MULTIOBJECTIVE PSO ALGORITHM IN PROJECT STAFF MANAGEMENT

Transcription

1 MULTIOBJECTIVE PSO ALGORITHM IN PROJECT STAFF MANAGEMENT I-Tung Yang Dept. of Cvl Engneerng Tamang Unversty 151 Yngchuan Road, Tamsu, Tape County, Tawan 251 Abstract: The assgnment of staff to projects s a regular actvty for many contractors, government agences and consultng frms. Whereas the prmary objectve of the assgnment s to maxmze the overall proft, ssues dealng wth manpower management must also be ncorporated to ensure strong morale and to enhance compettveness. Such ssues nclude to avod excessve overtme and to balance worloads. The present study develops a partcle swarm optmzaton (PSO) algorthm to handle the assgnment problem wth multple objectves, whch creates dffculty for conventonal optmzaton technques. The proposed algorthm s tested on an applcaton case to llustrate ts performance. It has also been compared to LINGO, a commercal optmzaton pacage. Keywords: PSO, Multobjectve optmzaton, Assgnment problem, Manpower management 1. INTRODUCTION One of the most mportant management tass facng the constructon ndustry s to assgn the worng staff to ncomng projects. Ths routne tas serves as the bass of proftablty and, f done properly, can greatly enhance compettveness. Yet, the staff-to-project assgnment s usually done under tme pressure. To address manpower management ssues, the real-world staff-to-project assgnment nvolves multple goals, such as (1) maxmzng the overall proft, (2) mnmzng the excessve overtme, and (3) mnmzng the varaton of worloads. The frst goal s self explanatory. The second serves to balance the worloads of employees whle the thrd ntends to avod excessve overtme hours. The latter two are consdered mportant because they ensure morale and address non-fnancal motvaton [5]. The assgnment problem wth multple objectves has been proved to be NP-complete and even #P-complete [4]. That s, nether does polynomal-tme algorthm exst nor s any determnstc algorthm own to fnd the approxmate answer wthn a reasonable error bound. To solve such a dffcult problem n lmted tme, ths study develops a partcle swarm optmzaton (PSO) algorthm, supported by stochastc and teratve search, to optmze all the objectves smultaneously. 2. BACKGROUND 2.1 Mono-objectve Assgnment Problem The orgnal assgnment problem s to assgn n people to n jobs so as to reach some overall level of competence,.e., to mnmze or maxmze an objectve. The mpled assumpton s that each person would be assgned a job and each job would utlze exactly one person. For example, f the cost of assgnng person to job j s a j, the objectve s to mnmze a j x j, j The bnary constrants are x j = 1 x j = 1 j where the assgnment varable x j s 1 f person s assgned to job j and 0 otherwse. 2.2 Soluton Procedure After ts ntal appearance n 1950s, the mono-objectve assgnment problem has been tacled by several approaches. On one hand, the problem can apparently be regarded as bnary nteger programmng because x j can only tae the value 0 or 1. On the other hand, t has been shown that by relaxng the nteger assumpton, lnear programmng technques can also lead to the desred result [8]. Although both nteger and lnear programmng models can be solved by commercal software pacages, the models could be tedously large snce t demands n 2 varables. It s more convenent to use a specalzed algorthm. One of the famous approaches s the Hungaran algorthm [7]. Also of help s the mnmum cost flow algorthm [1], whch sends flow from a set of supply nodes through the arcs of a networ, to a set of demand nodes, at mnmum total cost. Each arc n the networ denotes an assgnment between staff and projects. 2.1 Mult-objectve Assgnment Problem Despte the success n achevng the sngle goal for the assgnment problem, recent attentons have been shfted to optmzng multple objectves smultaneously. Whereas the prmary objectve s to mnmze costs or maxmze profts, managers are often concerned wth unbalanced worloads and excessve overtme. The former rases conflcts between heavly and lghtly loaded teams [9] and the latter creates (1) (2) (3) -890-

