4 Scheduling. Outline of the chapter. 4.1 Preliminaries

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1 4 Scheduling In this section, e consider so-called Scheduling roblems I.e., if there are altogether M machines or resources for each machine, a roduction sequence of all N jobs has to be found as ell as the determination of the time tables Consequently, e have to decide on The sequence of the resective jobs on each machine and its time table Business Comuting and Oerations Research 47 Outline of the chater. Preliminaries. Mathematical model. Objective functions. Single Machine Models. Sequencing roblem ith heads and tails 4. Multile stages. Use of riority rules. Elaborated heuristics. The Shifting Bottlenec Procedure. Tabu Search by Noici-Smutnici 5. Flo-sho roblems. The rocedure of Johnson. The multile-stage case Business Comuting and Oerations Research Preliminaries Production rogram is given Lot sizes are given Process sequence of each job is given Oerating times are given No oeration can be rocessed simultaneously on more than one machine At each oint of time every machine can rocess at most one job At the beginning of the lanning horizon all N jobs and their data are available (static roblem) Transorts and storage are never bottlenecs No maintenance and reair activities On each machine setu times are indeendent of the realized oeration sequence Business Comuting and Oerations Research 49

2 Given: Given and sought MS: Machine sequence matrix PT: Matrix of rocessing times Sought: JS: Job sequence matrix TT: Timetable lanning matrix ith: ( ) tm, n m M; n N [ ] Point of time here the rocessing of job n at machine m begins TU Business Comuting and Oerations Research Mathematical model Variables : t m, n y ( n N; m M ): see above ( m M ; n N; N ) m, n, sequence of jobs, i.e., : Binary variable defining the y m, n, if job n is rocessed on machine m before job = otherise Business Comuting and Oerations Research 5 Mathematical model Restrictions : [ m] ( MS) Machine sequence restrictions derived from the matrix : {,..., } : {,..., } : [ ] [ ] [ ] m M n N t t m,n m,n m,n defines in this connection the index of the machine that executes the m-th oeration of job n Business Comuting and Oerations Research 5

3 Mathematical model In case of the job sequence restrictions, the formulation deends on the structure of the found solution But, e have to ensure that there is no simultaneous rocessing of to jobs on any machine, herefore an arbitrary sequence of those jobs has to be realized Therefore, there are altogether to ossible cases: ( ) ( ) ( n ) First case before : t t m, n m, n m, m, m, m, n ( n) Second case before : t t Both ossibilities have to be considered in the model! Business Comuting and Oerations Research 5 Mathematical model Restrictions : ( ) Job sequence restrictions deends on the chosen solution : {,..., } :, {,..., } : m,n m,n m, ( m, n, ) {,..., } :, {,..., } : m M n N t t y C m M n N t t y C m, m, m,n m, n, C defines a big number, hich is larger than each definition of M N = m, n m= n= timetable variables t, e.g., C m,n Business Comuting and Oerations Research 54 Mathematical model Restrictions : m m {,..., M }: n, {,..., N}( n ) : y {,} {,..., M }: n {,..., N} : t m, n m, n, Business Comuting and Oerations Research 55

4 4.. Objective functions The model defined above can be regarded as a general starting oint for so-called job-sho scheduling roblems It abstains from the definition of a articular objective function but can be extended by a secific alication-deendent one A huge set of different objective functions is roosed in literature. These functions mainly influence the efficiency of alied solution rocedures In the folloing, e ill give some examles of ell-non objectives Business Comuting and Oerations Research 56 Minimization of cycle time Here, e consider the duration of roducing the total roduction quantities max, { } [ ], { [ M] n { }} Minimize Z = t = max C n,..., N ith: n,..., N : C : Point of time here the last M n rocessing of job n is finalized Business Comuting and Oerations Research 57 Minimization of machine aiting times Sum of all machine aiting times throughout all used resources M N Minimize Z = Cmax m, n m= n= Unused caacity of machine m M N = M Cmax m, n m= n= M N Since m, n and M are constants, Z and Z are m= n= equivalent Business Comuting and Oerations Research 58 4

5 Minimization of total comletion (lead) time This objective intends to minimize the total sum of all individual comletion or lead times Therefore, e comute the sum of dell times over all jobs N Minimize Z = C[ M ], n n= This objective is equivalent to the minimization of the aiting times of all jobs Minimize Z 4 = N M m, n n= m= the sum of Business Comuting and Oerations Research 59 Minimization of maximum lead time Here, e ant to minimize the dell time of the job hose rocessing taes the longest time among all N jobs This is the objective function Z Business Comuting and Oerations Research 6 Min. of the sum of due date deviations In this case comletion time and due date of each job are comared, hile the difference is taen as the result and summed u throughout all jobs to be rocessed As a consequence, an early comletion gets a bonus hile each lateness is unished N N N Minimize Z5 = C[ ] d, n = M n C[ M], n dn n n n = = = ith: d :Due date of job n n Since N n= d is a constant, this objective is equivalent to Z n Business Comuting and Oerations Research 6 5

6 Minimization of total lateness (or tardiness) Here, e ant to minimize the total lateness over all N jobs to be roduced in the considered roduction system Consequently, there is no longer comensation beteen early and late deliveries ossible { } N Minimize Z6 = max C[ M ],n dn, n= ith : d : Due date of job n n Business Comuting and Oerations Research 6 Minimization of maximum lateness By using this objective, e someho ant to balance the lateness equally among the different jobs in the found solution Thus, e try to minimize the maximum lateness of a job in the found solution Minimize Z7 = max{ max{ C[ ] d n, } n {,..., N M,n }} ith: d : Due date of job n n Business Comuting and Oerations Research 6 Minimization of eighted sum of lead times Here, each job gets an individual eight rating its delling time in the roduction system Altogether, by doing so e receive a combined eighted sum of lead times N Minimize Z8 = n C[ M], n n= ith: : Weight for roduct n n Business Comuting and Oerations Research 64 6

7 4.. Schedule classes In the folloing, e introduce some basic terms for secific tyes of schedules In scheduling, a distinction is frequently made beteen Sequence, Schedule and Scheduling olicy Sequence Corresonds to a secific ermutation of jobs to be rocessed on a given machine Schedule Usually corresonds to an allocation of jobs ithin a more comlicated setting of machines, hich could allo for reemtions of jobs by other jobs that are released at later oints in time. Comrises time tables Scheduling olicy Often used in stochastic settings; a olicy rescribes an aroriate action for any of the states the system may be in. In deterministic cases, usually only sequences or schedules are of imortance but can be extended by rule definitions Business Comuting and Oerations Research Definition Non-delay schedules A feasible schedule is called non-delay if no machine is et idle hen there is an oeration available for rocessing Business Comuting and Oerations Research Definition Active schedules A feasible schedule is called active if no oeration can be comleted earlier by starting earlier or changing the rocess sequence on machines ithout delaying any other oeration Business Comuting and Oerations Research 67 7

8 Attributes of active schedules 4... Lemma A non-delay schedule is alays active Business Comuting and Oerations Research 68 Proof of the lemma Let us assume there is a non-delay schedule that is not active Then, e no there is a machine here shifting an oeration i into an earlier osition at oint of time t results in an earlier comletion ithout delaying the other oerations But, if this is true, e no that during the rocessing of the schedule on machine m there is a constellation at oint of time t here the considered machine is idle but can rocess job i instead This is a contradiction to the assumtion that the schedule is non-delay Business Comuting and Oerations Research 69 Attributes of active schedules Note that the reverse is not necessarily true i.e., there are some active schedules that are not nondelay Examle: Schedule is active but not non-delay Machine Machine Machine Business Comuting and Oerations Research 7 8

9 Semi-active schedules Definition A feasible schedule is called semi-active if no oeration can be comleted earlier ithout altering the rocessing sequence on any of the machines Business Comuting and Oerations Research 7 Consequences Lemma An active schedule is alays semi-active The roof is trivial and immediately results from the definition Business Comuting and Oerations Research 7 Attributes of semi-active schedules Note that the reverse is not necessarily true i.e., there are some semi-active schedules that are not active Examle: Schedule is semi-active but not active Machine Machine Machine Business Comuting and Oerations Research 7 9

10 Schedule class hierarchy Class of non-delay schedules Class of active schedules Class of semi-active schedules Business Comuting and Oerations Research Single-Stage Systems No, e consider a single roduction stage only i.e., M=, herefore e have only one indexed rocessing times,, N No, the comlexity of the models only deends on the considered objective function There are some constellations that can be otimally solved in O(N log N) time using a simle riority rule as ell as models that are already NPcomlete roblems. And both haens desite the fact that besides their objective function, both roblems are comletely the same one-stage roblems Business Comuting and Oerations Research 75 Minimization of cycle time Trivial roblem Each solution leads to the same result Therefore, an arbitrary solution is already an otimal one Business Comuting and Oerations Research 76

