Single-machine scheduling problems with position- dependent processing times

Transcription

1 Single-machine scheduling problems with position- dependent processing times Julien Moncel Université Toulouse 1 IUT Rodez LAAS-CNRS

2 Joint work Gerd Finke G-SCOP Grenoble (France) Vincent Jost LIX Paris (France)

3 Outline About wearing and learning Some examples A literature review Generic results

4 About wearing and learning

5 «Classical» scheduling theory «Machine(s)» «Jobs» : processing time p i of job i A schedule (completion time : C i ) C 1 Objective functions C 2 C 3 C 4 Minimize makespan C max = max C i Minimize total flow time C i etc.

6 How does reality look like «Machines» = robots (NCMT), painters, surgeons, cooks, firemen, etc. «Jobs» = machining, painting, surgery, «Jobs» = machining, painting, surgery, meals, fires, etc.

7 Constant processing time? Legitimate to assume p i = constant? A robot machining? A painter painting? A surgeon doing surgery? A cook preparing meals? Firemen addressing fires? At first sight, why not But what if we want to refine a bit the model?

8 Refinements (1) Models with setup times Time for tool change for a NCMT Time for color change for a painter Preparation of operating block Picking of ingredients of the recipe Moving from a fire to another Often NP-complete ( traveling salesperson problem)

9 Refinements (2) Stochastic scheduling Processing times = random variables Similar results to the deterministic case when considering expected values Wide literature [M. Pinedo, Scheduling : Theory, Algorithms and Systems, Springer (2008)]

10 Refinements (3) Processing time depends on the moment when job is processed Example of fires : the more you wait, the harder it is to process! Essentially, this is neither due to a set-up time, nor to a stochastic phenomenon

11 Wearing / learning (1) Wearing = processing times increase as time passes Examples Fires (fire spreads) Painting, surgery, cooking, etc. Weariness The «machine» gets tired! Mechanical wear of tools etc.

12 Wearing / learning (2) Learning = processing times decrease as time passes Examples Cook : new recipes Surgeon : new techniques New team member New process New organization

13 Wearing and learning in the literature (1) Experimental psychology : proof of evidence in the XIX th century [H. Ebbinghaus, Memory. A contribution to experimental psychology, Dunker & Humblot, Leipzig (1885)]

14 Wearing and learning in the literature (2) In industrial management sciences : proof of evidence in the mid-xx th century [Yelle, The learning curve: historical review and comprehensive survey, Decision Science (1979)] [Dutton & Thomas, Treating progress functions as a managerial opportunity, Academy of Management Review (1984)]

15 The «80% hypothesis» unitary production cost [Wright, Factors Affecting the Cost of Airplanes, Journal of Aeronautical Sciences (1936)] c 0 C = C 0 (n/n 0 ) log c 0 n 0 2n 0 nb of units produced

16 Results of some surveys Ford T model ( ) : 87% Electronic chips industry ( ) : 67% Photovoltaic cells industry ( ) : 72% etc.

17 Wearing and learning in the literature (3) In scheduling Wearing : 1988 [Gupta & Gupta, Single facility scheduling with nonlinear processing times, Computers and Industrial Engineering (1988)] Learning : 1999 [Biskup, Single-machine scheduling with learning considerations, European Journal of Operational Research (1999)]

18 Wearing models (1) Model of Gupta & Gupta (1988) For each job i «Basic» processing time p i Wearing constant a i If i done at time t : «actual» processing time = a i t+p i p i

19 Wearing models (2) Model of Mosheiov (2005) : For each job i «Basic» processing time p i If i scheduled at rank r : «actual» processing time = rp i There exist many other models

20 Learning models (1) Model of Biskup (1999) : Common learning rate a<0 For each job i «Basic» processing time p «Basic» processing time p i If i scheduled at rank r : «actual» processing time = r a p i

21 Learning models (2) Model of Gordon et al (2008) : Common learning rate a ]0,1[ For each job i «Basic» processing time p i If i scheduled at rank r : «actual» processing time = a r-1 p i There exist many other models

22 Surveys on the topic [Biskup, A state-of of-the art review on scheduling with learning effects,, EJOR (2008)] [Cheng et al, A concise survey of scheduling with time-dependent processing times,, EJOR (2004)]

23 Some examples

24 Fires (1) Fires to deal with Fire n 1 : processing time = 5 Fire n 2 : processing time = 4+t (if starts at time t) Fire n 3 : processing time = 10+2t Fire n 4 : processing time = 6+3t Aim : maximum completion time (C max )

25 Fires (2) General problem : 1 p it = a i t+p i C max Dominance : jobs s.t. a i = 0 are scheduled at the end Interchange argument : when is it relevant to interchange two consecutive jobs i, j? i j?

