partial chedule on machine k, k {1, 2,..., m}, atifying Propertie 1 and 2 in the cae of Pm ind w C, and Propertie 1 and 3 in the cae of Pm dep w C. Let S = S 1 S 2... S m. For each S, we define, f = the total weighted completion time of the ob in the chedule, a = the number of time ob i covered in the chedule, c = the completion time of the maintenance in the chedule, x = 1 if the chedule i elected in the optimal olution and otherwie. Note that given a partial chedule S, the parameter f, a, and c are known. With the above defined notation, the two problem Pm ind w C and Pm dep w C can be formulated into the following et partitioning type program, denoted a [IND] and [DEP], repectively. [IND] Min f x.t. a x S S x S (5) = 1, N (6) = m (7) x = or 1, S (8) [DEP] Min f x c x S k + 1.t. a x S Sk x S (9) = 1, N (1) = 1, k = 1, 2,, m (11) c t, k = 1, 2,, m-1 (12) x S k x = or 1, S (13) In [IND], the obective (5) i to minimize the total weighted completion time of all the elected partial chedule. The contraint (6) enure that each ob i covered exactly once. The contraint (7) guarantee that exactly m partial chedule are elected ince we have m machine. In [DEP], the obective (9) and the firt contraint (1) have the ame meaning a (5) and (6) in [IND], repectively. The contraint (11) mean that 1

12 exactly one partial chedule i elected for each machine. Thi contraint i different from (7). In (7), we do not ditinguih different machine becaue the machine in the problem Pm ind w C are mutually independent and really identical and hence S 1 =S 2 =...=S m. On the contrary, the machine in the problem Pm dep w C are mutually dependent and not identical in term of maintenance cheduling. Without lo of generality, we aume that the machine are maintained following the order (1, 2,..., m). The contraint (12) enure that the maintenance activitie on different machine are not overlapped. Without (7), the formulation [IND] would be a et partitioning problem. Similarly, the formulation [DEP] would be a et partitioning problem without (12). So we call thee formulation et partitioning type formulation (with ide contraint). Note that each column in thee formulation repreent a chedule on a ingle machine and the number of column in thee formulation could be extremely large a the et S contain an exponential number of chedule. Thu it i not practical to olve thee formulation directly uing traditional branch and bound cheme. In the following ection, we develop nontraditional branch and bound algorithm to olve thee formulation Column Generation Approach Relaxing the contraint (8) and (13) by allowing x to take any value in the interval [, 1], we get the linear relaxation problem, denoted by [LIND] and [LDEP], repectively, of the et partitioning type formulation [IND] and [DEP]. In our branch and bound algorithm, each branch and bound node i a linear relaxation problem [LIND] or [LDEP] with additional contraint impoed by branching rule. We will dicu branching rule in Section General Framework To olve [LIND] and [LDEP], we ue the idea of the tandard column generation procedure (Ladon 197) for large cale linear programming problem. Thi procedure tart with a limited number of column in [LIND] or [LDEP] and generate neceary column iteratively. In each iteration, we firt olve the retricted verion of the problem [LIND] or [LDEP] (called retricted mater problem) with the column generated o far. 11

13 Then we generate column with mot negative reduced cot by olving one or more ingle-machine cheduling problem (called ingle-machine ubproblem) that are cloely related to the underlying parallel machine problem. Newly generated column are then added to the retricted mater problem which i then updated. When no new column with negative reduced cot can be found, the procedure terminate and the problem [LIND] or [LDEP] i olved In the column generation procedure, retricted mater problem are linear programming problem and can be olved very efficiently. Our computational experience ha hown that ingle-machine ubproblem uually conume much more computational time than retricted mater problem. Thu it i crucial to find an efficient algorithm for olving ubproblem. In [LIND], let the dual variable value be π for index in (6), and σ for (7). Similarly, in [LDEP], let the dual variable value be π for index in (1), σ k for index k in (11), and ρ k for index k in (12). Then the reduced cot r of the column correponding to a chedule S in [LIND] i given by the following formula (14), and that of the column correponding to a chedule S i in [LDEP] i given by the following formula (15). r = f - a π - σ (14) N r = f - N f - N f - N a π + c ρ1 - σ i, if i=1 a π - c ρ i 1 + c ρ i - a π - c ρ m 1 - σ i, if i=m σ i, if 1<i<m (15) There i only one ingle-machine ubproblem involved in the column generation procedure for [LIND], that i to find a chedule S with the mot negative reduced cot r defined by (14). More preciely, thi ubproblem i to elect a ubet of ob and find a oint chedule of thee elected ob and a maintenance activity on one machine uch that the maintenance i completed no later than T and the total weighted completion time of 12

