A Branch-and-Price Algorithm for Parallel Machine Scheduling with Time Windows and Job Priorities

Transcription

1 A Branch-and-Price Agorithm for Parae Machine Scheduing with Time Windows and Job Priorities Jonathan F. Bard, 1 Siwate Rojanasoonthon 2 1 Graduate Program in Operations Research and Industria Engineering, 1 University Station, C2200, University of Texas, Austin, Texas TMB Asset Management Co. Ltd., 990 Rama IV Road, Siom, Bangrak, Bangkok 10500, Thaiand Received 22 November 2004; revised 9 August 2005; accepted 20 August 2005 DOI /nav Pubished onine 10 November 2005 in Wiey InterScience ( Abstract: This paper presents a branch-and-price agorithm for scheduing n jobs on m nonhomogeneous parae machines with mutipe time windows. An additiona feature of the probem is that each job fas into one of ρ priority casses and may require two operations. The objective is to maximize the weighted number of jobs schedued, where a job in a higher priority cass has infinitey more weight or vaue than a job in a ower priority cass. The methodoogy makes use of a greedy randomized adaptive search procedure (GRASP) to find feasibe soutions during impicit enumeration and a two-cyce eimination heuristic when soving the pricing subprobems. Extensive computationa resuts are presented based on data from an appication invoving the use of communications reay sateites. Many 100-job instances that were beieved to be beyond the capabiity of exact methods, were soved within minutes Wiey Periodicas, Inc. Nava Research Logistics 53: 24 44, Keywords: parae machine scheduing; time windows; branch and price; GRASP; job priorities 1. INTRODUCTION Machine scheduing probems are characterized by sequencing and timing decisions and arise in situations where activities compete for scarce resources. Depending on the environment (e.g., singe machine or parae machines), the job characteristics (e.g., independent or precedence constrained), and optimaity criteria (e.g., makespan, tota tardiness), it is possibe to define a wide range of probem subcasses. In this paper, we address a parae machine scheduing probem in which each job requires either one or two operations that can be performed on one or more parae machines. Once the operations are competed, the machine becomes free and the job eaves the system. The three common casses of parae machine environments incude identica machines, uniform machines, and unreated machines. In the first environment, a job can be processed in the same amount of time on any of the machines. In the second, each machine may process a particuar job at different speed, so a job woud require ess processing time on a fast machine compared with a sower one. In genera, each machine has an associated speed factor. In the third environment, the processing time of jobs can be Correspondence to: J.F. Bard (jbard@mai.utexas.edu) competey arbitrary. One machine might be faster on one job but sower on another (Cheng and Sin [23]). Our probem fas into the third category and originated from a rea-word need to schedue the use of antennas on the Tracking and Data Reay Sateite System (TDRSS) (Rojanasoonthon, Bard, and Reddy [21]). In this system, sateites (machines) in geosynchronous orbit are used as reay patforms for communication between ow orbiting spacecraft and ground terminas. Spacecraft can communicate with the TDRSS ony when they are within ine-of-sight range. Users (spacecraft), with unique priorities, request bocks of contiguous antenna time by specifying the antenna preference, the contacting priority, and the time period in which contact is possibe (time window). Each of the six TDRSS antennas can hande at most one request (job) at a time. In some situations, users may request periodic contacts or mutipe simutaneous contacts. The objective is to aocate the antennas over the panning period to meet the users requests in such a way that some function of the schedued requests is maximized with strict priority enforcement. This means that a subset of jobs from one priority cass can be schedued independenty of a other jobs with ower priorities. In other words, scheduing a job in higher priority cass benefits infinitey more than scheduing any number of ower priority jobs Wiey Periodicas, Inc.

2 Bard and Rojanasoonthon: Branch-and-Price Agorithm 25 The purpose of this paper is to show how Dantzig Wofe decomposition can be combined with inteigent heuristics to sove extremey difficut parae machine scheduing probems with time windows and job priorities. Up unti now, instances with ony a handfu of jobs coud be soved to optimaity with exact methods. The proposed branch-and-price agorithm provides an order of magnitude improvement over current techniques. In the TDRSS appication, the panning period spans 24 h, the number of priority casses is about 20, and the number of requests may be as high as 400 per day. The orbita system consists of two sateites, each having two singe access antennas and one phased array mutipe access antenna (i.e., 3 distinct machines). Because the time windows of a job are a function of the reative positions of the user spacecraft and the sateites, the antennas can be viewed as six distinct machines, athough two on each sateite are identica. Processing times vary from a few seconds to 10 min whie the time windows can range up to severa hours. In some cases, the processing time of a job spans its entire time window. For those jobs that can be processed by either type of antenna, the phased array is aways faster. In the next section, we give a genera statement of the probem, foowed in Section 3 by an outine of the reevant iterature. In Section 4, the mathematica formuation of the probem is presented. Severa unsuccessfu attempts to sove the corresponding mixed-integer inear program with a commercia code ed to the deveopment of the branch-and-price agorithm detaied in Section 5. Test resuts are highighted in Section 6, where it is seen that instances ranging in size from 20 jobs and six machines up to 100 jobs and two machines can be soved in ess than 10 min in amost a cases. 2. PROBLEM DEFINITION We wish to schedue n jobs on m nonhomogeneous parae machines without preemption to maximize the tota weighted sum of processed jobs. Each job fas into one of ρ priority casses and must be processed within its specified time window. An optima schedue for this version of the parae machine scheduing probem with time windows (PMSPTW) is one that satisfies the foowing conditions: 1. Each job is processed within its time window; otherwise, it is considered unschedued. (In some cases, a job may have two time windows and so may be processed within either.) 2. If a job is seected, then it is processed exacty once without interruption on one of its permissibe machines if it requires ony one service (operation); if it requires two services, two different machines may be used, time windows permitting. 3. Each machine processes at most one job at a time and may require a sequence-dependent setup between jobs. 4. The tota contribution of the schedued jobs is maximized. Using the standard three-fied notation (Graham et a. [14]), PMSPTW may be written as Rm r j, deadine, M j, s jk w j U j, where Rmdenotes unreated parae machines, r j is the reease date, deadine impies a hard due-date constraint for each job, M j represents the machine-eigibiity restrictions, s jk is the sequence-dependent setup time, and U j is one if job j is processed and 0 otherwise. The objective is to maximize the weighted number of schedued jobs. The weights, w j, must be assigned so that the optima soution refects the strict enforcement of priorities. The PMSPTW possesses many of the characteristics of the mutipe traveing saesman probem with time windows (m-tsptw) which, itsef, is a specia case of the vehice routing probem (VRP) with time windows (e.g., see Bard, Kontoravdis, and Yu [3], Desrochers, Desrosiers, and Soomon [8]). When soving VRPs, one of severa objectives may be specified: Minimize the tota routing cost, minimize the number of vehices required, or minimize the distanced traveed. The objective of the probem addressed here is to maximize a function of customers visited (or minimize a function of unvisited customers). In generic terms, the VRP is stated as a covering probem whie the PMSPTW is stated as a packing probem. 3. LITERATURE REVIEW Among the many survey papers on machine scheduing, Bazewicz, Dror and Wegarz [24] primariy focused on mathematica formuations. They covered singe machine scheduing with severa different objective functions and a variety of operationa constraints. For parae machine probems, they discussed formuations for both nonpreemptive and preemptive scheduing of identica, uniform, and unreated machines. Lawer et a. [18] provided an overview of sequencing and scheduing probems and reated soution techniques. The iterature on parae machine scheduing has expanded rapidy in recent years argey due to the deveopment of inteigent heuristics. Our review concentrates on the use of exact methods to sove probems that are cosey reated to the PMSPTW. For a fu survey, see Rojanasoonthon [20] Scheduing Jobs with Fixed Start and End Times on Parae Machines Arkin and Siverberg [1] addressed a parae machine scheduing probem where each job i had a positive weight,

