Solving Multi mode Resource Constrained Project Scheduling with IC Algorithm and Compare It with PSO Algorithm

Transcription

1 Advances n Lfe Scences 2014, 4(3): DOI: /j.als Solvng Mult mode Resource Constraned Project Schedulng wth IC Algorthm and Compare It wth PSO Algorthm Sna Namaz 1,*, Mohammad Reza Zare Fard 2, Seyed Mohammad Reza Mousav 1, Mohammad Bahram Nasab 2 1 Department of Industral Engneerng, Islamc Azad Unversty, Karaj Branch, Iran 2 Department of Industral Engneerng, Islamc Azad Unversty, Qazvn Branch, Iran Abstract Well-tmed executon of project actvtes requres an ntegrated and effectve plannng and management n an accurate adjustment of the project actvtes executon tme. Although a lot of works has been done on MRCPSP (mult-resource constraned project schedulng problem) wth the goal of mnmzng the project makespan, but consderng other goals n current projects lke the project total cost s also mportant to the managers. So n ths artcle, a mathematcal model has been consdered for resource-constraned project schedulng wth two objectves: mnmzng both project total cost and makespan. And ths problem has been solved wth a meta-heurstc algorthm called IC (Imperalst Compettve) algorthm whch has never been used to solve ths type of problems. The results have been compared wth PSO algorthm n whch they are satsfactory. Keywords MRCPSP, Imperalst Compettve algorthm, MORCPSP, PSP 1. Introducton RCPSP (Resource-Constraned Project Schedulng Problem) s schedulng the project actvtes accordng to precedence relatons and constrant of resources. RCPSP conssts of a project wth j actvtes j=1,, j. the duraton of an actvty j s donated by DD jj. All actvtes can start just once, and they can be preemptve or not. Because of techncal needs, there s some precedence relatons among actvtes donated as a set of relatons PP jj. It shows that actvty j cannot be started unless all of ts precedence relatons and precedents ( PP jj ) are fulflled. Precedence relatons can be represented by an AON (Actvty on Node network); wth the presupposton of acyclc feature of the network. Each actvty needs a specal amount of resources for completon and executon [1]. In standard RCPSP, an actvty can only be executed n one way. So the executon tme and amount of needed resources wll be defnte [1]. Frst research on mult-state actvtes has been done by Elmaghraby [2]. An actvty, whch only could have been executed n one way n standard state, now can be executed n several states or methods. Each state needs ts own * Correspondng author: sna.namaz.e@outlook.com (Sna Namaz) Publshed onlne at Copyrght 2014 Scentfc & Academc Publshng. All Rghts Reserved executon tme and needed resources. Mult-state RCPSP s shown MRCPSP n ths problem. RCPSP problem has a presupposton of beng mult-state. Ths presupposton s defned as follows: "An actvty j must be started at one pont and must be executed n one of ts executon states whch s shown by 1,, MM jj. The actvty must be termnated n the same executon state whch t had been started. Preempton s not sutable n the basc model of MRCPSP [2]. The executon tme for actvty j n state m s shown by DD jjjj. The amount of resource needed for executon of actvty j n state m s shown by rr jjjjjj. Besdes the renewable resources, non-renewable resources are also beng used n mult-state models. A schedulng plan for mult-state RCPSP model s specfed wth a start tme SS jj and a state MM jj for each j actvty. The most well-known target functon n these knds of problems s mnmzng the project makespan. [2]" It's obvous that f there s only one state for each actvty, and there wll be no non-renewable resource, the problem wll turn nto a standard RCPSP problem. It should be mentoned that ncludng a non-renewable resource there s a chance that we couldn t have any feasble schedulng, and ths has been ndcated by Kolsh and Drexl [3]. The problem of fndng a feasble schedulng n ths problem s a knd of NP-Complete problem tself (n case there exst at least two non-renewable resources consdered for the problem). Accordng to Broker et al. symbolzaton, parameter α wll be started wth MPS [4].