11 Minimization of eighted sum of lead times 4.. Theorem The WSPT-rule leads to the otimal solution Weighted Shortest Processing Time First Rule: This rule rocesses all N jobs in the sequence of non-increasing order of the value j / j Business Comuting and Oerations Research 77 Proof of Theorem We ill sho the claim by contradiction Therefore, e assume that there is an otimal sequence of the roblem that does not fulfill all the restrictions of the WSPT olicy Consequently, there are to adjacent jobs, say job j folloed by job, such that j / j < / Business Comuting and Oerations Research 78 Proof of Theorem Assume job j starts its rocessing at time t Let us erform an interchange of j and Therefore, the modified schedule starts job no at t hile all other jobs remain in their original osition Consequently, their eighted objective value is not affected at all and, therefore, remains unchanged Call the old schedule S and the ne modified one T Business Comuting and Oerations Research 79

12 Business Comuting and Oerations Research 8 Proof of Theorem Under schedule S, the total eighted comletion of jobs j and is rated by: ( ) ( ) j j j t t Under schedule T, the total eighted comletion of jobs j and is rated by: ( ) ( ) j j t t Business Comuting and Oerations Research 8 Proof of Theorem IV ( ) ( ) ( ) ( ) ( ) / / > < < = = j j j j j j j j j j j j j j j j j j j j j t t t t t t t t holds : It T solution of value function Objective S solution of value function Objective Business Comuting and Oerations Research 8 Proof of Theorem V Consequently, solution T is better and, therefore, the roof is comleted

13 Minimization of total lead time 4.. Corollary The SPT-rule leads to the otimal solution Shortest Processing Time First Rule: This rule rocesses all N jobs in the sequence of non-decreasing rocessing times j Business Comuting and Oerations Research 8 Proof of the Corollary To rove the corollary, e may use again Theorem 4.. To do so, e easily derive that the objective function for minimizing the total lead time Z is a secial case of the more general eighted sum of lead time Z 8 In this case all eights are set to By alying Theorem 4.., e can derive that e receive the otimal sequence by using the WSPT-olicy, i.e., by resecting this secial setting, e sort all jobs in nonincreasing sequence of the j / j =/ j values Consequently, the jobs are sorted in an non-decreasing sequence of the j -values as defined by the ell-non SPT-rule. This comletes the roof of the corollary Business Comuting and Oerations Research 84 Precedence constraints Ho is the result affected by recedence constraints? In the folloing, e introduce additional recedence constraints limiting the solution sace through the exclusion of some ossible solutions. These constraints are very simle and can be described through arallel chains defining hich job has to be rocessed before another one This is a situation that frequently occurs during the rocessing of multi-stage systems First, e can rocess only entire chains. To solve these roblems otimally, e can use the folloing extended Theorem 4.. Business Comuting and Oerations Research 85

14 Parallel recedence chains 4 Chain 5 6 Chain Chain Chain Business Comuting and Oerations Research Theorem Entire chain roblem The entire chain roblem according to the objective of total eighted lead time minimization can be solved otimally by sorting the chains in non-increasing order of the value: j = j j = j is the value for the chain comrising the jobs,,..., Business Comuting and Oerations Research 87 Proof of the Theorem Again, e sho the claim by contradiction Therefore, e assume there is an otimal roduction lan violating the rule definition Therefore, there are to neighbored chains (,,) and (,,l) here the defined riority rule is not fulfilled Again, e can derive that there is no imact on the eighted lead time of the jobs not belonging to one of the to chains Moreover, e derive schedule T from the current schedule denoted S through the exchange of the to neighboring chains, i.e., in T e rocess (,,l) before (,,) is rocessed In hat follos, e comute the resective objective values of the both chains for the to ossible constellations S and T Business Comuting and Oerations Research 88 4

15 Schedule S Objective function value Objective function value under schedule S: ( ) ( ) t t... t j t j j = j = l l l j... l t j = t j j i j = j = j = i = Business Comuting and Oerations Research 89 Schedule T Objective function value Objective function value under schedule T: l ( t ) ( t )... l t j j = l l t j... t j j j = j = j = l l j l l = t j j i j i j i j = j = i = j = i = j = i = Business Comuting and Oerations Research 9 The ga beteen T and S l l j i j i j i = j = i = = We no: j j j j l l j = = j = j = l < l j j < j j j = j = j = j = j l l Consequently: j i j i < j = i = j = i = Business Comuting and Oerations Research 9 5

16 Consequence The objective value of schedule T is better than the result under schedule S and, therefore, the otimality of the rule defined above is shon This comletes the roof Business Comuting and Oerations Research Definition ρ-factor Let us consider a chain of jobs (,,). Then, the job * out of the chain is called the ρ-factor of the chain (,,) if i * i = i = * l i = l i = max l i i i = Business Comuting and Oerations Research 9 Alloing reemtion Assume no that the scheduler has the freedom to rocess any number of jobs in a chain (hile adhering to the recedence constraints) ithout necessarily having to comlete all the jobs in the chain before sitching to another chain In hat follos, e consider again the case of multile chains Moreover, total eighted lead time is assumed to be the objective function Then, e may aly the result given in the folloing Theorem 4..5 in order to derive an otimal roduction lan Business Comuting and Oerations Research 94 6

17 4..5 Lemma Subchain reemtion If job l* is the ρ-factor of the chain (,,), then there exists an otimal sequence that rocesses jobs,,l* one after another ithout interrution by jobs from other chains Business Comuting and Oerations Research 95 Proof of the Theorem Again, e rove this claim by contradiction Suose that under the otimal sequence, the rocessing of the subsequence,,l* is interruted by a job, say job v, from another chain that has to be rocessed simultaneously Thus, the otimal sequence contains the subsequence,,u,v,u,,l*, say subsequence S It suffices to sho that either ith subsequence v,,,l*, say S or ith,,l*,v, say S, the total eighted comletion time is less than ith subsequence S We no that the lead time of all other jobs besides,,l* and v is indeendent of the chosen subsequence S, S, and S In the folloing, e therefore assume S >S as ell as S >S Business Comuting and Oerations Research 96 Case : S >S S=(,,u,v,u,,l*); S =(v,,,u,u,,l*) Since S is better than S, e can aly the roof of Theorem 4.. to derive that it holds: v... u <... v If this is not true, e do not orsen the solution by alying the roof of Theorem 4.. and rocess v before the job sequence,,u It is trivial that e can choose S instead of S if it holds S S u Business Comuting and Oerations Research 97 7

18 8 Business Comuting and Oerations Research 98 Case : S >S S=(,,u,v,u,,l*); S =(,,u,u,,l*,v) Since S is better than S, e can again aly the roof of Theorem 4.. to derive that it holds: * * l u l u v v > If this is not true, e do not orsen the solution by alying the roof of Theorem 4.. and rocess v after the job sequence u,,l* It is trivial that e can again choose S instead of S if it holds S S Business Comuting and Oerations Research 99 Consequences Therefore, e no: * * l u l u v v u u v v > < In addition, l* is the ρ-factor of (,,). Therefore, it holds: u u l u l u u u l l > > * * * * Business Comuting and Oerations Research 4 Consequences * * * * But e assumed : l u l u v v v v u u l u l u > > > Therefore, the assumtions of Case and Case together have derived a contradiction Therefore, both cases cannot aly simultaneously. This obviously comletes the roof

19 Using the result The result derived above is intuitive. Its condition imlies that the ratios of the total eight divided by the total rocessing time of the jobs in the string,,l* must be decreasing in some sense If one had decided to start rocessing a stream, it maes sense to roceed until job l* is obtained By simultaneously using the result derived above, e can use the folloing algorithm for solving our roblem otimally Business Comuting and Oerations Research Algorithm Solution rocedure Whenever the machine is freed, select among the remaining chains the one ith the highest ρ- factor and rocess this chain ithout interrution u to the job that determines its ρ-factor. Note that this includes this job itself. Business Comuting and Oerations Research 4 Examle Consider the folloing to chains The eights and rocessing times of the jobs are given belo Jobs Weight Processing time Business Comuting and Oerations Research 4 9