3 26 Nava Research Logistics, Vo. 53 (2006) a fixed start time, a fixed end time (s i and t i, respectivey), and corresponding processing times t i s i. The objective was to maximize the tota vaue of the schedued jobs. Preemption was prohibited and jobs coud be processed at most once. This probem is simiar, but not identica, to ours. For the case where a the machines are identica, an O(n 2 og n) agorithm was presented based on formuating the probem as a minimum-cost fow network. Koen and Kroon [16] deat with a scheduing probem where each job had a fixed start time, a fixed end time, and a vaue representing its priority. Machines were avaiabe in specific time intervas (shifts), and a job coud be processed ony if the interva between the start and end time was a subinterva of a machine s shift. At most one job coud be processed at a time and preemptions were not aowed. The objective was to find a feasibe schedue for a jobs when such a schedue existed. Otherwise, the objective was to find a feasibe schedue in which the subset of processed jobs yieded the maximum tota vaue. Bouzinnd Emmons [5] studied interva scheduing where jobs with fixed start and end times were processed on identica parae machines. The objective was to find a feasibe schedue that maximized the number of competed jobs. When job weights were defined, the probem was to find a soution that maximized the sum of the weights of competed jobs. The unweighted and weighted versions were referred to as the maxima IS and maxima-weight IS, respectivey. For maxima IS, an O(nmax{og n, m}) agorithm was presented. For maxima-weight IS, the n-job probem was reformuated as a minima-cost fow probem with n+1 nodes and 2n arcs Scheduing Jobs with Time Windows and Job Priorities on Parae Machines Gabre [12] focused on the probem of scheduing n nonpreemptive jobs, each with a given processing time and an interva for the start time, on m identica parae machines. An additiona constraint imited each job to be processed ony on a subset of machines. Two variations of the probem were considered: (1) The fixed job scheduing probem where the interva of each job s start time was a point, and (2) the variabe job scheduing probem where the interva was a nonzero range. The majority of the work deat with deveoping upper and ower bounds for the fixed job case Orienteering Probem This probem is defined by a starting point, a terminus, and a number of intermediate ocations having associated scores. Aso incuded is a fixed panning horizon, which impies that it may be possibe to visit ony a subset of the ocations. The objective is to seect a feasibe subset of ocations such that the tota score coected achieves its maximum vaue. The orienteering probem is a generaization of the traveing saesman probem and sometimes is caed the traveing saesman s subtour probem, the maximum coection probem, or the seective traveing saesman probem (e.g., see Butt and Cavaier [6], Laporte and Marteo [17]). Appications incude finding a route for a speciaist who can provide support to ony a restricted number of customers (preferring those with higher saes potentia) due to his avaiabiity constraint, routing of oi tankers to service ships at various ocations, and scheduing the daiy operation of a stee roing mi (Baas [2], Goden, Levy, and Vohra [13]). The PMSPTW can be viewed of as a specia case a mutipe tour maximum coection orienteering probem with time windows. 4. MATHEMATICAL FORMULATION In this section, we define notation and present a constraintbased formuation for the PMSPTW when each job has one service. The mode is then extended to the case where a job may require two services. To begin, et n be the number of jobs and m the number of machines. Each job in the set J ={1,..., n} is a candidate to be processed without interruption on a set of parae, nonhomogeneous machines M ={1,..., m}. Associated with each job is a time window [a jk, b jk ] (for job j on machine k, the eariest time processing of j can begin on machine k is a jk and the atest time processing can begin is b jk ;0 a jk b jk ); a processing time p jk (the processing time of job j on machine k); and a set of machines on which job j can be processed M j M. Between jobs i and j there is a setup time, s ij k, if job i immediatey precedes job j on machine k. Aso, there are ρ priority casses such that J p J, p {1,..., ρ} is the set of jobs with priority p. Ifπ p is the contribution of a priority p job and β i is the reative benefit derived from processing job i with respect to the other jobs in its priority cass, then the tota contribution of priority p jobs is π p i J p β i. The strict enforcement of priorities i J q β i, p = 1,..., ρ 1. Note requires π p > ρ q=p+1 π q that some job j may have two time windows, (ajk 1, b1 jk ) and (ajk 2, b2 jk ), and may be processed in either one of them but not both. In the deveopments, we make use of the foowing additiona notation. Indices and sets: J 1 (J 2 ) = job sets with one (two) time windows, respectivey. F ij = set of machines on which both jobs i and j can be processed; F ij = M i M j, i, j J. 0 = dummy job that represents the starting and ending of the sequences on each machine, J 0 = J 0.

4 Bard and Rojanasoonthon: Branch-and-Price Agorithm 27 Parameters: P i = priority cass of job i J. n p = tota number of jobs with priority p(n p = J p ). β i = reative benefit associated with processing job i J with respect to the other jobs in P i. π p = contribution to the objective function of a priority p job, where cass 1 is the highest cass and ρ is the owest; π p > ρ q=p+1 π q i J q β i, p = 1,..., ρ 1, and π ρ = 1. µ ij = arge number; µ ij = max k Mi (b ik ) min k Mj (a jk ), i, j J ; i = j. Variabes: Fow variabes: xij k (i, j J 0 : i = j, k F ij ), where xij k = 1 if machine k processes job j immediatey after job i and xij k = 0 otherwise; note that xk ij = 0 for a the undefined indices. Time variabes: t j (j J)is the start time of job j. Time window variabes: yik 1 and y2 ik (i J 2, k M i ), where yik 1 = 1ifjobi is processed in its first time window on machine k and yik 1 = 0 otherwise; y2 ik = 1ifjobi is processed in its second time window on machine k and yik 2 = 0 otherwise. Mode 1: One Service Ony for Each Job Maximize π Pi β i subject to i J 0 \{j} i J xij k j J 0 \{i} k M x k ij x0j k j J i J 0 \{j} t i + k M(p ik + s ij k )x k ij j J 0 \{i} k M x k ij (1) 1, i J (2) 1, k M (3) x k ji = 0, j J 0, k M (4) ( 1 ) xij k µ ij t j, i, j J, i = j (5) k M y 1 ik + y2 ik k M i a ik j J 0 \{i} j J 0 \{i} x k ij = 0, i J 2, k M (6) x k ij t i, i J 1 (7) aik 1 y1 ik + aik 2 y2 ik t i, i J 2 (8) k M i k M i 1 1 k Mi xij k j J 0 \{i} k M i max k M i (b ik ) + k M i b ik j J 0 \{i} ( y 1 ik + yik) 2 max(b ik ) k M i x k ij t i, i J 1 (9) + k M i b 1 ik y1 ik + k M i b 2 ik y2 ik t i, i J 2 (10) xij k {0, 1} for i, j J 0, k M; y1 ik, y2 ik {0, 1} for i J 2, k M i ; and min k M j (a jk ) t j max k M j (b jk ) for j J (11) The objective function (1) maximizes the priority- and benefit-weighted number of schedued jobs. The weight vector π p must be specified so that, if job j is schedued, its contribution to the objective function exceeds the tota contribution of a ower cass jobs. Constraint (2) imits each job to be processed by at most one machine. It aso ensures that, if a job is schedued, it has no more than one successor, which might be the dummy job 0. Constraint (3) imits the number of initia jobs and, hence, the number of machines used, to at most m. In addition, it indirecty specifies that each machine can process at most one job at a time. Constraint (4) ensures that the same machine processes every job in a schedued sequence; i.e., if job j is assigned to machine k, both its predecessor and successor must be processed by machine k. This equaity is usuay referred to as the conservation of fow constraint. Inequaity (5) is the compatibiity requirements of consecutive jobs. It aso ensures subtour eimination because it forces the service initiation times, t i and t j, of any two consecutive jobs i and j in a sequence on the same machine to be stricty increasing. Therefore, the sequence cannot incude any of the previousy assigned jobs. Constraint (6) ensures that, if job i is assigned to machine k, then it is processed in ony one time window. Constraints (7) (10) enforce the time windows. The service initiation times must fa within two of these bounds depending on whether the job has one or two time windows. Note that t j sti has bounds of min k Mj (a jk ) and max k Mj (b jk ) even when it is not schedued. The reason derives from the definition of µ ij [see Constraint (5)], which is based on the argest and smaest vaues of the time windows of jobs i and j.