2 stress and fatgue, and ultmately causes poor-qualty products. To deal wth multple objectves, several methods are at hand. The most ntutve one s tang the weghted sum of all the objectves and optmzng the weghted sum nstead. The weghted sum approach, however, s not approprate here because the objectves of the assgnment problem are measured n dfferent unts and ther relatve weghts are often dffcult to assess. Another approach s goal programmng, whch ntends to mnmze the weghted sum of the absolute devatons between pre-specfed goals and objectve values. Agan, ths approach requres decson maers to devse goals for objectve values and set a proper weght for every objectve before the optmzaton. The tass rely on a pror artculaton of preference nformaton, whch may not be accessble to decson maers. The thrd approach s to optmze the most relevant objectve, and consderng other objectves as constrants bound by some allowable levels, ε. The major shortcomng of the ε-constrant approach s that t has to tedously repeat the optmzaton procedure for dfferent bounds of ε, f a full descrpton of optmal solutons s needed. The optmalty of solutons n multobjectve optmzaton s based on a tradeoff phlosophy. That s, we are really tryng to fnd good compromses (the trade-off surface n the search space) rather than a sngle soluton. The tradeoff surface s composed of non-domnated, also nown as Pareto optmal, solutons that are better than other solutons at some objectves whle beng at least as good as others for the other objectves. Havng the tradeoff surface, as opposed to a sngle soluton, offers decson maers the most flexblty for determnng the compromsed assgnment alternatve after a thorough evaluaton of exstng solutons. Thus, the ultmate am of the present study s to approxmate the tradeoff surface for the multobjectve assgnment problem. Detaled defntons of mult-objectve domnance can be found n [3]. 3. PROBLEM STATEMENT The targeted problem s to assgn n staff teams to m ncomng projects over a plannng tme horzon. Note that the numbers of teams and projects need not be equal. Hence, n addton to assgnng teams to projects, the soluton would also help determne whch project should be accepted. Fve sets of entres are needed. The man entry s the estmated man-hours for team to perform project j, denoted by t j. For the same project, the requred man-hours may vary. Ths s to accommodate an allowance for added productvty when a team s famlar wth a partcular project. Thus the optmal assgnment plan wll encourage teams to perform ther specalty, usng less tme. The vector r j denotes the revenue of project j. Three vectors, c, s, and g represent the hourly cost, the avalablty (upper lmt of worng hours), and the regular wor tme for team, respectvely. The avalablty dffers because some of the teams are concurrently worng on other n-house projects, whch shall not be reassgned to avod dsrupton. Worng hours beyond the regular wor tme are overtme and would ncur a hgher cost, e.g., 1.5 tmes the regular hourly cost. By defnton, the regular wor tme s not greater than the avalablty. As mentoned prevously, the decson varables are the bnary assgnment choces between team and project j x j = 0 or 1 (4) The present model conssts of three objectves, whose ndvdual reasonng and formulaton are gven below. The frst objectve s to maxmze the proft, whch s revenue R mnus the regular and overtme costs, C regu and C over. P = R C regu C over (5) The revenue can be expressed as R = r j ( tj ) (6) where t j x j stands for the actual worng hours spent on project j; r j s the revenue of project j. The regular-tme cost s C regu = c[mn( tj, g )] (7) where t j x j denotes the actual worng hours of team, whch s the mnmum between the actual worng hours and the regular tme g. It s then multpled by the regular hourly cost c to obtan the regular-tme cost. The overtme cost s * C over = c [max(0, tj g )] (8) * where c s the hourly rate of overtme pay. The element n square bracets calculates the overtme hours by deductng the regular wor tme from the actual worng hours, f the latter s greater. The second objectve s to mnmze the varaton of worloads. For each team, the worload s quantfed by a standardzed utlzaton rate: dvdng the team s actual worng hours by the regular wor tme: ( tj ) u = (9) g The utlzaton rate s less than 100% when the assgned projects can be completed wthn the regular tme; t s greater than 100% at the presence of overtme. On the foundaton of Eq. (9), the varaton of worloads s defned to be the standard devaton of utlzaton rates: 1 2 σ = ( u u) n (10) where u s the average utlzaton rate 1 u = u n (11) -891-