20 Solving the examle The ρ-factor of the first chain is (68)/(6)=4/9 and determined by job Theρ-factor of the second chain is (87)/(48)=5/ and determined by job 6 Therefore, e start rocessing the first chain (Schedule: ) The ρ-factor of the remaining first chain is ()/(6)= and determined by job Therefore, e roceed ith the second chain (Schedule: 5 6) The ρ-factor of the remaining second chain is (8)/()=.8 and determined by job 7 Hence, e roceed ith the first chain (Schedule: 5 6 ) The ρ-factor of the remaining first chain is (8)/(5) and determined by job 4 Consequently, e roceed ith the second chain (Schedule: 5 6 7) Resulting schedule is Business Comuting and Oerations Research 44 Maximum lateness In the folloing, e consider a general enalty function individually defined for a delayed rocessing of each job This leads to a situation here each job j has an individual enalty function h j (C j ) for a delayed rocessing of C j time units We assume that each enalty function is monotonous, i.e., the resulting values for increasing comletion times are non-decreasing In addition, e consider again existing recedence constraints beteen the different jobs Objective function is no: Minimize max{h (C ),,h n (C n )} Business Comuting and Oerations Research 45 Due-date-related enalty functions h j h j d j C j h j d j C j d j C j Business Comuting and Oerations Research 46

21 4..7 Algorithm Solution rocedure Ste : Set J=Ø; J c ={,,n}; J the set of all jobs ith no successors Ste : Let j* be such that h * min j i = h j i c j J c i J i J Add j* to J; Delete j* from J c In order to reresent the ne set of schedulable jobs, modify J accordingly Ste : If J c =Ø, then sto; otherise go to Ste Business Comuting and Oerations Research Theorem Consequences Algorithm 4..7 attains an otimal schedule for the considered roblem Business Comuting and Oerations Research 48 Proof of the Theorem Suose that there exists an iteration here job j** is selected from J but does not have the minimum comletion cost h j * j c j J among the jobs belonging to J at this moment The minimum cost job j* must then be scheduled at a later iteration, imlying that the resective job j* aears in the sequence before job j**. In addition, some jobs can aear beteen the jobs j* and j** Business Comuting and Oerations Research 49

22 Proof of the Theorem In order to comlete the roof, e move the sequence osition of job j* just behind job j**. What are the consequences for the objective value of the solution? All jobs that are located beteen j* and j** in the old schedule S are rocessed earlier, herefore the objective function value is not negatively affected What about job j*? This is the only job hose comletion time is increased through the alied modification But e no by assumtion that this modified value leads to a smaller enalty function value than the one caused by j** in schedule S Finally, e can state that the value for j** in the ne schedule is not increased. Therefore, the maximum of all lateness values in the ne schedule is not larger than the objective value of S This comletes the roof Business Comuting and Oerations Research Corollary Secial case Z 7 For the secial case h j =max{,c j -d j }, the alication of rule EDD (Earliest Due Date), hich schedules the different jobs in a nondecreasing sequence of due dates, results in the otimal solution. Business Comuting and Oerations Research 4 Proof of the Corollary In order to rove the claim of the corollary, e can aly Theorem 4..8 and Algorithm 4..7 Hence, jobs are scheduled in the first iterations and, therefore, at the end of the arising total sequence ith the loest enalty value These values only deend on individual due dates and, therefore, lead to a situation here jobs ith the highest due dates are referred (at the end of the schedule!) By referring the highest due dates for an inverted sequence, e aly the EDD-rule for the original one Business Comuting and Oerations Research 4

23 Total lateness This roblem is roven to be NP-hard in the ordinary sense, i.e., it exists a seudo-olynomial time algorithm based on dynamic rogramming Hoever, this roblem can be simlified by scheduling jobs hich are non-time-critical at the end, i.e., the total rocessing time of all jobs is loer or equal to their due date Business Comuting and Oerations Research 4 4. Sequencing roblem ith heads and tails In hat follos, e tae a ste toards multile stage roblems Therefore, e consider a single stage here a scheduling sequence has to be determined. Hoever, each job has receding and subsequent rocesses at other stages, hich are defined as head and tails Consequently, beside i, the rocessing time of the i-th job at the considered stage, there is a head a i and a tail q i As the ursued objective e consider the minimization of the maesan (lead time) Business Comuting and Oerations Research 44 Deriving a simle loer bound 4.. Lemma For all subsets I l of the set of jobs to be rocessed I, there exists the folloing loer bound on the otimal cycle time ( l ) = min { i l} i min { i l} lb I a i I q i I amin, Il i Il qmin, Il Business Comuting and Oerations Research 45

24 Proof of the Lemma Consider an arbitrary set of jobs I At least a min,i time units have to elase before the rocessing can start This rocessing taes altogether additional total,i time units Finally, there is alays one job rocessed at the last osition at the considered stage hose tail increases the total maesan of the rocessing. And this tail is larger than or equal to q min,i Thus, e have shon that the defined sum is a loer bound for the cycle time This obviously comletes the roof Business Comuting and Oerations Research Lemma Comlexity of the roblem The general scheduling roblem ith heads and tails is NP-hard In hat follos, e roose the ell-non Branch&Bound aroach of Carlier (98) in order to solve the roblem otimally Business Comuting and Oerations Research 47 Starting oint: The Schrage algorithm In this greedy aroach, e alays schedule the ready job ith the greatest tail ( i ) Set t = min i I ai; U = ; U = {,..., n} ; ( ii ) At time t, schedule amongst the ready jobs ( ai t ) of U, job j ith q j =max { qi i U ai t} ( or any one in the case of ties) ( iii ) Set: U = U { j} ; U = U \ { j} ; t j = t; t = max { t j j,min ai }; i U If U I = { n} otherise roceed ith ste ( ii ) is equal to,...,, the algorithm is finished; i Business Comuting and Oerations Research 48 4

25 Critical ath The critical ath of a solution of the roblem alays comrises, in the given sequence, the folloing arts: a head of some job, a sequence of jobs that are iteratively rocessed ithout interrution at the considered stage, and finally, a tail of some job that is rocessed at the last osition of the critical ath In hat follos, e derive the basic branching rule of the B&B rocedure of Schrage by analyzing the critical ath of the solution generated by the Schrage rocedure Business Comuting and Oerations Research Theorem Main result Let L be the maesan of the Schrage schedule (a). If this schedule is not otimal, there is a critical job c and a critical set J such that: lb ( J ) = minai i minqi > L c i J i J In an otimal schedule, either c is rocessed before all the jobs of J, or c ill be rocessed after all the jobs of J (b). If this schedule is otimal, there exists J such that LB(J)=L i J Business Comuting and Oerations Research 4 Proof of the Theorem Let G be the disjunctive grah defining the considered roblem ith source and sin s In addition, z is a critical ath assing through a maximal number of jobs We modify the numbering of the jobs according to the definition of this ath Therefore, the jobs rocessed on this ath are numbered from to, i.e., the critical ath is (,,,,,,s) Business Comuting and Oerations Research 4 5

26 Proof of the Theorem At first, e rove that there is no rocessing beteen the times a - and a If there is a job rocessed in this interval, it ould be finished at a since the rocessing of the first job starts just at this oint of time If so and there is a job j rocessed there and e as hether a j =t j. If so, e can extend the critical ath. Obviously, this is not ossible due to the assumtion of a maximal ath z Hoever, if a j <t j e no due to the rocessing of the Schrage rocedure that there is an additional job rocessed just before j Clearly, because of that cognitions, e no that there is alays a final job ith a =t. Note that this is at least the job firstly rocessed in the total schedule Hence, e have shon that there is no rocessing in the interval beteen a - and a due to the maximum choice of z Business Comuting and Oerations Research 4 Proof of the Theorem Secondly, e sho a =min{ a i i } The machine as idle just before job as rocessed Therefore, the Schrage rocedure schedules no job in this interval and job as scheduled subsequently I.e., all heads are larger than or equal to the head of job Thus, e obviously obtain a =min{ a i i } Business Comuting and Oerations Research 4 Proof of the Theorem Thirdly, if q is the minimal tail of all jobs,,, the length of the critical ath becomes L = a q i = i { a { } } i { q { } } = min,..., min,..., i = Hence, the loer bound and the solution value are equal, hich immediately roves the otimality of the generated solution Business Comuting and Oerations Research 44 6

27 Proof of the Theorem But, if otherise q is not the smallest tail of the jobs in {,,}, there is alays a job c ith largest index hose tail is smaller than q Let J={c,,} be the set of subsequently scheduled jobs on the critical ath We no q c <q r for all r in J and additionally a r >t c Why? If a r t c, then, oing to its larger tail, job r ould be scheduled before job c Hence, e derive a r >t c Business Comuting and Oerations Research 45 Proof of the Theorem Consequently, r J alays imlies a > t = a... r c c { a J} a c min { } additionally imlies { a J} { q J} min >..., and q = q J min c... min loer bound of the maesan lb( J ) > a q c c c c L = L c Business Comuting and Oerations Research 46 Proof of the Theorem Therefore, e have shon that the distance to the loer bound is smaller than c Thus, hat can e learn from this constellation about the searching rocess? Is it necessary to consider constellations here job c is rocessed among set J? Anser is NO! Why? If e rocess c somehere among the jobs of set J, the solution considered before cannot be imroved since job c cannot be rocessed until the oint of time t c (due to a c!). Therefore, the solution is deteriorated by at least one time unit since the osition for is otimal according to J Consequently, e have to decide about the scheduling osition of c either before or after the set J Business Comuting and Oerations Research 47 7