5 28 Nava Research Logistics, Vo. 53 (2006) From a practica point of view, the formuation needs to be repeatedy soved in the foowing way to avoid deaing with excessivey arge vaues of π p. Step 1. Let S 1 J 1. Sove the probem for jobs j S 1 ony and et the resuting objective function vaue be z 1. Set p = 2. Step 2. Let S p = S p 1 Jp. Sove the probem for jobs j S p ony with added constraints that require the objective function vaue for a jobs j S h (h = 1,..., p 1) to be at east as great as z h, where z h is the objective function vaue obtained when ony jobs j S h were considered. These constraints can be written as π Pi β i x k ij z h, i S h j J 0 \{i} k M h = 1,..., p 1 (12) Ca the resuting objective function vaue z p. Put p p + 1. Step 3. Repeat Step 2 unti p>ρ. If machines can be partitioned into subsets such that each subset is homogeneous, the number of constraints can be reduced by redefining k as an index of a machine type. To see this, et M k represent the set of machines of type k K and reca that M is the set of a machines. As such, M k K M k. The corresponding mixed-integer inear program is very simiar to (1) (11) except that M is repaced by K and constraint (3) is repaced by x0j k M k, k K (13) j J Mode 2: Two Services for Each Job In the TDRSS appication, some requests may be for two services. These do not necessariy have to be on the same machine and, in fact, must be schedued on different machines if they are required to be made simutaneousy. To extend the mode to account for this situation, et τ i be the offset time between the start of service 1 and service 2 for job i; τ can be negative, zero, or positive. For exampe, τ = 300 indicates that service 2 must start exacty 300 time units before the start of service 1, τ = 0 indicates that the two services must start simutaneousy, whie τ = 300 indicates that service 2 starts exacty 300 time units after the start of service 1. To hande this case, we spit each two-service job into two jobs, one for service 1 and the other for service 2. In addition, we reabe the job indices such that if job i represents service 1 of some two-service request, then job i + 1 must represent service 2 of that request. If i is service 1 of some two-service request, then the parameter β i can be interpreted as the benefit received when both jobs i and i + 1 are schedued. If ony one of the jobs can be schedued, the request is considered unsatisfied and no contribution is made to the objective function. Let S 1 and S 2 be the set of jobs representing service 1 and 2, respectivey. The augmented mode is Maximize π Pi β i x k ij (14) i J 1 j J 0 \{i} k M subject to constraints (2) (11), x(i 1)j k xij k = 0, i S 2 j J 0 \{i 1} k M j J 0 \{i} k M (15) t i 1 + τ i 1 = t i, i S 2 (16) Constraint (15) ensures that if a job requiring two services is seected, then both are schedued. Constraint (16) ensures the correct offset time between the start of service 1 and service BRANCH AND PRICE Branch and price (B&P) has been impemented successfuy for wide variety of vehice routing/scheduing probems (Desrochers, Desrosiers, and Soomon [8], Desrosiers et a. [10]) and machine scheduing probems (Chan, Murie, and Simchi-Levi [7]), Powe and Chen [19]) to name a few. In simpified terms, the genera procedure can be summarized as foows (Barnhart et a. [4]). The probem is first formuated as an integer program (IP) in which each restriction is represented by a set of agebraic constraints. Dantzig Wofe decomposition is then appied, resuting in a (restricted) set covering master probem that is soved with branch and bound. Bounds are computed at each node of the search tree by soving the inear reaxation of the set covering IPs; new coumns are generated by soving integra pricing subprobems. The benefit of the set covering reformuation is that, when the pricing subprobems are not totay unimoduar, the LP bounds are usuay much tighter than those obtained by soving the LP reaxation of the origina constraint-based formuation. The effectiveness of the approach, though, depends on the number and difficuty of the subprobems since they wi have to be soved many times. In the remainder of this section, we describe how the constraint-based mode defined by (1) (11) is transformed into a master probem and a series of subprobems one for each machine. We then discuss how each subprobem is

6 Bard and Rojanasoonthon: Branch-and-Price Agorithm 29 soved with our impementation of the generaized threshod agorithm. This is foowed by the detais of the branchand-price agorithm whose major components incude an intiaization scheme for the master probem, two branching rues for constructing the search tree, and a ower-bounding procedure. Severa exampes are aso given to iustrate the computations Set Packing Formuation After appying Dantzig Wofe decomposition to PMSPTW, a set packing rather than a set covering probem resuts. To see how the decomposition is carried out, define a partia schedue as a feasibe schedue on a singe machine formed by a subset of the avaiabe jobs in N. Accordingy, a feasibe soution to PMSPTW on m machines is simpy m partia schedues with no intersecting jobs. From the origina formuation of PMSPTW with one service ony for each job defined by (1) (11), the set of feasibe partia schedues on machine k is the set of points that satisfies (4) (11) for k fixed. By appying Dantzig Wofe to the origina formuation, the probem decomposes into a master probem consisting of (1) (3) and m subprobems with feasibe regions defined by (4) (11). Let k denote the set of a feasibe partia schedues for machine k. For each job j J and s k, et δjs k = 1if schedue s k processes job j, and 0 otherwise. Let ws k be the tota weight of the jobs processed on machine k using schedue s and define a binary variabe qs k = 1 if schedue s k is seected, and 0 otherwise. With this notation, the master probem MP is subject to Maximize δjs k qk s k M s k s k q k s k M s k w k s qk s (17) 1, j J (18) 1, k M (19) q k s {0, 1}, s k, k M (20) If θ j and φ k are the dua variabes associated with Eqs. (18) and (19), respectivey, then the reduced cost of a variabe qs k in MP is w s k = wk s θ j δjs k φ k (21) j J and the optimaity conditions are w k s j J θ j δ k js φ k 0, k M, s k (22) Note that, as previousy defined in Section 4, the priorityand benefit-weighted objective function coefficient of job j is π Pj β j. Therefore, the tota weight of the jobs processed on machine k by schedue s is w k s = j J π Pj β j δ k js (23) Using Eq. (23) and mutipying through by ( 1), the optimaity condition becomes (θ j π Pj β j )δjs k + φ k 0 (24) j J Because φ k is common to a partia feasibe schedues on k, it can be treated as a constant when soving the subprobems. Therefore, the subprobem associated with machine k is to find a feasibe partia schedue s k with minimum tota weight of jobs, where the weight of job j is (θ j π Pj β j ). Optimaity of the master probem is reached when the minimum tota weight partia schedue of each machine k M satisfies Eq. (24). To convert the variabes in the master probem back to the origina variabes, the foowing reationship can be used. x k ij = s k e s ij qk s (25) where eij s = 1 if both i and j are incuded in schedue s and j is the immediate successor of i; otherwise, eij s = 0. Because the origina formuation and the set packing reformuation are both vaid for PMSPTW, an optima soution to one is optima to the other. Nevertheess, when the reaxed version of the set packing mode is soved, it is ikey to produce a better bound, ca it z CG, than the LP reaxation, z LP. This foows from the fact that, to get z CG, we are optimizating over the convex hu of feasibe points of (4) (11) rather than over the LP reaxation of these constraints (see Wosey [22]). Working with the set packing mode is aso more ikey to yied feasibe soutions because ony convex combinations of feasibe sequences are permitted. During branch and bound, the definition of a feasibe partia schedue needs to be extended to account for both expicit and impicit job-ordering restrictions. The former are those imposed by the precedence constraints that are part of the origina probem, whie the atter are those imposed by the branching constraints The Subprobem: Shortest Path Probem with Time Windows and 2-Cyce Eimination Procedure The subprobem associated with machine k is to find a feasibe partia schedue whose tota job weights are

