State of art salbp


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State of art salbp

  1. 1. European Journal of Operational Research 168 (2006) 666–693 Invited Review State-of-the-art exact and heuristic solution procedures for simple assembly line balancing Armin Scholl *, Christian Becker Fakultat fur Wirtschaftswissenschaften, Friedrich-Schiller-Universitat Jena, Carl-Zeiß-Straße 3, D-07743 Jena, Germany ¨ ¨ ¨ Available online 11 September 2004Abstract The assembly line balancing problem arises and has to be solved when an assembly line has to be configured or rede-signed. It consists of distributing the total workload for manufacturing any unit of the product to be assembled amongthe work stations along the line. The so-called simple assembly line balancing problem (SALBP), a basic version of thegeneral problem, has attracted attention of researchers and practitioners of operations research for almost half acentury. In this paper, we give an up-to-date and comprehensive survey of SALBP research with a special emphasis on recentoutstanding and guiding contributions to the field.Ó 2004 Elsevier B.V. All rights reserved.Keywords: Assembly line balancing; Mass-production; Literature survey; Combinatorial optimization; Branch-and-bound; Heuristics1. Introduction systems, assembly line balancing problems are important tasks in medium-term production Assembly lines are flow-oriented production which are still typical in the industrial pro- An assembly line consists of (work) stationsduction of high quantity standardized commodi- k = 1, . . ., m arranged along a conveyor belt or aties and even gain importance in low volume similar mechanical material handling equipment.production of customized products. Among the The workpieces (jobs) are consecutively launcheddecision problems which arise in managing such down the line and are moved from station to sta- tion. At each station, certain operations are repeatedly performed regarding the cycle time * Corresponding author. Fax: +49 3641 943171. (maximum or average time available for each E-mail addresses: (A. Scholl), workcycle). The decision problem of (C. Becker). partitioning (balancing) the assembly work among0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.ejor.2004.07.022
  2. 2. A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693 667the stations with respect to some objective is scribed and surveyed in Becker and Scholl (thisknown as the assembly line balancing problem issue).(ALBP). Manufacturing a product on an assembly linerequires partitioning the total amount of work into 2. Simple assembly line balancing problema set of elementary operations named tasks (SALBP)V = {1, . . ., n}. Performing a task j takes a tasktime tj and requires certain equipment of machines Most of the research in assembly line balancingand/or skills of workers. Due to technological and has been devoted to modelling and solving the sim-organizational conditions precedence constraints ple assembly line balancing problem (SALBP). Thisbetween the tasks have to be observed. classical single-model problem contains the follow- These elements can be summarized and visual- ing main characteristics (cf. Baybars, 1986a;ized by a precedence graph. It contains a node Scholl, 1999, Chapter 2.2):for each task, node weights for the task timesand arcs for the precedence constraints. Fig. 1 • mass-production of one homogeneous product;shows a precedence graph with n = 10 tasks having • given production process;task times between 2 and 9 (time units). The prec- • paced line with fixed cycle time c;edence constraints for, e.g., task 5 express that its • deterministic (and integral) operation times tj;processing requires the tasks 1 and 4 (direct prede- • no assignment restrictions besides the prece-cessors) and 3 (indirect predecessor) be completed. dence constraints;The other way round, task 5 must be completed • serial line layout with m stations;before its (direct and indirect) successors 6, 8, 9, • all stations are equally equipped with respect toand 10 can be started. machines and workers; Any type of ALBP consists in finding a feasible • maximize the line efficiency E = tsum/(m Æ c) with Pnline balance, i.e., an assignment of each task to ex- total task time tsum ¼ j¼1 tj .actly one station such that the precedence con-straints and possibly further restrictions are Several problem versions arise from varying thefulfilled. The set Sk of tasks assigned to a station objective as shown in Table 1. SALBP-F is a feasi-k (=1, . . ., m) constitutes its station load, the cumu- P bility problem which is to establish whether or notlated task time tðS k Þ ¼ j2S k tj is called station a feasible line balance exists for a given combina-time. When a fixed common cycle time c is given, tion of m and c. SALBP-1 and SALBP-2 have aa line balance is feasible only if the station time dual relationship, because the first minimizes m gi-of neither station exceeds c. In case of t (Sk) < c, ven a fixed c, while the second minimizes c (maxi-the station k has an idle time of c À t (Sk) time mizes the production rate) given m. SALBP-E isunits in each cycle. the most general problem version maximizing the In the following, we consider the simple assem- line efficiency thereby simultaneously minimizingbly line balancing problem (SALBP). More gen- c and m considering their interrelationship.eral ALBP with additional characteristics like For the example of Fig. 1 with tsum = 48, a fea-cost objectives, paralleling of stations, equipment sible line balance for the SALBP-F instance withselection, and mixed-model production are de- Table 1 Versions of SALBP 6 6 4 Cycle time c 1 2 7 2 9 2 Given Minimize 8 9 10 5 5 4 5 No. m of stations 3 4 5 6 Given SALBP-F SALBP-2 Minimize SALBP-1 SALBP-E Fig. 1. Precedence graph.
  3. 3. 668 A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693cycle time c = 9 and m = 7 stations is given by the Due to the complexity of SALBP, formulating afollowing sequence of station loads (S1 = {1}, mathematical model and solving it by standardS2 = {2}, S3 = {3,7}, S4 = {4,5}, S5 = {6,8}, optimization software is no realistic choice forS6 = {9}, S7 = {10}). While no idle time occurs finding an optimal solution in case of real-worldin stations 3, 4, and 6, the other stations have idle instances of ALBP (cf. Section 3.6). Thus, we dotimes of 3, 3, 2, and 7 time units. without describing such models but refer to Scholl One of the optimal solutions to the SALBP-1 (1999, Chapter where different models areinstance with c = 11 is ({1,3}, {2,4}, {5,6}, {7,8}, presented. Moreover, see Ugurdag et al. (1997),{9,10}) with the minimal number of m* = 5 sta- Pinnoi and Wilhelm (1997a), Bockmayr and Pis-tions. An optimal SALBP-2 solution given m = 6 aruk (2001), and Peeters and Degraeve (this issue).stations is ({3,4}, {1,5}, {2,7}, {6,8}, {9}, {10}) with In what follows, we describe solution proceduresminimal cycle time c* = 10. Given a SALBP-E in- for the different versions of SALBP using the nota-stance with the number of stations restricted to 5 tions of Table 2. SALBP-1 is one of the classicalto 7 stations, the optimal line efficiency E* = 48/ optimization problems that has been studied inten-(5 Æ 11) = 0.87 is achieved by the SALBP-1 solu- sively since almost 50 years. The most relevanttion given above. developments are effective branch-and-bound pro- The versions of SALBP may be complemented cedures, including intelligent branching schemesby a secondary objective which consists of smooth- and a variety of bounding procedures (Section 3),ing station loads, i.e., equalizing the station times flexible priority rule based procedures and modern(vertical balancing, cf. Merengo et al., 1999; equal meta-heuristics like tabu search and genetic algo-piles problem, cf. Rekiek et al., 1999). One rithms and their problem-specific application (Sec-may minimize the smoothness index SX ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tion 5). Solution procedures for SALBP-2 and Pm 2 SALBP-E are mostly search methods based on k¼1 ðc À tðS k ÞÞ provided that the combination repeatedly solving SALBP-F instances by(m, c) is optimal with respect to line efficiency (see, SALBP-1 procedures (Section 4), others are di-e.g., Moodie and Young, 1965; Rachamadugu and rectly applied heuristics similar to those forTalbot, 1991). SALBP-1 (Section 5). Modifying assumptions of In our example with given combination SALBP leads to generalized problems, solution(m, c) = (5,11), the smoothness index favors the procedures for which are usually based on suchsolution ({3,4}, {1,5}, {2,7}, {6,8}, {9,10}) with for SALBP (cf. Becker and Scholl, this issue).SX = 4.36 versus the other optimal SALBP-1 solu-tions (including the one with SX = 5.39 specified Table 2for c = 11 above), because the stations of the first Notationssolution are more equally loaded. n Number of tasks; index j = 1, . . ., n Since SALBP-F is an NP-complete feasibility V Set of tasks; V = {1, . . ., n}problem, the optimization versions of SALBP, m Number of stations; index k = 1, . . ., mthat may be solved by iteratively examining several m* Optimal number of stationsinstances of SALBP-F, are NP-hard (cf. Wee and LM, UM Lower, upper bound on m c, c* Cycle time, optimal cycle timeMagazine, 1982; Scholl, 1999, Chapter LC, UC Lower, upper bound on cDespite the principal NP-hardness of the problem tj Task time of task j = 1, . . ., nclass, there are considerable differences concerning tmin, tmax, tsum Minimal, maximal, total task timethe difficulty of solving single instances. In order to pj Station requirement of task j; pj = tj/cdistinguish different classes with respect to the Pj ðP Ã Þ j Set of direct (all) predecessors of task j Fj ðF Ã Þ Set of direct (all) followers of task jcomplexity of instances, different measures like j Sk, t (Sk) Station load, station time of stationthe order strength or the task time variability ratio P k; tðS k Þ ¼ j2S k tj , k = 1, . . ., mhave been developed (for surveys see Scholl, 1999, Ik Idle time of station k; Ik = cÀt (Sk)Chapter; Driscoll and Thilakawardana, aj, nj Head, tail of task j2001). Ej, Lj Earliest, latest station for task j