28 Branching scheme This leads directly to the folloing branching scheme of the algorithm Alays roceed ith the node resulting in the loest bound found so far. The Loer Bound of a node S f(s) is alays derived from the maximum of f(f) (F=Father node of S), LB(J), and LB({c} U J) A ne node is added to the tree only if its loer bound is less than the uer bound f found so far Aly the Schrage rocedure in each node If the solution is otimal, the rocedure can be finished and the otimal result is generated Otherise, comute c and J Generate the to additional subsequent nodes c before J (=Node ) and c after J (=Node ) This can be easily conducted through an aimed modification of the considered instance Business Comuting and Oerations Research 48 Node After determining node c and the subsequent set J, e modify the tail of c in the folloing ay: qc = max qc, r q r J By doing so, the execution of the Schrage rocedure alays results in a constellation here c is rocessed before all jobs laced in set J. Additionally, the algorithm "nos" the extended tail of c to rocess this job otentially earlier Business Comuting and Oerations Research 49 Node After determining node c and the subsequent set J, e modify the head of c in the folloing ay: ac = max ac,min { ar r J} r r J By doing so, the execution of the Schrage rocedure alays results in a constellation here c is rocessed after all jobs laced in set J. Additionally, the algorithm "nos" the extended head of c to rocess other jobs otentially earlier Business Comuting and Oerations Research 4 8

29 Bound comutation and Schrage rocedure Carlier rooses a secific technique to be able to execute the Schrage rocedure in O(n. log n) time Uer bound comutation: Every time the Schrage rocedure is alied, the generated maesan is comared ith the current uer bound f. Moreover, an alternative constellation conserving the order of all jobs excet for job c, hich is rocessed after J, is additionally comared ith this uer bound An additional loer bound is derived from the alication of the Schrage rocedure ith alloed reemtions Business Comuting and Oerations Research 4 Preemtion The reemtion version of the Schrage algorithm maes use of the greedy rule of the original Schrage rocedure, but can additionally reemt each rocessed job henever another one arrives ith a larger tail It is trivial to sho that the generated solution is alays otimal and is therefore a Loer Bound of the original roblem In addition, for examle, by using hea data structures, this rocedure can be executed again in O(n. log n) stes Business Comuting and Oerations Research 4 Comutational results This rocedure as coded in FORTRAN on an IRIS 8 and initially tested on roblems For each roblem ith n jobs,. n integers ith uniform distributions beteen and a max, and max as ell as and q max ere resectively dran different values for n ere tested; n=5,, 5,,,. Further details can be found in Carlier (98) 999 roblems ere solved otimally One roblem ith n=85 as not solved (but the distance to bound as!). The loer bound as 9.8 (UB=9.8) In most cases the solution rocess taes only a small amount of time (extreme small-sized solution trees) Business Comuting and Oerations Research 4 9

30 Branch&Bound of Carlier Examle We consider the folloing examle Jobs i Job Job Job Job 4 Job 5 Job 6 Release dates a i Processing times i Tails q i Job 7 Business Comuting and Oerations Research 44 Alying Schrage Nr. Job Tail Start End Comleted Av None 7 5, , , None Critical Path: -4-s Business Comuting and Oerations Research 45 Analyzing the constellation c= and J={,,4} LB(J)=min{,,}674min{6,4,}= 7=49 LB({,,,4})=7=9 UB=5 No, e have to branch c before J c after J Business Comuting and Oerations Research 46

31 Enumeration tree UB=5 LB=49 Business Comuting and Oerations Research 47 c before J Ne roblem constellation Jobs i Job Job Release dates a i Processing times i Job Job 4 Job 5 Job 6 Tails q i 7= Job 7 Ne loer bound: LB({,,,4})==5 Hence, this node can be fathomed Business Comuting and Oerations Research 48 Enumeration tree UB=5 LB=49 c before J LB=5=UB is fathomed Business Comuting and Oerations Research 49

32 c after J Ne roblem constellation Jobs i Job Job Release dates a i 7 =8 Job Job 4 Job 5 Job 6 Job 7 Processing times i Tails q i Ne loer bound: LB({,,,4})=7=4 Hence, this node has to be exlored Business Comuting and Oerations Research 44 c after J Alying Schrage Nr. Job Tail Start End Comleted Av None , None Critical Path: s Business Comuting and Oerations Research 44 Analyzing the constellation c= and J={} LB(J)=min{}6min{6}=45 LB({,})=674=48 Therefore, e inherit the Loer Bound of the father node. This is LB=49 UB=5 No, e have to branch again c before J c after J Business Comuting and Oerations Research 44

33 Enumeration tree UB=5 LB=49 c before J LB=5 is fathomed UB=5 LB=49 Business Comuting and Oerations Research 44 c before J Ne roblem constellation Jobs i Job Job Job Job 4 Job 5 Job 6 Release dates a i 8 Processing times i Tails q i Job 7 Ne loer bound: LB({,})=6=5=UB Hence, this node is fathomed Business Comuting and Oerations Research 444 Enumeration tree UB=5 LB=49 c before J LB=5 is fathomed UB=5 LB=49 LB=5=UB is fathomed Business Comuting and Oerations Research 445

34 c after J Ne roblem constellation Jobs i Job Job Job Job 4 Job 5 Job 6 Release dates a i 8 9 Processing times i Tails q i Job 7 Ne loer bound: LB({,})=4=5=UB Hence, this node is fathomed Business Comuting and Oerations Research 446 Enumeration tree UB=5 LB=49 Otimal solution c before J UB=5 LB=49 LB=5 is fathomed LB=5=UB is fathomed LB=5=UB is fathomed Business Comuting and Oerations Research 447 Otimal solution Therefore, e obtained an otimal solution This otimal solution is given by Maesan is: 5 Business Comuting and Oerations Research 448 4

35 4.4 Multile stages If M>, each objective function itemized in Section 4. leads to an NP-comlete roblem for the general job-sho system case Therefore, a huge set of different heuristics can be found in literature Oing to its simle reresentation in disjunctive grahs, the minimization of the cycle time or the maesan is frequently ursued In comarison to other NP-comlete roblems, the jobsho roblem belongs to the most comlex ones. This results from the fact that most efficient exact rocedures are not able to solve even small-sized roblems in a reasonable time (e.g., jobs on machines) Business Comuting and Oerations Research Use of riority rules A more intuitive aroach can be the alication of dynamic rules deciding about the sequence on every machine Therefore, in case of an idle machine, this rule decides about the next job to be scheduled by selecting one of the aiting jobs Note that this aroach is very flexible since it can also be alied to dynamic roblems hile its comlexity only deends on the defined comutation of the integrated riority rule Frequently, the SPT and its variants integrated into secific hierarchies are alied Business Comuting and Oerations Research Elaborated heuristics Well-non aroaches are for examle The Shifting Bottlenec Procedure (SBP) of Adams, Balas and Zaac The Tabu Search rocedure of Noici and Smutnici Business Comuting and Oerations Research 45 5

36 4.4.. The Shifting Bottlenec Procedure This rocedure can be alied to arbitrary M- staged job-sho systems to minimize the cycle time It maes use of the Branch&Bound algorithm of Carlier as a subroutine The roblem descrition is defined as a disjunctive grah The bottlenec machine of the total schedule is considered to be lanned more accurately in each ste Business Comuting and Oerations Research 45 Basic attributes The machines are sequenced one at a time, consecutively In order to do so, a one-machine scheduling roblem ith head and tails is otimally solved for each not yet sequenced machine This result is taen as ran of the machine to decide about its necessity to sequence it ermanently. After sequencing the current machine, all machines sequenced before are resequenced otimally due to the modified heads and tails The one-machine roblems ith head and tails are constructed out of the modified disjunctive grah Business Comuting and Oerations Research 45 Deriving one-machine roblems Let M M be the set of machines that have already been sequenced by choosing selections ( ). For any \, let ( (, )) S M M M P M be the roblem obtained from the original roblem ( ) definition relacing each disjunctive arc set E M by the corresonding selection S ( M ) ( ) each disjunctive arc set E M \ M, and deleting Business Comuting and Oerations Research 454 6