7 30 Nava Research Logistics, Vo. 53 (2006) minimized; i.e., Minimize ( ) θj π Pj β j x k ij + φ k (26) j J subject to (4) (11) (27) This is equivaent to the eementary shortest path probem with time windows (ESPPTW) which is NP-hard in the strong sense (Dror [11]) so the existence of a pseudo-poynomia agorithm is unikey. Nevertheess, the reaxed version, the shortest path probem with time windows (SPPTW), where each node may be visited more than once, is NP-hard in the ordinary sense. Athough cyces may exist in a soution to this probem, the time window constraints guarantee that a feasibe paths are finite. Powe and Chen [19] deveoped an O(D 3 ) abe correcting procedure for SPPTW that they caed the generaized threshod agorithm (GTA). Empiricay, it was shown to run faster than the O(D 2 ) abe setting agorithm of Desrochers and Soumis [9], where D = i V (b i a i + 1). Because of its better average performance, we chose GTA to sove the subprobems at each node of our B&P agorithm. According to Desrochers, Desrosiers and Soomon [8], the quaity of the LP bound of the set covering formuation of the master probem may be much improved by eiminating subprobem soutions that contain a 2-cyce. A path contains a 2-cyce if it revisits a node after visiting exacty one other node; i.e., at some point the path visits node x and then visits some other node y, and then immediatey returns to node x. To improve the quaity of the subprobem soutions, we have extended GTA to incude 2-cyce eimination ogic. The agorithm is described beow. Let G = (V, A) be a directed graph, where V = N {p, q} is the set of nodes such that p is the source node and q is the sink node, and A is the set of arcs. Each arc (i, j) A has a positive duration t ij, which is the time to trave from i to j, and a cost c ij. Each node i V has a time window [a i, b i ] within which a visit can start. The objective is to find the east cost path between p and q whie respecting the time windows of each node visited. In the agorithm, a abe (Ti α, Ci α ) is assigned to each node, where Ti α is the starting time of service at node i for path α and Ci α is the cost of the path. When comparing two abes, there are two types of ordering reationship: Dominance ordering and exicographic ordering. DEFINITION 1: (T 1, C 1 ) dominates (T 2, C 2 ), written as (T 1, C 1 ) (T 2, C 2 ), if and ony if T 1 T 2, C 1 C 2 and at east one of the inequaities is strict. DEFINITION 2: (T 1, C 1 ) is exicographicay ess than L (T 2, C 2 ), written as (T 1, C 1 ) < (T 2, C 2 ), if and ony if T 1 <T 2 ;ort 1 = T 2 and C 1 <C 2. DEFINITION 3: (T 1, C 1 ) is exicographicay ess than or equa to (T 2, C 2 ), denoted by (T 1, C 1 ) L (T 2, C 2 ), if and ony if (T 1, C 1 ) <(T L 2, C 2 ) or (T 1, C 1 ) = (T 2, C 2 ). A abe ( ˆT, Ĉ) Q is the exicographic minimum abe of Q if ( ˆT, Ĉ) L (T, C); (T, C) Q\{( ˆT, Ĉ)}. DEFINITION 4: Let Q be a set of abes. A abe (T, C) Q is efficient with respect to Q if no other abe in Q dominates it. A abe for a node is said to be efficient if it is efficient with respect to the set of abes at the node. The path associated with the efficient abe is the efficient path for the node. Define R i = α {(Ti α, Ci α )} be the set of efficient abes at node i. Given an efficient abe (T i, C i ), an efficient path is the shortest path arriving at node i at time no ater than T i. Let (T thres, C thres ) be the threshod abe. At each iteration of the agorithm, the foowing updating procedure is appied. (Tthres 0, C0 thres ) = (T basic, C basic ) (T n+1 thres, Cn+1 thres ) = (T thres n, Cn thres ) + (1, 1) + (T basic, C basic ) where such that (T basic, C basic ) = (T avg, C avg ) α 1 /α 2 ( α 2 = min 50, A ) V T avg = 1 A C avg = 1 A (i,j) A (i,j) A 5 α 1 10 Note that (T basic, C basic ) is computed from the characteristics of the network with an adjustment factor α 1. Based on empirica resuts, choosing a vaue of α 1 between 5 and 10 is recommended. t ij c ij 5.3. GTA Agorithm Let Q 1, Q 2, and Q 3 be the three queues used by the agorithm for maintaining a candidate ist L of abes. A queue is a first-in first-out data structure where new objects are paced at one end and removed from the other. The genera steps of the agorithm foow.

8 Bard and Rojanasoonthon: Branch-and-Price Agorithm 31 Step Construct Ɣ(i) ={j : (i, j) A} i V ; i.e., the set of nodes that immediatey foow node i in the graph G Initiaize the set of efficient abes for each node i V : R i ={(T 1 i 0.3. Initiaize the three queues: R p ={(T 1 p = a p, C 1 p = 0)} = a i, C 1 i = )}, i V \{p} Q 1 ={(T 1 p = a p, C 1 p = 0)} Q 2 = Q 3 = 0.4. Initiaize the threshod abe: (T thres, C thres ) = (T basic, C basic ) Step 1. Seect (Ti α, Ci α) from the bottom of Q 1. Step 2. (Treat the abe) For each j Ɣ(i) If Ti α + t ij b j and seecting j does not create a 2-cyce in the efficient path associated with (Ti α, Ci α), (Tj new, Cj new ) = (max{a j, Ti α + t ij }, Ci α + c ij ), Cj new ) is not dominated by any abe in R j, For a abes (T j, C j ) R j,if(tj new, Cj new ) (T j, C j ), remove (T j, C j ) from R j, Q 1, Q 2, and Q 3 R j = R j {(Tj new, Cj new )} 2.2. If (T new j If(Tj new, Cj new ) L (T thres, C thres ), add (Tj new, Cj new ) to the bottom of Q 2. Otherwise, add (Tj new, Cj new ) to the bottom of Q 3. Step 3. Uness Q 1 =, go to Step 1. Step 4. (Partition abes) If Q 2 =, transfer a abes in Q 2 to Q 1, Q 2 =. Otherwise, ifq 3 =, update the threshod abe and get (Tthres new, Cnew thres ). If there is no abe (T, C) Q 3 : (T, C) L (Tthres new, Cnew thres ), find (T exmin, C exmin ) Q 3 : (T exmin, C exmin ) L (T, C), (T, C) Q 3 \{(T exmin, C exmin )}, then update (Tthres new, Cnew thres ) = (T exmin, C exmin ) + (T basic, C basic ). Remove a abes (T, C) Q 3 : (T, C) L (Tthres new, Cthres new ) and put them in Q 1. Step 5. Uness a three queues are empty, go to Step 3. THEOREM 1: The modified generaized threshod agorithm with the 2-cyce excusion Step (2.1) yieds an optima soution to the SPPTW without 2-cyces. PROOF: The agorithm terminates with the optima soution as ong as no abes generated are dominated. Therefore, the proof wi show that the agorithm wi not choose a job for processing that wi ead to a dominated schedue. Assume that the partia schedue (a, b, c,..., i, j, k, j,,...) is feasibe, where (j, k, j)is the first 2-cyce in the schedue. Let (a, b, c,..., i, j, k, )be path P 1 and (a, b, c,..., i, k, j, ) be path P 2. Assume that the agorithm generated P 1 but (T P 2, C P 2 ) (T P 1, C P 1 ). Because the agorithm generated P 1, (T P 1 j, C P 1 j ) (T P 2 k, C P 2 k ). Aso because the set of jobs preceding j in P 1 and k in P 2 are the same, C P 1 j = C P 2 k. From Definition 4, it foows that T P 1, which impies j <T P 2 k that the finish time of k in P 1 is ess than the finish time of j in P 2 ; therefore, T P 1 T P 2. Now, because the set of jobs preceding is the same in both P 1 and P 2, C P 1 = C P 2. But from the assumption that (T P 2, C P 2 ) (T P 1, C P 1 ), wehave T P 1 >T P 2, which is a contradiction GTA Exampe An instance with three jobs is used to iustrate the 2-cyce eimination procedure. The data for the exampe are contained in Tabe 1, where a i is the eariest start time of i and b i is the atest start time of i. First, the basic abe is cacuated as ( , ). Let p and q be the dummy source node and sink node, respectivey. A directed graph is created when an arc exists between jobs i and j if job i can be processed immediatey before job j. The resuting adjacency ist of the graph is p : {1, 2, 3},1:{p},2:{p},3:{1, p}. ITERATION 1: Seect the abe (0, 0) of node p from Q 1 (Step 1). After treating the other nodes (Step 2), the resuting abes are as foows: Node 1 has a new abe (7, 0) Node 2 has a new abe (6, 0) Node 3 has a new abe (5, 0) Each of these abes is efficient and therefore paced in Q 3. ITERATION 2: Seect abe (5, 0) of node 3 from Q 1 (Step 1). Node 1 and p are treated (Step 2) and the new abes are (7, 1) and (10, 1), respectivey. Both are efficient and therefore, paced in Q 3 (Step 2.2). Tabe 1. Data for GTA exampe. i p i a i b i

