# Scheduling Single Machine Scheduling. Tim Nieberg

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1 Scheduling Single Machine Scheduling Tim Nieberg

2 Single machine models Observation: for non-preemptive problems and regular objectives, a sequence in which the jobs are processed is sufficient to describe a solution Dispatching (priority) rules static rules - not time dependent e.g. shortest processing time first, earliest due date first dynamic rules - time dependent e.g. minimum slack first (slack= d j p j t; t current time) for some problems dispatching rules lead to optimal solutions

3 Single machine models: 1 w j C j Given: n jobs with processing times p 1,...,p n and weights w 1,...,w n Consider case: w 1 =... = w n (= 1):

4 Single machine models: 1 w j C j Given: n jobs with processing times p 1,...,p n and weights w 1,...,w n Consider special case: w 1 =... = w n (= 1): SPT-rule: shortest processing time first Theorem: SPT is optimal for 1 C j Proof: by an exchange argument (on board) Complexity: O(n log(n))

5 Single machine models: 1 w j C j General case WSPT-rule: weighted shortest processing time first, i.e. sort jobs by increasing p j /w j -values Theorem: WSPT is optimal for 1 w j C j Proof: by an exchange argument (exercise) Complexity: O(n log(n)) Further results: 1 tree w j C j can be solved by in polynomial time (O(n log(n))) (see [Brucker 2004]) 1 prec C j is NP-hard in the strong sense (see [Brucker 2004])

6 Single machine models: 1 prec f max Given: n jobs with processing times p 1,...,p n precedence constraints between the jobs regular functions f 1,...,f n objective criterion f max = max{f 1 (C 1 ),...,f n (C n )} Observation: completion time of last job = p j

7 Single machine models: 1 prec f max Given: n jobs with processing times p 1,...,p n precedence constraints between the jobs regular functions f 1,...,f n objective criterion f max = max{f 1 (C 1 ),...,f n (C n )} Observation: Method completion time of last job = p j plan backwards from p j to 0 from all available jobs (jobs from which all successors have already been scheduled), schedule job which is cheapest on that position

8 Single machine models: 1 prec f max S set of already scheduled jobs (initial: S = ) J set of all jobs, which successors have been scheduled (initial: all jobs t time where next job will be completed (initial: t = p j ) Algorithm 1 prec f max (Lawler s Algorithm) REPEAT select j J such that f j (t) = min k J f k (t); schedule j such that it completes at t; add j to S, delete j from J and update J; t := t p j ; UNTIL J =.

9 Single machine models: 1 prec f max Theorem: Algorithm 1 prec f max is optimal for 1 prec f max Proof: on the board Complexity: O(n 2 )

10 Single machine models: Maximum Lateness Problem 1 L max : Earliest due date first (EDD) is optimal for 1 L max (Jackson s EDD rule) Proof: special case of Lawler s algorithm

11 Single machine models: Maximum Lateness Problem 1 L max : Earliest due date first (EDD) is optimal for 1 L max (Jackson s EDD rule) Proof: special case of Lawler s algorithm Problem 1 r j C max : 1 r j C max 1 L max define d j := K r j, with constant K > maxr j reversing the optimal schedule of this 1 L max -problem gives an optimal schedule for the 1 r j C max -problem

12 Single machine models: Maximum Lateness Problem 1 prec L max : if d j < d k whenever j k, any EDD schedule respects the precedence constraints, i.e. in this case EDD is optimal defining d j := min{d j,d k p k } if j k does not increase L max in any feasible schedule

13 Single machine models: Maximum Lateness Problem 1 prec L max : if d j < d k whenever j k, any EDD schedule respects the precedence constraints, i.e. in this case EDD is optimal defining d j := min{d j,d k p k } if j k does not increase L max in any feasible schedule Algorithm 1 prec L max 1 make due dates consistent: set d j = min{d j,min k j k d k p k } 2 apply EDD rule with modified due dates

14 Single machine models: Maximum Lateness Remarks on Algorithm 1 prec L max leads to an optimal solution Step 1 can be realized in O(n 2 ) problem 1 prec L max can be solved without knowledge of the processing times, whereas Lawler s Algorithm (which also solves this problem) in general needs this knowledge (Exercise), Problem 1 r j,prec C max 1 prec L max