37 The rocedure. M =Ø (set of already sequenced machines). Identify a bottlenec machine m among the machines in M\M and sequence it otimally by alying the Carlier algorithm. Set M =M U{m}. Reotimize the sequence of each critical machine in M in turn hile eeing the other sequences fixed, i.e., set M =M \{} and solve P(,M ). Then, if M =M, sto; otherise go to ste Business Comuting and Oerations Research 455 Reotimization rocesses The reotimization rocess is reeated at most three times for sets M < M in every iteration Every time a full cycle is comleted, the elements of M are reordered according to the nonincreasing values of the solutions of the resective one-machine roblems ith heads and tails In the last ste, hen M = M, e continue the local reotimization rocess to the oint here no more imrovement for a full cycle occurs Business Comuting and Oerations Research 456 To versions To different versions of the SBP are roosed by Balas et al. The first version oerates as described above The second one alies the SBP to the nodes of a searching tree generating several solutions simultaneously, i.e., in each branching ste, alternative constellations are generated to increase algorithm diversification Business Comuting and Oerations Research 457 7

38 The second SBP-version The rocess starts again ith the node defined by M =Ø In each branching ste, in a node on level l= M, the f(l) machines ith the largest resective machine objective value out of M\M are rocessed as alternative child nodes. Note that f is a monotonous decreasing function reducing the branching degree in the levels that are generated later. A second instrument for limiting the size of the branching tree is a enalty function defined for every node that enalizes the choices made at different levels in generating the node in question, in roortion to their deviation from the bottlenec, and ith a eighting that is heavier for the higher than for the loer levels of the tree. Whenever the value of the enalty function for a node exceeds a redetermined limit, the node is discarded Business Comuting and Oerations Research 458 Combined breadth-first / deth-first For the first l*=m / levels, the breadth first search is used to roduce all successors according to the function f, i.e., a full tree over l* is generated In the second art of the rocedure, the nodes are clustered into grous of size f(l*), containing the successors of level l*. Subsequently, the deth-first searching hase starts. In this hase, the highest raning member of one of the grous is chosen and exlored straight to the bottom of the search tree, or as far as the enalty function ermits The current best solution is alays stored as an uer bound. Hence, branches, hich reach the uer bound, are fathomed. After ending this exloration, the highest ran member of another grou of nodes is chosen Business Comuting and Oerations Research 459 Examle MS = ; PT = 4 Business Comuting and Oerations Research 46 8

39 Disjunctive grah Job s 4 Oeration Oeration Oeration Business Comuting and Oerations Research 46 One-machine roblems Machine Job Job Job Head 5 4 Processing time Tail 5 Business Comuting and Oerations Research 46 One-machine roblems Machine Job Job Job Head Processing time 4 Tail 6 4 Business Comuting and Oerations Research 46 9

40 One-machine roblems Machine Job Job Job Head 6 7 Processing time Tail Business Comuting and Oerations Research 464 Schrage rocedure Scheduling Machine Process job first. Start:; End:; Tail:8 Process job next. Start:4; End:7; Tail:8 Process job at last. Start:7; End:; Tail: Objective function value: Otimal solution since the loer bound is min{5,4}min{,}=46= Business Comuting and Oerations Research 465 Schrage rocedure Scheduling Machine Process job first. Start:; End:; Tail:8 Process job next. Start:; End:6; Tail: Process job at last. Start:6; End:9; Tail: Objective function value: Otimal solution since the loer bound is min{,,}4min{,6,4}=9= Business Comuting and Oerations Research 466 4

41 Schrage rocedure Scheduling Machine Process job first. Start:; End:5; Tail:8 Process job next. Start:6; End:8; Tail:8 Process job at last. Start:8; End:9; Tail:9 Objective function value:9 Otimal solution since the loer bound is min{6,7}min{,}=6=9 Business Comuting and Oerations Research 467 Bottlenec machine Machine : Comletion time is Machine : Comletion time is Machine : Comletion time is 9 Consequently, the bottlenec machine is Machine ith Z= Therefore, e fix the sequence: on this machine Business Comuting and Oerations Research 468 Disjunctive grah Job s 4 4 Oeration Oeration Oeration Business Comuting and Oerations Research 469 4

42 One-machine roblems Machine Job Job Job Head 5 6 Processing time Tail 5 Business Comuting and Oerations Research 47 One-machine roblems Machine Job Job Job Head 9 9 Processing time Tail Business Comuting and Oerations Research 47 Scheduling Machine The Schrage rocedure rovides the folloing schedule: Process job first. Start:; End:; Tail:8 Process job next. Start:5; End:8; Tail:8 Process job at last. Start:8; End:; Tail: Objective function value: Cannot be roven to be otimal since the loer bound is min{5,6}min{,}=56= J={}, c=; Business Comuting and Oerations Research 47 4

43 Modified Branching roblem (c before J) Machine (bold means modified value) Job Job =c Job =J Head 5 6 Processing time Tail 5 4= Business Comuting and Oerations Research 47 Rescheduling Machine c before J Schrage rocedure Process job first. Start:; End:; Tail:8 Process job next. Start:5; End:8; Tail: Process job at last. Start:8; End:; Tail: Objective function value: Is the otimal solution in the considered sub-tree since the loer bound=min{5,6}min{4,}=56= But already dominated by the solution considered before Business Comuting and Oerations Research 474 Modified Branching roblem c after J Machine (bold means modified value) Job Job =c Job =J Head 9=6 6 Processing time Tail 5 Business Comuting and Oerations Research 475 4

44 Rescheduling Machine c after J Schrage rocedure Process job first. Start:; End:; Tail:8 Process job next. Start:6; End:9; Tail: Process job at last. Start:9; End:; Tail: Objective function value: Is the otimal solution in the considered sub-tree since the loer bound amounts to min{9,6}min{,}=66= Business Comuting and Oerations Research 476 Schrage rocedure Scheduling Machine Process job first. Start:; End:5; Tail:8 Process job next. Start:9; End:; Tail: Process job at last. Start:; End:; Tail: Objective function value: Otimal solution since the loer bound is min{9,9}min{,}=9= Business Comuting and Oerations Research 477 Bottlenec machine Machine : Comletion time is Machine : Comletion time is Consequently, the bottlenec machine is machine ith Z= Therefore, e fix the sequence: on this machine Business Comuting and Oerations Research

45 Disjunctive grah Job s 4 4 Oeration Oeration Oeration Business Comuting and Oerations Research 479 Rescheduling Machine No, e have to reotimize the sequence on Machine according to the otentially modified head and tails Therefore, e erase the fixed disjunctive arcs in the grah to derive the modified scheduling roblem ith head and tails Business Comuting and Oerations Research 48 Job Disjunctive grah s 4 Oeration Oeration Oeration Business Comuting and Oerations Research 48 45

46 Modified one-machine roblem Machine (bold means modified value) Job Job Job Head Processing time 4 Tail 4 Business Comuting and Oerations Research 48 Schrage rocedure Rescheduling Machine Process job first. Start:; End:; Tail: Process job next. Start:; End:6; Tail: Process job at last. Start:6; End:9; Tail: Objective function value: Otimal solution since the loer bound for set s={} is min{}min{}== The sequence on Machine is et unchanged! Business Comuting and Oerations Research 48 Disjunctive grah Job s 4 4 Oeration Oeration Oeration Business Comuting and Oerations Research

47 One-machine roblems Machine (bold means modified value) Job Job Job Head 9 Processing time Tail 7 Business Comuting and Oerations Research 485 Scheduling Machine Schrage rocedure Process job first. Start:; End:5; Tail: Process job next. Start:9; End:; Tail: Process job at last. Start:; End:; Tail: Objective function value: Otimal solution since the loer bound is min{9,}min{,}=9= Fixing sequence on Machine to Business Comuting and Oerations Research 486 Job Disjunctive grah s 4 4 Oeration Oeration Oeration Business Comuting and Oerations Research

48 Resequencing Subsequently, e have to resequence the already scheduled Machines and The current objective values of these machines are: Machine : Machine : We tae Machine as the first machine to be rescheduled Business Comuting and Oerations Research 488 Rescheduling Machine No, e have to reotimize the sequence on Machine according to the otentially modified head and tails Therefore, e erase the fixed disjunctive arcs in the grah to derive the modified scheduling roblem ith head and tails Business Comuting and Oerations Research 489 Job Disjunctive grah s 4 Oeration Oeration Oeration Business Comuting and Oerations Research 49 48

49 Modified one-machine roblem Machine (bold means modified value) Job Job Job Head Processing time 4 Tail 4 Business Comuting and Oerations Research 49 Schrage rocedure Rescheduling Machine Process job first. Start:; End:; Tail: Process job next. Start:; End:6; Tail: Process job at last. Start:6; End:9; Tail: Objective function value: Otimal solution since the loer bound for set s={} is min{}min{}== The sequence on Machine is et unchanged! Business Comuting and Oerations Research 49 Job Disjunctive grah s 4 4 Oeration Oeration Oeration Business Comuting and Oerations Research 49 49

