SETTING GATES IN THE STOCHASTIC PROJECT SCHEDULING PROBLEM USING CROSS ENTROPY

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1 19 th Iteratioal Coferece o Productio Research SETTING GATES IN THE STOCHASTIC PROJECT SCHEDULING PROBLEM USING CROSS ENTROPY I. Bedavid, B. Golay Faculty of Idustrial Egieerig ad Maagemet, Techio Israel Istitute of Techology, Haifa 32000, Israel Abstract This paper addresses the problem of schedulig the start times of activities i projects with stochastic duratios. The solutio approach is to set for each activity a gate, i.e. a time before it the activity caot begi. The resources of a activity are scheduled to arrive accordig to its gate. I this stochastic eviromet, the actual start ad fiish times of each activity are ukow. We may the icur extra costs if: (1) a activity is ready sice its predecessors are fiished, but it caot actually start because the resources required for it are scheduled to arrive at a later time; (2) the resources are ready but the activity caot start because of precedece costraits. We associate with each activity a ``holdig" ad a ``shortage" cost, icurred i previous cases, respectively. Our objective is to set the gates so as to miimize the sum of the expected holdig ad shortage costs. We employ the Cross Etropy method. The paper describes the implemetatio of the method, a compariso to other heuristic methods ad some isights. Keywords: Cross etropy, stochastic project schedulig, gates. 1 INTRODUCTION The project schedulig problem gave rise to a extesive literature that addresses the umerous variatios of the basic problem. The various project schedulig problem formulatios differ from each other by the characterizatio of the activities duratio, the existece of resource costraits ad most importatly by their objective fuctio. The basic techiques for project schedulig have evolved sice the 1950s. A detailed descriptio of these methodologies ca be foud i textbooks such as [1], [2], ad [3]. More recetly, a ew approach to project schedulig uder ucertaity has emerged. This schedulig approach cosists of determiig i advace a gate for each activity, i.e. a time before it the activity caot begi. The motivatio for this approach exists also i the schedulig problem of jobs o a sigle facility, see [4], [5]. Trietsch [6], [7] explais the motivatio for gates i a eviromet where resources are booked ad plaed to be ready for a activity at a predetermied time. If the resource is ready but the activity is ot, i the terms of precedece feasibility, tha a cost per time uit is icurred. If for a specific activity, this cost is equal to zero, i.e.\ the fact that resources have to wait for the activity causes o cost, the the gate of this activity ca be set to its early start. However, for the other activities, o specific methods were developed to determie these gates. The goal of this paper is to develop a methodology that will determie a gate for each activity, with the objective to miimize the expected pealty costs. To do so, we first eed to uderstad the reaso for this approach. The idea behid this approach is that, i a eviromet with stochastic duratios, we caot kow for sure the start time ad the fiish time of each activity; this fact ca lead to oe of two outcomes: (1) a specific activity is potetially ready to start its processig sice all its predecessors are fiished, but caot actually start because the resources required for this activity were plaed to arrive at a subsequet time. (2) The resources required for a specific activity are ready sice they were plaed to arrive i a earlier time but the activity is ot ready to start its processig because of precedece costraits. I both cases pealty costs are icurred. I the first case, the pealty cost is similar to a holdig cost icurred by the fact that the activity is ready sooer tha plaed. This holdig cost ca result from a alterative cost due to the fact that we ivest moey for processig precedig activities sooer tha eeded. It ca also result from a idirect cost i cases where the products of precedig activities deteriorate if ot used immediately or withi a specific time iterval. I the secod case, the pealty cost is similar to a shortage cost icurred by the fact that the activity is ready later tha plaed. Agai, this shortage cost ca result from a alterative cost due to the fact that we ivest moey for orderig the resources sooer tha eeded. It ca also result from a idirect cost i cases where the resources deteriorate if ot used immediately or withi a specific time iterval. The schedulig problem is to determie the vector of gates that miimizes the expected sum of holdig ad shortage costs. Our goal i this paper is to cotrol the schedule of the project by determiig start times costraits with the idea that startig activities as soo as feasible is ot always optimal. The rest of this paper is orgaized as follows. I the ext sectio we describe the problem eviromet. I Sectio 3, we preset the Cross-Etropy approach ad its implemetatio to our particular problem. I Sectio 4, we preset the compariso of its results to other heuristic methods ad i Sectio 5, we preset our cocludig remarks ad provide isights towards actual applicatios. 2 PROBLEM DESCIPTION We cosider a project composed of N activities. These activities are liked by precedece relatioships, ofte due to techological costraits. The liks betwee the activities produce a directed graph composed of paths that ca be i parallel or coected by some activities. We defie P i to be the set of all the predecessors of activity i ad S i to be the set of all the successors of activity i. We cosider the case where the duratio of each activity is a

