Grey Critical Chain Project Scheduling Technique and Its Application

Transcription

1 Canadian Soial Siene Vol.3 No.3 June 2007 Grey Critial Chain Projet Sheduling Tehnique and Its ppliation TECHNIQUE DE PROGRMMTION DE L CHINE CRITIQUE GRISE DU PROJET ET SON PPLICTION Gao Peng 1 Feng Junwen 2 Wang Huating 3 bstrat: Based on the idea of Grey System and interval number oeffiient notation, a Grey Critial Chain sheduling approah is studied. ording to Grey system theory, the time of projet or task ompletion an be onsidered as the objet that extension is definite but intension is unertain, whih is oinident with the harater of the projet management. The Grey Critial Chain Sheduling Tehnique mainly aims at the single projet time management, but the management idea an also be applied to the other knowledge areas of the projet management. In this Tehnique, we improve the seletion method of the buffer time in the Critial Chain, in order to obtain reasonable Feeding Buffer time and Projet Buffer time. In this paper, we will use an example to disuss the Grey Critial Chain Sheduling Tehnique, ompare Grey Critial Chain with Program Evaluation and Review Tehnique, Critial Chain and Fuzzy Critial Chain, analyze the advantages, disadvantages and appliable sope of their own. Key words: Critial Chain, Grey System, Interval Number, Shedule Management, Projet Management Résumé: Sur la base de l idée de Système Gris et la notation du oeffiient de nombre d intervalle, l approhe de pragrammtion d une Chaîne Critique Grise est étudiée. Selon la théorie du Système Gris, le temps du projet ou de la tâhe peut être onsidéré omme l objet dont l extention est définitive mais l intention est inertaine, qui est onforme au aratère du management de projet. La Tehnique de Programmation de la Chaîne Critique Grise vise essentiellement le management du temps du projet simple, mais l idée de management peut aussi être appliquée dans d autres domaines du management de projet. ve ette tehnique, nous améliorons la méthode de séletion du temps d amortissement dans la Chaîne Critique afin d obtenir le temps d amortissement de l alimentation raisonnable et le temps d amortissement de projet. Dans l artile présent, nous allons utiliser un exemple pour disuter la Tehnique de Programmation de la Chaîne Critique Grise, omparer la Chaîne Critique Grise ave l Evaluation du Programme et la Tehnique de révision, la Chaîne Critique et la Chaîne Critique Floue, et analyser leurs avantages, désavantages et hamp d appliation. Mots-Clés: Chaîne Critique, Système Gris, nombre d intervalle, management de programme, management de projet 1. INTRODUCTION The Critial Chain (CC) proposed by Goldratt [1] is different from those traditional projet management tehniques, for example, Critial Path Method (CPM), Program Evaluation and Review Tehnique (PERT), Graphial Evaluation and Review Tehnique (GERT), Venture Evaluation and Review Tehnique (VERT) and so on. Traditional projet management tehniques less embody the physial and psyhial fators of the projet 1 Shool of Eonomis and Management, Nanjing University of Siene and Tehnology, China. 2 Shool of Eonomis and Management, Nanjing University of Siene and Tehnology, China. 3 Shool of Eonomis and Management, Nanjing University of Siene and Tehnology, China. *Reeived 5 November 2006 ; aepted 9 Marh

2 managers and the projet team members. The main idea of the Critial Chain is to reflet the whole psyhial and environment fators in projet management. Based on this, some harmful habits, suh as Student Syndrome (Wait until the last minute to start a task) and Parkinson s Law (Work expands to fill the available time), led by physial and psyhial fators of the projet managers and the projet team members, an be eliminated in projet management rapidly. Consequently, by using Critial Chain Projet Management tehnique [2], the feasibility would be improved greatly for us to aomplish the projet on shedule. Originally, Critial Chain Tehnique was mainly used in projet shedule management, then was also applied to other knowledge areas of projet management. But there are some diffiulties in the appliation of Critial Chain Sheduling Tehnique, beause the determination of the Buffer time is very subjetive. Previously, the Critial Chain Sheduling Tehnique was based on Stohasti nalysis and Probability Theory. Later, based on Fuzzy Theory, some sholars disussed Critial Chain Projet Sheduling Method based on Fuzzy Theory [3], but the disadvantages that buffer time is hard for quantitative desription and analysis still exists. Stohasti Theory requires that the estimate of the assignment aomplishment time obeys some probability distribution but every projet management ativity only ourred one, so it is diffiult to simulate the time distribution aording to the historial data. Besides, aording to Fuzzy Theory, the assignment aomplishment time is onsidered as objet whose extension is indefinite, whih