15 Single machine models: Maximum Lateness Problem 1 r j L max : problem 1 r j L max is NP-hard Proof: by reduction from 3-PARTITION (on the board)

16 Single machine models: Maximum Lateness Problem 1 pmtn,r j L max : preemptive EDD-rule: at each point in time, schedule an available job (job, which release date has passed) with earliest due date. preemptive EDD-rule leads to at most k preemptions (k = number of distinct release dates)

17 Single machine models: Maximum Lateness Problem 1 pmtn,r j L max : preemptive EDD-rule: at each point in time, schedule an available job (job, which release date has passed) with earliest due date. preemptive EDD-rule leads to at most k preemptions (k = number of distinct release dates) preemptive EDD solves problem 1 pmtn,r j L max Proof (on the board) uses following results: L max r(s) + p(s) d(s) for any S {1,..., n}, where r(s) = min j S r j, p(s) = j S p j, d(s) = max j S d j preemptive EDD leads to a schedule with L max = max S {1,...,n} r(s) + p(s) d(s)

18 Single machine models: Maximum Lateness Remarks on preemptive EDD-rule for 1 pmtn,r j L max : can be implemented in O(n log(n)) is an on-line algorithm after modification of release and due-dates, preemptive EDD solves also 1 prec,pmtn,r j L max

19 Single machine models: Maximum Lateness Approximation algorithms for problem 1 r j L max : a polynomial algorithm A is called an α-approximation for problem P if for every instance I of P algorithm A yields an objective value f A (I) which is bounded by a factor α of the optimal value f (I); i.e. f A (I) αf (I)

20 Single machine models: Maximum Lateness Approximation algorithms for problem 1 r j L max : a polynomial algorithm A is called an α-approximation for problem P if for every instance I of P algorithm A yields an objective value f A (I) which is bounded by a factor α of the optimal value f (I); i.e. f A (I) αf (I) for the objective L max, α-approximation does not make sense since L max may get negative for the objective T max, an α-approximation with a constant α implies P = N P (if T max = 0 an α-approximation is optimal)

21 Single machine models: Maximum Lateness The head-body-tail problem (1 r j,d j < 0 L max ) n jobs job j: release date r j (head), processing time p j (body), delivery time q j (tail) starting time S j r j ; completion time C j = S j + p j delivered at C j + q j goal: minimize max n j=1 C j + q j

22 Single machine models: Maximum Lateness The head-body-tail problem (1 r j,d j < 0 L max ), (cont.) define d j = q j, i.e. the due dates get negative! result: min n j=1 C j + q j = min n j=1 C j d j = min n j=1 L j = L max head-body-tail problem equivalent with 1 r j L max -problem with negative due dates Notation: 1 r j,d j < 0 L max an instance of the head-body-tail problem defined by n triples (r j,p j,q j ) is equivalent to an inverse instance defined by n triples (q j,p j,r j ) for the head-body-tail problem considering approximation algorithms makes sense

23 Single machine models: Maximum Lateness The head-body-tail problem (1 r j,d j < 0 L max ), (cont.) L max r(s) + p(s) + q(s) for any S {1,...,n}, where r(s) = min r j, p(s) = p j, q(s) = min q j j S j S j S (follows from L max r(s) + p(s) d(s))

24 Single machine models: Maximum Lateness Approximation ratio for EDD for problem 1 r j,d j < 0 L max structure of a schedule S t = r(q) C c Q c critical job c of a schedule: job with L c = maxl j critical sequence Q: jobs processed in the interval [t,c c ], where t is the earliest time that the machine is not idle in [t,c c ] if q c = min j Q q j the schedule is optimal since then L max (S) = L c = C c d c = r(q) + p(q) + q(q) L max Notation: L max denotes the optimal value