9 32 Nava Research Logistics, Vo. 53 (2006) ITERATION 3: Seect abe (6, 0) of node 1 from Q 1 (Step 1). The ony efficient abe generated from node 1 is (10, 1) for node p (Step 2). Pace it in Q 3 (Step 2.2). ITERATION 4: Seect abe (7, 0) of node 1 from Q 1 (Step 1). One efficient abe is created which is (10, 1) for node p (Step 2). ITERATION 5: Q 1 is empty but Q 3 is not. Transfer a the abes from Q 3 to Q 1 (Step 4). The abes are (10, 1) at p from node 1, (10, 1) at p from node 2, (10, 1) at p from node 3, and (8, 1) at node 1 from node 3. Note that no abes were put in Q 2 so it has aways been empty. ITERATION 6: Treating a four abes at node p yieds no new abe (Step 2). Next, abe (8, 1) at node 1 is seected and treated (Step 2). A new abe (10, 2) is created at p and paced in Q 3 (Step 2.2). ITERATION 7: Labe (10, 2) at p is seected and treated (Step 2). No new abe is created. ITERATION 8: A queues are empty (Step 5). The fina path is (p,3,1,q) with the ength of 2. The foowing notation is used to describe the agorithm. N = set of partia probems sti open; i.e., set of a nodes currenty in the search tree. A = set of active nodes, A N. P = set of a possibe partia probems, where P i P is the i th partia probem in P. z best = objective function vaue of the incumbent soution. I = incumbent soution. s = search function; i.e., the procedure for seecting an active node for immediate exporation. u = upper bound. G(P i ) = objective function vaue of the GRASP soution for P i. ψ = number of iterations without any improvement in z best. = frequency of GRASP cas. The agorithm is summarized in Figure 1. In the foowing subsections, we further describe the branching schemes, how to appy the branching restrictions to the subprobems, soving the master probem, what information to store at each node, how a node is seected from the poo of active nodes, 5.5. Branch-and-Price Impementation The B&P agorithm starts by soving the restricted master probem; i.e., the inear programming reaxation of a restricted version of the origina set packing mode Eqs. (17) (20). In the deveopment and exporation of the search tree, if the soution at some node is not integra or contains a cyce, an active probem P i is chosen using the search function s. If the current upper bound u(p i ) is ess than or equa to the objective vaue of the incumbent soution, z best, the node is fathomed and the next active probem is seected. Otherwise, the probem is set up with the branching restriction of the current node P i. If no improvement in the incumbent is reaized in consecutive iterations, our greedy randomized adaptive search procedure (GRASP) is run in hopes of finding a new ower bound (Rojanasoonthon et a. [21]). The incumbent is updated if a better soution is found. Next, the upper bound is recacuated by soving the master probem using coumn generation. If the origina variabes xij k in the soution are a integra, we check to see whether there are any cyces in the schedues. If no cyces are found and the objective function vaue is better than the incumbent z best, then the incumbent is updated. In contrast, if the soution is not integra or some cyces exist and the upper bound shows that this node can sti ead to a better soution than the incumbent, then the node is partitioned and the set of active nodes A is updated. The two nodes created by partitioning inherit the upper bound of their parent. Figure 1. Overview of branch-and-price ogic for PMSPTW.

10 Bard and Rojanasoonthon: Branch-and-Price Agorithm 33 the data structure used, and, finay, the ower-bounding procedure Branching Schemes Both time window branching and SOS branching are used by the B&P agorithm. Each serves a unique purpose and has proven to be an effective compement of the other Time Window Branching In the current impementation, time window branching is appied when an active coumn in the master probem at node κ in the search tree (MP κ ) contains a cyce. An active coumn is one whose associated qs k variabe is nonzero; i.e., the coumn is incuded in the current soution of MP κ. Given a soution to the LP reaxation, LP κ, the agorithm searches for a job (1) with the highest priority weight, (2) that is processed the most times, and (3) that has the highest number of time windows (maximum of 2), in this order. Given such a job, its time window is partitioned using the foowing rues. If the job has two discrete time windows, create two descendent nodes, one for each time window. If the job j has ony one time window on machine k, (a jk, b jk ), check for the foowing two cases. Let σ j = b jk a jk be the sack in the time window of job j. Case I: the current time window has sack σ j = 1. Now create the eft chid with the time window [a jk, a jk ] and the right chid with the time window [b jk, b jk ]. Case II: the time window has sack greater than 1. Create the eft chid with the time window [a jk, a jk + σ j ] and the right chid with the time 2 window [a jk + σ j 2, b jk]. Figure 2. Overview of time window branching for PMSPTW. partition of the set J such that J 1 J 2 = J and J 1 J 2 =. SOS branching resuts in a eft descendent with the restriction j J 1 xk ij = 0, which impies j J 2 xk ij 1, and a right descendent with the restriction j J 2 xk ij = 0, which impies j J 1 xk ij 1. To keep these two nodes baanced, J is partitioned so that j J 1 xk ij = j J 2 xk ij and the number of free xij k s whose current vaue is zero is approximatey the same in J 1 and J 2. For the specia case in which the inequaity j xk ij 1 has ony one component, the eft node is created with xij k = 0 and the right node is created with xij k = 1 if either i or j is processed on machine k. SOS branching for job j on machine k is exhausted when the corresponding inequaity becomes empty. A more detaied discussion on how to enforce the branching requirements whie soving the subprobem is presented in Section 5.7. Figure 3 summarizes the procedure. Time window branching for job j on a machine is exhausted once the sack σ j = 0. The method is summarized in Figure Specia Ordered Set Branching Specia ordered set (SOS) branching is based on the current vaues of the components of Eq. (2), j J xk ij 1, which imits the number of predecessors of i to at most 1. If the soution to the LP reaxation is fractiona, the agorithm searches for j xk ij that is most fractiona; i.e., the agorithm searches for the i and k indices whose j xk ij is cosest to 0.5. Given such an i and k, two branches are created. Let J be a subset of J in which a variabes xij k are free, j J ; i.e, none are fixed to zero due to previous restrictions on the path from the root node to the current node. Let J 1 and J 2 be a Figure 3. Overview of SOS branching for PMSPTW.

