A discrete differential evolution algorithm for multi-objective permutation flowshop scheduling

Transcription

1 A dscrete dfferental evoluton algorthm for mult-objectve permutaton flowshop schedulng M. Baolett, A. Mlan, V. Santucc Dpartmento d Matematca e Informatca Unverstà degl Stud d Peruga Va Vanvtell, 1 Peruga, Italy Abstract. Real-world versons of the permutaton flowshop schedulng problem (PFSP) have a varety of objectve crtera to be optmzed smultaneously. Mult-objectve PFSP s also a relevant combnatoral mult-objectve optmzaton problem. In ths paper we propose a multobjectve evolutonary algorthm for PFSPs by extendng the prevously proposed dscrete dfferental evoluton scheme for sngle-objectve PF- SPs. The noveltes of ths proposal resde on the management of the evolved Pareto front and on the selecton operator. A prelmnary expermental evaluaton has been conducted on three b-objectve PFSPs resultng from all the possble b-objectve combnatons of the crtera makespan, total flowtme and total tardness. Introducton The Permutaton Flowshop Schedulng Problem (PFSP) s an mportant type of schedulng problem whch has many applcatons n manufacturng and large scale product fabrcaton. In ths problem there are n jobs J 1,..., J n and m machnes M 1,..., M m. Each job J s composed by m operatons O 1,..., O m. The generc operaton O j can be executed only by the machne M j and ts gven processng tme s p j. Moreover, the executon of any operaton cannot be nterrupted (no pre-empton) and job passng s not allowed,.e., the jobs must be executed usng the same order n every machne. The goal of PFSP s to fnd the optmal job permutaton π = π(1),..., π(n) wth respect to a gven objectve functon. Three mportant crtera are to mnmze the total flowtme (TFT), the makespan (MS) and the total tardness (TT) defned as follows: MS(π) = T T (π) = T F T (π) = n C(π(), m) (1) =1 max C(π(), m) = C(π(n), m) (2) =1,...,n n max{c(π(), m) d π(), 0} (3) =1

2 where C(h, j) s the completon tme of the operaton O hj and s computed by the followng recursve equaton C(π(), j) = p π(),j + max{c(π( 1), j), C(π(), j 1)} for, j 1, whle the termnal cases are C(π(0), j) = C(, 0) = 0. In equaton (3), for each job h, also a gven delvery date d h s consdered. The mnmzaton of each one of these crtera s computatonally hard. Indeed, both the TFT and TT problems are NP-hard for m 2, whle the MS mnmzaton becomes NP-hard when m > 2. Many sngle-optmzaton algorthms exsts, ether exact or approxmate, for nstance: heurstc technques, local searches or evolutonary algorthms [2]. Anyway, n ths paper we nvestgate the PFSP problem as a mult-objectve optmzaton problem, n whch the goal s to fnd a set of job permutatons whch are good enough wth respect to two or more contrastng crtera,.e. a set of Pareto optmal solutons. Gven k objectve functons f 1,..., f k, a soluton x domnates a soluton x (denoted by x x ) f f (x) f (x ) for = 1,..., k, and there exsts at least an ndex j {1,..., k} such that f j (x) < f j (x ). A soluton x s Pareto optmal f there exst no other soluton x such that x x. The Pareto optmal set s the set of all the Pareto optmal soluton. If two solutons x and x are such that nether x x nor x x, then x and x are ncomparable. Snce the Pareto set s n general very large, the goal s to fnd an approxmaton of ths set,.e., a set composed by ncomparable solutons whch s as close as possble to the Pareto optmal set. One of the most promsng approaches to solve mult-objectve optmzaton problems s to use evolutonary algorthms [1]. In the context of mult-objectve PFSP, many approaches have been proposed. The surveys [3, 7] descrbe and compare many algorthms for PFSP wth all the three possble combnatons of two objectves among TFT, MS and TT. In ths paper we descrbe an algorthm for mult-objectve optmzaton whch s based on Dfferental Evoluton for Permutaton (DEP) [6]. DEP s a dscrete dfferental evoluton algorthm whch drectly operates on the permutatons space and hence s well suted for permutaton optmzaton problems lke PFSP. Indeed, n [6] and n [5], t was shown that DEP reaches state-of-the-art results wth respect to total flowtme and makespan sngle objectve optmzaton. Here, DEP has been extended n order to handle mult-objectve problems. The prelmnary expermental results show that ts performances are comparable wth state-of-the-art algorthms. The rest of the paper s organzed as follows. The second secton descrbes the classcal Dfferental Evoluton algorthm. Its extenson to the permutatons space and mult-objectve PFSP s ntroduced n the thrd secton. An expermental nvestgaton of the proposed approach s provded n the fourth secton, whle conclusons are drawn at the end of the paper. The Dfferental Evoluton algorthm In ths secton we provde a short ntroducton to Dfferental Evoluton (DE) algorthm. For more detal see [4]. Dfferental Evoluton (DE) s a powerful populaton-based evolutonary algorthm for optmzng non-lnear and even non-dfferentable real functons