50 Rescheduling Machine No, e have to reotimize the sequence on Machine according to the otentially modified heads and tails Therefore, e erase the fixed disjunctive arcs in the grah to derive the scheduling roblem ith modified heads and tails Business Comuting and Oerations Research 494 Disjunctive grah Job s 4 4 Oeration Oeration Oeration Business Comuting and Oerations Research 495 One-machine roblems Machine (bold means modified value) Job Job Job Head 5 6 Processing time Tail 6 Business Comuting and Oerations Research 496 5

51 Rescheduling Machine Schrage rocedure Process job first. Start:; End:; Tail:9 Process job next. Start:5; End:8; Tail:8 Process job at last. Start:8; End:; Tail: Objective function value: Cannot be roven to be otimal since the loer bound is min{5,6}min{,}=56= J={}, c=; Business Comuting and Oerations Research 497 Modified Branching Problem c before J Machine (bold means modified value) Job Job =c Job =J Head 5 6 Processing time Tail 6 4= Business Comuting and Oerations Research 498 Rescheduling Machine c before J Schrage rocedure Process job first. Start:; End:; Tail:9 Process job next. Start:5; End:8; Tail: Process job at last. Start:8; End:; Tail: Objective function value: Is the otimal solution in the considered sub-tree since the loer bound=min{5,6}min{4,}=56= But already dominated by the solution considered before Business Comuting and Oerations Research 499 5

52 Modified Branching Problem c after J Machine (bold means modified value) Job Job =c Job =J Head 9=6 6 Processing time Tail 6 Business Comuting and Oerations Research 5 Rescheduling Machine c after J Schrage rocedure Process job first. Start:; End:; Tail:9 Process job next. Start:6; End:9; Tail: Process job at last. Start:9; End:; Tail: Objective function value: Is the otimal solution in the considered sub-tree since the loer bound=min{9,6}min{,}=66= Business Comuting and Oerations Research 5 Rescheduling Machine The sequence of Machine is et unchanged! is the chosen sequence! Business Comuting and Oerations Research 5 5

53 Disjunctive grah of the final solution Job s 4 4 Oeration Oeration Oeration Business Comuting and Oerations Research 5 Objective function value The resulting maesan is determined by the length of the longest ath from to s This ath has the total length of, hich defines the resulting cycle time This is illustrated by the final grah Business Comuting and Oerations Research 54 Longest ath Job s 4 4 Length== Oeration Oeration Oeration Business Comuting and Oerations Research 55 5

54 Priority rule alication In order to rate the solution quality of the SBP versions, different riority rules are alied in a secific constellation. The alied rules are FCFS (=First Come First Serve) LST (=Late Start Time) EFT (=Early Finish Time) LFT (=Late Finish Time) MINSLK (=Minimum Slac) SPT (=Shortest Processing Time) LPT (=Longest Processing Time) MIS (=Most Immediate Successors) FA (=First Available) RANDOM Business Comuting and Oerations Research 56 Comutational results Procedures ere imlemented in FORTRAN on a VAX 78/ on 4 roblems taen from ellnon benchmars In hat follos, e deict the results resented by Balas et al. They tested the SBP in its both variants against some simle riority rules The consumed CPU time is illustrated in the tables beside the solution quality Business Comuting and Oerations Research 57 Priority rule alication First, the riority rule algorithms are alied in a straightforard fashion Second, the riority rules are alied in a random fashion by alying all rules The randomized rule is to select one of the available oerations to be rocessed next randomly This is done by alying a robability distribution hich maes the odds of being selected roortional to the riority assigned to each oeration by the given disatching rule The run is reeated until ten consecutive runs roduce no imrovement, and the best result obtained is reorted as the rocedure s outut Business Comuting and Oerations Research 58 54

55 Performance results Number of SBI SBII Instance Machines Jobs Oerations Value CPU Sec Micro-runs Value CPU Sec Macro-runs LB 5 4 * * *# * * * * * * Value: maesan of the best schedule obtained Micro-runs: number of the one-machine roblems solved Macro-runs: number of times SBI as run *: value non to be otimal #: otimal value found after seconds LB: loer bound given by solution value for the first bottlenec roblem Business Comuting and Oerations Research 59 Performance results ith 5 machines Priority Disatching Rule SBI SBII Imrovement Straight Randomized Value CPU Sec Value CPU Sec LB SBI % SBII % Problem Value CPU Sec Value CPU Sec 5 machines, jobs * * machines, 5 jobs * * * * * * 7 959* machines, jobs * * * * * Business Comuting and Oerations Research 5 Performance results ith machines Priority Disatching Rule SBI SBII Imrovement Straight Randomized Value CPU Sec Value CPU Sec LB SBI % SBII % Problem Value CPU Sec Value CPU Sec machines, jobs ** ** ** ** ** machines, 5 jobs ** ** * 5** ** ** machines, jobs ** ** ** ** * 55** Business Comuting and Oerations Research 5 55

56 Further erformance results Priority Disatching Rule SBI SBII Imrovement Straight Randomized Value CPU Sec Value CPU Sec LB SBI % SBII % Problem Value CPU Sec Value CPU Sec machines, jobs * * * * * machines, 5 jobs ** ** ** ** , ** Value: LB: maesan of the best schedule obtained loer bound given by solution value for the first bottlenec roblem Imrovement:imrovement (in ercent) in solution value over that found by the randomized riority disatching rule *: value roved to be otimal **: time limit set to time required by randomized riority disatching rule. Business Comuting and Oerations Research 5 Main results Priority rules: No domination beteen the rules can be identified Eight of the ten rules shoed best result on at least one roblem To rules (LPT and FA) never Priority rules vs. SBP I/II In 8 cases SBP I finds better solutions than the constellations generated by the riority rule rocedure hether in the straight or randomized version Furthermore, Version finds substantially imroved solutions for many constellations most of the time Altogether, it can be stated that SBP II is alays ithout excetion at least as good as the randomized riority rule Moreover, in the vast majority of the considered cases, it is considerably better Tyical average imrovement rates ere beteen 4 and ercent Business Comuting and Oerations Research 5 Pros SBP Pros and Cons Elaborated rocedure Desite the fact that the rocedure uses a Branch&Bound rocedure to tacle an NP-hard roblem as a frequently called subroutine, it is quite fast in comarison to ell-non meta strategies, as for examle, the Tabu Search rocedure of Noici and Smutnici SBP I is frequently used as an initial rocedure to generate a first solution ith quite good quality Cons Solution quality is oorer than non from elaborated meta strategies Single riority rules are much faster Business Comuting and Oerations Research 54 56

57 4.4.. Tabu Search by Noici-Smutnici Besides the SBP as ell as various Branch&Bound-rocedures, meta heuristics have recently been develoed for the job-sho scheduling roblem ith maesan objective A very efficient and relatively easy to imlement algorithm is the Tabu Search (TS) rocedure introduced by Noici and Smutnici in 996 The algorithm bases on the disjunctive grah and tries to reduce the roblems maesan iteratively by changing the job sequence ithin the critical ath Business Comuting and Oerations Research 55 Neighborhood Search Tabu Search methods are based on Neighborhood Search, a local search method. Given a solution s, a Neighborhood Search creates out of a solution π a ne solution π by maniulating π; this oeration is called a move. The set of moves alicable on a given solution s is called the neighborhood N(π). Neighborhood Search selects the best move in N(π) and alies it. If a solution can be reresented as a ermutation of numbers, common Neighborhood Search moves are sas and shifts ithin this ermutation. Business Comuting and Oerations Research 56 Critical ath Let u = (u,,u ) denote the critical ath, ith the number of oerations on a longest ath ithin the directed disjunctive grah The ath can be divided into blocs B,, B r ith the folloing attributes: B i = (u a i, u ai,, u b i ) and =a b <b =a b <b =a a r b r = B i contains all oerations rocessed on the same machine (i =,, r) To consecutive blocs contain oerations rocessed on different machines, i.e., µ(b i ) µ(b i ), i =,, r- Business Comuting and Oerations Research 57 57

58 Critical ath and bloc reresentation s 4 4 Critical ath Bloc Business Comuting and Oerations Research 58 Idea Permuting the job sequence ithin a bloc can yield to a schedule ith smaller maesan! But ho? Business Comuting and Oerations Research 59 The alied neighborhood The size of the neighborhood lays an imortant role. Thus, Noici and Smutnici introduced a reduced neighborhood ith the folloing moves: In bloc B the last to oerations are ermutated In bloc B to B r- the first to oerations and, if a i <b i, the last to oerations are ermuted In bloc B r the first to oerations are ermutated Business Comuting and Oerations Research 5 58