3 The thrd objectve s to mnmze excessve overtme, whch s the maxmal amount among all teams overtme hours. h = max[max(0, tj g )] (12) 4. PSO ALGORITHM 4.1 Concepts The orgnal PSO scheme was desgned to mmc the cooperaton wthn a bologcal populaton, such as a group of brds or a swarm of nsects [6]. Wthn the populaton, mult-dmensonal partcles, each a possble soluton, are flown through the problem space, n search of optma. Each partcle has ts own velocty, whch s determned by (1) the local best: the memory of the best soluton t has obtaned thus far and (2) the global best: the best soluton found by the entre populaton. It has been shown that PSO s able to converge to global optma fast wthout beng trapped n local optma, especally when the problem space s complex and rregular. The capablty of PSO algorthms has been testfed by a wde varety of recent applcatons, such as bomedcal mage regstraton [11], neural networ tranng [2], and resource-constraned schedulng [14]. However, the orgnal PSO scheme focuses on only one objectve and hence requres further enhancements n the context of multobjectve optmzaton. In what follows, we ntroduce the mproved PSO algorthm 4.2 Proposed Algorthm In the proposed PSO algorthm, the poston of the th partcle s m-dmensonal (m equals the number of projects) and expressed n a vector form: Y = y, y,..., y..., y ], [ 1 2 d m where all y d [0,1]. The component n dmenson d of the poston determnes whch team s assgned to project d by a mappng process. The process s vrtually mappng a contnuous varable y d to a bnary varable x j as graphcally depcted n Fg. 1 where m equals 3. Y Fg. 1. Mappng process where yd (t) s the component n dmenson d of partcle at teraton t; vd ( t + 1 ) s the velocty at teraton t+1, whch s determned by vd ( t + 1) = wvd ( t) + (14) c1r1 ( Lbestd yd ( t)) + c2r2 ( Gbestd yd ( t)) where w s the nerta weght; vd (t) s the velocty n the prevous teraton; c 1 and c 2 are learnng constants; r 1 and r 2 are random factors n the [0,1] nterval; Lbest d s the component n dmenson d of the local best; Gbest d s the component n dmenson d of the global best. Whle the swarm sze, w, c 1, and c 2 are specfable algorthm parameters, r 1 and r 2 provde the mperatve randomness for fndng better solutons along the drecton guded toward the local and global best. The algorthm parameters n Eq. (14) should be fne-tuned to ensure performance. However, prevous experments have suggested the followng confguratons. The swarm usually contans 10 to 50 partcles. The nerta weght w s used to control the mpact of the prevous hstory of veloctes on the current velocty, thus to nfluence the trade-off between global exploraton and local explotaton abltes of the partcles. It s therefore better to ntally start at a large value (around 1), n order to promote global exploraton, and gradually decrease t (no less than 0) to get more refned solutons [10]. Both learnng constants c 1 and c 2 range from 1 to 4, whereas relatve mportance can be gven to stress the nfluence of one s own memory (cognton learnng) or that of the entre populaton (socal learnng). The velocty s constraned wthn a specfed bound to avod vcous oscllaton vd ( t + 1) max max vd ( t + 1) = v f vd ( t + 1) > v (15) vd ( t + 1) where the velocty bound max v s often smaller than the doman of the search space. Besdes the velocty bound, the partcle postons are constraned wthn the feasble range [0,1]. Ths s accomplshed by a new bouncng routne llustrated n Fg. 2. Once a partcle s moved beyond the feasble range, t wll be automatcally sent bac wth a dstance equal to that beyond the lmt. The bouncng routne s consdered superor to the conventonal absorbng routne,.e., the partcle s ept at the boundary as ts velocty s absorbed (shown as the gray crcle), because t provdes more exploraton capablty. The proposed algorthm teratvely changes the partcle postons as follows yd ( t + 1) = yd ( t) + vd ( t + 1) (13) -892-