2 radom variable. Each activity i has a duratio Yi with a realizatio y i, a probability desity fuctio f i ad a cumulative desity fuctio F i It is assumed that the distributio of Y i is bouded i the iterval [ ai, bi] ( 0 ai b i ). A gate for a activity is the scheduled time for startig it. The gates of the activities are the decisio variables. Oce they are determied, the processig of a activity caot start before its gate but it may start later (if its predecessors are late). The recursio that determies the startig time ti of activity i whose gate was set at g i ad whose predecessors j P i have started at time t is { } give by: ti = max j P gi, tj + y j. i As explaied before, i such a stochastic eviromet, two outcomes are possible: (1) a specific activity is potetially ready to start its processig but caot actually start because the resources required for this activity were plaed to arrive at a subsequet time. (2) The resources required for a specific activity are ready sice they already were scheduled to arrive but the activity is ot ready to start its processig because of precedece costraits. I both cases pealty costs are icurred. I the first case, the pealty cost is similar to a holdig cost icurred by the fact that the activity is ready sooer tha plaed. I the secod case, the pealty cost is similar to a shortage cost icurred by the fact that the activity is ready later tha plaed. Cosequetly, we defie for each activity i a holdig cost per uit time h i ad a shortage cost per uit time p i. A ideal situatio would be to fiish processig all the predecessors of activity i exactly at time g i. This argumet motivate us to defie for each activity i a due date d i equal to the gate of its immediate successor such that if the processig of activity i eds before the time d i, holdig costs are icurred ad if it eds after the time d i, shortage costs are icurred. The etire project has a due date d (geerally imposed by exteral agets) that is also the due date of the last activities of the project. 3 A CROSS ENTROPY APPROACH 3.1 Descriptio of the Cross Etropy (CE) method The CE method, developed by Rubistei ad Kroese [8], is a geeral heuristic method for solvig estimatio ad optimizatio problems. This method has bee applied to solve a variety of estimatio ad optimizatio problems, especially determiistic ad stochastic (oisy) combiatorial problems. For example, Alo et al. [9] presets a applicatio of the cross-etropy method to the buffer allocatio problem, i.e. the problem of allocatig buffer spaces amogst machies i a serial productio lie, so as to optimize some performace measure. The CE method has also bee applied i the field of project maagemet. Cohe et al. [10] used the CE method to determie the optimal loadig of a stochastic ad dyamic multi-project eviromet by keepig a costat umber of projects performed curretly i the system ad i order to miimize the average stay-time of the projects. For optimizatio problems, a geeral outlie of the CE method may be described as follows: traslatio of the uderlyig optimizatio problem ito a meaigful estimatio problem called the associated stochastic problem. For the example of a miimizatio problem, the estimatio problem may be the expected umber of times the objective fuctio gives a lower value tha a specific threshold value. Next, the CE algorithm ivolves the followig two phases: (1) geeratio of a sample of radom data (demads, duratios, j trajectories, etc.) accordig to a specified radom mechaism, ad simultaeous calculatio of the objective fuctio, (2) updatig the parameters of the radom mechaism (o the basis of the data collected) i order to produce a better" sample i the ext iteratio, a sample that will improve the value of the objective fuctio. The applicatio of the CE method to our schedulig problem is explaied i the ext sectio. 