26 Fires (3) Optimal schedule for 1 p it = a i t+p i C max Schedule jobs s.t. p i = 0 at the beginning Schedule jobs s.t. a i p i 0 by decreasing order of a i /p i (Schedule jobs s.t. a i = 0 at the end) O(n log n) [Gupta & Gupta, Single facility scheduling with nonlinear processing times, Computers and Industrial Engineering (1988)]

27 Fires (4) Fire n 1 : 5 ratio : 0/5=0 Fire n 2 : 4+t 1/4 = 0.25 Fire n 3 : 10+2t 2/10 = 0.2 Fire n 4 : 6+3t 3/6 = C max * = 63

28 Weariness (1) Jobs to be done Job n 1 : processing time 4r (if at rank r) Job n 2 : processing time 6r Job n 3 : processing time 2r Job n 4 : processing time 3r Aim : total flow time ( C i )

29 Weariness (2) General problem : 1 p ir = rb i C i Interchange argument : when is it relevant to interchange two consecutive jobs i, j? i j?

30 Weariness (3) Optimal schedule for 1 p ir = rb i C i Put jobs having a small b i in the middle of schedule Put jobs having a large b i on the side (beginning or end) of schedule O(n log n) «V-shaped» optimal schedule : If b 1 b 2 b 3 b n Then an optimal schedule is :

31 Weariness (4) Optimal schedule Job n 1 : processing time 4r Job n 2 : processing time 6r Job n 3 : processing time 2r Job n 4 : processing time 3r C i* = 70

32 Remarks Same argument shows that SPT is optimal for 1 p ir = rb i C max Problems 1 p ir = rp i C max 1 p ir = rp i C i 1 p it = a i t+p i C max all reduce to sorting (SPT) («V -shaped») (non-increasing a i /b i )

33 A literature review

34 Time-dependent models 1 p it =p i t C max : O(n) 1 p it =p i t w i C i : O(n log n) 1 p it =p i t+a i C max : O(n log n) 1 p it =a i-p it C i : O(n log n) 1 d i,p it =p i t+a i C max : O(n 5 ) [Mosheiov (1994)] 1 d i,p it =p i t+a i C i : O(n 5 ) 1 r i,p it =a i -p i t C max : O(n 6 log n) 1 r i,p it =a i +p i t C max : NP-C [Mosheiov (1994)] [Gupta & Gupta (1988)] [Ng et al (2002)] [Cheng & Ding (2000)] [Cheng & Ding (2000)] [Cheng & Ding (1998)]

35 Rank-dependent models 1 p ir =r a p i C i : O(n log n) 1 p ir =r a p i C max : O(n log n) 1 p ir =r a(i) p i C i : O(n 3 ) 1 p ir =r a(i) p i C max : O(n 3 ) 1 p ir =p i -v i r C i : O(n 3 ) 1 p ir =p i -v i r C max : O(n 3 ) 1 p ir =p i (w-vr) C max : O(n log n) 1 p ir =p i (w-vr) C i : O(n log n) 1 r i,p ir =r a p i C max : NP-C [Biskup (1999)] [Mosheiov (2001)] [Mosheiov & Sidney (2003)] [Mosheiov & Sidney (2003)] [Bachman & Janiak (2004)] [Bachman & Janiak (2004)] [Wang & Xia (2005)] [Bachman & Janiak (2004)]

36 Mixed models 1 p ir =(p i +wt)r a C i : O(n log n) 1 p ir =(p i +wt)r a C max : O(n log n) [Wang (2006)] etc [Biskup, A state-of of-the art review on scheduling with learning effects,, EJOR (2008)] [Cheng et al, A concise survey of scheduling with time-dependent processing times,, EJOR (2004)]

37 Generic results

38 Framework Single machine Decomposable rank-dependent processing times ; notation : p r i Decomposable objective functions (like C max and C i ) Schedule described by a permutation π π(r) = index of job scheduled at rank r Actual processing time of job scheduled at rank r is p π(r) r

39 Definition of decomposability (1) An objective function is decomposable if it can be written as υ r p r π(r) with p r π(r)r denoting the actual processing time of job scheduled at rank r Examples : C max = p π(r) r C i = (n+1-r) p π(r) r etc.