14 thee elected ob (i.e. the quantity f ) minu the total dual variable value of thee elected ob (i.e. the quantity aπ N in (14) becaue it i identical for all chedule in S. ) i minimized. Here we can ignore the third term For [LDEP], there are m ingle-machine ubproblem involved, one correponding to each machine. The ith ubproblem correponding to machine i i to find a chedule S i on machine i with the mot negative reduced cot r defined by (15). Thi ubproblem i imilar to the one involved in [LIND] except that one more quantity (given by a linear function of the completion time of the maintenance c ) i involved in the obective function Solving the Subproblem Involved in [LIND] Without the maintenance activity, thi ubproblem would be equivalent to the ubproblem involved in the column generation procedure for the claical problem P m w C tudied in Chen and Powell (1996). They howed the NP-hardne of their ubproblem. Thu, the ubproblem aociated with [LIND] i alo NP-hard. In order to find an effective algorithm to olve thi ubproblem, we need to modify it in a way that it can be more tractable while the optimality of the integer problem [IND] i not affected by thi modification. In any chedule in the original et S, a ob can be covered at mot once, either before the maintenance or after the maintenance. We relax thi retriction and allow a ob to be covered both before and after the maintenance on a machine. We firt re-index ob in N in the WSPT order, and denote the maintenance on a given machine a ob n+1. For notational convenience, we denote N(1) = N = {1, 2,..., n}. Then we create another n ob, indexed a n+2,, 2n+1, uch that ob n++1 i identical to ob, i.e. p n++1 =p, w n++1 =w, and π n++1 =π, for =1, 2,, n. Denote N(2) ={ n+2,, 2n+1}. We precribe that on any machine, ob from N(1) can only be cheduled before the maintenance (i.e. ob n+1), and ob from N(2) can only be cheduled after the maintenance. Now we conider the modified ubproblem of finding a ubet of ob from N(1) to be cheduled in the ame order a their indice before the maintenance and a ubet of ob from N(2) to be cheduled in the ame order a their 13

15 indice after the maintenance uch that (i) the maintenance i completed no later than time T, and (ii) the total weighted completion time of thee ob minu the total dual variable value of thee ob i minimized. It i eay to ee that the olution pace of thi modified ubproblem contain the et S. Given a chedule to thi problem, if, for each =1, 2,, n, at mot one of the ob and n++1 i covered, then thi chedule i feaible to the original ubproblem. For each ob {1, 2,, 2n+1}, we denote B a the et of ob that can be cheduled immediately before ob in an optimal chedule to the modified ubproblem, that i, B ={1, 2,, -1} for N(1) {n+1}, and B ={n+1, n+2,, -1} for N(2). Note that in our branch and bound algorithm, ome B may be changed by branching procedure. In the following, we give a peudo-polynomial dynamic programming algorithm for optimally olving thi modified ubproblem. DP Algorithm 1 Define R 1 and R 2 by (1) and (2). Define F(, x) a the minimum reduced cot of a partial chedule of the modified ubproblem, containing a ubet of ob {1, 2,, }, where (i) ob i the current lat ob and completed at time x, and (ii) if n+2, then the maintenance (i.e. ob n+1) i already covered and completed before or at time T. Initial condition: F(, ) = -σ ; F(, x) = for all x ; and F(n+1, x) = for all x>t. Recurive relation: For = 1, 2,, n; x =, 1,..., R 1, F(, x) min{ F( k, x p ) + w x π k B {}} = For = n+1; x =,..., R 2, F(, x) min{ F( k, x t) k B {}} = For = n+2,,2n+1; x =, 1,..., R 2, F(, x) = min{ F( k, x p ) + w x π k B } 14