does not onform with the harater that projets have time limit. Grey Systems method proposed by Deng [4] and refined by Liu [5] is a useful sientifi systemati approah to dealing with the unertainty that the extension is deterministi buy the intension is unsure. In this paper, based on the Grey System theory, a Grey Critial Chain (GCC) sheduling approah is studied. This approah mainly aims at the single projet time management, but the basi idea an also be applied to the other knowledge areas of the projet management. In this approah, we improve the seletion method of the buffer time in the Critial Chain, and use an example to disuss the Grey Critial Chain Sheduling Tehnique, ompare Grey Critial Chain with Program Evaluation and Review Tehnique, Critial Chain and Fuzzy Critial Chain, analyzes the advantages, disadvantages and appliable sope of their own. 2. CRITICL CHIN PROJECT MNGEMENT 2.1 Basi Problem nalysis of Projet Management 4 The basi idea of the Theory of Constraints (TOC) is that there exist onstraints or limitation in any organization or system. The onstraints are the fators that limit the goal realization of the organization or system. If an organization has no onstraint, its output would be unlimited. That is impossible in reality. The theory of onstraints proposed a systemati thinking proess to help deision makers to develop the feasible and appliable solutions in a logial dedution manner. The appliation of the theory of onstraints to the projet management leads to the ritial hain development. The spirit of the ritial hain projet management is that it works from the pratial aspet of the projet to solve the problems that happen frequently during the projet implementation. In this setion, the ommon happened problems in pratial projet management are analyzed, and their possible solutions are suggested based on the TOC logial thinking proess. Three problems are ommon in the pratie of the projet management. Firstly, projet ompletion time delay. Seondly, projet ost overrun. Thirdly, tasks redue in the projet planning. Regularly, projet team members attribute these problems to the unertainty in the projet management. In the opinion of Goldratt, these three problems are due to the following three effets that exist in the physial and psyhial side of the projet team members. 1st. Student Syndrome. Student Syndrome means that no matter how long the projet or task takes, endeavor happens just before the ompletion time approahes. For example, in the university, students often begin to work their homework reports just several days ahead the deadline. If this syndrome is used to the projet ase, it is easy to understand why the projet or task durations are often overdue. It is beause that endeavor of the projet team members to work the projet or task begins just before the ompletion time is ahead. 2nd. Multiple Task Effet. Suppose that a projet team member has three tasks on hands,, B and C. Eah task needs 10 days to omplete. The projet team member works by the sequene B C. If the member first works 5 days for, then 5 days for B, then 5 days for C, then 5 days for, then 5 days for B, then 5 days for C. ll the tasks need 20 days to finish. Then the predetermined time for all the three tasks, B, C beomes 20 days, whih will definitely affet the their suessor tasks. 3rd. Transmission Differene Effet. 4 M.Chen, S.Hse. (2004). Fuzzy-ritial-hain-based projet management [J].Journal of the Chinese Institute of Industrial Engineers, 21(2):

3 Suppose, B, C are three parallel tasks, task D is the suessor of the tasks, B, C, and ompletion time is 5 days for, B,C, and 10 days for D. Regularly, we need 15 days to omplete this projet. Suppose that task B has an unforeseen, whih leads to 3 days behind the shedule. Then the whole projet ompletion time will beome 18 days. But thinking reversely, if task, B, or C ould be finished 3days ahead the shedule, then the whole projet ompletion time ould be 13 days? Unfortunately, the answer often is negative. The reason is as follows. Firstly, if the projet team member finish the task ahead of time, then the duration estimate for the following tasks will be redued, whih does no good to him or her. So the projet team member is unwilling to finish the task ahead of time. Seondly, even though the tasks, B, and C are finished ahead of time, task D may not be ready for work, beause the resoures suh as the people that task D needs may not be idle at this time, so the projet an not be finished ahead. ritial hain. 1st. Inrease the ritial hain resoure buffer and ritial hain feeding buffer to protet the ritial hain. 2nd. Follow the above steps to solve the new onflits inurred by adding buffers 3rd. Chek the whole projet to assure that all the resoure onflits are solved. Step 4. Enhane the resoure utilization effiieny or inrease new resoures to shorten projet predetermined time. Step 5. Go bak to Step 1, and do not let the inertia or laziness beome the new onstraint. 