2 Advances n Lfe Scences 2014, 4(3): Proposed B-objectve MRCPSP Problem Accordng to the lterature that has been revewed, the proposed model has been consdered n B-objectve state [5], mult-mode state, wth renewable or non-renewable resources and dual resources whch s a partcular state of resources. Also for the goals, two well-known and practcal goals has used whch the frst one s mnmzng project makespan and the second one s mnmzng project total cost. The mathematcal model for the proposed problem s as follows: llll nn MMMMMM FF 1 = tt. xx nn1tt tt=eeee MMMMMM FF 2 = cc. XX nn AA mm MM M ls X mt = 1 m = 1t = es n M m= 1t= es M j ls ls j m= 1t= es j ( t + d + FS ) X tx (, j) E jm t m j m t M mn{ t 1, ls } r x α mk ms k = 1 m= 1 s= max{ t d, es } m n M ls ϑ ϑ rmk xms αk = 1m= 1 s= es xmt {0,1} m = 1,.., M = 1,.., n ; t = 1,.., T Constrant 1 s lke a basc MRCPSP problem whch causes the actvty to be started n any executve mode and specfed tme that had been chosen and other modes wll not be used untl the end of the executon. Constrant 2 s the end-to-start precedence relatons constrant. If ths constrant s used between actvtes and j, actvty j won't start unless the actvty fnshed. Constrant 3 s the constrant of renewable resources whch s the basc model of MRCPSP. Constrant 4 s the constrant of non-renewable resources whch s the basc model of MRCPSP. fs Constrant 5 s the constrant of decson varable beng bnary. Now we should menton two mportant notes about project cost functon and dual-resource constrant. Accordng to the fact that the cost constrant of each actvty s a knd of nborn non-renewable resource, so consderaton of ths constrant has been gnored and t s consdered n non-renewable resource constrant, whch completely dscussed n sample problems posed for ths proposed problem [4]. Accordng to the defnton of dual-resource constrant whch mentoned before [4], some tems had been consdered such as project budget. If the project fnance won't be avalable at the begnnng of the project, and some amount of the budget wll be allotted to the managers perodcally, n ths case, the project budget s a knd of dual-resource. On the one hand t's a non-renewable resource snce the project budget has a specfc amount and t wll be fnshed; on the other hand t must be consdered a renewable resource because some amount of t s avalable n each perod. So for each b-resource n ths proposed problem, two more renewable & non-renewable constrants must be consdered, whch s also completely descrbed n sample problems posed for ths proposed problem. 3. Proposed Algorthm to Solve the Proposed Problem ICA (Imperalst Compettve Algorthm) s a method n the feld of evolutonary computaton whch s used to fnd the optmum soluton n dfferent optmzaton problems. Ths algorthm represents an algorthm to solve the mathematcal optmzaton problems wth the use of mathematcal modelng of the socal-poltcal evoluton process. From the usage perspectve, ths algorthm s grouped wth evolutonary optmzaton algorthms lke genetc algorthms, ant colony optmzaton, smulated annealng and etc. lke all algorthms of ths group, ICA algorthm forms a basc set of probable answers. These basc answers are known as "chromosomes" n genetc algorthm, "partcle" n partcle swarm algorthm and "country" n ICA algorthm. ICA algorthm mproves these basc answers (countres) gradually wth a specal method whch wll be dscussed later, and fnally wll fnd the approprate answer (desred country) for the optmzaton problem. In the begnnng, the steps of the appled algorthm are presented to solve the proposed RCPSP problem. The process of the algorthm for these knds of problems s as follows: Step 1: n ths step, a seres of possble orders consderng the constrants (precedence consderaton and resource constrant wll be created). The process s as follows: at frst, for each project actvty,