16 Optimal olution: min{f(, x) = n+1,..., 2n+1; x =, 1,..., R 2 }. Jutification: By principle of optimality. Time complexity: O(n 2 R 2 ). Thi i becaue in the DP, there are a total of O(nR 2 ) tate and the value of each tate i calculated in O(n) time. We note that in our branch and bound algorithm, branching rule (ee detail in Section 3.3) may impoe retriction on which ob can be cheduled immediately before a given ob before the maintenance or after the maintenance, which i equivalent to updating the et B or B n++1. Clearly, Algorithm 1 till work after B are updated becaue we have explicitly incorporated B into the algorithm. Alo note that if both ob and n++1 are covered in an optimal chedule of the modified ubproblem of [LIND], the parameter a (ued in the formulation [LIND]) i 2, and thu by the contraint (6) it i guaranteed that thi chedule will not be elected a part of the optimal chedule for the original integer problem [IND]. However, in the olution of the linear relaxation problem [LIND], the value of x may be poitive Solving the Subproblem Involved in [LDEP] Clearly, each ubproblem involved in [LDEP] i at leat a hard a the one involved in [LIND] and hence i NP-hard. In order to olve a given ubproblem of [LDEP] efficiently, we modify the problem exactly the ame way a what we have done for the ubproblem of [LIND] in Section We ue the ame notation a in Section The following algorithm olve the modified ubproblem of the ith ubproblem of [LDEP] correponding to machine i. DP Algorithm 2 Define P, and B imilarly to the repective one in Algorithm 1. Define R 3 and R 4 by (3) and (4). Define F(, x) a the minimum reduced cot of a partial chedule of the modified ubproblem of machine i, containing a ubet of ob {1, 2,, }, where (i) ob i the 15

17 current lat ob and completed at time x, and (ii) if n+2, then the maintenance (i.e. ob n+1) i already covered and completed before or at time T. Initial condition: F(, ) = -σ i ; F(, x) = for all x ; and F(n+1, x) = for all x>t. Recurive relation: For = 1, 2,, n; x=, 1,, R 3, F(, x) min{ F( k, x p ) + w x π k B {}} = For = n+1; x=, 1,, T, F(, x) min{ F( k, y) + g( i) x y min( x t, R3 ) ; k B {}} = where ρ i if i = 1 g(i) = ρ i -ρ i-1 if 1<i<m -ρ m-1 if i= m For = n+2, n+3,, 2n+1; x =, 1,, R 4, F(, x) = min{ F( k, x p ) + w x π k B } Optimal olution: min{ F( n + 1, x) x =,1,..., T} min min{ F(, x) = n + 2,...,2n + 1; x =,1,..., Time complexity: O(n 2 R 4 +nr 3 T). Thi i becaue in the DP, there are a total of O(nR 4 ) tate (, x) with n+1, the value of each calculated in O(n) time, and O(T) tate, the value of each calculated in O(nR 3 ) time. Jutification: By principle of optimality. We note that, imilar to Algorithm 1, Algorithm 2 may generate chedule where both ob and n++1 are covered. In thi cae, the parameter a (ued in the formulation [LDEP]) i 2. The contraint (1) guarantee that uch a chedule will not be elected in the optimal olution of the integer problem [DEP]. R 4 } 16

18 3.3. Branching Strategie In the branch and bound algorithm, each node in the branch and bound tree i a linear relaxation problem, [LIND] or [LDEP], with additional contraint impoed by branching rule. The linear relaxation problem i olved uing the column generation approach propoed in Section 3.2. The olution of uch a problem i ued to meaure the percentage of each ob covered in the olution, and quantify ordering relation of ob. Then branching procedure are performed baed on thee value. Given the olution of a branch and bound node, x for S, we define the value of ordering relation, λ k for each ob pair (k, ), for k B, and N(1) N(2), a follow: λ k = S b (16) k x where b k i 1 if in the chedule ob k i cheduled immediately before ob, and otherwie. Roughly peaking, the bigger the value λ k, the tronger the ordering relation of the ob pair (k, ). Similarly, we define the value of coverage, v for each ob N(1) N(2) a follow: v = λ k (17) k B Since the formulation [LIND] and [LDEP] guarantee that each ob i covered exactly once in any olution, thu it mut hold that v +v n++1 =1 for each N. Alo it i eay to ee that λ k [, 1] for each ob pair (k, ), for k B, and N(1) N(2). Given the olution to a branch and bound node, we only need to conider the following three cae where branching procedure may be performed. Cae 1: The olution i fractional, the integer part of it obective value i le than that of the bet integer olution generated o far, and ome of the v are fractional. In thi cae, we branch down thi node a follow. We firt chooe a ob N(1) N(2) with the value v cloet to.5. Then we create two child node (called Child 1 and Child 2), i.e. two new [LIND] (or [LDEP]) problem, uch that the olution of Child 1 will reult in v =1 and that of Child 2 will reult in v =. Note that, if N(1), then ob i forced to be 17