25 Single machine models: Maximum Lateness Approximation ratio for EDD for problem 1 r j,d j < 0 L max structure of a schedule Q b c... 0 r(q) S b C c Q interference job b: last scheduled job from Q with q b < q c Lemma: For the objective value L max (EDD) of an EDD schedule we have: (Proofs on the board) 1 L max (EDD) L max < q c 2 L max (EDD) L max < p b Theorem: EDD is 2-approximation algorithm for 1 r j,d j < 0 L max

26 Single machine models: Maximum Lateness Approximation ratio for EDD for problem 1 r j,d j < 0 L max Remarks: EDD is also a 2-approximation for 1 prec, r j, d j < 0 L max (uses modified release and due dates) by an iteration technique the approximation factor can be reduced to 3/2

27 Single machine models: Maximum Lateness Enumerative methods for problem 1 r j L max we again will use head-body-tail notation Simple branch and bound method: branch on level i of the search tree by selecting a job to be scheduled on position i if in a node of the search tree on level i the set of already scheduled jobs is denoted by S and the finishing time of the jobs from S by t, for position i we only have to consider jobs k with r k < min j / S (max{t, r j} + p j ) lower bound: solve for remaining jobs 1 r j, pmtn L max search strategy: depth first search + selecting next job via lower bound

28 Single machine models: Maximum Lateness Advanced b&b-methods for problem 1 r j L max node of search tree = restricted instance restrictions = set of precedence constraints branching = adding precedence constraints between certain pairs of jobs after adding precedence constraints, modify release and due dates apply EDD to instance given in a node critical sequence has no interference job: EDD solves instance optimal backtrack critical sequence has an interference job: branch

29 Single machine models: Maximum Lateness Advanced b&b-methods for problem 1 r j L max (cont.) branching, given set Q, critical job c, interference job b, and set Q of jobs from Q following b Q b c... 0 r(q) S b C c Q L max = S b +p b +p(q )+q(q ) < r(q )+p b +p(q )+q(q ) if b is scheduled between jobs of Q the value is at least r(q ) + p b + p(q ) + q(q ); i.e. worse than the current schedule

30 Single machine models: Maximum Lateness Advanced b&b-methods for problem 1 r j L max (cont.) branching, given sequence Q, critical job c, interference job b, and set Q of jobs from Q following b Q b c... 0 r(q) S b C c Q L max = S b +p b +p(q )+q(q ) < r(q )+p b +p(q )+q(q ) if b is scheduled between jobs of Q the value is at least r(q ) + p b + p(q ) + q(q ); i.e. worse than the current schedule branch by adding either b Q or Q b

31 Single machine models: Maximum Lateness Advanced b&b-methods for problem 1 r j L max (cont.) lower bounds in a node: maximum of lower bound of parent node r(q ) + p(q ) + q(q ) r(q {b}) + p(q {b}) + q(q {b}) using the modified release and due dates upper bound UB: best value of the EDD schedules discard a node if lower bound UB search strategy: select node with minimum lower bound

32 Single machine models: Maximum Lateness Advanced b&b-methods for problem 1 r j L max (cont.) speed up possibility: let k / Q {b} with r(q ) + p k + p(q ) + q(q ) UB if r(q ) + p(q ) + p k + q k UB then add k Q if r k + p k + p(q ) + q(q ) UB then add Q k

33 Single machine models: Number of Tardy Jobs Problem 1 U j : Structure of an optimal schedule: set S 1 of jobs meeting their due dates set S 2 of jobs being late jobs of S 1 are scheduled before jobs from S 2 jobs from S 1 are scheduled in EDD order jobs from S 2 are scheduled in an arbitrary order Result: a partition of the set of jobs into sets S 1 and S 2 is sufficient to describe a solution

34 Single machine models: Number of Tardy Jobs Algorithm 1 U j 1 enumerate jobs such that d 1... d n ; 2 S 1 := ; t := 0; 3 FOR j:=1 TO n DO 4 S 1 := S 1 {j}; t := t + p j ; 5 IF t > d j THEN 6 Find job k with largest p k value in S 1 ; 7 S 1 := S 1 \ {k}; t := t p k ; 8 END 9 END

35 Single machine models: Number of Tardy Jobs Remarks Algorithm 1 U j Principle: schedule jobs in order of increasing due dates and always when a job gets late, remove the job with largest processing time; all removed jobs are late complexity: O(n log(n)) Example: n = 5; p = (7,8,4,6,6); d = (9,17,18,19,21)