3. UNCERTIN CPM ND UNCERTIN CRITIL CHIN 2.2 Critial hain thinking proess [3] Goldratt applied the five steps of TOC to overome the above three projet onstraints, whih beomes the ritial hain thinking proess for the projet management. The thinking proess is in the following. Step 1. Critial Chain Identifiation 1st. Chek the projet network or diagram, inluding the duration estimates for all the tasks and the resoures they need, and do best to bakward shift the starting time of the projet tasks. 2nd. Identify the resoure onflits, and determine whih onflit needs to solve first, that is, make priority ranking. The basi priniple is that the tasks lose to the projet ompletion time have high priority. 3rd. Rearrange all the tasks that have resoure onflits, or supply additional resoures to solve the onflits. For the onflit resoures, forward shift the tasks with highest priority. 4th. Follow the (2) and (3) to ontinue identifying and solving the resoure onflits until all the resoure onflits are solved. 5th. Find the longest hain for the relevant tasks of the projet. Step 2. Best Employment of the Critial Chain Redue all the duration estimates of the tasks on the ritial hain by half, whih will be putt together to use as the projet buffer time to deal with the unertainty effets on the ritial hain and shorten the predetermined time of the tasks. Step 3. Full Coordination with the Critial Chain Exept the ritial hain, all the other tasks, paths and resoure in the projet do best to ooperate with the Probability and statistis, fuzzy mathematis and grey system theory are three ommon researh methodologies for the unertainty systems. But the unertainties dealt with in these three subjets are different. Probability and statistis studies the stohasti unertainty phenomena, its fundamental starting point is large sample and objet following ertain typial distribution. The appliation of the probability and statistis in the projet management leads to the generalization of the CPM to PERT and traditional ritial hain tehniques [2]. Fuzzy mathematis studies ognitive unertainty problems, its fundamental starting point is the lear intension and unlear extension. Unertainties in the fuzzy theory are desribed by the membership funtions proposed by deision makers. The utilization of the fuzzy theory in the projet management leads to the development of the fuzzy CPM [6] and fuzzy ritial hain sheduling tehnique [3]. Grey system theory fouses on the unertain problems with the small sample and inadequate information. Its starting point is small data modeling by information overage and operator funtion. The differene between fuzzy and grey is that the latter emphasizes on the lear extension and unlear intension. Table 1. Comparison of three unertain methodologies Comparison items Grey system Probability and Fuzzy mathematis Study objet Fundamental set Methodology basis Manner Data requirement Emphasis Goal Inadequate information Grey set Information overage Grey series operator ny distribution Intension statistis Stohasti unertain Cantor set Mapping Frequeny statistis Typial distribution Intension Historial Cognitive unertain Fuzzy set Mapping Cutting set Known membership Extension Cognitive expression 37

4 features Reality rule Small sample statistial rule Large sample Experiene No matter that what kinds of unertainties are desribed, when they are applied to the ritial hain projet management, we need onsider the unertainty from the three aspets: task duration estimate, revision of the task duration, unertainty evaluation and buffer alulations. We will disuss these four key issues in the following setion in detail. 4. GREY CRITICL CHIN PROJECT MNGEMENT TECHNIQUE Traditionally, buffer in the ritial hain is very subjetive, whih makes the ritial hain approah diffiulty in pratie. We use the grey system theory to improve the taken way of the buffer time. Buffer has relationship with the task duration and its unertainty. 4.1 Grey Task Duration Estimate ny projet has a definite ompletion time, so in essene, the duration for any task in the projet has lear bound, that is, the extension of the task duration is lear and deterministi. Lower bound of the duration is greater than zero, and upper bound is less than the projet ompletion time. Therefore, we an suppose the task duration is an interval grey number with some kind of whitening weight funtion. typial interval grey number for task =(, a; b, d) may be expressed in the following whitening weight funtion showing in Figure 1. f L(, t [, a) = 1, t [ a, b] R(, t ( b, d] f ( 1 L( R( a b d t Fig 1. whitenization grey funtion L( is alled the left inrease funtion, and R( is alled the right derease funtion, [a, b] is the peak area, is the beginning point, d ending point, and a, b turn points. In pratie, for onveniene in programming and alulation, L( and R( are often simplified as the straight lines