3.2 Applicatio of the CE method to serial projects We first apply the Cross-Etropy (CE) method to the etwork with activities i series (serial project). We use the followig otatio: the umber of activities i the project N the umber of vectors geerated at each step G = ( g1,, g ) the vector of the gates, where g k is the gate of activity k Gi = ( gi1,, gi) the ith vector of gates geerated ˆ ˆ ˆ the distributio for the geeratio of a Pt = ( Pt,1,, Pt, ) vector of gates i step t + 1 P the probability that the gate of activity ˆt, kj k gets the value j i step t + 1 ρ the percetage of best results we wat to use i each step ˆt γ the sample ( 1 ρ ) quatile of the performaces (total cost) i step t S( G i ) the performace of the ith vector geerated The algorithm of the CE for this specific problem is: Step 1 ˆ Set t = 1, P0, k = U gk 1, d au for k = 2,, u= k ˆ ad P0,1 = U 0, d au. u= 1 1 Step 2 Geerate a sample G1,, GN from Pˆt ad calculate the performace (the total cost of the project) for S G. Order these performaces from each vector: ( ) i smallest to biggest ad let ˆt γ = S G. ( ρ N ) Step 3 Calculate P ˆ ( ˆ ˆ t Pt,1,, Pt, ) Pˆ tkj, N i= 1 = = = accordig to: I { ( ) ˆ I SGi γ } { g t ik = j}. N I { SG ( ) ˆ i γ t} i 1 Step 4 If for some t, for some k ad for all j, P ˆ tkj, > 0.95, the stop; otherwise set t = t + 1 ad reiterate from Step 2. I the first step, we set the couter to 1 ad defie the iitial distributio of the gates. Sice we have o a priori iformatio o the value of the gates, we could have start from a uiform distributio betwee 0 to the due date of the project d. The the umber of possible vectors would be: ( d + 1). Sice there is a direct relatioship betwee the umber of possibilities ad N, the umber of vectors we have to geerate i each step, we have to limit the umber of possibilities for the vector of gates. The first way to limit

3 19 th Iteratioal Coferece o Productio Research this umber is to use the costrait: gk gk 1 for all k = 2,,. The secod way to limit this umber is to fix the gate of activity k such that there exists a chace to fiish the remaiig activities o time. For example, the gate of activity should ot be greater tha d a, otherwise with probability 1 we will ed the last activity after the due date. For activity k, k = 1,,, the miimal time to complete the remaiig activities is: a u, therefore the gate should ot be greater tha d au. I the secod step, we geerate a sample of N vectors of gates ad we calculate the cost of the project for each vector. To calculate the cost of a specific vector of gates Gi, we geerate N vectors of activity duratios, calculate for each realizatio the cost of the project, calculate the average of these N costs. This average is the estimator of the expected cost of the project for the specific vector of gates G i. For a specific realizatio of activity duratios ( ) y = y1,, y, the cost is: ( ) hk( gk+ 1-tk - yk) + pk( tk + yk -gk+ 1) where is k t k = the start time of activity k, t1 = g1, g+ 1 = d ad { } tk = max gk, tk 1+ yk 1. The, these costs are ordered from the smallest to the biggest. I the third step we update the distributio of the gates accordig to the results of the last step. For each good performace, i.e. lowest costs, we ote the value of the gates. The probability for each value of the gates will be the frequecy of this value i the best performaces, i.e. the umber of time we get this value out of all the best performaces. I the fourth step, we will stop if for each gate, the probability to be equal to a specific value is close to 1. Otherwise we cotiue aother iteratio. 3.3 Extesio of the CE method to o-serial projects Direct extesio of the CE method to o-serial projects The applicatio of the CE method for o-serial projects is based o the same algorithm that was developed i Sectio 3.2 for serial projects. The differeces betwee the serial ad o-serial algorithms are i the iitial distributio of the vector of gates ad i the calculatio of the cost: The iitial distributio of the vector of gates: to limit the umber of possibilities for the geerated vectors, we ca use the costrait adapted to the etwork structure of the project: gk g j for all j Pk, for all k = 2,,. I serial projects, the secod way to limit this umber was to fix the gate of activity k such that there exists a chace to fiish the remaiig activities o time. For activity k, k = 1,,, the miimal time to complete the remaiig activities was: a u, therefore the gate was ot be greater tha d au. I o-serial projects, it is more complicated to kow, from a specific activity the remaiig time util the completio of the projects. Usig the fact that there is oe termial activity, we bouded the gates of all the activities by d a. The cost calculatio: i o-serial projects, for a specific realizatio y = ( y1,, y ), the cost is: ( hk( gj -tk - yk) pk( tk yk -gj) ) k= j Sk + + where t k is the start time of activity k, t1 = g1, g+ 1 = d ad tk = max j P { gk, tj + j } k y. Heuristic method usig the serial CE method Whe we apply the direct extesio of the CE method to o-serial projects with a large umber of activities, the umber of possible vectors of gates grows. Therefore, the umber N of geerated vectors should also grow, which icreases cosiderably the ruig time of the algorithm. To reduce this problem, we propose the followig heuristic method: Fid the critical path of the o-serial project, for example the path with the largest expected duratio. Apply to the critical path the algorithm for serial projects developed i Sectio 3.2 ad fid the gates for all activities i the critical path. For each o-critical path, apply the algorithm for serial projects developed i Sectio 3.2 whe the due date of this path will be the gate of the activity i which the path merges with the critical path. This heuristic algorithm allows reducig the size of the problem. 