  4. 4. A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693 669 Former surveys covering SALBP procedures 3.1.1. Bin packing boundsare given by Buxey et al. (1973), Baybars SALBP-1 reduces to the bin packing problem(1986a), Shtub and Dar-El (1989), Ghosh and BPP-1 (distributing a number of items to a mini-Gagnon (1989), Erel and Sarin (1998), Scholl mum number of fixed-capacity bins; cf. Martello(1999, Chapter 1) as well as Rekiek et al. (2002b). and Toth, 1990) by ignoring the precedence rela- tions. That is, BPP-1 is a relaxation of SALBP-1 and lower bounds for BPP-1 are valid lower3. Exact solution procedures for SALBP-1 bounds for SALBP-1, too. We specify only some of those bounds (further ones can be found in Many operations researchers have been en- Martello and Toth, 1990; Scholl et al., 1997; Fek-gaged with developing effective solution proce- ete and Schepers, 2001; Alvim et al., 2003).dures for exactly solving SALBP-1. This has Total capacity bound (Baybars, 1986a). Theresulted in about two dozens of procedures which most obvious bound LM1 follows from the ine-can be subdivided into branch and bound (B&B) quality m Æ c P tsum, i.e., the total time (total capac-procedures and dynamic programming (DP) ap- ity) available on the line must be no smaller thanproaches. In the following, we summarize the most the total work content:effective key developments without giving closed & ’statements of single procedures. Further (proce- Xn LM1 :¼ dtsum =ce ¼ pj : ð1Þdure oriented) surveys are given by Baybars j¼1(1986a), Ghosh and Gagnon (1989), and Scholl(1999, Chapter 4.1). Simple counting bounds (Johnson, 1988). A Due to their importance for solving SALBP-1 bound LM2 is obtained by counting the numberexactly, we begin with methods for computing of tasks j with tj > c/2 (i.e., pj > 1/2), because alllower bounds on the number of stations (though of those tasks have to be assigned to different sta-they are usually not used in DP approaches) tions. LM2 can be strengthened by adding half ofbefore different enumeration schemes and possibil- the number of tasks (rounded up to the next inte-ities for reducing the effort of enumeration are ger if necessary) with task time c/2, because two ofdiscussed. them may share one station. A further bound LM3 generalizes LM2 with re-3.1. Lower bounds spect to thirds of the cycle time and is computed by adding up weights which are determined as fol- SALBP-1 is to minimize the number m of sta- lows: All tasks j with pj > 2/3 are given the weighttions given the cycle time c. Due to the integrality 1, because they cannot be combined with any otherof m, only a relatively small number of objective of the tasks considered. Tasks with pj 2 (1/3, 2/3)values are possible. Therefore, it can be expected get the weight 1/2, because two of them may sharethat a large number of line balances is feasible a station. The tasks with pj = 1/3 and pj = 2/3 arefor each feasible value of m. That is, finding a line weighted with 1/3 and 2/3, respectively.balance given the optimal number m* of stations Extended bin packing bounds. The logic behind(solving the corresponding SALBP-F instance) is the simple counting bounds is combined and ex-often easier than proving a certain number of sta- tended in several manners in order to define moretions m < m* to be infeasible. As a consequence, sophisticated bound arguments (cf. Martello andknowing sharp lower bounds LM on m* is helpful ´ Toth, 1990; Labbe et al., 1991; Berger et al.,in solving SALBP-1 instances provided that the 1992; Scholl et al., 1997).bounds can be computed efficiently. We describe and categorize some of the most 3.1.2. One-machine scheduling boundimportant bound arguments for SALBP-1 (a more The one-machine scheduling bound LM4comprehensive survey is given by Scholl, 1999, (Johnson, 1988) relies on relaxing SALBP-1 to a one-Chapter, and Sprecher, 1999). machine scheduling problem. Tasks are interpreted
  5. 5. 670 A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693as jobs j = 1, . . ., n with ‘‘processing times’’ pj = tj/c information given by heads and tails provides thewhich have to be successively performed on a sin- overall bound LM4 = max{daj + pj + nje j j =gle machine. After processing job j, a certain 0, . . ., n + 1} which covers Z1 and Z2 for j = 0amount of time (called tail nj) is necessary before and j = n + 1, respectively.the job is terminated. The objective of the one-ma-chine problem is to find a sequence of all jobs for 3.1.3. Destructive improvement boundswhich the makespan, i.e., the time interval from the The bound arguments described so far may bestart of the first to the termination of the last job, classified as being constructive, because they findis minimized. An optimal solution for this problem (construct) optimal solutions to relaxed achieved by processing the jobs in order of non- A further class of bounds can be defined as beingincreasing tails. Let hh1, . . ., hn] be such a job order- destructive because they try to contradict trialing and tails nj been given, the minimum makespan objective values (numbers of stations) in order tois: successively improve on an initial lower bound. Therefore, this general approach for computingMS ¼ maxfph1 þ nh1 ; ph1 þ ph2 þ nh2 ; . . . ; ph1 lower bounds by a successive process of contra- þ Á Á Á þ phn þ nhn g: ð2Þ dicting potential bound values has been called destructive improvement by Klein and Scholl In the case of SALBP-1, a tail nj of a task j is a (1999).lower bound on the number of stations (not neces- Earliest and latest stations bound LM5 (Saltz-sarily integral) needed by its successors in F Ã . j man and Baybars, 1987; Scholl, 1999, p. 48). ByThus, any bound argument for SALBP-1 can be means of heads and tails (determined as describedapplied to the subproblem defined by F Ã for com- j for LM4), we may compute earliest and latest sta-puting the tails nj (without rounding off the bound tions to which a task j can be assigned in order notvalues). The computation requires a fictitious to exceed a certain number m of stations:source node j = 0 added to the precedence graph(with t0 = 0 and directed arcs to all original source Ej ¼ daj þ pj e and Lj ðmÞ ¼ m þ 1 À dpj þ nj enodes). The tails are recursively computed in the for j ¼ 1; . . . ; n: ð3Þreverse topological task numbering j = n, . . ., 0.Johnson (1988) proposes to apply LM1, LM2, The destructive improvement concept for com-and LM3 as well as (2) to the subproblem with puting a bound LM5 works as follows: We starttasks F Ã (due to the reverse computation the tails j with a valid lower bound m computed by any con-of those tasks are already known). The largest of structive method. If for any task j the relationthose (unrounded) bound values defines the tail Ej > Lj (m) holds, j is not assignable to any of thenj of a task j considered. Whenever the conditions m stations and m is increased by 1. This is repeatednj < dnje and pj + nj > dnje are fulfilled, task j can- until Ej 6 Lj (m) is true for all tasks. The currentnot share the first of the dnje stations required by value of m defines the bound LM5.its followers such that nj can be rounded up to dnje. SALBP-2 based bound LM6 (Scholl and Klein,This rounding process is the core logic of LM4 1997). Due to the ‘‘duality’’ of SALBP-1 andresponsible for the quality of the computed value. SALBP-2 (cf. Section 2), one may use lowerThe final value Z1 = dn0e defines a first lower bounds for SALBP-2 in order to compute boundsbound for SALBP-1. for SALBP-1 in the destructive improvement By applying the same logic in a forward ori- framework: For each trial value m, a lower boundented manner (using heads instead of tails), a sec- c (m) on the cycle time is computed by a boundond bound Z2 = dan+1e may be computed for a argument for SALBP-2 (see Section 4.1). If c (m)fictitious sink node n + 1 (cf. Scholl, 1999, p. 47). is larger than the given cycle time c, m is increasedA head aj is a lower bound on the number of sta- by 1, and the process is repeated. Otherwise, thetions required by all predecessors of task j and is current value of m provides a lower bound LM6computed analogously to tails. Combining the on the number of stations for SALBP-1.