expressed in the following formula. But in priniple, any kinds of orresponding funtions are appliable. f t L( =, t [, a) a = 1, t [ a, b] d t R( =, t ( b, d] d b Using the f ( expression, we an alulate the following expeted grey duration to simulate the task duration for task : EG d = d f f tdt dt If only task =[,a;b,d] is known and the f ( expression unknown, we an use the following formula to alulate the expeted grey duration for task : a + 2b d EG = Revision for the Grey Task Duration Estimate In traditional projet management, projet manager often adds buffer to all the tasks involved in the projet, whih may lead to the waste of the buffers. Goldratt emphasized that the buffers must be putt together to manage by the projet manager. Therefore, we must revise the former alulated duration for all the tasks in order to leave some buffers to the projet manager. ording to Goldratt [1], the projet manager often taken the twie estimate as the alulating task duration, hene we divide EG by 2 to restore the original estimate, and take the result as the revised duration for task (RD). RD =EG /2 4.3 Unertainty evaluation Goldratt suggested taking the half of the task duration as its buffer, whih had ertain experiene bakground. The unertainty ould be from the following two aspets: (1) degree of greyness for the task, and (2) the distane between the starting time of the projet and the task. The greater the greyness degree of the task, the more diffiulty the task duration desribed orretly. Hene, we an take the degree of the greyness of the task as one of the basis to measure the task unertainty. 38

5 Let interval grey number be =(,a;b,d), the degree of greyness of is alulated by the following formula DG, whih is a quite revision of the formula from [4],[5]: DG d a f dt L( dt + ( b a) + R( dt b = = d d Middle part of the formula indiates the effet of the peak region, and the other two parts the effet of the overed area of the L( and R(. The greater DG, the bigger the unertainty in the task. Regularly, the farther the future, the more diffiult to foreast the happen. Therefore, the farther the distane between the projet starting time and the task, the greater the unertainty involved in the task. This priniple an be used to evaluate the unertainty of the task. This fator affeting the unertainty of task we all the position weight, denoted by PW, whih is the index that measures the unertainty that is aused by the distane fator. In number, PW equals the distane between the projet starting time and the middle point of the duration for task divided by the projet ompletion time. 4.4 Buffer Determination We use a onept alled α-ut set to determine the buffer of the tasks. d We use FI to denote the area bounded by the f ( for task =[,a;b,d], whih is easy to alulate: FI =(b+d--a)/2. For the linear ase, given α, we an easily find m and n: m=a-(a-)(1-α) n=b+(d-b)(1-α) Hene, after α-ut operation for task, its new FI we denote by α FI, an be determined as follows: α FI =(d+n-m-)* α /2 Finally, we an use the following formula to alulate the buffer for task denoted by BF : FI α BF = FI EG 4.5 Thinking Proess of the Grey Critial Chain Projet Management To summarize the steps for the grey ritial hain projet management, we get following thinking proesses: Step 1. Use the interval grey number to estimate the durations for all the tasks in the projet The deision maker or makers may apply their experiene to find these durations. Step 2. Transfer the interval grey number of the task duration into a deterministi evaluation DG Step 3. Calulate the revised grey duration RD Step 4. Identify the ritial path based RD, and determine the projet ompletion time and starting time. To hange α from 0 to 1, interval grey number hange from the known to grey. We an also ut the area bounded by f( to two parts, upper and down. The upper part denotes the buffer for task. The size of the buffer is determined by α, and buffer size is inversely proportional to α. Therefore, (1-α) an be used to denote the unertainty of tsak. This unertainty is determined by two fators, that is, the degree of grey and the position weight. We use the following formula to determine the α value for task, denoted by α: α =1-(PW *β+dg *(1-β)) Here, β is a deision parameter whih an be used to reflet the deision preferene of the deision maker for the buffer. The bigger the β, the more unertain the task duration. Step 5. Find the α values for all the tasks in the projet by their position weights and degrees of grey: 1st. alulate the position weight for eah task in the projet 2nd. alulate the degree of grey for eah task in the projet 3rd. selet proper deision parameter β eah task 4th. find α value for eah task Step 6. Determine the buffers for all the tasks in the projet 1st. alulate FI for eah task 2nd. alulate α FI for eah task 3rd. determine the buffer BF for eah task in the 39