40 General model for a decomposable objective function Objective = υ r p π(r) r Associated bipartite graph 1 1 i υ r p i r r n n jobs ranks

41 Matching We seek for a minimum weight perfect matching in this graph! (linear assignment problem) Algorithm O(n 3 ) Captures some results of the literature [Mosheiov & Sidney (2003)] [Bachman & Janiak (2004)] [Gordon et al (2008)]

42 Definition of decomposability (2) Rank-dependent processing times are decomposable if they can be written as p ir = f(r)p i Examples : r a b i a r-1 b i etc.

43 A preliminary lemma Product lemma Let a, b, x, y be positive numbers such that : a > b x > y Do we have : ax + by > ay + bx? ax + by < ay + bx? It depends? [Hardy & Littlewood (1988)]

44 A preliminary lemma Product lemma Let a, b, x, y be positive numbers such that : a > b x > y Then we always have ax + by > ay + bx

45 A preliminary lemma Product lemma Let a, b, x, y be positive numbers such that : a > b x > y Then we always have ax + by > ay + bx To get small, it is better to do small big + big small than to do big big + small small

46 A preliminary lemma Product lemma Let a, b, x, y be positive numbers such that : a > b x > y Then we always have ax + by > ay + bx Proof ax + by (ay + bx) = (a - b)(x - y) > 0

47 A generic result Generic interchange argument When both the objective function Z and the processing times are decomposable, then the problem 1 p ir =f(r)p i Z can be solved by an O(n log n) double-sorting algorithm

48 Generic case of C max max C max = p π(r)r = f(r)p π(r) We wish to multiply the f(r) and the p r as in the product lemma Generic solution : Sort the f(r) : f(σ(1)) f(σ(2)) f(σ(n)) Sort the p r : p τ(1) p τ(2) p τ(n) Optimal schedule : π = τ σ -1 So as f(σ(r)) is multiplied by p τ(r)

49 Generic case of C i C i = C π(i) = i ( j=1 i f(j)p π(j) ) = nf(1)p π(1) + (n-1)f(2)p π(2) + +f(n)p π(n) υ 1 υ 2 υ n We wish to minimize the product of the υ i with the p i! We apply again the product lemma

50 Example 1 : 1 C i nf(1)p π(1) + (n-1)f(2)p π(2) + +f(n)p π(n) υ 1 υ 2 υ n We have f 1 In this case υ 1 = n, υ 2 = n-1,, υ n = 1 Hence we have σ(i) = n-i Thus π(i) = τ(n-i) This is Smith s SPT rule!

51 Example 2 : 1 p ir =rp i C i nf(1)p π(1) + (n-1)f(2)p π(2) + +f(n)p π(n) υ 1 υ 2 υ n We have f(i) = i In this case υ i = (n+1-i)i

52 Example 2 : 1 p ir =rp i C i nf(1)p π(1) + (n-1)f(2)p π(2) + +f(n)p π(n) υ 1 υ 2 υ n We have f(i) = i In this case υ i = (n+1-i)i This is Mosheiov s «V-shaped» schedule! (n+1)/2

53 Sample of existing results that are captured by this approach 1 p ir =p i r a C i or 1 p(i,r)=p i r a C max 1 p ir =p i (M+(1-M)r a ) C max (a 0 and M [0,1]) 1 p r ir =p i (a-br) C i and 1 p ir =p i (a-br) C max (a 0 integer, b 0 rational, and a-(n+1)b>0) 1 p ir =p i γ r-1 C i (γ ]0,1[ [2,+ [) 1 p ir =p i r a C i -C j (a<0) 1 p ir =f(r)p i C max (f monotone)

54 Sample of existing results that are captured by this approach [Biskup (1999)] [Mosheiov (2001)] [Mosheiov (2005)] [Wang, Wang & Xia (2005)] [Wang & Xia (2005)] [Gordon, Potts, Strusevich, Whitehead (2008)] [Yang & Kuo (2010)] [Rustogi & Strusevich (2011)]

55 Recent similar generic results Burkhard et al (2009) Rustogi & Strusevich (2011)

56 Characterization? Assume you have a decomposable objective function υ r p r π(r) Decomposability of processing times sufficient, but not necessary, for the sorting procedure to apply Seeing p ir as f r (p i ), what conditions on f r ensure that the double-sorting procedure applies?

57 Comparability Let P be the set of possible p i s Define g r =υ r f r and set G={g 1,...,g n } We say g r g s iff g r (p)-g r (q) g s (p)-g s (q) for all p,q P G is a totally preordered set iff functions g r are all pairwise comparable

58 Characterization G is a totally preordered set iff there exists a permutation π s.t. for any instance (p 1,...,p n ) of 1 p ir =f r (p i ) υ r p [r] s.t. p 1... p n, assigning job i to rank r if and only if π(r)=i leads to an optimal schedule Related to so-called Monge matrices

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