  6. 6. A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693 671Table 3Heads and tails of tasksj 0 1 2 3 4 5 6 7 8 9 10 11pj 0 0.6 0.6 0.5 0.5 0.4 0.5 0.4 0.2 0.9 0.2 0aj 0 0 1 0 0.5 1.6 2 1.6 3.5 4 5 5.2nj 5.7 4.1 3 4 3.4 3 2.2 2.2 2 1 0 0 Interval bound LM7. A further destructive tsum = 48, tmax = 9, and tmin = 2 as well asbound is proposed by Fleszar and Hindi (2003) c = 10. The following bound values are obtained:who define station intervals [m1, m2] with1 6 m1 6 m2 6 m for a current lower bound value • LM1 = d48/10e = 5.m. If a solution with m stations exists, the tasks in • LM2 = 3 + d3/2e = 5, because there are threethe set {j 2 VjEj P m1 and Lj (m) 6 m2} must be tasks (1, 2, 9) exceeding half of the cycle timeassigned to at most m2 À m1 + 1 stations. If apply- and three further tasks (3, 4, 6) having task timeing any SALBP-1 bound to this subproblem re- c/2. LM3 = d1 Æ 1 + 1/2 Æ 7e = 5 due to p9 > 2/3veals that this is not possible for any interval and pj 2 (1/3, 2/3) for j 2 VÀ{8, 9, 10}.[m1, m2], m is contradicted as a feasible number • We do without explaining the computation ofof stations and is increased by 1. The process is re- LM4 in detail but give the heads and tails inpeated until no ‘‘destruction’’ of the current m is Table 3. We get LM4 = d5.9e = 6 (defined bypossible which then defines LM7. task 9). The rounding logic of LM4 becomes Bin packing applications of destructive obvious considering, e.g., task 2 whose headimprovement are also applicable to SALBP-1 as a2 is set to 1 because it is not possible to assignmentioned earlier (cf. Scholl et al., 1997; Alvim 2 to the same station as its direct predecessor 1et al., 2003). though the station requirement of task 1 is only Computational experiments show that among p1 = 0.6.the bound arguments presented so far LM4 is • LM5 is computed starting with, e.g., m =the most powerful one (cf. Scholl, 1999, p. 242). LM1 = 5. Table 4 contains the values for EjHowever, due to the diversity of problem instances and Lj (5). Due to Ej > Lj (5) for j = 8, 9, 10,to be solved applying all arguments available is these tasks cannot be assigned to any of the 5recommendable. stations, and m is increased to 6. Since the con- A further class of bounds is based on solving dition Lj (6) P Ej holds for all tasks jrelaxations of mathematical model formulations (Lj (6) = Lj (5) + 1; cf. Table 4), LM5 gets thefor SALBP-1 by means of standard optimization value 6. Beyond the bound computation, wepackages. Peeters and Degraeve (this issue) present get the information that tasks 8, 9, and 10 musta Dantzig-Wolfe type formulation of SALBP-1 the be assigned to station 4, 5, and 6, respectively,LP-relaxation of which is solved using column in order to find a solution with no more thangeneration combined with subgradient optimiza- 6 stations.tion. Computational experiments show that theresulting bound values are close to optimality.However, this must be paid by a high computa-tional effort which is often higher than for exactly Table 4solving the unrelaxed problem. Earliest and latest stations j 1 2 3 4 5 6 7 8 9 103.1.4. Example for bound computation Ej 1 2 1 1 2 3 2 4 5 6 For illustration purposes, we reconsider our Lj (5) 1 2 1 2 2 3 3 3 4 5example of Fig. 1 which is characterized by Lj (6) 2 3 2 3 3 4 4 4 5 6
  7. 7. 672 A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693• For LM6 we may also start with m = 5. A sim- assignable tasks is built for a station k, before ple bound for SALBP-2 consists of considering the next one k + 1 is considered. the m + 1 largest tasks. Provided that they are- • Task-oriented assignment. Procedures which are sorted in non-increasing order of task times, task-oriented iteratively select a single available c (m) = tm + tm + 1 is a lower bound on the cycle task and assign it to a station, to which it is time given m stations. In our example, we get assignable. c (5) = 5 + 5 = 10 = c and, hence, LM6 = 5.• Starting the computation of LM7 with m = 5, one finds a contradiction considering the sub- 3.3. Dominance rules problem with station interval [1, 2] and the task set {1, . . ., 5} due to LM1 = d26/10e = 3 > 2. No Most procedures use dominance rules to reduce further contradiction arises for m = 6 such that the enumeration effort. Such a rule compares par- LM7 = 6 is determined. tial solutions (and corresponding residual prob- lems) in order to find dominance relationships which allow for excluding (dominated) partial3.2. Construction schemes solutions without explicitly completing them to full solutions. The concept of dominance is based Most exact algorithms are based on enumerat- on the fact that all partial solutions which cannoting feasible solutions by successively assigning be completed to a solution with minimal numbertasks or subsets of tasks to stations. Therefore, of stations and even all of those but one whichthese algorithms consider partial solutions contain- can be completed to an optimal solution coulding a number of already assigned tasks and (par- be eliminated from further consideration. Since ittial) station loads Sk (k = 1, 2, . . .), while the is usually not known in advance into which cate-remaining tasks and station idle times constitute gory the partial solutions fall, a dominance rela-a residual problem. tionship between two partial solutions P1 and P2 For explaining solution procedures we need is established whenever it can be proved that thesome further terms and definitions: optimal completion of P1 does not require more stations than that of P2. Whenever such a domi-• A yet unassigned task j is available if all preced- nance relationship is found, P2 can be excluded ing tasks h 2 P Ã have already been assigned to a j from further consideration. station. Due to their importance in developing effec-• An available task is assignable to a station k if tive procedures for SALBP-1 we explain some all preceding tasks have already been assigned of the most powerful dominance rules (further- to station k or an earlier station 1, . . ., k À 1 more, see Scholl, 1999, Chapter 4.1, and Sprecher, and the current idle time is sufficient. 1999).• A station load Sk is maximal if no available task • Maximum load rule (Jackson, 1956). It excludes is assignable to station k. each partial solution P2 which contains one or• A subset of tasks S V is feasible if S also more completed but non-maximal station loads, contains the predecessors of each task included. because there exists at least one another partial solution P1 with the same number of maximally Almost every solution procedure for SALBP-1 loaded stations, where additional tasks are as-is based on either of the two following construction signed. In such case the completion of P1 doesschemes, which define the principle way of assign- not require more stations than that of P2. Theing tasks to stations: maximum load rule does not require explicit com- parisons between several partial solutions, because• Station-oriented assignment. In any step of a sta- it is sufficient to examine the maximality of station tion-oriented procedure a complete load of loads while these are constructed (Johnson, 1988).