59 Mathematical reresentation of the moves Let V(π) = (V (π),, V r (π)) denote the set of moves that are alicable to a given job sequence π. {( ub, ub )} if a < b and r > V ( π ) = else {( u, u a ),( ub, u )} b if < bi ai a i i i i Vi ( π ) = else {( ua, u )} < and > r a if a r r br r Vr ( π ) = else Business Comuting and Oerations Research 5 Alied to our examle s 4 4 Critical ath Bloc Only the exchange of these to oerations is examined Business Comuting and Oerations Research 5 Remar to the critical ath One might argue that the critical ath is not ell-defined. But numerical results shoed that the selection of one critical ath has a minor influence in regard to the solutions quality. An arbitrary critical ath can be chosen. Business Comuting and Oerations Research 5 59

60 Tabu List A major drabac of local search rocedures, such as hill-climbing, is cycling beteen to solutions and only returning a local otimal solution In order to avoid cycling ithin the searching rocess, Tabu Search algorithms use a short time memory of bloced moves, called Tabu List If a move v=(x,y) is erformed, the inverse move v =(y,x) is added to the Tabu List The Tabu List has a given size maxt, and it contains the inverse moves of the moves alied in the last maxt iterations Business Comuting and Oerations Research 54 Asiration criterion To secure that romising moves are not bloced, Noici and Smutnici divide the set oerations in the Tabu List into subsets UP and UNP UP contains all bloced moves leading to a better solution than the ones visited in all ast iterations (rofitable moves) UNP consists of all bloced non-rofitable moves A criterion, called asiration criterion, allos the search to erform a rofitable move although it belongs to the Tabu List. Business Comuting and Oerations Research 55 Long time memory If a solution ossesses a good objective function value, it is liely that its neighbors contain good objective function values as ell Since from a given solution ith small objective function value only the best move as chosen, the observation of other neighbors as discarded although they could guide into regions ith good solutions, too To tae this thought into account, Noici and Smutnici roosed to embed their rocedure into a guided suer routine by storing the solutions ith the loest objective function values ithin a list L The elements of L consist of the ermutation π for the given solution, a modified neighborhood N(π )\{v } (v is the already alied move), and the Tabu List T Business Comuting and Oerations Research 56 6

61 Performance analysis test sets Noici and Smutnici tested their algorithms on grous of ell-non job-sho scheduling instances Grou I: 45 instances ith 6 to oerations Grou II: 8 instances ith 5 to oerations 4 instances created by a random generator ith 5 to oerations Business Comuting and Oerations Research 57 Performance analysis - results Test set Number of instances C* better than bestnon value Otimality roven Grou I 45 In of unnon cases* Grou II a) 8 b) 4 No references available In of 6 unnon cases* No references available *Unnon u to Noici and Smutnici (996) Business Comuting and Oerations Research 58 Noici and Smutnici Summary The rocedure is a solution method for solving the jobsho scheduling roblem ith maesan objective hich is relatively easy It substantially intensifies the searching rocess in romising regions, evaluates the neighborhoods in the single stes very fast, and the authors can imrove the best non maesan for difficult roblem instances in many cases In 5, Noici and Smutnici roose a further advanced Tabu Search rocedure ith imroved diversification (long term behavior) Additionally, they roose an imroved starting heuristic in order to construct a suitable starting solution Business Comuting and Oerations Research 59 6

62 4.5 Flo-sho roblems In the folloing, e consider flo-sho roblems as a secial case of job-sho systems In this secial case, each job has an identical machine sequence in hich it is rocessed Therefore, e can define a definite numbering (,,M) of the used resources that determine the rocessing sequence of each job Desite the fact that the total solution sace still consists of altogether (N!) M constellations, this roblem seems to be someho relaxed in comarison to the general jobsho roblem Business Comuting and Oerations Research 5 The dominance criteria The first dominance criterion: In an M-staged flo-sho system, there is alays an otimal solution minimizing the maesan here the scheduling on the first to machines is identical. This is also true for the minimization of the total lead time. Business Comuting and Oerations Research 5 Proof of the first dominance criterion Let us assume there are unequal sequences that are rocessed on the first to machines Let N be the job sequence alied to machine We define t as the first (loest numbered) osition in this job sequence here a difference beteen the sequences on machine and machine arises Therefore, e have the sequence t- l ith l>t at the first stage In hat follos, e consider an alternative constellation by exchanging t and l on machine Business Comuting and Oerations Research 5 6

63 Illustration,,,,l,t,t,l Alternative rocessing.,,,,t,l,t,l c Business Comuting and Oerations Research 5 Consequences for jobs Job t The exchange on machine can imrove only the subsequent constellation by an earlier rocessing at stage Therefore, the remaining schedule is of better or at least of an equal quality Job l Firstly, note that the end of rocessing of job l at stage in the modified constellation is equal to the oint of time the rocessing of job t ends at stage in the original constellation. Let c denote this oint of time in both schedules Therefore, job l can not be rocessed at stage before t is rocessed. Note that this alies to both schedules Schedule : Reason: Processing of job t at stage before job l. In addition, job t at stage has to ait for its rocessing at stage, hich is not ended before c Schedule : Reason: Processing of job l at stage. In detail, job l at stage has to ait for its rocessing at stage, hich is not ended before c Business Comuting and Oerations Research 54 Conclusions The rocessing times of job t and job l at stage are not influenced by the executed exchange No effect on the total maesan as ell as on the resulting cycle time This comletes the roof Business Comuting and Oerations Research 55 6

64 The dominance criteria The second dominance criterion: In an M-staged flo-sho system there is alays an otimal solution generating the minimal maesan here the scheduling on the last to machines is identical Business Comuting and Oerations Research 56 Proof of the second dominance criterion Again, e assume that there is no equal sequence at the last to stages In detail, e assume that the sequence N is rocessed on machine M- Let t be the minimal number of a job here the sequences at stages M- and M differ, i.e., l, ith l>t is rocessed on machine M No, e consider an alternative constellation here e exchange job l and job t at the last stage Business Comuting and Oerations Research 57 Illustration y M- M-, M-, M-, M-,t M-,l M M,l M,t M- M Alternative rocessing. M-, M-, M-, M-,t M-,l x M,t M,l c Business Comuting and Oerations Research 58 64

65 Consequences Otimal Schedule Let x be the beginning of the rocessing of job l at stage M hile y denotes the end of rocessing of job l at stage M- We no x y We have cycle time C Alternative Schedule In order to comute the maesan for this schedule, e no that nothing is lost due to the executed exchange of l and t Moreover, there is no side-effect on the jobs rocessed after job t, herefore the ne cycle time C is loer than or equal to c This comletes the roof Business Comuting and Oerations Research The rocedure of Johnson In hat follos, e consider the secial case of the flo-sho roblem here only roduction stages are given Note that already this quite simle constellation is an NP-comlete roblem for the general job-sho case In contrast to this, e ill resent a very efficient algorithm generating an otimal schedule for the -staged flo-sho roblem in O(N log N) stes: the so-called Johnson algorithm. This algorithm determines the schedule ith the minimal maesan Business Comuting and Oerations Research 54 Changing the sequence in /-FS Theorem For the to- or three-staged flo-sho ith the objective of maesan minimization, there is alays an otimal solution that has equal scheduling sequences on all machines Business Comuting and Oerations Research 54 65

66 Proof of the Theorem Clearly, e may aly the to dominance criteria Thus, the claim of the Theorem follos immediately as a corollary of both criteria This comletes the roof Business Comuting and Oerations Research Algorithm (Johnson) { N} Initialization: R =,,,..., is the list of all jobs to be scheduled. Determine job nˆ by: {, nˆ, nˆ } = {, n, n n N}, nˆ = {, nˆ, nˆ } min, min,. If min,, then nˆ is laced on the next available osition at the to of the current schedule; otherise nˆ is laced on the next available osition at the end of the current schedule. Delete nˆ from the list of jobs to be rocessed in the schedule 4. Proceed ith ste until the list of remaining jobs R is emty Business Comuting and Oerations Research Theorem The roof of otimality Algorithm generates an otimal solution for the maesan minimization roblem in a tostaged flo-sho roduction system Business Comuting and Oerations Research