4 EXAMPLES 4.1 First isights To get some isights from the CE method ad its applicatio to our schedulig problem, we studied a specific umerical example of a etwork with four activities i series. We programmed the CE method applied to our problem i MatLab. The data used for the example is preseted i Table 1. Table 1: The data for the illustratio of the CE method for a etwork with four activities i a row Activity 1 Activity 2 Activity 3 Activity 4 a i b i h i p i We solved the above example for three differet due dates of the project: d = 38,35,30. The resultig gates for each activity are preseted i Table 2. Table 2: The results of the CE method for a etwork with four activities i a row g 1 g 2 g 3 g 4 d = d = d = From the results preseted i Table 2, we observe the followig: Whe the due date of the project is sooer, the gates are set earlier. This is due to the fact that for each activity, the holdig cost per time uit is lower tha the shortage cost per time uit. Settig the gates earlier meas that we are ready to take the risk of icurrig higher holdig costs but o shortage costs. It is iterestig to ote that decreasig the project due date from 35 to 30 hardly iflueces the gate vector obtaied. This is due to the high shortage cost per time uit of activity 2. Recall that the gate of a activity is the due date of the precedig oe. Thus, settig a tardier gate for activity 3 meas that we do ot wat to take the risk of fiishig activity 2 after its due date ad the icurrig high pealty costs.

4 4.2 Applicatio of the CE algorithm for serial projects ad compariso to other heuristic methods We applied the CE algorithm to a project with seve activities i series. To study the performace of the CE algorithm we compare it to two other heuristic methods: Early Start (ES): i this method, the gates are determied accordig to their early start time. Therefore, g 1 = 0 ad gk = gk 1+ E( Yk 1) for all k = 2,,. Late Start (LS): i this method, the gates are determied accordig to their late start time with respect to the due date of the project. Therefore, g+ 1 = d ad gk = gk+ 1 E( Yk) for all k = 1,,. The data used for the example is preseted i Table 3. Table 3: The data for a project with seve activities i series Activity a i b i h i p i For this example the due date was set to 60. The gates ad the estimator for the expected cost obtaied i each method are preseted i Table Applicatio of the CE algorithm for o-serial projects ad compariso to other heuristic methods For o-serial projects, we compared the performace of the ES heuristic method, the LS heuristic method, the direct extesio of the CE method to o-serial projects (DCE) ad the heuristic method usig the serial CE algorithm for each path (CEP). The first example of o-serial project is preseted i Figure 1. Start Ed Figure 1: No-serial project with seve activities ad two paths This project icludes seve activities ad two paths. The due date was set to 42. The data used for this example is preseted i Table 6. Table 4: The results for a project with seve activities i a row with d = 60 Table 6: The data for the o-serial project with seve activities ad two paths g 1 g 2 g 3 g 4 g 5 g 6 g 7 Mea cost ES LS CE We ca see from Table 4 that for this specific due date: d = 60, the CE method gave the best results ad the results of the ES method were close to those of the CE method. This is because the due date was close to the expected duratio of the project ad that the gates i the ES method are determied as the early start time of each activity accordig to the expected duratios. For the same example, we checked the ifluece of postpoig the due date. We used the example preseted i Table 3 with a due date d = 70. Of course, just the gates obtaied from the CE ad the LS method chaged. The gates of the ES method were ot chaged sice they do ot deped o the due date. The gates ad the estimator for the expected cost obtaied i each method are preseted i Table 5. Table 5: The results for a project with seve activities i a row with d = 70 g 1 g 2 g 3 g 4 g 5 g 6 g 7 Mea cost ES LS CE We ca see that the advatage of the CE method i the example with d = 70 is obvious. This ca be explaied by the fact that the CE method adapts the gates of the activities to the cost whereas the other methods determie the gates accordig to the duratios oly. Activity a i b i h i p i For the CEP method, we first apply the CE method o the serial project with d = 42, the we apply the CE method o the serial project 4-5 with d = g 6. The gates ad the estimator for the expected cost obtaied i each method are preseted i Table 7. Table 7: The results for the o-serial project with seve activities ad two paths g 1 g 2 g 3 g 4 g 5 g 6 g 7 Mea cost ES LS DCE CEP The secod example of o-serial project is preseted i Figure 2. This project icludes te activities ad four paths. The due date was set to 62. The data used for this example is preseted i Table 8.