  8. 8. A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693 673 Example. Given a cycle time of 10 and the prec- m1 6 m2 stations, P2 is dominated by P1 andedence graph of Fig. 1, S1 = {1} and S1 = {3, 4} excluded.are maximal loads for station 1, while S1 = {3} is In order to apply the feasible set dominancenon-maximal. Provided that S1 = {3, 4} has al- rule, it is necessary to store all feasible subsets ofready been built, S2 = {1, 5} is the only maximal tasks already enumerated together with their min-load available for station 2. imum station requirements in an efficient manner• Jackson dominance rule (Jackson, 1956, strength- with respect to storage space and retrieval speed.ened by Scholl and Klein, 1997). This rule is ap- For this purpose several approaches are available:plied to reduce the number of alternative loads Held et al. (1963) propose a recursive addressingwhich have to be considered for a certain station algorithm which assigns a unique address to eachk. It is based on potential dominance. feasible subset using the address space compactly. A task h potentially dominates a task j that is not Because this addressing algorithm is very time-related to h by precedence, if F j F Ã and tj 6 th h consuming, Schrage and Baker (1978) develop ahold. If tj = th and Fj = Fh, the lower-numbered simpler procedure based on task labels which havetask is defined to be dominating. to be summed up to quickly compute the address The Jackson rule excludes a maximal station of a subset. However, this addressing is not com-load Sk from consideration if at least one task pact and therefore requires a lot of storage space.j 2 Sk can be replaced by an available not yet as- In order to overcome this disadvantage, Nouriesigned task h which potentially dominates j, and and Venta (1991) define a tree structure for effi-the resulting station time t (Sk) À tj + th does not ciently storing all feasible subsets which outper-exceed the cycle time c. forms the previous approaches. A similar This rule utilizes the fact that all successors of approach is proposed by Sprecher (1999). Alltask j are successors of h as well and cannot start those methods are restricted to examining thebefore h is finished. Hence, the sequence of j dominance relation for the special case T2 = T1,and h is not important for the successors of j such the more general case T2 T1 is examined in athat replacing h by j does not exclude later task solution procedure for the related resource con-assignments which would be possible when j re- strained scheduling problem (cf. Klein and Scholl,mained. The condition tj 6 th guarantees that the 2000).station utilization will not decrease if h replaces j Example. The partial solution P1 = ({3, 4},in Sk. {1, 5}, {2, 7}) assigns the feasible set T1 = {1, . . ., Example. In Fig. 1, the following pairs of poten- 5, 7} to m1 = 3 stations. Another partial solutiontial dominance (h, j) are present: (1, 4), (2, 6), (3, 7), P2 = ({1}, {3, 4}, {2, 5}) corresponds to(4, 7), (5, 7), (6, 7). This results, e.g., in the par- T2 = {1, . . ., 5} and requires m2 = 3 stations, too.tial solution P1 = ({1}, {3, 4}, {2, 5}) dominating Thus, P2 is dominated by P1.P2 = ({1}, {3, 4}, {2, 7}) and also P3 = ({1}, {3, 4}, • Station ordering rules. Because finding a single{5, 6}). optimal solution is sufficient, it is not necessary• Feasible set dominance rule (Schrage and Baker, to consider different partial solutions which have1978; Nourie and Venta, 1991; Scholl and Klein, the same station loads but differ only in the se-1999). It extends the logic of the maximal load rule quence these loads are assigned to stations. Severalto feasible subsets of tasks. Provided that a partial rules which attempt to find respective dominancesolution P2 is constructed in forward direction relationships in an efficient manner are proposed(from early to late stations) and the maximum load by Johnson (1988), Scholl and Klein (1997) as wellrule is applied to all m2 stations already loaded, the as Sprecher (1999).corresponding feasible subset T2 of tasks is as- Example. The partial solutions P1 = ({1},signed to these m2 stations. Whenever another par- {2, 7}, {3, 4}) and P2 = ({1}, {3, 4}, {2, 7}) are iden-tial solution P1 has already been considered which tical except for the sequence of the second andassigns a feasible task set T1 (with T2 T1) to third load. Thus, one can be excluded.
  9. 9. 674 A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693 Computational experience reveals that applying • Additional precedence relations (Fleszar and Hin-(computationally inexpensive) dominance rules is di, 2003). Consider two tasks h and j which are notvery important for the computation times of exact related by precedence. If m (j, h) = daj + pj +procedures (cf. Johnson, 1988; Nourie and Venta, ph + nhe UM À 1, an additional arc (h, j) is1991, 1996; Scholl and Klein, 1997; Scholl, 1999, p. added to the precedence graph, because perform-242). ing j before h cannot lead to an improved solution. If m (h, j) UM À 1 is additionally true, no solu- tion with less than UM stations exists. That is,3.4. Reduction rules examining precedence relations in both directions is a means that may be applied within destructive Besides dominance rules several procedures use improvement bounds (cf. Klein and Scholl, 1999).reduction rules to lower the computational effort Example. Again starting with UM = 7, we canfor solving SALBP-1. Such rules try to modify add the arc (3, 2) due to m (2, 3) = d6.1e 6 andproblem data, i.e., task times or precedence rela- m (3, 2) = d4.1e 6 (cf. Table 3). Trying to contra-tions, such that the number of solution alternatives dict m = 5 or, equivalently, trying to improve oncan be reduced or bounds may get improved val- UM = 6, a contradiction is found consideringues. However, the reduced problem must have at the tasks 1 and 3 because of m (1,3) 5 andleast one optimal solution in common with the m (3,1) 5.original problem. • Task conjoining rule (Fleszar and Hindi, 2003). If• Task time incrementing rule (Johnson, 1988; all (potential) maximal loads containing a particu-Scholl, 1999, p. 117). The task times of all tasks lar task j also contain another task h, both nodeswhich cannot share a station with any other task can be merged into one node thereby uniting theare set to c. A simple sufficient condition for incre- sets of predecessors and successors.menting the time tj of a task j to c is tj + tmin c. Example. Provided that the arc (3, 2) has beenImproved conditions are given by Talbot and Pat- added as described in the rule above, the only max-terson (1984), Sprecher (1999), and Fleszar and imal load containing task 3 is {3, 4} such that bothHindi (2003). tasks can be merged. Example. In the instance given by Fig. 1 and Fleszar and Hindi (2003) define further rulesc = 10, we find that the time of task 9 can be in- which may allow for decomposing the prece-creased to 10. Inspecting the precedence structure dence graph and improving on lower bounds byreveals that the time of task 10 can be set to 10, successively adding dummy tasks and increasingtoo. Doing so, immediately leads to the improved task times. Their computational experimentsLM1 = d57/10e = 6. show that reduction rules have a great potential• Prefixing (Scholl and Klein, 1999). Whenever it for improving solutions found by any heuristic ifbecomes clear that a task can only be assigned to they are applied prior to the heuristic. Further-one specific station in order to find an improved more, they allow for computing sharper lowersolution (or one with a predefined number of sta- bounds.tions) this task can be assigned (prefixed) to therespective station. Provided that UM denotes a va-lid upper bound on the number of stations, 3.5. Dynamic programming proceduresEj = Lj (UM À 1) is a sufficient condition which al-lows for prefixing task j to station Ej because an As already mentioned, exact solution proce-improved solution must not have more then dures for SALBP-1 are based either on dynamicUM À 1 stations. programming or branch and bound. Example. Let us assume that a feasible solution The first dynamic programming (DP) procedurewith UM = 7 stations is already known. Then the developed by Jackson (1956) and modified by Heldtasks 8, 9, and 10 can be prefixed to the stations 4, et al. (1963) subdivides the solution process in5, and 6, respectively (cf. Table 4). stages which correspond to stations, i.e., it follows