11 34 Nava Research Logistics, Vo. 53 (2006) Figure 4. Exampe of subprobem network: (a) before appying x k ij = 1 requirement; (b) after appying xk ij = 1 requirement Appication of the Branching Restrictions This section discusses how the branching restrictions associated with a given master probem MP κ are enforced when soving the SPPTW subprobems. We must consider the foowing two cases Time Window Branching The time window branching restriction is appied in a straightforward manner by temporariy modifying the time window information of jobs. The agorithm coects the reevant branching restrictions of a predecessor nodes and updates the time windows of the associated jobs before soving SPPTW or running the GRASP Specia Ordered Set Branching When the SOS inequaity has more than one free variabe, the branching requirements are to set some xij k to 0. Setting xij k = 0 can be impemented directy in the subprobem by removing the corresponding arc in the network of machine k. As a resut, machine k wi not be abe to process job i immediatey before j. For the specia case where the SOS inequaity has ony one free variabe, the subprobem is modified in exacty the same manner as when the restriction is xij k = 0. The compementary restriction of setting xij k = 1 is more difficut to hande. In this case, job i must be the immediate predecessor of job j on machine k, a restriction that cannot be enforced in an efficient way in the subprobem. For exampe, soving two SPPTWs, one from the source node p to node i and the second from node j to the sink node q does not necessariy provide the shortest path from p to q through (i, j). Instead, we modify the subprobem network by removing a arcs (i, ) for = j and (, j) for = i. This is equivaent to setting xi k = 0 for = j and xj k = 0 for = j. Hence, if either i or j is processed by k, both have to be processed and i must immediatey precede j. Athough this impementation is ess restrictive than setting xij k = 1, it eads to a fu partition of the feasibe region. As the tree grows, a feasibe sequences of interest wi eventuay be generated. At some point in the enumeration the remaining arcs in a subprobem wi consist ony of those in a particuar sequence. Because no feasibe sequence is excuded by this branching scheme, an optima soution wi eventuay emerge. Its vaidity is formay stated beow. Figure 4(a) shows the subprobem network before appying the restriction xij k = 1 and Figure 4(b) shows the modified network after the restriction is appied. PROPOSITION 1: The branching restriction xij k = 1 imposed indirecty by removing a arcs (i, ) for = j and (, j)for = i from the subprobem network, eads to a compete partition of the feasibe region when couped with it compement xij k = 0. This scheme is equivaent to setting xi k = 0 for = j and xk j = 0 for = j. Note that when running the GRASP to find feasibe soutions at some node in the search tree, the SOS restrictions are enforced ony during phase 1 construction. Because the idea is to perturb the GRASP in an attempt to find a feasibe soution better than the incumbent, no checking is performed when running the oca search in phase 2 improvement; i.e., oca search moves are not required to respect the SOS branching restrictions Master Probem Initiaization and Coumn Modification At the root node of the search tree, a feasibe soution is first obtained by running the GRASP. The corresponding schedues, one per machine, are used to initiaize the LP associated with MP 1. At subsequent nodes, the initia coumns of MP κ are those inherited from its immediate predecessor; however, some of them may turn out to be infeasibe due to the new

12 Bard and Rojanasoonthon: Branch-and-Price Agorithm 35 branching restrictions. Rather than removing the entire coumn to regain feasibiity, a sequentia search is conducted to find and remove those jobs that are the cause of the probem. Note that a coumn wi never become infeasibe with respect to the time window constraints when jobs are removed. For time window branching, if the new restrictions at a node cause a coumn to be infeasibe, then the job j associated with the vioation is removed. This guarantees a feasibe sequence since the coumn was originay feasibe for the parent node and ony became infeasibe when the time window of j was changed. Removing j from the sequence aows the remaining jobs to shift back to their origina starting times. For SOS branching, the agorithm sequentiay removes a job j that is an immediate successor to i on k if the restriction is xij k = 0. The process is repeated unti no vioations exist. In the rare case in which a the jobs are removed from a coumns, an initia coumn is generated with a singe job whose weight is the argest among those that are not restricted from being the ast job in the sequence. Another parameter of the upper-bounding procedure is the number of coumns added to the master probem each time a subprobem is soved. This number is set to 2 in the impementation so at most 2 M coumns can be added at each node of the search tree Information Stored at Each Node Storing detaied information at a node can make the B&P agorithm more efficient, but it can become a disadvantage in terms of memory usage as the size of the search tree grows. To achieve a baance between speed and memory, we store the foowing information at each node κ. 1. Parent and chidren node information κ ={λ κ p, λκ, λ κ r }, where λκ p is the parent node, λκ is the eft chid, and λ κ r is the right chid. 2. The job sequence σk κ for each coumn associated with machine k in MP κ. 3. The compete LP associated with the node, LP κ. This is ony stored temporariy and removed once node κ is partitioned. 4. Branching restrictions B κ ={βtw κ, βκ SOS }, where βtw κ is the time window branching restriction [aik TW, bik TW ] for job i on machine k and βsos κ is the SOS branching restriction associated with j J xij k 1. To track the status of each variabe in this inequaity, we define a vector ν κ = (ν1 κ, νκ 2,..., νκ J ), where νκ j takes on three vaues: νj κ = 0ifxk ij is free, νκ j = 1ifxk ij = 0, and νκ j = 2 if xij k = The upper bound u κ. 6. Node identification κ, where κ = 1 indicates the root node. In summary form, the information stored at node κ is ( κ, σ κ, LP κ, B κ, u κ ). Note that the branching information at a node ony contains the restriction imposed by the partitioning decision at its parent node. To obtain the entire set of restrictions for a node, those of a its predecessors are coected. Let N κ be the set of a the nodes on the path that joins κ and the root node. The restrictions at κ are {B κ i : i N κ } Node Seection The best bound criterion is used to seect the next node for processing from the active node poo A. This method was chosen because it is computationay inexpensive and offers the prospect of finding good feasibe soutions quicky. It aso is more baanced and with fewer eves compared to, say, depth-first search Data Structures To do the best bound search efficienty during the node seection step, another data structure is used to store the set of active nodes A. These nodes are sorted according to their upper bound vaues and stored as an AVL tree (see Horowitz, Sahni, and Anderson-Freed [15]). The compexity of inserting, deeting, and finding a node in an AVL tree is O(og n). Retrieving the node associated with the maximum or minimum bound is aso O(og n) Lower-Bounding Procedure The GRASP is caed periodicay in an attempt to convert a fractiona soution at a node into a feasibe schedue. The ogic used for this purpose is based on the frequency with which the incumbent is updated during B&P. When no improvement in the incumbent is reaized in consecutive iterations, the GRASP is caed with the foowing parameters: an RCL size of 3, 1 iteration of phase 1, and 100 iterations of phase 2. Currenty, = 1 due to the ow computationa burden of the procedure Branch-and-Price Exampe A 20-job, 6-machine instance is used to demonstrate the B&P computations. Tabe 2 incudes a the input data, where p i is the processing time of job i, and a ik and b ik are the eariest and atest start times of job i on machine k, respectivey. If the a ik and b ik are not defined for job i, it means that i cannot be processed by machine k. Branch and price starts by soving the restricted master probem with an initia set of coumns provided by the GRASP, one for each machine. For the exampe, the six coumns in terms of job sequences are: machine 1: (5,13), machine 2: (7,4), machine 3:(8,15,16), machine 4:(6,2,9),

13 36 Nava Research Logistics, Vo. 53 (2006) Tabe 2. Input data for 20-job, 6-machine exampe. i p i a i1 b i1 a i2 b i2 a i3 b i3 a i4 b i4 a i5 b i5 a i6 b i Tabe 3. Computationa resuts for 20-job, 2-machine instances (B&P). LP Coumns Average Optima % gap generated coumns Node with first Tota GRASP GRASP Name a soution at root at root per node integer soution nodes improves cas ptw ptw ptw ptw ptw pttw pttw pttw pttw pttw rand rand rand rand rand sptw sptw sptw sptw sptw spttw spttw spttw spttw spttw Average a ptw, ong processing time and oose time windows; pttw, ong processing time and tight time windows; rand, random processing time and time windows; sptw, short processing time and oose time windows; spttw, short processing time and tight time windows.