3 n R n. The man pecularty of DE s to explot the dstrbuton of the solutons dfferences n order to probe the search space. DE ntally generates a random populaton of NP canddate solutons x 1,..., x NP unformly dstrbuted n the solutons space. At each generaton, DE performs mutaton and crossover n order to produce a tral vector u for each ndvdual x, called target vector, n the current populaton. Each target vector s then replaced n the next generaton by the assocated tral vector f and only f the produced tral s ftter than the target. Ths process s teratvely repeated untl a stop crteron s met (e.g., a gven amount of ftness evaluatons has been performed). The dfferental mutaton s the core operator of DE and generates a mutant vector v for each target ndvdual x. The most used mutaton scheme s rand/1 and t s defned as follows: v = x r0 + F (x r1 x r2 ) (4) where r 0, r 1, r 2 are three random ntegers n [1, NP ] mutually dfferent among them. x r0 s called base vector, x r1 x r2 s the dfference vector, and F > 0 s the scale factor parameter.in [4] t s argued that the dfferental mutaton confers to DE the ablty to automatcally adapt the mutaton step sze and orentaton to the ftness landscape at hand. After the mutaton, a crossover operator generates a populaton of N P tral vectors,.e. u, by recombnng each par composed by the generated mutant v and ts correspondng target x. The most used crossover operator s the bnomal one that bulds the tral vector u takng some components from x and some other ones from v accordng to the crossover probablty CR [0, 1]. Fnally, n the selecton phase, the next generaton populaton s selected by a oneto-one tournament among x and u for 1 NP. Dscrete Dfferental Evoluton for Mult-Objectve Optmzaton In ths secton we descrbe the proposed Mult-Objectve Dfferental Evoluton for Permutaton (MODEP) whch drectly evolves a populaton of NP permutatons π 1,..., π NP. Wth respect to the classcal DE, mportant varatons have been made to the genetc operators of mutaton, crossover and selecton. Moreover, an addtonal archve of solutons s ntroduced to mantan the evolved Pareto front. To smplfy our descrpton, let us restrct to the case of two objectve functons f 1 and f 2. A populaton of NP permutatons π 1,..., π NP s randomly generated at the begnnng. At each teraton, a secondary populaton of tral elements υ 1,..., υ NP s generated by means of the mutaton and crossover operators. Then, a selecton operator selects, for = 1,..., NP, whch element among υ and π should be part of the populaton for the next teraton. The pseudo-code of MODEP s depcted n Alg. 1. Dfferental Mutaton The mutaton operator used s the same of DEP [6]. It produces a mutant ν for each populaton element π usng some algebrac concepts related to the symmetrc group of permutatons. Here we brefly recall ts structure:

4 Algorthm 1 MODEP 1: Intalze Populaton 2: Update ND 3: whle num ft eval max ft eval do 4: for 1 to NP do 5: ν DfferentalMutaton() 6: υ (1), υ (2) Crossover(π, ν ) 7: Update ND 8: υ SelectChld(υ (1), υ (2) ) 9: end for 10: for 1 to NP do 11: π Selecton (π, υ ) 12: end for 13: end whle 1 Fnd r 0, r 1, r 2 dfferent to and to each other 2 δ πr 1 2 π r1 3 S RandBS(δ) (S s a sequence of adjacent swaps) 4 L Length(S) 5 k F L 6 ν π r0 7 for j = 1,..., k apply S j to ν where s the ordnary permutaton composton operator, 1 denotes the nverse of a permutaton, and RandBS s the randomzed bubble sort procedure whch allows to decompose a permutaton n a sequence of adjacent swaps (that are themselves smple permutatons). For more detals, see [6]. It s worth to notce that ths operator works drectly wth permutatons, smulatng from an algebrac pont of vew, the expresson of equaton (4). Crossover The crossover operator for permutaton representatons s the same of DEP and produces two chldren υ (1) and υ (2) from π and ν. The detals are descrbed n [6]. The two permutatons υ (1) and υ (2) are compared wth respect to both f 1 and f 2. If υ (1) domnates υ (2), then the tral υ s υ (1). Analogously, f υ (2) domnates υ (1), then the tral υ s υ (2). When υ (1) and υ (2) are ncomparable, then one of them s randomly selected to become the tral υ. Selecton The selecton operator chooses the new populaton element π between the old element π and the tral ν. If π ν, then π becomes π,.e., π remans n the populaton. Otherwse, f ν π or t s equal to π, then π becomes ν, that s ν replaces π n

5 the next generaton populaton. However, f π and ν are ncomparable, then we use a probablstc method somehow smlar to the α-selecton descrbed n [6]. Suppose frst that f 1 (ν ) < f 1 (π ) but f 2 (ν ) f 2 (π ). Then, π becomes ν wth probablty max{0, α 2 (2) }, otherwse t retans the old element π, where (2) = f 2(ν ) f 2 (π ) f 2 (π ) s the relatve worsenng of ν wth respect to π accordng to f 2. Analogously, f f 1 (ν ) f 1 (π ) and f 2 (ν ) < f 2 (π ). Then, π becomes ν wth probablty max(0, α 1 (1) ), where = f 1(ν ) f 1 (π ). f 1 (π ) (1) The ratonale behnd ths selecton operator s that ν enters the populaton f t domnates or s equal to π or, wth a small probablty, f t s not too worse than π n one of the objectve functons, whle t s better than π n the other objectve functon. Moreover, note that the probablty of acceptng a slghtly worsenng populaton element lnearly shades from α h, when (h) = 0, to 0, when (h) = α h, for h = 1, 2. Therefore, the parameters α h regulates how worse ν can be n order to be accepted n the new populaton: f α 1 = α 2 = 0 only better elements (n the Pareto sense) can replace old elements n the populaton. Pareto Front The algorthm keeps updated the approxmated Pareto front N D, whch contans all the non-domnated elements ever generated and evaluated. Intally N D contans all the non-domnated populaton elements created durng the random ntalzaton. Then, at each generaton, all the couples of chldren υ (1) and υ (2) are used to update ND. A new element υ enters ND f t s not domnated by any element of ND. Moreover, all the elements of ND whch are domnated by υ are removed. Expermental Results In ths secton we report some prelmnary expermental results obtaned wth an mplementaton of MODEP. The experments have been performed by solvng the well known Tallard s nstances wth the addtonal due tmes gven n [3]. These nstances are dvded n 11 groups of 10 nstances wth the same values of n and m. The values of n are n the set {20, 50, 100, 200}, whle m les n {5, 10, 20}. The combnaton (n = 200, m = 5) s not consdered. The processng tme p j of each nstance are randomly generated n {1,..., 99}, whle the due date of each job J are generated by multplyng the value m j=1 p j for a random factor n [1, 4]. MODEP has been run 10 tmes for each nstance and the adopted stoppng crteron s the maxmum number of evaluatons, whch