  10. 10. A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693 675the station-oriented construction scheme. States Table 5are given by the feasible subsets of tasks already Possible componentsassigned at a given stage. The optimal solution LMi Lower bound LMi (i = i, . . ., 6)is searched for stage-by-stage in a forward recur- ML Maximum load rule FS Feasible set dominance rulesion, i.e., the procedure enumerates all first station JD Jackson dominance ruleloads before it considers all additional second sta- SO Station ordering dom. ruletion loads and so on (breadth-first search). In or- TI Task time incrementing ruleder to restrict the enumeration effort, only PF Prefixingmaximal station loads not being dominated by RN Task renumbering HS Additional heuristicthe Jackson dominance rule (cf. Section 3.3) are FW/BW Constructing solutions inbuilt. forward/backward direction If in JacksonÕs approach the states are repre-sented by nodes and the station loads, by arcswhich are weighted with the corresponding stationidle times, SALBP-1 is transformed to an equiva- 3.6.1. Search strategieslent shortest path problem. Each path in this graph Besides the construction scheme (station- orcorresponds to a feasible solution and each short- task-oriented), the BB procedures differ with re-est path to an optimal solution of SALBP-1 (cf. spect to search strategy.Gutjahr and Nemhauser, 1964). Since the number • In a depth-first search (DFS), a single branch ofof nodes grows exponentially with the number of the tree is developed until a leaf node is reachedtasks, it is generally not possible to construct the or the current node is fathomed. On its way backcomplete graph. Therefore, Easton et al. (1989) to the root, the search follows the first possibleincorporate lower and upper bounds as well as alternative branch, i.e., each node is completelydominance rules to reduce the size of the graph. developed before its ancestor nodes are revisited.Furthermore, their procedure is designed to be ap- Most commonly, DFS is organized as laser searchplied in forward and backward direction, (LS), i.e., in each node of the current branch, onlyrespectively. one descending node is built and developed at a Further DP procedures apply task-oriented time. Alternatively, all descending nodes may beconstruction schemes. The procedures of Held generated and sorted by a priority rule (e.g.,et al. (1963) and Schrage and Baker (1978) per- according to non-decreasing lower bound values).form a forward recursion and store states together The one with the highest priority is branched first.with their minimum station requirement in a The remaining nodes are stored in a candidate listmemory using different addressing schemes (see and are developed according to the priority orderfeasible set dominance rule; cf. Section 3.3). at each revisit of the node. This derivative is calledHowever, both procedures have large memory DFS with complete node development (DFSC).requirements because they do not strictly work • A minimal lower bound strategy (MLB) alwaysstage-by-stage. The procedures of Lawler (1979) chooses a not yet developed node which has theand Kao and Queyranne (1982) improve on this minimum value of a lower bound (see Sectiondrawback and get by with considerably less 3.1) from a candidate list. This node is completelymemory. branched by constructing all descending nodes. The latter are stored in the list which is sorted3.6. Branch and bound procedures according to non-decreasing bound values, while the current node is dropped. At the beginning, In the last decades, a considerable number of the list only contains the root node.branch and bound (BB) approaches have been Irrespective of the strategy used, a node is fath-proposed in the literature. Table 6 gives a survey omed when its (local) lower bound (computedof these procedures and characterizes them briefly explicitly or inherited from the father node) isby means of the notation in Table 5. not smaller than the global upper bound UM
  11. 11. 676 A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693(number of stations in the currently best known earlier. In order to reduce this negative effect, mostsolution), because it does not have an optimal procedures using LS renumber the tasks concern-solution with less than UM stations. ing some priority rule such that the first subprob- While DFS considers (and stores) only a small lems generated contain promising taskpart of the enumeration tree at a time, MLB has assignments which have the potential of being partto manage a large number of not yet developed of an improved solution.nodes considerably restricting its applicability to DFSC explicitly tries to avoid the drawback ofSALBP-1 and other combinatorial optimization LS by generating all subproblems of a currentproblems. Therefore, the most effective algorithms node before they are examined in order of non-for SALBP-1 presented recently are based on some decreasing lower bound values. So, promisingtype of DFS strategy. partial solutions are considered first, but all sub- However, both extreme versions of DFS dis- problems have to be generated irrespective of thecussed above have potential disadvantages, too: difficulty in finding an improved UM value. ThatLS considers the different subproblems in an (arbi- is, even in case of finding the optimal solution totrary) sequence induced by the task labeling taking a node in the first branch followed, all subprob-the risk to examine large subtrees before promising lems have been generated of the enumeration tree are entered and im- The so-called local lower bound method (LLBM)proved solutions (with decreased UM) are found. aims at finding a reasonable compromise betweenThat is, LS tends to spend a lot of time in investi- these extreme positions (cf. Scholl and Klein,gating many solutions which could have been fath- 1997). The subproblems of a current node are sub-omed if an improved UM value was determined divided into two classes: class I contains all sub-Table 6Survey of branch and bound procedures for SALBP-1Station-oriented Task-orientedDFS• Mertens (1967): FW, LS, LM1, RN • Talbot and Patterson (1984), based on additive• Johnson (1981): FW, DFSC, LM1, modified LM2,3, ML algorithm of Balas (1965): FW, LS, reduction• Betts and Mahmoud (1989): similar to Johnson (1981), tests based on LM1, TI enumeration based on Hoffmann (1963) • Saltzman and Baybars (1987): parallel FW-BW,• Berger et al. (1992): similar to Hackman et al. (1989) LS, RN, LM1,5, LM2,3 modified like Johnson restricted to out-tree type precedence graphs, (1981), TI, HS LM1 and extended bin packing bound • FABLE, Johnson (1988, 1993): FW, LS, RN,• EUREKA, Hoffmann (1992, 1993): FW-BW (successively), LM1-4, ML, JD, FS (restricted), TI, SO LS, LM1; lower bound method: repeated application for • Nourie and Venta (1991, 1996): branching like increasing trial numbers of stations; application of heuristic FABLE, FW, LS, RN, LM1, ML, FS, HS of Hoffmann (1963) if no feasible solution is found • Sprecher (1999): branching like FABLE, FW-BW (static direction switching), LS, RN, LM1-6, ML, JD, FS, TI, SO • Sprecher (2003): distributed version of Sprecher (1999) for parallel computersMLB• Jaeschke (1964): FW, LM1, ML• Van Assche and Herroelen (1979): FW, LM1, ML, FS (restricted)• Hackman et al. (1989): FW, LM1, ML, FS (restricted), HSLLBM• SALOME-1, Scholl and Klein (1997, 1999): FW-BW (bidirectional), RN, LM1-6, ML, JD, FS (Nourie and Venta, 1991), PF, TI, SO• Bock and Rosenberg (1998), Bock (2000): distributed version of SALOME-1