14 Bard and Rojanasoonthon: Branch-and-Price Agorithm 37 Tabe 4. Computationa times for 20-job, 2-machine instances (s). Time to Time Time Average time Name Tota time best soution at root for GRASP per node ptw ptw ptw ptw ptw pttw pttw pttw pttw pttw rand rand rand rand rand sptw sptw sptw sptw sptw spttw spttw spttw spttw spttw Average machine 5: (20,12,18), and machine 6: (14,17,11). These resuts impy that z best = 16. ITERATION 1: At node 1 (root), a soution to the master probem MP 1 was found after 40 more coumns were added. The objective function vaue of the LP reaxation is u 1 = 17. Some of the variabes in the soution are fractiona but there are no cyces in any of the active coumns so SOS branching is invoked. The inequaity j J x1 2,j 1 is seected and two descendent nodes are created based on the fact that x2,4 1 = 0.5 is the ony nonzero variabe in the inequaity and a the other variabes are free. The eft chid, indexed as node 2, has the branching restriction x2,j 1 = 0 for a j {1, 2, 3, 5, 6, 7, 8, 9, 10} and the right chid, indexed as node 3, has the restriction x2,j 1 = 0 for a j {4, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}. Both nodes 2 and 3 are put in the active poo A. Finay, the GRASP is run but did not provide an improved feasibe soution. ITERATION 2: Examining the active poo A ={2, 3}, we see that u 2 = 17 and u 3 = 17 so there is no preference between nodes 2 and 3. The search function s(a) arbitrariy seected node 3. MP 3 is then constructed and soved to optimaity after adding 36 more coumns bringing the tota to 50. The rounded objective function of the LP reaxation is u 3 = 17. Athough the soution is fractiona, there are no cyces in any of the active coumns so SOS branching is invoked. The inequaity j J x5 12,j 1 is seected and two descendent nodes are created based on the fact that x12,18 6 = 0.5 is the ony reevant nonzero variabe. The eft chid, indexed as node 4, has the branching restriction x12,j 6 = 0 for a j {1, 2, 3, 4, 5, 6, 7, 8, 9} and the right chid, indexed as node 5, has the restriction x12,j 6 = 0 for a j {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}. Both node 4 and 5 are put in A and the GRASP is run producing a soution of 16, which is equa to z best.

15 38 Nava Research Logistics, Vo. 53 (2006) Tabe 5. Computationa resuts for 30-job, 6-machine instances (B&P). LP Coumns Average Optima % gap generated coumns Node with first Tota GRASP GRASP Name soution at root at root per node integer soution nodes improves cas ptw ptw ptw ptw ptw pttw pttw pttw pttw pttw rand rand rand rand rand sptw sptw sptw sptw sptw spttw spttw spttw spttw spttw Average b a Branch and price ran out of memory before producing an optima soution. Vaues in coumn 2 and 3 are based on the initia GRASP soution. b Average vaues based ony on instances that were soved. ITERATION 3: Node 5 is seected next by the search function, s(a) = node 5. MP 5 is constructed and soved to optimaity with 54 coumns. The objective function vaue is 17, a the variabes are integra, and no cyce exists in the soution. Therefore, the incumbent is updated by setting z best = 17 and the node is fathomed. The corresponding schedue is (2, 17, 13) on machine 1, (3, 1) on machine 2, (19, 15, 16) on machine 3, (6, 8, 9) on machine 4, (20, 18) on machine 5, and (14, 17, 11, 12) on machine 6. ITERATION 4: The remaining nodes in A are 2 and 4. Both have an upper bound of 17 so they are fathomed and the agorithm terminates. The fu probem was soved in 0.27 s. As we sha see in the next section, this type of performance is typica of probem instances with up to 50 jobs and 2 machines. 6. COMPUTATIONAL RESULTS The B&P agorithm was impemented in C++ using the gcc compier version 3.3 on a De Precision 530 workstation running SuSE inux 8.2. The De has dua 1.8 GHz Xeon processors with 1 GB of RAM but ony one processor was used in the computations. CPU times were coected using the C function cock() and CPLEX version 7.5 was used to sove the inear program master probems Data Sets For testing purposes, five data sets containing five instances each were randomy generated for different probem sizes using the same scheme described in Rojanasoonthon, Bard, and Reddy [21]. The data sets fe into the foowing categories. 1. Short processing time and oose time windows (sptw) 2. Short processing time and tight time windows (spttw)

16 Bard and Rojanasoonthon: Branch-and-Price Agorithm 39 Tabe 6. Computationa times for 30-job, 6-machine instances (s). Time to Time Time Average time Name Tota time best soution at root for GRASP per node ptw ptw ptw ptw a 0.42 ptw a 0.32 pttw pttw pttw pttw pttw rand rand rand rand rand sptw sptw sptw sptw a 0.5 sptw spttw spttw spttw spttw spttw Average b a Branch and price ran out of memory before producing an optima soution. b Average vaues based ony on instances that were soved. 3. Long processing time and oose time windows (ptw) 4. Long processing time and tight time windows (pttw) 5. Random instances (rand) Athough categories 2 and 4 best refect the rea data, our intent was to estabish the effectiveness of the agorithm over a wide range of parameter vaues. We were aso interested in determining how arge a probem the agorithm coud sove. The probem sizes investigated were: 20 jobs and 2 machines, 20 jobs and 6 machines, 30 jobs and 2 machines, 30 jobs and 6 machines, 50 jobs and 2 machines, 50 jobs and 3 machines, and 100 jobs and 2 machines. In a instances, the objective function coefficient β i wasfixedat1forai J, impying that each job was equay important within its priority cass. Depending on the scenario, other settings might be more appropriate. When the benefit of a job is proportiona to its ength, for exampe, β i shoud be defined accordingy. In the generation process, each job had a 10% chance of having two time windows, which were either oose or tight depending on the data set. The panning period was set at 3600 s and the processing times and time windows were scaed to refect the number of machines in a particuar instance. In genera terms, we first determined the tota time avaiabe for processing to be 3600 number of machines, and then divided the resut by the number of jobs. This gave us the time per job. We then created a range that covered the time per job and randomy seected a processing time based on a probabiity distribution derived from the origina data. As an exampe, if there were 20 jobs and 2 machines, then the tota time avaiabe = = 7200 s and the time per job = 360 s. The range used in this instance was 10 to 720 s. A simiar procedure was used to generate the time windows. In a data sets, the machines were assumed to be heterogeneous so the processing time and time window(s) of a job were a function of the machine. The computations were hated when a zero optimaity gap was reached or when an out of memory error message was reported Output The resuts are summarized in Tabes 3 to 8. Agorithmic performance for the 20-job, 2-machine instances is reported in Tabe 3. The coumn headings are (1) probem name, which

17 40 Nava Research Logistics, Vo. 53 (2006) Tabe 7. Computationa resuts for 100-job, 2-machine instances (B&P). LP Coumns Average Optima % gap generated coumns Node with first Tota GRASP GRASP Name soution at root at root per node integer soution nodes improves cas ptw ptw ptw ptw ptw pttw pttw pttw pttw pttw rand rand rand rand rand sptw sptw sptw sptw sptw spttw spttw spttw spttw spttw Average b a Branch and price ran out of memory before producing an optima soution. Vaues in coumn 2 and 3 are based on the initia GRASP soution. b Average vaues based ony on instances that were soved. reveas some of its characteristics, (2) optima objective function vaue, (3) percentage gap between the LP objective vaue at the root node and the optima objective vaue, (4) number of coumns generated at the root node to sove the LP reaxation, (5) average number of coumns generated per node, (6) index of the node at which the first integer soution was found either by GRASP or by the B&P agorithm, (7) tota number of nodes in the search tree, (8) number of times GRASP produced a new incumbent soution, and (9) tota number of times GRASP was caed after initiaization. With respect to heading (4), an entry of 1 indicates that the initia GRASP soution was optima and that B&P was abe to confirm this at the root node after soving the LP reaxation. Tabe 4 contains the computation times (seconds) for the various components of the agorithm for the 20-job, 2- machine instances. The coumn headings are (1) probem name, (2) tota CPU time, (3) time to find best feasibe (optima) soution, (4) time required to sove the LP reaxation at the root node, (5) time spent on GRASP, and (6) average time spent at each node. Tabes 5 to 8 present identica information for some of the more difficut data sets. Generay speaking, probems with oose time windows tended to be more difficut to sove. For the arger instances, this was evidenced by the size of the search trees associated with the data sets ptw and sptw compared to the others. In addition, a instances not soved beong to these two groups. Looking at Tabe 6, for exampe, we see that no soutions were found for three instances. In a cases, the computations terminated abrupty within 2 h due to memory imitations. Resuts for 50-job, 2-machine instances in which the jobs are eveny divided into two priority casses are summarized in Tabes 9 and 10. The optima vaues in this case are the weighted number of jobs schedued, where the weights refect the smaest possibe vaues that ensure strict priority enforcement. For the priority 1 jobs, the weights are 26; for priority 2 jobs the weights are 1. As can be seen, the resuts are simiar to those obtained for the singe priority data sets. This was to be expected because the difference between the two modes is ony in the definition of the cost coefficients. Tabe 11 summarizes the computationa resuts for a singe priority instances investigated. Probems with up to 100 jobs and 2 machines were soved to optimaity, most within a