19 cheduled before maintenance in Child 1 and after maintenance in Child 2. Similarly, if N(2), then ob k (with k=-n-1) i forced o. Thi can be accomplihed by the following procedure. For Child 1, (1) Suppoe that ob h i identical to ob, i.e. h=n++1, if n, and =n+h+1, if n+2. Then ob h cannot be included in the olution of Child 1. Update B k :=B k \{h}, for each k N(1) N(2), and do not conider ob h in the algorithm for ubproblem. (2) Form the initial retricted mater problem [LIND] (or [LDEP]) of Child 1 by electing all the column already generated in the parent node except the one that contain ob h. For Child 2, we perform the ame procedure except that everything done to ob h in the cae of Child 1 i now done to ob intead. Cae 2: The olution i fractional, the integer part of it obective value i le than that of the bet integer olution generated o far, all the v are integral, but ome of the λ k are fractional. In thi cae, we branch down thi node a follow. We firt chooe a pair of ob (k, ) with k B, and N(1) N(2) uch that the value λ k i cloet to.5. Then we create two child node, Child 1 and Child 2, uch that the olution of Child 1 will reult in λ k =1 and that of Child 2 will reult in λ k =. Note that, in Child 1, ob k i forced to be cheduled immediately before ob, while in Child 2, ob k i prohibited to be cheduled immediately before ob. To thi end, we perform the following procedure. For Child 1, (1) Suppoe that ob l and h are identical to ob k and, repectively. Then ob l and h cannot be included in the olution of Child 1. Update B k :=B k \{l, h}, for each k N(1) N(2), and do not conider ob l and h in the algorithm for ubproblem. Alo, we need to make ure that in the olution of Child 1, λ ku = for all u, and λ q = for all q k. Thi can be achieved by updating B u :=B u \{k}, for all u, and B :={k}. (2) Form the initial retricted mater problem [LIND] (or [LDEP]) of Child 1 by electing all the column already generated in the parent node except the one that contain the ob l, or ob h, or any pair of ob (k, u) with u, or (q, ) with q k. 18

20 For Child 2, we imply update B :=B \{k} and elect all the column in the parent node except the one that contain the ob pair (k, ) a the initial column in the retricted mater problem. Cae 3: The olution i fractional, the integer part of it obective value i le than that of the bet integer olution generated o far, and all the v and λ k are integral. We claim that thi cae will not happen to the problem [LIND], which can be proved following directly the proof given in Chen and Powell (1996) for a imilar claim for another parallel machine cheduling problem. However, for the problem [LDEP], thi cae could happen in theory. For example, if, in the formulation, there are two column covering the ame et of ob but with different completion time of maintenance, then it i poible that both column are with fractional value in the olution. In our computational experiment, however, thi cae never happened. In other word, in every problem we teted for the problem Pm dep w C, if all the v and λ k are integral in a branch and bound node, then the olution of thi node x are all integral. The reaon why thi did not happen in computation i probably due to the fact that only good column are generated in the dynamic programming algorithm. It i worth noting that one difference of [LIND] and [LDEP] i that in the earlier problem there i no idle time before the maintenance while there may be ome idle time before maintenance in the latter problem. Thu, in the problem [LIND], different column mut cover different et of ob, while in [LDEP], two column could cover the ame et of ob (but with different completion time of maintenance). We note that, if thi cae doe happen, we can branch on completion time of ob and maintenance activitie. Denote C a the completion time of ob N(1) {n+1} N(2) in a given chedule S. Given the olution of a branch and bound node, x for S, we define the completion time of ob correponding to thi olution a C = S x C. It i eay to how that when thi cae happen, there mut exit ome ob or/and maintenance activitie with fractional C. We can elect one fractional C and branch on it by creating two child node, one requiring that in it olution the completion time of mut be greater than or equal to C, and the other requiring that mut be le 19