36 Single machine models: Number of Tardy Jobs Remarks Algorithm 1 U j Principle: schedule jobs in order of increasing due dates and always when a job gets late, remove the job with largest processing time; all removed jobs are late complexity: O(n log(n)) Example: n = 5; p = (7,8,4,6,6); d = (9,17,18,19,21) d 3 20

37 Single machine models: Number of Tardy Jobs Remarks Algorithm 1 U j Principle: schedule jobs in order of increasing due dates and always when a job gets late, remove the job with largest processing time; all removed jobs are late complexity: O(n log(n)) Example: n = 5; p = (7,8,4,6,6); d = (9,17,18,19,21) d 5

38 Single machine models: Number of Tardy Jobs Remarks Algorithm 1 U j Principle: schedule jobs in order of increasing due dates and always when a job gets late, remove the job with largest processing time; all removed jobs are late complexity: O(n log(n)) Example: n = 5; p = (7,8,4,6,6); d = (9,17,18,19,21) Algorithm 1 U j computes an optimal solution Proof on the board

39 Single machine models: Weighted Number of Tardy Jobs Problem 1 w j U j problem 1 w j U j is NP-hard even if all due dates are the same; i.e. 1 d j = d w j U j is NP-hard Proof on the board (reduction from PARTITION) priority based heuristic (WSPT-rule): schedule jobs in increasing p j /w j order

40 Single machine models: Weighted Number of Tardy Jobs Problem 1 w j U j problem 1 w j U j is NP-hard even if all due dates are the same; i.e. 1 d j = d w j U j is NP-hard Proof on the board (reduction from PARTITION) priority based heuristic (WSPT-rule): schedule jobs in increasing p j /w j order WSPT may perform arbitrary bad for 1 w j U j :

41 Single machine models: Weighted Number of Tardy Jobs Problem 1 w j U j problem 1 w j U j is NP-hard even if all due dates are the same; i.e. 1 d j = d w j U j is NP-hard Proof on the board (reduction from PARTITION) priority based heuristic (WSPT-rule): schedule jobs in increasing p j /w j order WSPT may perform arbitrary bad for 1 w j U j : n = 3;p = (1,1,M); w = (1 + ǫ,1,m ǫ);d = (1 + M,1 + M,1 + M) wj U j (WSPT)/ w j U j (opt) = (M ǫ)/(1 + ǫ)

42 Single machine models: Weighted Number of Tardy Jobs Dynamic Programming for 1 w j U j assume d 1... d n as for 1 U j a solution is given by a partition of the set of jobs into sets S 1 and S 2 and jobs in S 1 are in EDD order Definition: F j (t) := minimum criterion value for scheduling the first j jobs such that the processing time of the on-time jobs is at most t F n (T) with T = n j=1 p j is optimal value for problem 1 w j U j Initial conditions: F j (t) = { for t < 0; j = 1,...,n 0 for t 0; j = 0 (1)

43 Single machine models: Weighted Number of Tardy Jobs Dynamic Programming for 1 w j U j (cont.) if 0 t d j and j is late in the schedule corresponding to F j (t), we have F j (t) = F j 1 (t) + w j if 0 t d j and j is on time in the schedule corresponding to F j (t), we have F j (t) = F j 1 (t p j )

44 Single machine models: Weighted Number of Tardy Jobs Dynamic Programming for 1 w j U j (cont.) if 0 t d j and j is late in the schedule corresponding to F j (t), we have F j (t) = F j 1 (t) + w j if 0 t d j and j is on time in the schedule corresponding to F j (t), we have F j (t) = F j 1 (t p j ) summarizing, we get for j = 1,...,n: { min{f j 1 (t p j ),F j 1 (t) + w j } F j (t) = F j (d j ) for 0 t d j for d j < t T (2)

45 Single machine models: Weighted Number of Tardy Jobs DP-algorithm for 1 w j U j 1 initialize F j (t) according to (1) 2 FOR j := 1 TO n DO 3 FOR t := 0 TO T DO 4 update F j (t) according to (2) 5 wj U j (OPT) = F n (d n )