67 Proof of the Theorem First of all, e have to introduce some additional arameters to determine ho the maesan is affected by the chosen schedule Since rocessing of the first machine is never a bottlenec, e concentrate on the second one Idle times and the total rocessing time at this stage determine the sought maesan Therefore, I j should determine the amount of idle times on machine before rocessing job j, i.e., if the machine as not idle, the arameter is set to zero Note that these values deend on the generated solution hile the total rocessing time at stage is alays fixed Machines,,,,4,5,6 I, I 6,,,4,5,6 Business Comuting and Oerations Research 545 Calculation of the occurring idle times I = I = max I = max..., j {,, I,,} { I I,}, {,..., N},, : I j = max j i=, i, j I i i=, j i=, i, Business Comuting and Oerations Research 546 The objective value Z = N, i i= Constant roduction time at stage N Ii i = Additional solution -deendent aiting time at stage First, e have to comute the total sum of aiting times, i.e., N I i i= Business Comuting and Oerations Research

68 Comuting the total sum of aiting times N N i= I i i= = N i= Lemma : max Ii = max i= i =, i, i= i, i I = i =, N, Business Comuting and Oerations Research 548 Proof of the Lemma We sho the claim by induction: Start: N = I = max I, = max I, i i i i,,,, i i = = = = = = = = { } = max, =,, N N N It holds: Ii = max, i, i i = i = i = N Business Comuting and Oerations Research 549 Proof of the Lemma We comute I = I max I, N N N N N i i, i i, i i = i = i = i = i = N N N, i, i N, i Ii, i i = i = i = i = i = = max max, Business Comuting and Oerations Research 55 68

69 Proof of the Lemma Case : N N N > I, i, i i i i i = = = N N N The difference I is added hile the resulting, i, i i i i i = = = N N value is equal to max N since is the ne maximum. Therefore, the lemma gives the correct calculation in this case.,,,, i i i i i i i i = = = = Business Comuting and Oerations Research 55 Proof of the Lemma Case : N N N I, i, i i i i i = = = N N N N N The difference I is at most zero and, therefore,, i, i i i i i = = = zero is added to the sum. In addition, it holds: max N, herefore the, i, i, i, i i i i = = = i = total sum is equal to max, i, i N. i = i = Therefore, the lemma gives the correct calculation also in this second case. Business Comuting and Oerations Research 55 Intermediate summary A found solution can only influence aiting times on machine to minimize the maesan We have generated a comact form for comuting all resulting aiting times on machine Business Comuting and Oerations Research 55 69

70 Notation In hat follos, e mae use of the folloing additional arameters: {,..., N} : Y, i {,..., N} : Ii = max{ Yt t N} i= = i= i=, i Business Comuting and Oerations Research 554 Transformation No, the reliminary or is done to start the roof of the theorem by transforming an otimal solution into a ne one resecting the claimed attributes Therefore, let us assume e have an otimal schedule S found ith sequence ( N) Furthermore, let us assume e have a solution not fulfilling the construction rules of the algorithm of Johnson If so, there is a minimally chosen index i ith: {, i, i } > {, i, i } min, min, Business Comuting and Oerations Research 555 Transformation Note that in the algorithm of Johnson it ould have been rocessed after i No, e generate schedule T out of S by exchanging the jobs i and i Oing to this simle modification, e can easily derive the ne udated objective function by comuting the ne Y -values Business Comuting and Oerations Research 556 7

71 Comaring schedules S and T T Let us consider Y as the value for job under schedule T i i i,, = = i i T i,,, i = = i i,, = = ( ) Y = i i i N ( ) Y = i i i N Y = i i T i =,, i, = = Y i All other values are not affected! Business Comuting and Oerations Research 557 max = = i = i = i = i = Comaring schedules S and T T T { Y, Y },, max = = { Y, Y },, i i i = i = i i = i i = = max,,,, max, i = max, i i = {, } i = max, i, {,} {, }, i, max max, i i =, i i =, i, i {, }, i,, i, i, i, =, i,, i, i i, i =, Business Comuting and Oerations Research 558 i =,,, i i =, Comaring schedules S and T Case i i { } T : max,,,, i, i, i = = i i { } S : max,,,, i, i, i = = i {, i, i } > {, i, i } { } { } We no: min, min, Case : = min, > min, T :, i, i, i, i, i i,, = = i i, i, i, i =,,, i, i, i = = Business Comuting and Oerations Research 559 7

72 Comaring schedules S and T Case i i { } S : max,,,, i, i, i = = i i { i i} { } { } T S : max,,,,, <, i min, i,, i max,, i, i = = if, i, i =, if, i, i =, i, i, otherise =, i, i, otherise Business Comuting and Oerations Research 56 Comaring schedules S and T i i { } T : max,,,, i, i, i = = i i { } S : max,,,, i, i, i = = {, i, i } > {, i, i } We no: min, min, Business Comuting and Oerations Research 56 Comaring schedules S and T Case i i { } { } { } Case : = min, > min,, i, i, i, i, i T :,,, i = = i i { } S : max,,,, i, i, i = = { } {, i, i} T S : max, < min,, i, i, i, i max, =, i, i, i = if,, =, if,, i i i i =,,, otherise =,,, otherise i i i i Business Comuting and Oerations Research 56 7

73 Conclusion T is not orse than S in both cases As a consequence, each otimal schedule can be transformed into a Johnson schedule ithout losing its otimality This comletes the roof Business Comuting and Oerations Research 56 Examle Given: Machines A and B, and 5 jobs to be rocessed Processing times Machine Jobs 4 5 A 5 7 B Business Comuting and Oerations Research 564 Stes of Johnson s algorithm. Minimum is 4 on A Consequence: First ossible osition (4,-,-,-,-). Minimum is on B Consequence: Last ossible osition (4,-,-,-,). Minimum is on A Consequence: First ossible osition (4,,-,-,) 4. Minimum is 5 on B Consequence: Last ossible osition (4,,-,5,) 5. Comlete otimal schedule is (4,,,5,) Business Comuting and Oerations Research 565 7

74 4.5. The multile-stage case No, e consider the general case M> Unfortunately, it as shon that these roblems are NP-hard Therefore, e introduce a simle heuristic aroach in the folloing Business Comuting and Oerations Research 566 Palmer s heuristic The guideline suggested by Palmer as a very first heuristic for sequencing M-staged flo-sho systems is as follos Give riority to jobs ith the strongest tendency to rogress from short times to long times in the sequence of oerations In detail, Palmer rooses the folloing riority calculation to measure this attribute in each job Business Comuting and Oerations Research 567 Palmer s heuristic {,..., } : M j ( ( ) ) j N s = M t M = : = ( ) ( ) j, M s = t t = t t j j j j j,,,, M = : ( ) ( ) ( ) s = t t t j j j j,,, = t t t M = 4 : j j j,,, ( ) ( ) ( ) s = 4 t 4 t 4 t 4 t j j j j,4,, = t t t t j j j j,4,,, ( ) j, Business Comuting and Oerations Research

75 Solution The jobs are scheduled in sequence of nonincreasing riority Generates only solutions ith an equal sequence at all stages Business Comuting and Oerations Research 569 Examle Processing time of job j on machine Jobs 4 t j, 7 t j, 4 9 t j, 5 Business Comuting and Oerations Research 57 Priorities Processing time of job j on machine t j, - t j, t j, Jobs Priority Business Comuting and Oerations Research 57 75

76 Solution Is 4 Business Comuting and Oerations Research 57 CDS Heuristic CDS= Cambel Dude Smith, the authors of the resective aer Extension of the Johnson algorithm for multile-stage cases Considers only solutions ith equal sequences at all stages Note that it starts from at least four stages Generates artificial -staged roblems out of the general constellation and solves them otimally by the alication of the Johnson algorithm For M= the CDS rocedure becomes the Johnson algorithm generating an otimal solution Otherise, the rocedure generates M- iterations reresenting an additional to-staged flo-sho roblem Can be used for the minimization of cycle time or total lead time Business Comuting and Oerations Research 57 CDS rocedure. Establish NxM-matrix of rocessing times t j,i, here t j,i is the rocessing time of j-th job on machine i. Establish number of auxiliary n-job, -machine roblems,, to be calculated, here M-. Set = for first auxiliary roblem 4. Comute the rocessing time for all jobs j=,,n on the to machines in the -th auxiliary roblem: θ j, = i= t j, i Business Comuting and Oerations Research

77 CDS Procedure 5. Comute the rocessing time for all jobs j=,,n on the to machines in the -th auxiliary roblem: θ j, = M t j, i i= M 6. Solve the roblem ith the Johnson algorithm 7. Chec if <. If so, set =, go to ste ; Otherise roceed ith ste 8 8. Use the original roblem to comute the objective value of all generated solutions 9. Select best result as the outut of the rocedure Business Comuting and Oerations Research 575 Examle Given: 4 Machines A, B, C and D, as ell as 5 jobs to be rocessed Processing times Machine Jobs 4 5 A B 4 9 C D 6 8 Business Comuting and Oerations Research 576 First iteration Machine Jobs Business Comuting and Oerations Research