6 projet Step 7. Sheduling the projet by the ritial hain thinking proess Find the ritial hain and shedule the projet aording to the ritial hain thinking proess. 5. CSE STUDY Consider a projet onsisting of eleven tasks. The fat is as follows. Table 2. ase study projet fat Task name Immediate predeessor Resoures / shy B / David C David D B Chart E C,D David The implementation proess of the grey ritial hain projet management is as follows: Step 1. Evaluate the task durations by interval grey numbers. See Table 3. Step 2. Transfer the grey duration to the deterministi expeted grey value EG. See Table 3. Step 3. Find the revised duration RD. See Table 3. Step 4. Identify the ritial hain based on the RD. See Table 3 Table 3. implementing results in steps 1-3. Task Task duration EG RD Critial name (2,4;6,8) No B (6,8;10,12) Yes C (3,5;8,10) No D (2,3;4,5) Yes E (3,6,10,13) Yes The projet starting time supposed to be zero, so the projet ompletion time is The ritial hain is B D E. Tasks B,C and D use the same resoure David, so the resoure appliation sequene is B,C,E. We may solve the resoure onflit aording to this resoure sequene. fter solving the resoure onflit, we get a new projet shedule. We will ontinue our work from this no-resoure-onflit shedule. The new projet ompletion time is The starting time for C is 4.5 instead of 2.5, and the starting time for E is 7.75 instead of The starting times for other tasks remain unhanged. Step 5. Find α values for all the tasks. Suppose the whitenization weight funtion is pieewise linear. Firstly, based on the new shedule, alulate the position weights and degrees of grey for all the tasks. Seondly, selet the deision parameter β for eah task. Finally, alulate the α values. See Table 4 for the results. Table 4. implementing results in step 5 Task PW DG β α name B C D E Step 6. Determine the buffer for eah task in the projet. See Table 5 for the results. Table 5. implementing results in step 6 Task name FI α FI BF B C D E Step 7. Sheduling the projet by the ritial hain thinking proess Find the ritial hain and shedule the projet aording to the ritial hain thinking proess. In the following, steps of the ritial hain thinking proess need to implement. Limited to the page numbers, Disussion is omitted. Interested readers may finish the remaining for pratie. 5. COMPRISONS BETWEEN FUZZY CPM, CRITICL CHIN ND GREY CRTICL CHIN Table 6. omparisons between fuzzy CPM, CC and GCC Comparison items Fuzzy CPM Critial Chain Grey Critial Chain Unertainness Fuzzy Stohasti Grey Psyhologial No Yes Yes Consideration Unertain Yes No No Effet Buffer No Intuitive and Unertain Calulation Unertain Solution Response on resoure obtainment Constraint hange Fuzzy Expetation experiene No fators Developed thinking proess No Yes Yes Yes No No Generally, the buffer alulated aording to the ritial hain approah often is greater than the buffer 40

7 alulated aording to the grey ritial hain tehnique, whih makes the projet manager more perfetly ontrol the projet when applying the grey ritial hain management tehnique. Besides, the unertainty onsideration is more sientifi. 6. CONCLUSIONS Critial hain is a quite new onept in the modern projet management, whih supplies many new thinking ways for the management of both single projet and multiple projets or projet portfolio. In this paper, based on the grey system theory, a generalization development of the ritial hain approah is proposed, whih overome parts of the disadvantages existing in the traditional ritial hain ase based on the stohasti unertainty. We use degree of grey and the position weight to reflet the unertainness of the tasks in the projet. n advaned alulation method is developed to find the task buffer, whih greatly improves the buffer manner in the traditional ritial hain and makes the projet ontrol more easily. ase study shows that the advaned method is appropriate and appliable for the pratial projet management. The grey ritial hain method not only an be used in the projet time management, its thinking proess also an be applied in the other areas of the projet management. REFERENCES [1] E.M. Goldratt. Critial hain [M]. Grant Barrington: The North River Press, US, [2] L.P. Leah. Critial Chain Projet Management. Boston: rteh House, US, [3] M.Chen, S.Hse. Fuzzy-ritial-hain-based projet management [J].Journal of the Chinese Institute of Industrial Engineers, 2004, 21(2): [4] P. Chanas, and J. Kamburowski. The use of fuzzy variables in PERT [J]. Fuzzy Sets and Systems, 1981, (5): [5] S.F.Liu. Grey systems theory and appliations [M]. Beijing: Siene Press, 3 rd version, China, [6] P. Chanas, and J. Kamburowski. The use of fuzzy variables in PERT [J]. Fuzzy Sets and Systems, 1981, (5):11-19 [7] J.L. Deng. Fundamentals of grey systems [M]. Wuhan: Huangzhong University of Siene and Tehnology Press, China, THE UTHORS Gao Peng, Feng Junwen, Wang Huating: Shool of Eonomis and Management, Nanjing University of Siene and Tehnology, Nanjing, , P.R. China 41