18 Bard and Rojanasoonthon: Branch-and-Price Agorithm 41 Tabe 8. Computationa times for 100-job, 2-machine instances (s). Time to Time Time Average time Name Tota time best soution at root for GRASP per node ptw a ptw ptw a ptw ptw pttw pttw pttw pttw pttw rand rand rand rand rand sptw sptw sptw sptw sptw spttw spttw spttw spttw spttw Average b a Branch and price ran out of memory before producing an optima soution. b Average vaues based ony on instances that were soved. matter of seconds or minutes. Our experience with the B&P agorithm confirmed the effectiveness of both the upper- and ower-bounding procedures to imit the size of the search trees to no more than a handfu of nodes on average. For the 20- job, 2-machine data sets, a 25 instances were soved at the root node and, in a but one case (rand4), both the GRASP and the LP reaxation produced the same soutions (upper and ower bounds were equa). This observation is consistent with the resuts obtained by Bard, Kontorovdis, and Yu [3], who embedded a GRASP in a branch-and-cut procedure for the VRPTW. Using GRASP to find feasibe soutions at a node was instrumenta in soving 18 of the 175 instances. Some additiona testing showed that, without the GRASP, severa of those probems woud not have been soved. With regard to probem difficuty, increasing the number of machines appears to have a pronounced effect on agorithmic performance. Examining the resuts for the 30-job, 6-machine instances, for exampe, we see that 22 of 25 instances were soved to optimaity whie a 25 of the 50-job, 2-machine instances were soved. In addition, the size of the search tree grew much more rapidy with the number of machines than with the number of jobs. These observations can be partiay expained by the fact that the soution space is somewhat symmetric with respect to the definition of the data sets and that its size grows exponentiay with the number of machines. A second indicator of probem difficuty is the average number of coumns per node. For probems with oose time windows, many simiar coumns may be generated before the reaxed master probem is optimized. As is the case with standard branch and bound, branch and price strugges when it is not abe to find a near-optima soution within the first severa dozen nodes. For those instances in which the optima soution was never found, no nodes were ever fathomed due to integraity; i.e., the LP soutions to the reaxed master probem were never integra. 7. SUMMARY AND CONCLUSIONS Parae machine scheduing with time windows is an extremey difficut combinatoria optimization probem, posing an array of chaenges for the research community. For the variant addressed in this paper, the novety was the presence of priorities, two services, and doube time windows. Each of these features added a degree of compexity that required significant deveopmenta work.

19 42 Nava Research Logistics, Vo. 53 (2006) Tabe 9. Computationa resuts for 50-job, 2-machine, 2-priority instances (B&P). LP Coumns Average Optima % gap generated coumns Node with first Tota GRASP GRASP Name soution at root at root per node integer soution nodes improves cas ptw ptw ptw ptw ptw pttw pttw pttw pttw pttw rand rand rand rand rand sptw sptw sptw sptw sptw spttw spttw spttw spttw spttw Average Tabe 10. Computationa times for 50-job, 2-machine, 2-priority instances (s). Time to Time Time Average time Name Tota time best soution at root for GRASP per node ptw ptw ptw ptw ptw pttw pttw pttw pttw pttw rand rand rand rand rand sptw sptw sptw sptw sptw spttw spttw spttw spttw spttw Average

20 Bard and Rojanasoonthon: Branch-and-Price Agorithm 43 Tabe 11. Summary of branch-and-price computationa resuts. Jobs Probem size Machines Average a Probems Probems Probems Average a Average a Average a optima Average a soved to soved at soved ater tota coumns CPU soution LP-IP gap optimum (of 25) root by GRASP nodes per node time (s) a Average vaues based ony on instances that were soved. Initia efforts were aimed at trying to sove 20-job, 6- machine instances with CPLEX. Lack of success ed to the design of a B&P agorithm based on Dantzig Wofe decomposition. When couped with a GRASP for obtaining ower bounds, instances with up to 100 jobs and 2 machines were soved to optimaity, often within a few dozen seconds and within the first few nodes of the search tree. For the more difficut 100-job instances in which the agorithm faied to converge after a arge number of nodes were expored, ack of sufficient memory was aways the reason. In a ikeihood, though, no amount of memory woud have changed these resuts. Reativey speaking, we found that increasing the number of machines caused the probem to become much harder than increasing the number of jobs. Because the agorithm staed when it was not abe to sove a probem eary in the tree, the design of more effective branching schemes might represent one area of future research. Ony time window and SOS branching were considered here. Other possibiities for future research incude more efficient coumn management procedures and the identification of cuts that coud be incuded in the master probem. The question remains, though, of how to find good soutions to instances arising in the TDRSS appication that motivated this work. Reca that such instances incude up to 400 jobs, 6 machines, and 20 priority casses. One approach woud be to use a roing horizon framework. For each priority cass, the jobs woud be ordered by their eariest start times, and a manageabe number, say the first 30, woud be schedued with the agorithm. Given a soution, a fraction of the 30 jobs, say the first 20, woud be fixed. The remaining 10 pus the next 20 woud then be schedued and the first 20 again fixed. With suitabe modifications to account for the priority casses, a compete schedue coud be obtained by repeating the process a sufficient number of times. Athough the resuts woud most ikey be suboptima, the ength of the time windows reative to the 24-h panning horizon suggests that the oss in optimaity woud be more than offset by the expected gain in computationa efficiency. Difficut optimization probems often require this type of tradeoff. In summary, the B&P agorithm combined with the GRASP was abe to sove many instances that were beieved to be beyond the capabiities of exact methods. The success of the approach is mainy due to the consistency with which it can generate extremey tight upper and ower bounds throughout the search tree. REFERENCES [1] E.M. Arkin and E.B. Siverberg, Scheduing jobs with fixed start and end times, Discrete App Math 18 (1987), 1 8. [2] E. Baas, The prize coecting traveing saesman probem, Networks 19 (1989), [3] J.F. Bard, G. Kontoravdis, and G. Yu, A branch-and-cut procedure for the vehice routing probem with time windows. Transport Sci 36 (2002), [4] C. Barnhart, E.L. Johnson, G.L. Nemhauser, M.W.P. Savesbergh, and P.H. Vance, Branch-and-price: coumn generation for soving huge integer programs, Oper Res 46 (1998), [5] K.I. Bouzinnd H. Emmons, Interva scheduing on identica machines, J Goba Optimization 9 (1996), [6] S.E. Butt and T.M. Cavaier, A heuristic for the mutipe tour maximum coection probem, Comput Oper Res 21 (1994), [7] M.A. Chan, A. Murie, and D. Simchi-Levi, Parae machine scheduing, inear programming and parameter ist scheduing heuristics, Oper Res 46 (1998), [8] M. Desrochers, J. Desrosiers, and M.M. Soomon, A new optimization agorithm for the vehice routing probem with time windows, Oper Res 40 (1992), [9] M. Desrochers and F. Soumis, A generaized permanent abeing agorithm for the shortest path probem with time windows, INFOR 26 (1988), [10] J. Desrosiers, Y. Dumas, M.M. Soomon, and F. Soumis, Time constrained routing and scheduing, Handbook in operations research and management science: Network routing, Vo. 8, M.O. Ba, T.L. Magnanti, C.L. Monma, and G.L. Nemhauser (Editors), North-Hoand, New York, 1995, pp