4 Y (orgnal) Y (new) Fg. 2. Bouncng routne A ey requrement for locatng the non-domnated solutons s to preserve soluton dversty durng optmzaton. To do so, we mantan an elte archve and alter the defnton of the global best n the orgnal PSO scheme. As teratons progress, the global best s selected dynamcally from the elte archve that stores all the non-domnated solutons. The rule of selecton gves preference to the non-domnated solutons that domnates the fewest partcles n the current teraton. The underlyng concept s to preserve dversty by promotng movements to the extremes and unrepresented areas. The local best s the best poston ever acheved by the partcle. Every partcle compares ts current poston to the prevous local best and chooses the non-domnated one as the new local best. The proposed PSO algorthm s not sensble to the scales of the objectves because t by no means calculates the dstances between solutons. Thus ts performance s ndependent from the choce of scales. For nstance, convertng the proft from local currences to US dollars does not affect the solutons beng found. Ths s of partcular mportance to nternatonal frms. The proposed PSO algorthm s composed of the followng steps 1. Randomly ntalze the postons for all the partcles. 2. Intalze the velocty of each partcle. 3. Map the partcle postons to assgnment choces. 4. Evaluate the four objectve values of the assgnment choces accordng to Eqs. (5), (10), and (12). 5. Store the postons representng non-domnated solutons n the elte archve. 6. Intalze the memory of each partcle. 7. WHILE the maxmal number of teratons has not yet been reached For each partcle DO (a) Compute the velocty as descrbed n Eq. (14). (b) Constran the velocty so that t does not exceed the bound by Eq. (15). (c) If the velocty would cause nfeasblty, adjust t usng the bouncng routne. (d) Compute the new poston by addng the velocty to the prevous poston usng Eq. (13). (e) Evaluate the objectve values of the current poston. (f) Compare the new poston wth members n the archve. (g) Update the elte archve by nsertng all the currently non-domnated postons and elmnate any domnated locatons from the archve. (h) When the new poston domnates the local best, replace the local best. () Locate the archve member that domnates the fewest partcles n ths teraton as the global best. (j) Increase the loop counter. 8. Return the archve as the non-domnated soluton set. 5. APPLICATION An actual case was used to demonstrate the applcaton of the proposed algorthm. Ths case s adopted from a prevous study of an envronmental consultng engneerng frm [13]. Sx engneerng teams are avalable to be assgned to ffteen projects. Table 2 (at the end of ths artcle) enumerates nput data n the followng sets: (1) t j : estmated man-hours for team to perform project j; (2) r j : revenue for project j; (4) c : hourly cost of team ; (5) s : avalablty of team ; (6) g : regular wor tme for team. The overtme hourly cost s 1.5 tmes of the regular hourly cost. There are three objectves beng optmzed: (1) to maxmze the overall proft, (2) to mnmze the maxmal overtme, and (3) to mnmze the varaton of worloads. Detaled formulas of these objectves can be found n Secton 3. The multobjectve assgnment problem s solved usng the followng algorthm parameters: (1) Swarm sze s 20; (2) Inerta weght starts at 1.0 and lnearly decreases to 0.4; (3) Learnng constants are 1 and 2 (socal learnng s twce more mportant than cognton learnng); (4) Velocty bound s 0.5. A plot study s conducted to set the teraton number. In summary, the maxmal teraton number s set to 10,000 as t yelds the most non-domnated solutons per CPU tme. Ths crteron s one of the performance metrcs n multobjectve optmzaton [15]. A set of non-domnated solutons are found wthn 15 mnutes at a Pentum IV 2.4 GHz machne. Fg. 3 plots the non-domnated solutons, whch reflect a three-dmensonal -893-