3 142 Sna Namaz et al.: Solvng Mult mode Resource Constraned Project Schedulng wth IC Algorthm and Compare It wth PSO Algorthm a random number between 0 and 1 s produced. Then these random numbers are arranged and saved n a lst. The sequence number of these actvtes are saved n another lst. Step 2: the results are arranged dependng on ther sutablty and the frst 10 results are chosen as mperalst countres. Then the normalzed cost and power of an mperalst s defned by C = max{ C } C (1) n n PP nn = CC nn NN =1 CC (2) Step 3: the normalzed power of an mperalst s the porton of colones that should be possessed by that mperalst. Then the number of colones of an empre wll be { } NC = round P N (3) n n col Step 4: the best mperalst s saved n BestSol. The major crcle of the algorthm, steps 5 to 7, s repeated for a specfc number of tmes (here, the number of repettons s the crtera to make the algorthm to stop). Step 5: mperalsts and ther colones are nvoked. Colones move toward mperalsts on the bass of ths formula ( ) B = Rand A B θ (4) After the movement, f a colony s more powerful than ts mperalst, the colony and the mperalst change ther places. Step 6: the operaton n step 5 s done for all mperalsts. The power of mperalsts s calculated one more tme. A pre-determned number of colones go under the control of the most powerful mperalst. Step 7: ths s the last step. Imperalsts are arranged dependng on ther sutablty and the best mperalst s saved n bestsol. Table 1. Results ICO algorthm for MRCPSP problem 4. The Results of the Proposed Imperalst Compettve Algorthm for MRCPSP In ths secton, RCPSP problems wth mult-mode (mult-state) actvtes are beng consdered. There are j10, j12, j14, j16, j18, j20 and j30 sets n the lbrary of these problems. One more feature of MRCPSP problems s that besde the renewable resources, they have non-renewable resources as well. Problems wth names of c15, c21, m1, m2, m4, m5, n0, n1, n3, r1, r3, r4 and r5 can also be found among the standard MRCPSP problems whch have some dfferences n the number of resources and actvtes. For nstance, group c problems have 18 actvtes, 2 renewable resources and 2 non-renewable resources. Group m problems have also 18 actvtes, 2 renewable resources and 2 non-renewable resources. Group n0 problems have 14 actvtes and only 2 renewable resources and n1 problems have 18 actvtes, 2 renewable resources and 1 non-renewable resource, and n3 problems also have 18 actvtes wth 2 renewable and 3 non-renewable resources. Group r problems also dffers n the number of renewable resources n the way that r1 problems have 18 actvtes wth 1 renewable and 2 non-renewable resources, r3 have 3 renewable and 2 non-renewable resources, r4 have 4 renewable and 2 non-renewable resources and r5 have 5 renewable and 2 non-renewable resources [6]. In ths study, only group j problems have been used for comparson. Between group j problems, j10, j12, j14, j 20 and j30 problems has been used. These problems have 2 renewable and 2 non-renewable resources. In the table (1), the average devaton from the best answers n j10 up to j20 problem seres s calculated, whch the optmum answer s avalable. So the average devaton from these optmum answers has been compared n the table (1), usng the best and newest ever-appled algorthms, n whch the proposed algorthm holds the 3 rd place n ths study. Av Dev % J10 J12 J14 J16 J18 J20 SFLA [7] EFEA [8] Ths paper RSS [9] AGA [10] JSA [11] The second crtera whch s evaluated n these knds of problems s the percentage of the optmum answers and another one s the process tme of the CPU, whch has been evaluated n the table (2).

4 Advances n Lfe Scences 2014, 4(3): Table 2. Result CPU tme for solvng MRCPSP problem wth ICO algorthm Optmal rate % CPU tme (sec) J10 J12 J14 J16 J18 J20 SFLA [7] EFEA [8] Ths paper RSS [9] AGA [10] JSA [11] The Results of the Proposed ICA for Solvng the Proposed Problem Fgure 1. Pareto Answers wth ICO algorthm Accordng to the fact that a b-objectve model has been developed for RCPSP problems n ths study, at frst a problem s solved as an nstance n order to show the algorthm performance n producng parto answers n the form of fgure (1). It s obvous from the chart that the algorthm could produce the sutable parto to solve ths sample problem whch the answers and dversty are desrably sutable. The algorthm has been used to solve some b-objectve problems from the MRCPSP problems and results has been evaluated through 2 mportant crtera namely dversty and spacng metrc to check the qualty of the produced parto answers n mult-objectve problems. 1- Dversty Crteron Ztzler defned a specfc measurement crteron n whch the length of a dameter s measured. Ths dameter s the dameter of the space used by fnal value of the objects for parto answer set [12]. 2- Spacng Metrc crteron It's another crteron for checkng parto's charts. Ths crteron studes dstance of the answers from each other n order to fnds the dstance between answers of one pont wth ts nearest neghbor. Ths crteron could be consdered ether ordnarly or normalzed. Formulas of each group are brought later.