  12. 12. A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693 677 6 {3,4} 6 6 7 {9} 7 {5,6} {8} {10} 2 3 4 5 6 7 7} 6 {2 , UM=7 6 {6,7} 6 {8} 7 1 } ,5 } 9 10 11 16 {1 {3 {2 ,4 } 6 6 {5,6} 6 {8} 7 LM1=5 {2,7} 8 12 13 14 19 0 {5 7 {3 ,6} 6 {2,7} 6 {8} ,4 15 16 17 22 } 5 18 5 {6,8} 6 {9} 6 {10} {1 ,5} 5 {2,7} 20 21 22 23 UM=6 19 6 {6} 24 29 Fig. 2. Enumeration tree with laser search and bound LM1.problems which have the same local lower bound precedence graph of Fig. 1. In order to concen-value LLB as the current node, class II contains trate on these strategies, we only apply the simpleall remaining subproblems which have an in- lower bound LM1 and do without dominance andcreased LLB. Following the basic principle of reduction rules except for the maximum load rule.DFSC, the subproblems of the first class are As construction scheme, we use the station-ori-branched before those of the second class. This is ented assignment. Fig. 2 shows the enumerationimplemented as follows: the subproblems are gen- tree obtained by applying the LS strategy, whereerated following a task renumbering as in case of the node numbering indicates the sequence of nodeLS. Each subproblem is judged concerning its generation (the tree is developed from left to rightmembership to class I or II. Class II-problems and from top to bottom). The initial bound of theare dropped and re-enumerated later if necessary, root node 0 is given by LM1 = d48/10e = 5, the lo-class I-problems are chosen for branching. After cal lower bounds of the other nodes are given ashaving examined the corresponding subtrees, the node weights. In the first branch, a feasible solu-following situations may occur in the current node: tion with UM = 7 is found. This upper bound allows for fathoming the shaded nodes 11, 14,(1) If an improved upper bound UM = LLB has and 17. been found, the current node can be fathomed In node 23, an improved solution with UM = 6 immediately without generating further stations is found, such that node 24 can be fath- subproblems. omed. Since all nodes with LM1 = 5 are now com-(2) If an improved upper bound UM = LLB + 1 pletely branched, the procedure stops with the has been found, all class II-problems can be optimal solution S* = ({3, 4}, {1, 5}, {2, 7}, {6, 8}, dropped, i.e., the search stops after having {9}, {10}) with 6 stations. Solving this problem in- examined all class I-problems. stance to optimality requires examining 25 nodes(3) If all class I-problems have been examined, of the enumeration tree. LLB is increased by 1 (the current node has The DFSC strategy starts with completely no solution with at most LLB stations) start- branching node 0 thereby building (and storing) ing the examination of the class II-problems if the nodes 1 and 18. Computing LM1 for both LLB remains smaller than UM. Otherwise, nodes indicates that the search should continue the current node is fathomed. with branching node 18. This generates only node 19 whose branching results in the nodes 20 and 24 with lower bound values 5 and 6, respectively.3.6.2. Comparison of search strategies Therefore, the search continues with node 20 We compare the different search strategies by finally leading to the optimal solution with 6 sta-means of our example defined by c = 10 and the tions. Backtracking leads to fathoming the nodes
  13. 13. 678 A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–69324 and 1 and the procedure stops. In total, DFSC completed even in cases where the finished stationhas to consider 9 nodes. load is recognized as being inferior due to much The LLBM only builds the branch leading to the idle time. That is, the search can less strictly be di-optimal solution (nodes 0 and 18–23) without stor- rected to finding promising partial solutions sooning the nodes 1 and 24. After having found the as in case of other search strategies.solution with UM = 6, no further branching is re- Similarly to station-oriented LS procedures (seequired because the loads {1} in node 0 and {6} in above), additionally applying bound argumentsnode 19 would lead to class II-subproblems. So, and dominance rules partially removes this disad-LLBM requires the minimal number of 7 nodes. vantage. However, this comes along with addi- The MLB strategy might produce the same tree tional expense in computation time and storageas DFSC. However, due to several nodes having space. This tradeoff can best be seen comparingthe same bound value at a time it might perform FABLE (Johnson, 1988) and EUREKA (Hoff-much worse. For example, the following sequence mann, 1992), because the first performs a task-ori-of selecting nodes for further development or fat- ented LS with a lot of bounding, dominance andhoming is possible which requires examining 23 reduction rules, while the latter performs a sta-nodes: 0, 18, 19, 20, 1, 2, 3, 4, 8, 9, 10, 12, 13, tion-oriented LS almost without such rules. Exper-15, 16, 21, 22, 23, 24, 5, 11, 14, 17. That is, the per- imental tests show that both procedures whichformance of the procedure depends on resolving were the benchmark procedures in the early nine-such tie break situations. ties get similar results when EUREKA is restricted Of course, one may find other problem in- to the forward direction (Johnson, 1993; Hoff-stances where MLB or DFSC or even LS per- mann, 1993; Scholl, 1999, p. 238) despite their veryform better than LLBM but experimental results different approaches. The tests have been per-indicate that LLBM is the most effective formed using benchmark data sets of Talbotscheme in many situations (cf. Scholl and Klein, et al. (1986), Hoffmann (1990, 1992), and Scholl1997). (1993) that are downloadable from http:// When these basic strategies are enriched with and described inadditional bound arguments, dominance and Scholl (1999, p. 234).reduction rules the differences between the search Some improvements are obtained by includingstrategies are reduced. For example, the nodes 12 the static backward planning of EUREKAand 15 are avoided by the Jackson dominance rule, (Scholl, 1999, p. 239) which indicates that somebecause task 7 is potentially dominated by task 5 problem instances are easier solved in the reverseand task 6 by task 2. Moreover, node 11 would direction. Significant further improvements arebe avoided by the feasible set dominance rule due to the LLBM connected with the flexible bidi-due to node 5 and node 24 due to node 15. Never- rectional branching of SALOME-1, which at-theless, 17 nodes remain for LS. tempts to find the preferable planning direction in any node of the tree by some type of priority3.6.3. Comparison of construction schemes and rule (Scholl and Klein, 1997). Additional compo-complete procedures nents like dynamic prefixing and dynamic task Comparing the different construction schemes renumbering as well as the FS rule together with(task- versus station-oriented assignment) reveals the storage scheme of Nourie and Venta (1991,a methodical drawback of the task-oriented ap- 1996) considerably speed up SALOME-1 (Schollproach, because it successively fills stations with and Klein, 1999).tasks without controlling their final idle times. This The only (serial) procedure which seems to bedrawback becomes apparent most clearly when the competitive to SALOME-1 is that of Sprechertask-oriented assignment is connected with LS (as (1999) who applies many of the components alsois the case with all task-oriented procedures cur- contained in SALOME-1 but performs a task-ori-rently on hand; cf. Table 6), because each partial ented LS. The potential disadvantage of such typestation load and the following subtree must be of enumeration (see above) is compensated by
  14. 14. A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693 679utilizing improved knowledge on characteristics of for SALBP-2 is given by LC1 := max{tmax, dtsum/optimal solutions. me}. Significant speed ups are obtained by parallel- Bound LC2 (Klein and Scholl, 1996). SALBP-2ized versions of SALOME-1 and the Sprecher passes into a parallel machine problem with thealgorithm (cf. Bock and Rosenberg, 1998; Bock, objective of minimizing the makespan by omitting2000; Sprecher, 2003). This is due to the fact that the precedence relations. A lower bound for thesimultaneously searching in different parts of the parallel machine problem and, thus, for SALBP-solution space enlarges the probability of finding 2 is obtained as follows provided that the tasksan optimum soon. are renumbered according to non-increasing task SALBP-1 can be formulated as a special case times, i.e., tj P tj+1 for j = 1, . . ., n. Consider theof the generalized resource-constrained project m + 1 largest tasks 1, . . ., m + 1. A lower boundscheduling problem (GRCPSP) as shown in De on the cycle time for this reduced problem isReyck and Herroelen (1997) and Sprecher (1999). tm + tm+1, the sum of the two smallest task times,However, applying exact procedures for GRCPSP because at least one station contains two not competitive to specialized SALBP proce- For the 2m + 1 largest tasks, a lower bound is gi-dures (cf. De Reyck and Herroelen, 1997). ven by the sum of the three smallest operation Recently, Bockmayr and Pisaruk (2001) devel- times, t2mÀ1 + t2m + t2m+1. In general, a loweroped a branch and cut procedure based on integer bound LC2 is defined asprogramming formulations with additional valid ( ) Xkinequalities and constraint programming tech- LC2 :¼ max tkÁmþ1Ài j k ¼ 1; . . . ; bðn À 1Þ=mcniques. Pinnoi and Wilhelm (1997a,b, 1998) also i¼0propose branch and cut procedures for SALBP-1 ð4Þconnected with vertical balancing (cf. Section 2)as well as for more general problems. Bound LC3 (Scholl, 1999, p. 56). This bound is Though these researchers invest a lot of work in based on earliest and latest stations and destruc-finding sharp valid inequalities, their computa- tive improvement. Given an initial lower bound ctional experiments clearly show that such proce- on the cycle