5 tradeoff between proft, overtme, and varaton of worloads. Proft ($) x Maxmal overtme (hr) STD of utlzaton rate (%) Fg. 3. Non-domnated solutons Due to lmted space, Table 3 lsts only the frst 10 solutons, sorted by proft. A close observaton confrms that a hgher proft comes at the expense of more excessve overtme or a greater varaton of worloads. Table 3. Non-domnated solutons (partal lst) soluton of $241,751, whch s apparently domnated by our soluton: $243,616. In addton to the effectves, the proposed algorthm s notable for ts effcency. For the applcaton case, a full permutaton would requre O(10 12 ) trals. The huge number of trals s obvously expensve and almost nfeasble n computaton. In contrast, the proposed algorthm tres only O(10 5 ) possble solutons (20 partcles for 10,000 teratons). In a nutshell, the proposed algorthm searches a very small porton of the soluton space (n the order of 10-7 ) and yet, fnds very compettve non-domnated solutons. 6. CLOSING REMARKS The present study develops a new PSO algorthm to facltate the mult-objectve staff-to-project assgnment, whch cannot be solved n polynomal tme usng conventonal optmzaton technques. Thus, the am of the proposed algorthm s to locate good non-domnated solutons under tme pressure. To optmze multple objectves smultaneously, the proposed algorthm mantans an elte archve and uses the archve members to dynamcally lead the partcle swarm n searchng for more and better non-domnated solutons. The performance of the proposed algorthm has been valdated through an applcaton case. It has been shown that the proposed algorthm can fnd very compettve solutons wth consderable effcency. Proft ($) Maxmal overtme (hr) SD of utlzaton rate (%) $244, $243, $241, $239, $236, $234, $234, $232, $229, $225, The non-domnated soluton set llustrates the advantage of the proposed algorthm: gvng decson maers a set of compromsed solutons to choose from when tme s lmted. For nstance, whereas the second hghest proft (mared n bold) s about $1300 less, t can decrease the maxmal overtme from 785 to 383 hours and also lead to a more balanced worload. It s therefore chosen to be the optmal soluton after all. To valdate the effectveness of the proposed algorthm, t s compared to LINGO 8.0, a powerful optmzaton pacage [12]. Snce LINGO cannot handle multple objectves, we used t to maxmze the proft under the followng two constrants: (1) the maxmal overtme s less than 383 hours, and (2) the standard devaton of utlzaton rates s less than 49.79%. LINGO too 6 mllon teratons, usng the branch-and-bound solver, to fnd the local optmal REFERENCES [1] Ahuja, B.K., Magnant, T.L., & Orln, J.B. (1993). Networ flows: theory, algorthms, and applcatons, Prentce-Hall, Englewood Clffs, NJ, [2] Chatterjee, A., Pulasnghe, K., Watanabe, K., & Izum, K. (2005). A partcle-swarm-optmzed fuzzy-neural networ for voce-controlled robot systems. IEEE Transactons on Industral Electroncs, 52(6), [3] Coello Coello, C.A., Veldhuzen, D.A., & Lamont, G.B. (2002). Evolutonary algorthms for solvng mult-objectve problems, Kluwer Academc Publshers, NY. [4] Ehrgott, M. (2005). Multcrtera optmzaton, 2 nd edton, Sprnger, New Yor, [5] Hecer, P. (1996). Human resources strateges for successful consultng engneerng frms. Journal of Management n Engneerng, 12(5), [6] Kennedy J., Eberhart R.C., & Sh Y. (2001). Swarm ntellgence, Morgan Kaufmann Publshers, San Francsco, CA. [7] Kuhn, H.W. (1955). The Hungaran method for the assgnment problem. Naval research Logstcs Quarterly, 2, [8] Luenberger, D.G. (1984). Lnear and nonlnear programmng, 2 nd edton, Addson-Wesley, Readng, MA, [9] Molleman, E. (2005). The multlevel nature of team-based wor research. Team Performance Management, 11(3),

6 [10] Sh, Y. & Eberhart, R. (1998). Parameter selecton n partcle swarm optmzaton. Proceedngs of the Seventh Annual Conf. on Evolutonary Programmng, [11] Wachowa, M. P., Smolova, R., Zheng, Y., Zurada, J. M., & Elmaghraby, A. S. (2004). "An approach to multmodal bomedcal mage regstraton utlzng partcle swarm optmzaton. IEEE Transactons on Evolutonary Computaton, 8(3), pp [12] Wnston, W.L. & Venataramanan, M. (2002). Introducton to mathematcal programmng - applcatons and algorthms, 4 th edton, Duxbury Press, Boston, MA. [13] Zanas, S.H. (1983). A staff to job assgnment (parttonng) problem wth multple objectves. Computer & Operatons Research, 10(4), [14] Zhang, H., L, X., L, H., & Huang, F. (2005). Partcle swarm optmzaton-based schemes for resource-constraned project schedulng. Automaton n Constructon, 14(3), pp [15] Ztzler, E., Deb, H., & Thele, L. (2000). Comparson of mult-objectve evolutonary algorthms: Emprcal results. Evolutonary Computaton, 8(2), Table 2. Input data Project t j c ($) s (hr) g (hr) r j ($1000) Team -895-