5 144 Sna Namaz et al.: Solvng Mult mode Resource Constraned Project Schedulng wth IC Algorthm and Compare It wth PSO Algorthm 6. Checkng the Performance of the Proposed Algorthm from the Pont of Evaluaton Crtera wth PSP Algorthm Accordng to the dscussed crtera n prevous secton, the algorthm s beng used to solve dfferent MRCPSP problems and ts results have been compared wth PSO algorthm whch the results of these two crtera are brought n the table(3). At the end, two algorthms are beng evaluated from the aspect of problems' solvng tme. Results ndcated satsfactory performance of the ICA algorthm. Table 3. Compare ICO algorthm wth PSO algorthm Algorthms Evaluatng facts Proposed problem Frst crteron (dversty) Second crteron (spacng metrc) Ths paper ICO algorthm PSO algorthm J J J J J J J J J Numercal Example for Showng ICA Algorthm Performance n Problem Solvng Table 4. Colonal power and the number of samples n problem solvng Imperl Cost power NC Table 5. The best example of elegance n problem solvng Imperl Cost power NC To show the algorthm performance better, we're usng a sample example for step-by-step functonalty of ICA algorthm. Problem defnton s stated n bee's algorthm secton. So n ths secton, we'll only dscuss about the functonalty of ICA algorthm. 10 mperalst are beng chosen at frst. Proprety, power and number of colones wll be specfed for each mperalst and brought n the table (4). As t s obvous from the table, mperalst number 1 has the best proprety and owns the most colones. Ths mperalst wll be stored n BestSol as the best answer. After that each colony changed toward ts mperalst, the power of each mperalst wll be calculated agan and the best mperalst wll be specfed. The results are presented n the table (5). As t s obvous, mperalst number 1 has the best proprety and the most empre power. So 10 weakest colones wll be added to the colones of ths mperalst. Ths mperalst wll be stored n BestSol as the best result. In addton, as t s clear, empres 8, 9 and 10 fal, because the number of ther colones become zero. Ths process wll be repeated for a specfc number of tmes. On more mportant crteron for ths algorthm to stop s when there s only one mperalst and all other empres had fal. 8. Conclusons In ths paper, a b-objectve problem was suggested for MRCPSP problem seres, and then MRCPSP problem seres and a proposed problem were solved wth ICA algorthm. Results of MRCPSP problem soluton have compared wth the newest algorthm used to solve these seres of problems

6 Advances n Lfe Scences 2014, 4(3): and the results of proposed problem have compared wth PSO algorthm whch the results ndcated a successful performance of ths algorthm to solve these knds of problems. Ths algorthm can be used to solve other RCPSP problems n future researches. REFERENCES [1] Usng Bees Algorthm to Solve the Resource Constraned. Project Schedulng Problem n PSPLIB. Amr Sadegh, Abolfazl Kalanak, Azadeh Noktehdan, Azamdokht Saf Samghabad and Farnaz Barznpour,Theoretcal and Mathematcal Foundatons of Computer Scence Communcatons n Computer and Informaton Scence, 2011, Volume 164, [2] Actvty networks: project plannng and control by network models, Elmaghraby, S.E Wley. [3] Adaptve search for solvng hard project schedulng problems, Raner Kolsch* and Andreas Drexl, Naval Research Logstcs (NRL), February 1996,Volume 43, Issue 1, pages [4] Resource-constraned project schedulng: Notaton, classfcaton, models, and methods, Brucker, Drexl A, Mohrng R, Neumann K, Pesch E. European Journal of Operatonal Research, 1999, 112,3 41. [5] B-objectve resource-constraned project schedulng wth obustness and makespan crtera, B. Abbas, S. Shadrokh, Appled Mathematcs and Computaton, 2006, J. Arkat 180 (1), [6] A random key based genetc algorthm for the resource constraned project schedulng problem, Mendes JJ, Goncalves JF, Resende MG,. Computers and Operatons Research 2009;36(1), [7] An effcent hybrd algorthm for resource-constraned project schedulng, Chen W, Sh YJ, Teng HF, Lan XP, Hu LC. Informaton Scences 2010;180(5), [8] A decomposton-based genetc algorthm for the resource-constraned project schedulng problem, Debels D, Vanhoucke M. Operatons Research 2007;55(3), [9] A hybrd genetc algorthm for the resource-constraned project schedulng problem, Valls V, Ballestn F, Quntanlla S. European Journal of Operatonal Research 2008; 185(2), [10] A hybrd estmaton of dstrbuton algorthm for solvng the resource-constraned project schedulng problem, 2012, Expert systems and applcaton, 39, [11] A neurogenetc approach for the resource-constraned project schedulng problem, Agarwal, A., Colak, S., & Erenguc, S. Computers & Operatons Research, 2011, 38, [12] Qualty assessment of Pareto set approxmatons. In Multobjectve Optmzaton. Ztzler, E., Knowles, J., Thele, L., Sprnger Berln Hedelberg,