time, earliest and latest stations aredures cannot compete with state-of-the-art obtained from heads and tails (depending on c)enumeration based BB algorithms. The same is as follows (cf. Section 3.1):true for DP procedures (cf. Section 3.5). Ej ðcÞ :¼ daj ðcÞ þ pj ðcÞe and Lj ðcÞ :¼ m þ 1 À dpj ðcÞ þ nj ðcÞe for j ¼ 1; . . . ; n:4. Exact solution procedures for SALBP-2 and ð5ÞSALBP-E LC3 is computed by successively increasing c4.1. Lower bounds for SALBP-2 until Ej (c) 6 Lj (c) is obtained for all j = 1, . . ., n. Bound LC4 (Scholl, 1999, p. 57). By subdividing In what follows, we describe only a few argu- the interval [1, m] of all stations into two subinter-ments for computing lower bounds for SALBP-2, vals [1, i] and [i + 1,m], we obtain two aggregatefurther ones are proposed by Scholl (1999, Chap- ‘‘stations’’. For a valid lower bound c, these ‘‘sta-ter As in case of SALBP-1, these bounds tions’’ have to observe the ‘‘cycle times’’ i Æ c andutilize relationships to other problems and the (m À i) Æ c, respectively. This modified problem isdestructive improvement approach (Section 3.1). solved as follows: Bound LC1 (McNaughton, 1959). By analogy All tasks with Lj (c) 6 i are assigned to the firstwith LM1, a simple lower bound on the cycle time ‘‘station’’ and those with Ej (c) i are assigned tofollows from the necessary feasibility condition the second one. The remaining tasks and them Æ c P tsum. Furthermore, the indivisibility of remaining idle times of the two aggregate ‘‘sta-tasks requires that c P tmax. Hence, a lower bound tions’’ constitute a residual problem. This problem
  15. 15. 680 A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693can be formulated as the feasibility version of the tsum = 51 and tmax = 9, we get LC1 = max{9,bin packing problem with two bins and capaci- d51/5e} = 11. Because the six largest tasks are 9,ties equal to the remaining idle times of the two 1, 2, 3, 4, and 6, LC2 = t4 + t6 = 10 is obtained.aggregate ‘‘stations’’ (2Bin-F for short). The ques- Taking c = max{LC1,LC2} = 11 as initialtion of this feasibility problem reads as: ‘‘Does bound for computing LC3, we get the correspond-there exist a partition of the remaining tasks (items) ing station requirements, heads and tails given ininto at most two bins with given capacities?’’ Even the first rows of Table 7. Due to a11 (11) 5, it isthough 2Bin-F is NP-complete, it can often be clear that c = 11 is no feasible cycle time and is in-solved in a reasonable computation time, because creased to c = 12. The further rows of Table 7 con-the residual problem is usually much smaller than tain the respective station requirements, heads andthe original SALBP-2 instance and only two bins tails as well as earliest and latest stations. Obvi-are considered. The problem may be solved by ously, the condition Ej (12) 6 Lj (12) holds forany exact procedure for the bin packing problem j = 1, . . ., 10 and LC3 = 12 cannot be ‘‘destroyed’’(cf., e.g., Martello and Toth, 1990, Chapter 8; as a potential cycle time.Scholl et al., 1997). If the remaining idle times of For computing LC4, we start with the bestthe two ‘‘stations’’ are different, equal-sized bins known lower bound c = max{LC1, LC2, LC3} =are obtained by pre-assigning a fictitious item, 12 and use the values Ej (12) and Lj (12) of Table 7.whose weight is equal to the idle time difference, For i = 1 we get two ‘‘stations’’ with ‘‘cycleto form a bin with smaller capacity as required. times’’ c1 = 12 and c2 = 48. The tasks 5–10 are as- Following the destructive improvement ap- signed to ‘‘station’’ 2 ðS 02 ¼ f5; . . . ; 10gÞ and task 1proach, LC4 is given by the minimal trial value c to ‘‘station’’ 1 ðS 01 ¼ f1gÞ. The two bins have resid-for which 2Bin-F has a positive outcome for all ual capacities j1 = 6 and j2 = 19, and the tasks 2,possible subdivisions of [1, m] into two aggregate 3, and 4 remain. A feasible solution of 2Bin-F is‘‘stations’’ [1, i] and [i + 1,m], i.e., for i = 1, . . ., given by the sets S 00 ¼ f2g and S 00 ¼ f3; 4g. 1 2m À 1. Example. We consider a SALBP-2 instance with i = 2. c1 ¼ 24; c2 ¼ 36; S 01 ¼ f1; 3; 4g; S 02 ¼ f6; 8;the precedence graph of Fig. 3 and m = 5. With 9; 10g; j1 ¼ 8; j2 ¼ 15; S 00 ¼ f2g; S 00 ¼ f5; 7g. 1 2 i = 3. c1 = 36, c2 = 24, S 01 ¼ f1; . . . ; 7g; S 02 ¼ 6 6 4 f8; 9; 10g; no residual problem. 1 2 7 i = 4. c1 = 48, c2 = 12, S 01 ¼ f1; . . . ; 9g; S 02 ¼ f10g; 2 9 5 8 9 10 no residual problem. 5 5 4 5 3 4 5 6 Since no infeasibility is found, LC4 = 12 cannot be Fig. 3. Precedence graph. disproved as a feasible cycle time.Table 7Heads and tails of tasks for c = 11 and c = 12j 0 1 2 3 4 5 6 7 8 9 10 11pj (11) 0 0:54 0:54 0:45 0:45 0:36 0:45 0:36 0:18 0:81 0:45 0aj (11) 0 0 1 0 0:45 1:45 2 1:54 3:18 4 5 5:45nj (11) 5 4 3 4 3:36 3 2:18 2:18 2 1 0 0pj (12) 0 0.5 0.5 0:41 6 0:41 6 0:33 0:41 6 0:33 0:1 6 0.75 0:41 6 0aj (12) 0 0 0.5 0 0:41 6 1:33 2 1 3 3:1 6 4 4:41 6nj (12) 4:91 6 3:58 3 2:33 3:41 6 3 2:41 6 2 2 1.75 1 0 0Ej (12) – 1 1 1 1 2 3 2 4 4 5 –Lj (12) – 1 3 2 2 3 3 3 4 4 5 –
  16. 16. A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693 6814.2. Exact solution procedures for SALBP-2 to c + 1. The search stops with the optimal cycle time UC when UC = LC. While a large variety of exact solution proce-dures exist for SALBP-1, only a few have been Two further search methods are the Fibonaccideveloped which directly solve SALBP-2. Most re- binary search and the upper bound search. Com-search has been devoted to considering search putational experiments indicate that the binarymethods which are based on repeatedly solving search is the best compromise in case of restrictedSALBP-1. computation time (cf. Scholl, 1999, p. 256). When the SALBP-F instances are solved heuris-4.2.1. Iterated search methods tically, the whole search procedure becomes a heu- Following our argumentation in Section 2, ristic (cf. Section 5.1).SALBP-2 (minimize c for given m) can be formu-lated as the problem of finding the smallest cycletime c for which the corresponding SALBP-F in- 4.2.2. Direct solution approachesstance (m, c) has a feasible solution. Hence, In recent years, only two BB proceduresSALBP-2 can be solved by successively solving which directly solve SALBP-2 have beenSALBP-F instances with m stations and various developed.trial cycle times. The latter are restricted to an • Scholl (1994) proposes the task-oriented BBinterval [LC, UC] which is bounded by a lower procedure TBB-2 which is based on a successiveand an upper bound on the cycle time. While lower reduction of the precedence graph to a stationbounds may be computed by means of the argu- graph (for extensions see Scholl, 1999, Chapterments described in Section 4.1, upper bounds are 4.2.3). At the beginning, the precedence graph isderived from any heuristic solution for SALBP-2 complemented by an initial station graph contain-(cf. Section 5). ing an empty node Nk for the stations k = 1, . . ., m. Any exact procedure for SALBP-1 (cf. Section The nodes N1 and Nm serve as fictitious source and3) can easily be modified for SALBP-F instances sink node of the combined graph, respectively.(m, c) by considering the given cycle time c and Each task which is assigned to a station k byby excluding (fathoming) all solutions which have branching, prefixing or another logical test isa larger number of stations than m. This is merged into the station node Nk. A complete solu-achieved by setting UM = m + 1. tion is obtained when only the station graph re- The efficiency of solving a SALBP-2 instance mains. The graph representation of partialdepends on the sequence in which the trial cycle solutions allows for applying logical tests which re-times are examined. Therefore, several search duce the BB tree greatly.methods have been proposed and tested by, e.g., • Klein and Scholl (1996) develop an adaptationDar-El and Rubinovitch (1979), Hackman et al. of SALOME-1 to SALBP-2 which is called SAL-(1989), Klein and Scholl (1996). Two common OME-2. It also utilizes the LLBM, a bidirectionalmethods are the following: branching strategy, and several dominance and reduction rules which have to be modified to fit• Lower bound search. Starting with the lower the conditions of SALBP-2. A distributed version bound LC, the trial cycle time c is successively of SALOME-2 is examined by Bock (2000, Chap- increased by 1 until the respective SALBP-F ter 6.2). instance (m, c) is feasible. In opposite to SALOME-1, more than two• Binary search. The search interval [LC, UC] is classes of subproblems have to be considered in successively subdivided into two subintervals each node of the enumeration tree depending on by choosing the mean element c = b(LC + the current value of the local lower bound LLC UC)/2c. If SALBP-F is feasible for c, the upper on the cycle time. After examining all maximal sta- bound UC is set to the maximal station time in tion loads feasible for the lower bound cycle time the corresponding solution. Otherwise, LC is set LLC, its value is increased such that at least one