21 than or equal to C, where the notation z i the leat integer greater than or equal to z, and z i the larget integer le than or equal to z. Thee retriction can be eaily incorporated into Algorithm Computational Reult In thi ection, we report computational reult of the branch and bound algorithm developed in Section 3 for the problem Pm ind w C and Pm dep w C. All the algorithm involved are coded in C and teted on a Sun Ultra 1 worktation (164 MHZ). Linear programming problem involved in the column generation approach are olved by CPLEX Verion 5.. To generate a tet problem, there are ix parameter to be determined: number of ob n, number of machine m, proceing time p, weight p, maintenance length t, and the maintenance deadline T by which all maintenance activitie mut be finihed. The firt four parameter are generated independent of one another a follow: (1) Three cae for the number of ob, n {2, 3, 4}. (2) Four cae for the number of machine, m {2, 4, 6, 8}. (3) Two cae for proceing time, p =U[1, 1], or p =U[1, 3], where U[x, y] i an integer uniformly drawn from the interval [x, y]. (4) One cae for weight, w =U[1, 1]. The parameter t and T are generated baed on n, m, and p. In a tet problem, denote the average proceing time of a ob a p avg =( p )/n, and the average total N proceing time on a machine a P avg =( p )/m. Then t and T are generated a follow: N (5) Three cae for the maintenance length, t=αp avg, where the parameter α {1, 2, 4}. (6) Three cae for the maintenance deadline, T=βP avg +t, for the problem Pm ind w C, and T=βP avg +mt, for the problem Pm dep w C, where the parameter β {.2,.4,.8}. 2

22 We note that t and T generated thi way atify the baic feaibility requirement: T t for Pm ind w C, and T mt for Pm dep w C. The parameter α control the maintenance length t, and β control the maintenance deadline T. Our computational experiment how that for fixed n, m, p, w, and α, the cpu time required for olving a tet problem doe not vary much with the parameter β. Thu we will not report tet reult directly baed on the parameter β. Table 1 and 2 how the tet reult for the problem Pm ind w C where proceing time of ob drawn uniformly from the interval [1, 3] and [1, 1], repectively. Table 3 and 4 how the tet reult for the problem Pm dep w C with the ame cae of proceing time, repectively. For each given combination of n, m, and α, a total of thirty tet problem, ten for each of the three cae of β, are generated and olved. In thee table, each row ummarize the computational reult baed on the thirty tet problem olved. The performance meaure hown in thee table are a follow. Column (4) and (5) are, repectively, the average gap and the maximum gap in percentage between the linear relaxation olution value of the root node in the branch and bound tree and the optimal integer olution value. Thi percentage reflect the tightne of the lower bound obtained by olving the linear relaxation of the original integer problem. Column (6) i the number of problem out of the thirty tet problem olved at the root node without any branching. Column (7) and (8) repreent, repectively, the average number and the maximum number of node explored in the branch and bound tree for olving the problem. Note that at leat one node (the root node) mut be explored to olve any problem. Column (9) and (1) repreent, repectively, the average and the maximum cpu time in econd conumed to olve the problem to optimality. From thee table, we can make the following obervation: (1) The integrality gap i extremely mall. Every problem teted ha an integrality gap within 1%. Due to thi, few node need to be explored in the branch and bound tree, and many tet problem are olved at the root node without any branching. Thi reult i conitent with the reult obtained by van den Akker (1995), and Chen and Powell 21