46 Single machine models: Weighted Number of Tardy Jobs DP-algorithm for 1 w j U j 1 initialize F j (t) according to (1) 2 FOR j := 1 TO n DO 3 FOR t := 0 TO T DO 4 update F j (t) according to (2) 5 wj U j (OPT) = F n (d n ) complexity is O(n n j=1 p j) thus, algorithm is pseudopolynomial

47 Single machine models: Total Tardiness Basic results: 1 T j is NP-hard preemption does not improve the criterion value 1 pmtn T j is NP-hard idle times do not improve the criterion value Lemma 1: If p j p k and d j d k, then an optimal schedule exist in which job j is scheduled before job k. Proof: exercise this lemma gives a dominance rule

48 Single machine models: Total Tardiness Structural property for 1 T j let k be a fixed job and Ĉk be latest possible completion time of job k in an optimal schedule define ˆd j = { d j max{d k,ĉ k } for j k for j = k Lemma 2: Any optimal sequence w.r.t. ˆd 1,..., ˆd n is also optimal w.r.t. d 1,...,d n. Proof on the board

49 Single machine models: Total Tardiness Structural property for 1 T j (cont.) let d 1... d n let k be the job with p k = max{p 1,...,p n } Lemma 1 implies that an optimal schedule exists where {1,...,k 1} k Lemma 3: There exists an integer δ, 0 δ n k for which an optimal schedule exist in which {1,...,k 1,k+1,...,k+δ} k and k {k+δ+1,...,n}. Proof on the board

50 Single machine models: Total Tardiness DP-algorithm for 1 T j Definition: F j (t) := minimum criterion value for scheduling the first j jobs starting their processing at time t by Lemma 3 we get: there exists some δ {1,...,j} such that F j (t) is achieved by scheduling 1 first jobs 1,...,k 1, k + 1,...,k + δ in some order 2 followed by job k starting at t + k+δ l=1 p l p k 3 followed by jobs k + δ + 1,...,j in some order where p k = max j l=1 p l

51 Single machine models: Total Tardiness DP-algorithm for 1 T j (cont.) {1,...,j} {1,...,k 1, k + 1,...,k + δ} k {k + δ + 1,...,j} Definition: J(j, l, k) := {i i {j, j + 1,...,l}; p i p k ; i k}

52 Single machine models: Total Tardiness DP-algorithm for 1 T j (cont.) {1,...,j} {1,...,k 1, k + 1,...,k + δ} k {k + δ + 1,...,j} Definition: J(j, l, k) := {i i {j, j + 1,...,l}; p i p k ; i k} {1,...,j} J(1, k + δ, k) k J(k + δ + 1, j, k)

53 Single machine models: Total Tardiness DP-algorithm for 1 T j (cont.) {1,...,j} J(1, k + δ, k) k J(k + δ + 1, j, k) Definition: J(j, l, k) := {i i {j, j + 1,...,l}; p i p k ; i k} V(J(j, l, k), t) := minimum criterion value for scheduling the jobs from J(j, l, k) starting their processing at time t

54 Single machine models: Total Tardiness DP-algorithm for 1 T j (cont.) t t J(j, l, k) J(j, k + δ, k ) k J(k + δ + 1, l, k ) C k (δ) we get: V (J(j,l,k),t) = min δ where p k = max{p j j J(j,l,k)} and C k (δ) = t + p k + j V(J(j,k +δ,k ) p j {V (J(j,k + δ,k ),t) + max{0,c k (δ) d k } +V (J(k + δ + 1,l,k ),C k (δ)))} V (,t) = 0, V ({j},t) = max{0,t + p j d j }

55 Single machine models: Total Tardiness DP-algorithm for 1 T j (cont.) optimal value of 1 T j is given by V ({1,...,n},0) complexity: at most O(n 3 ) subsets J(j, l, k) at most p j values for t each recursion (evaluation V(J(j, l, k), t)) costs O(n) (at most n values for δ) total complexity: O(n 4 p j ) (pseudopolynomial)

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