  17. 17. 682 A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693further load becomes feasible. This process of examined by Zapfel (1975, p. 53), Klenke (1977, ¨branching a node terminates when a feasible com- p. 47) as well as Scholl and Voß (1995). A similarplete solutions with cycle time LLC has been approach is given for a cost-oriented problem byfound or no further load is available. Rosenblatt and Carlson (1985). Computational experiments indicate that SAL- The lower bound search of Scholl and VoßOME-2 gets much better results than TBB-2 when (1995) starts with a list L of combinationsapplied to standard benchmark data sets (cf. (m, LC (m)) with m 2 [mmin, mmax]. That is, for eachScholl, 1999, p. 258). Comparing SALOME-2 with feasible value of m the smallest possible cycle timea binary search applying SALOME-1 to each trial (given by a lower bound LC (m); cf. Section 4.1) iscycle time shows that the search method is compet- considered. In case of LC(m) cmax, m is removeditive (cf. Klein and Scholl, 1996). This gives a cer- from the station interval. If LC (m) cmin, we havetain foundation for the concentration on finding to use LC (m) = cmin. The (m, c)-combinations in Leffective procedures for SALBP-1 in the past 50 are sorted in non-decreasing order of their lineyears though SALBP-2 is a problem which may capacities T = m Æ c. The search examines theeven have the greater practical relevance, because SALBP-F instances in this order, where ties areit arises whenever an existing line has to be rebal- broken in favor of the smaller m because the cor-anced, while SALBP-1 is relevant mainly in case of responding SALBP-F instances are often easierthe first installation of a line. to solve than those with larger m. If the combina- tion (m, c) is identified as being infeasible, m is re-4.3. Search methods for SALBP-E moved from the station interval in case of c = cmax. Otherwise, the next possible cycle time As is the case with SALBP-2, instances of the c 0 (normally c 0 : = c + 1) is defined for m and themore general SALBP-E may be solved by some resulting combination is sorted into the list L. Thissearch method. Such a method has to find a feasi- process continues until a feasible (=optimal) com-ble combination (m, c) of the number m of stations bination is found.and the cycle time c such that the line efficiency is Examination of the SALBP-F instances may bemaximized or, equivalently, the required line done by applying a (correspondingly modified)capacity T = m Æ c is minimized (cf. Section 2). SALBP-1 or SALBP-2 procedure. The latter have A SALBP-E instance is defined by an interval the advantage that larger increments than 1 may[cmin, cmax] of possible cycle times and/or an inter- be identified accelerating the search (cf. Scholl,val [mmin, mmax] of possible numbers of stations. 1999, Chapter 4.3). Furthermore, they more oftenThe intervals have to observe the feasibility condi- find feasible solutions in case of restricted compu-tions T = m Æ c P tsum and cmin P tmax as well as tation time (cf. Scholl, 1999, p. 272).space and productivity constraints. Procedures directly solving SALBP-E are not An obvious solution procedure for SALBP-E available. This may be due to the difficulty to di-consists of considering all possible SALBP-F in- rect the search in promising regions of the solutionstances (=(m, c)-combinations with m 2 [mmin, space when neither m nor c is fixed.mmax] and c 2 [cmin, cmax]) and examining their fea-sibility. A feasible combination with minimal valueof T is optimal. This may be done with modified 5. Heuristic approaches for different versions ofprocedures for SALBP-1 or SALBP-2, respec- SALBPtively. However, usually a lot of (m, c)-combina-tions exist. Furthermore, the SALBP-F instances A large variety of heuristic approaches to differ-to be considered are NP-complete. Thus, the out- ent versions of SALBP have been proposed in thelined approach is usually very inefficient. last decades. While constructive procedures con- In order to find an optimal (m, c)-combination structing one or more feasible solution(s) weremore efficiently, search methods similar to those developed until the mid nineties, improvementdefined for SALBP-2 (Section 4.2) have been procedures using metastrategies like tabu search
  18. 18. A. Scholl, C. Becker / European Journal of Operational Research 168 (2006) 666–693 683and genetic algorithms have been in the focus of For the rule MaxPW, this procedure is calledresearchers in the last decade. ranked positional weight technique by Helgeson and Birnie (1961).5.1. Constructive procedures • Task-oriented procedures. Among all available tasks, one with highest priority is chosen and The majority of constructive procedures have assigned to the earliest station to which it isbeen proposed for SALBP-1 and are based on pri- assignable.ority rules, others are restricted enumerative pro- Depending on whether the set of available taskscedures. The most recent comprehensive surveys is updated immediately after assigning a task orof those approaches are given by Talbot et al. after assigning all currently available tasks,(1986) and Scholl (1999, Chapter 5.1.1). Further- task-oriented methods can be subdivided intomore, see Boctor (1995) and Ponnambalam et al. immediate-update-first and general-first-fit(2000). methods (cf. Wee and Magazine, 1982; Hack- man et al., 1989).5.1.1. Priority rule based procedures for SALBP-1 Those procedures use priority values computed Theoretical analyses show that both schemesfor the different tasks based on the task times and obtain the same solution when the used prioritythe precedence relations given. Some of the most rule is strongly monotonous, i.e., the priority valueeffective ones are given in Table 8 (cf. Talbot of any task j is smaller than that of each predeces-et al., 1986; Hackman et al., 1989; Scholl and sor h 2 Pj. This is, e.g., true for MaxPW, MaxF,Voß, 1996). In any case, the tasks are sorted and MaxCPW (cf. Scholl, 1999, p. 182). Computa-according to non-increasing priority values to get tional experiments indicate that, in general, sta-a priority list. tion-oriented procedures get better results than By analogy with exact solution procedures (cf. task-oriented ones though no theoretical domi-Section 3.2), two construction schemes are relevant nance exists (cf. Scholl and Voß, 1996). This sup-for priority rule based approaches. They differ ports the analogous finding in case of exactwith respect to the manner in which the tasks to procedures (cf. Section 3.6).be assigned are selected out of the set of available These classical priority rule based procedurestasks (cf. Talbot et al., 1986; Hackman et al., 1989; work unidirectionally in forward direction andScholl and Voß, 1996): construct a single feasible solution. Improvements are obtained by following approaches.• Station-oriented procedures. They start with the • Flexible bidirectional construction. The stations first station (k = 1). The following stations are to be loaded are considered in forward and back- considered successively. In each iteration, a task ward direction, simultaneously (Scholl and Voß, with highest priority which is assignable to the 1996). That is, a station-oriented procedure con- current station k is selected and assigned. When siders the earliest and the latest unloaded station station k is loaded maximally, it is closed, and at a time. Besides selecting a (forward or backward the next station k + 1 is opened. assignable) task by some priority rule the (earliest or latest) station to be considered next is chosen.Table 8 Task-oriented procedures simultaneously considerPriority values forward and backward available tasks and alwaysName Priority value choose the one with highest priority. Both ap- proaches require defining reversed priority rulesMaxT Task time tj PMaxPW Positional weight pwj ¼ tj þ h2F à th (cf. Scholl, 1999, p. 184).MaxF à Number of followers j F j j j • Dynamic priority rules iteratively adapt the prior-MaxTL Task time over latest station tj/Lj ities depending on the current partial solutions (cf.MaxTS Task time over slack tj/(LjÀEj + 1) P Boctor, 1995; Scholl and Voß, 1996). For example,MaxCPW Cumulated positional weight pwà ¼ tj þ h2F à pwà j j h MaxTS can be applied dynamically (in a uni- or