23 (1996, 1998, 1999) in olving other parallel-machine cheduling problem uing the column generation approach. (2) When n and α are fixed, a problem with a larger m can uually be olved fater than a problem with a maller m. Thi reult i alo conitent with thoe obtained in van den Akker (1995), and Chen and Powell (1996, 1998, 1999). Thi reult i probably caued by the fact that, for fixed n, if m i maller, then an ideal partial chedule on a machine hould contain more ob, and hence there are more poible ideal partial chedule and more column may have to be generated in the coure of the column generation, which thu take more time. (3) Our algorithm are capable of olving both problem Pm ind w C and Pm dep w C with 4 ob and any number of machine within a reaonable amount of cpu time. The algorithm are very promiing, particularly, for problem with the ratio n/m no more than Concluion We have analyzed the complexity of the problem Pm ind w C and Pm dep w C and propoed branch and bound exact olution algorithm for olving thee problem to optimality. The computational reult have demontrated that the algorithm are capable of olving problem of medium ize. An intereting topic for future reearch i to invetigate other parallel machine cheduling problem with maintenance activitie, for example, problem with due date related obective function. 22

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26 Table 1: Computational Reult for the Problem P ind w C with Proceing Time Drawn from the Uniform Ditribution U[1, 1] Problem n m α (1) (2) (3) Integrality gap avg max (4) (5).9%.29%.38%.82%.62%.9%.1%.55%.18%.56%.3%.88%.9%.3%.35%.88%.27%.54% % % <.1% <.1% <.1% <.1%.3%.19%.21%.6%.5%.87%.1%.23%.33%.85%.26%.96% <.1%.2%.8%.49%.31%.87% <.1%.3%.5%.17%.8%.23%.6%.15%.14%.25%.2%.47%.1%.24%.14%.31%.27%.58%.9%.2%.24%.53%.42%.94%.3%.13%.7%.26%.33%.62% Problem olved at root node (6) b&b node avg max (7) (8) CPU time () avg max (9) (1)

27 Table 2: Computational Reult for the Problem P ind w C with Proceing Time Drawn from the Uniform Ditribution U[1, 3] Problem n m α (1) (2) (3) Integrality gap avg max (4) (5).2%.48%.4%.9%.53%.99%.9%.4%.27%.96%.25%.49%.7%.39%.8%.27%.14%.45%.1%.7% % %.1%.5%.5%.14%.14%.27%.3%.63%.11%.49%.22%.86%.45%.9% <.1%.8%.5%.26%.2%.91% <.1%.2%.15%.28%.15%.79%.8%.23%.12%.38%.24%.53%.7%.18%.17%.38%.38%.62%.5%.15%.17%.4%.38%.67%.2%.8%.6%.18%.33%.76% Problem olved at root node (6) b&b node avg max (7) (8) CPU time () avg max (9) (1)

28 Table 3: Computational Reult for the Problem P dep w C with Proceing Time Drawn from the Uniform Ditribution U[1, 1] Problem n m α (1) (2) (3) Integrality gap avg max (4) (5).19%.31%.21%.37%.59%.91%.1%.26% <.1% <.1%.9%.19% <.1%.25% <.1% <.1%.17%.53% <.1%.1% % %.6%.13%.4%.8%.11%.24%.13%.25%.5%.9%.1%.4%.6%.14%.2%.16% <.1% <.1% <.1% <.1% <.1% <.1%.1%.3%.8%.26%.15%.43%.18%.67%.48%.9%.4%.12%.3%.9%.11%.31% <.1%.4% <.1% <.1%.2%.7% <.1% <.1%.2%.8%.9%.25% Problem olved at root node (6) b&b node avg max (7) (8) CPU time () avg max (9) (1)

29 Table 4: Computational Reult for the Problem P dep w C with Proceing Time Drawn from the Uniform Ditribution U[1, 3] Problem n m α (1) (2) (3) Integrality gap avg max (4) (5).22%.5%.21%.62% <.1% <.1%.1%.1% <.1% <.1% <.1%.2% % % <.1% <.1% <.1% <.1% <.1% <.1%.1%.7% <.1% <.1%.21%.67%.4%.14%.27%.59% <.1%.8% <.1% <.1% <.1%.2%.2%.9% % %.7%.12%.1%.7%.3%.1%.1%.39%.19%.73%.8%.31%.23%.67% <.1%.8%.1%.1%.3%.22%.4%.31%.4%.24% <.1%.7% <.1% <.1%.5%.29%.13%.52% Problem olved at root node (6) b&b node avg max (7) (8) CPU time () avg max (9) (1)