6 has been set to 2000 n m. Three combnatons of objectves have been consdered: (MS, T F T ), (MS, T T ), and (T F T, T T ). For each executon the obtaned Pareto front (correspondng to ND) has been analyzed by computng two performance ndces: the hypervolume I H and the unary multplcatve epslon I 1 ɛ. I H s computed as the area delmted by the solutons of ND and a reference pont. I 1 ɛ compares ND wth the best known Pareto front B and s computed as I 1 ɛ = max x B mn max y ND j=1,2 f j (y) f j (x). The ndces have been computed by averagng over the multple executons and nstances for every combnaton of n m. The value for the parameter NP has been set to 100 after some prelmnary experments. The parameter F used n the mutaton operator s, as n [6], self-adapted durng the evoluton. Instead, the values for the selecton parameters α 1 and α 2 have been set after a calbraton phase accordng to Table 1. Table 1. Calbraton values for α 1 and α 2 Opt. α 1 α 2 (MS, T F T ) (MS, T T ) (T F T, T T ) The results of the optmzaton of (MS, T F T ) are shown n Table 2. MODEP works well on ths problem and the values of the second ndex I 1 ɛ (whose optmal value s 1) are qute good, whle the values for I H (whose optmal value s 1.44) are however good, compared to those reported n [3].It s worth to notce that, fxng n, I H seems to have a decreasng behavor as m ncreases (except when n = 20). Table 2. Results for (MS, T F T ) n m I H I 1 ɛ

7 The results of the optmzaton of (MS, T T ) are shown n Table 3 and are smlar to those for (MS, T F T ), even f the decreasng behavor of I H wth respect to m s not so apparent. Table 3. Results for (MS, T T ) n m I H I 1 ɛ Fnally, the results of the optmzaton of (T F T, T T ) are shown n Table 4. Here, whle the performances as measured by I 1 ɛ are stll satsfactory, the results of I H are slghtly worse than n the prevous cases. Table 4. Results for (T F T, T T ) n m I H Iɛ Concluson and Future Work In ths paper we have descrbed an algorthm for optmzaton of mult-objectve permutaton flowshop schedulng problems. Some prelmnary expermental results show that ths approach s promsng and reaches results whch are comparable to the stateof-the-art algorthms. As a future lne of research, we would lke to add to our algorthm

8 some method to enhance the dversty of the populaton, as done n other evolutonary mult-objectve algorthms, lke crowdng dstance or nchng technques. References 1. Carlos A. Coello Coello, Arturo Hernández Agurre, and Eckart Ztzler. Evolutonary multobjectve optmzaton. European Journal of Operatonal Research, 181(3): , J. Gupta and J.E. Stafford. Flowshop schedulng research after fve decades. European Journal of Operatonal Research, (169): , Gerardo Mnella, Rubén Ruz, and Mchele Cavotta. A revew and evaluaton of multobjectve algorthms for the flowshop schedulng problem. INFORMS Journal on Computng, 20(3): , K.V. Prce, R.M. Storn, and J.A. Lampnen. Dfferental Evoluton: A Practcal Approach to Global Optmzaton. Sprnger, Berln, Valentno Santucc, Marco Baolett, and Alfredo Mlan. Solvng permutaton flowshop schedulng problems wth a dscrete dfferental evoluton algorthm. submtted to AI Communcaton. 6. Valentno Santucc, Marco Baolett, and Alfredo Mlan. A dfferental evoluton algorthm for the permutaton flowshop schedulng problem wth total flow tme crteron. In Parallel Problem Solvng from Nature - PPSN XIII - 13th Internatonal Conference, Ljubljana, Slovena, September 13-17, Proceedngs, pages , M.M. Yensey and B. Yagmahan. Mult-objectve permutaton flow shop schedulng problem: Lterature revew, classfcaton and current trends. Omega, (45): , 2014.