5 19 th Iteratioal Coferece o Productio Research Start Ed Figure 2: No-serial project with te activities ad four paths Table 8: The data for the o-serial project with te activities ad four paths Activity a i b i h i p i For the CEP method, we first apply the CE method o the path with d = 62. This path is cosidered as the critical path sice its expected duratio is the larger. The we apply the CE method o the path that icludes activity 8 with d = g 10 ad o the path 4-5 with d = g7. The gates ad the estimator for the expected cost obtaied i each method are preseted i Table 9. Table 9: The results for the o-serial project with te activities ad four paths g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 g 10 Mea cost ES LS DCE CEP REFERENCES I the example preseted i Figure 2 ad i Table 8, we ca see from the results preseted i Table 9 that the CE method gave better results tha the ES ad LS heuristic methods. I additio, we ca see a et advatage of the CEP versus the DCE method. This ca be explaied by the fact that for the same umber of geerated vectors, the size of the problem is smaller i the CEP method ad this allows the CE method to coverge ito vector of gates that offer lower costs. 5 CONCLUDING REMARKS This paper deals with the stochastic project schedulig problem through a gatig approach. The objective is to determie the gates of each activity i order to miimize the expected holdig ad shortage costs. For achievig this, we choose a suboptimal approach which is the use of a geeral heuristic method. For this purpose, we chose the Cross Etropy method. We applied the method for serial ad o-serial projects. To check the performace of the algorithms developed, we compared them to two simple heuristic methods: the Early Start (ES) heuristic method ad the Late Start (LS) heuristic method. We applied the algorithms developed o small projects. I all the examples, the algorithms based o the CE method developed i this paper gave the best performaces. A larger computatioal experimet should be coducted. 6 ACKNOWLEDGMENTS The authors are grateful to Prof. Reuve Rubistei for his costructive suggestios. [1] Elmaghraby S.E., 1977, Activity etworks: project plaig ad cotrol by etwork models, New York: Wiley. [2] Demeulemeester E.L., Herroele W.S., 2002, Project schedulig a research hadbook, Kluwer Academic Publishers, Bosto. [3] Shtub A., Bard J.F., Globerso S., 2005, Project maagemet Processes, methodologies, ad ecoomics, Pearso Pretice Hall, NJ. [4] Elmaghraby S.E., Ferreira A.A., Tavares L.V., 2000, Optimal start times uder stochastic activity duratios, Iteratioal Joural of Productio Ecoomics, 64, [5] Elmaghraby S.E., 2001, O the optimal release time of jobs with radom processig times, with extesios to other criteria, Iteratioal Joural of Productio Ecoomics, 74, [6] Trietsch D., 2006, Ecoomically balaced criticalities for robust project schedulig ad cotrol, Workig paper, ISOM Departmet, Uiversity of Aucklad, New Zealad. [7] Trietsch D., 2006, Optimal feedig buffers for projects or batch supply chais by a exact geeralizatio of the ewsvedor result, Iteratioal Joural of Productio Research, 44(4), [8] Rubistei R.Y., Kroese D.P., 2004, The Cross- Etropy method A uified approach to combiatorial optimizatio, Mote-Carlo simulatio, ad machie learig, Spriger, New York. [9] Alo G., Kroese D.P., Raviv T., Rubistei R.Y., 2005, Applicatio of The Cross-Etropy method to the buffer allocatio problem i a simulatio-based

6 eviromet, Aals of Operatios Research, 134, [10] Cohe I., Golay B., Shtub A., 2005, Maagig stochastic, fiite capacity, multi-project systems through the Cross-Etropy methodology, Aals of Operatios Research, 134,