VIEWS: 49 PAGES: 10 CATEGORY: Business POSTED ON: 1/11/2011 Public Domain
A FUZZY PERT APPROACH FOR MANAGING THE PETROLEUM CHEMICAL PLANT CONSTRUCTION PROJECT Chun-Wei R. Lin* and Hsian-Jong Hsiau, Department of Industrial Management, National Yunlin University of Science and Technology, 123 University Road, Section 3, Touliu, Yunlin, Taiwan, 640, ROC lincwr@yuntech.edu.tw* ABSTRACT The petroleum chemical plant construction project generally involves lots of activities. Unfortunately precise information about the activity duration is seldom available, due to the complication, long time frame, and less identical experiences. It is not suitable to use crisp data to evaluate the project scheduling. This paper presents an extended fuzzy PERT (Program Evaluation and Review Technique) approach which includes three major improvement aspects to support the project scheduling management and the project bidding process: 1) Adopting a maximal method to compare fuzzy numerical construction information to determine the reasonable earliest start date of each construction task, 2) Using fuzzy algebra method instead of fuzzy substraction method to compute the fuzzy latest start dates, and 3) Developing a project risk level index to assist the decision maker for bidding a potential project. Simulations experiments are conducted and demonstrated satisfactory results. Keyword: Fuzzy PERT approach, project management INTRODUCTION The scheduling of petroleum chemical plant construction is not easy to control due to various uncertain factors. In industrial practice, the decision makers usually use crisp value to estimate the project time while bidding a potential project. But when they get the orders or contracts, frequently they can’t complete construction on time and the resulting cost always exceeds original expectations. In order to reduce construction project bidding risk, it is necessary to use a project management method. Owing to the capability to describe the uncertain nature of real industrial practice in project management, fuzzy PERT (program evaluation review techniques) is widely used. There is a vast literature devoted to research about the fuzzy PERT theory and application. For example, Mon et al. (1995) applied fuzzy distributions on project management to analysis schedule and cost. Wang (2002) used a fuzzy project scheduling approach to minimize schedule risk for product development. Dubois et al. (2003a) studied on latest starting times and floats in activity networks with ill-known durations. Dubois et al. (2003b) also research fuzzy scheduling with incomplete knowledge. Slyetsov et al. (2003) researched fuzzy temporal characteristics of operations for project management based on the network models. However, there are still several unsolved issues in fuzzy PERT applications: 1 (1). In large scale of project management, the operation time of each activity is not easy to be known even using fuzzy number. If the decision maker directly assumes that operation time of activity is known to compute the scheduling of project, the result of scheduling may be imprecise. (2). There are many ranking methods for fuzzy number. However a suitable method to compute the earliest starting time of each activity in project network is yet to developed. (3). There is no overall time risk index to assist the decision maker to make decisions while bidding a potential construction project. In coping with the aforementioned issues, this paper presents an extended fuzzy PERT (Program Evaluation and Review Technique) approach which includes three major improvement aspects to support the project scheduling management and the project bidding process: 1) Adopting a maximal i level method to compare fuzzy numerical construction information to determine the reasonable earliest start date of each construction task, 2) Using fuzzy algebra method instead of fuzzy substraction method to compute the fuzzy latest start dates, and 3) Developing a project risk level index to assist the decision maker for bidding a potential project. In order to resolve the plant construction project duration and time risk, major assumptions are made as follows: (1). There is a project with n item activities. The precedence or succeed relations between each activities are available. (2). Suppose that the working volumes of each activity Wn are available from bidding information. (3). Suppose that project manager can decide the resource quantity K n for each activity. (4). Suppose that project manager can get the information about the fuzzy working capacity of resources, the fuzzy working capacity of resources can be shown as ~ trapezoid fuzzy number Vn (v1 , v2 , v3 , v4 ) . ~ The membership of Vn , the fuzzy working capacity of resources at nth activity, can be defined as a trapezoid fuzzy number： A ( x v1 ) /(v2 v1 ) L ~ , v1 x v 2 ~ 1 , v 2 x v3 Vn ( x ) R A ( x v4 ) /(v3 v 4 ) ~ , v3 x v 4 0 ,o t h e r v1 ： the most pessimistic fuzzy working capacity of resources v2 ,v3 ：the most possible fuzzy working capacity of resources v 4 ： the most optimistic fuzzy working capacity of resources THE EXTENDED FUZZY PERT APPROACH In this section we use the modified fuzzy PERT method to establish the computing procedure model of project time and risk level index for plant construction. 2 The modified fuzzy PERT method includes: (1). Compute operation fuzzy time for each activity by available working volumes and fuzzy capacity of working resources. (2). Using Max i level method to compare fuzzy number, in order to determine the reasonable earliest startting date of each activity. (3). Using fuzzy algebra method instead of fuzzy substraction method to compute the fuzzy latest dates. (4). To create an appropriate project risk level index for decision makers to make a decision. The computing procedure model of plant construction project time and risk level index can be stated as this: Step1. To decide project time computing parameters The project time computing parameters includes： (1). Items of project activity n (2). Precedence or succeed relations between activities (3). Working volumes of each activity Wn ~ (4). Fuzzy capacity of resources for each activity Vn (v1 , v2 , v3 , v4 ) (5). Quantity of resources for each activity K n (6). Overall project contract time Tcon and the maximum project risk level index m ax which decision maker can accept. (7). Numbers of i level cut p . Step 2. Compute operation fuzzy time of each activity In large scale of plant construction, the operation time is difficult to know directly. In order to compute every kind of fuzzy time for activities in project network, it is necessary to compute the operation time by working volume and fuzzy capacity of resources. The operation fuzzy time of each activity can be computed by fuzzy divided method as following: An an , bn , cn , d n = n ~ W 1 1 1 1 , , , Kn v 4 v3 v 2 v1 Wn : The working volume of nth activity ~ Vn : The fuzzy working capacity of resources shown by trapezoid fuzzy number, Vn v1 , v2 , v3 , v4 . ~ K n : Quantity of resources for nth activity. Step 3. Compute the membership of fuzzy time for each activity at i level cut In this paper, we propose the Max i level cut method to compute the every kind of time for each activity. Therefore, we have to compute the membership of fuzzy time for each activity at i level cut. The memberships of fuzzy time for each activity at i level cut are computed base on value, which is decided by decision maker. From step 2, we can get the operation fuzzy time of each activity 3 An an , bn , cn , d n . Suppose decision maker set numbers of i level cut p ,then ~ 1 can be used to compute the membership of fuzzy time for each activity at p ~ i i i i level cut，the membership at i level cut Ani AnL , AnR ，the value of AnL and i AnR can be calculated as following equation︰ i AnL an (bn an ) i i AnR d n (d n cn ) i i i, i P 0,1,2,..., p ~ From above equation can show the membership of each activity An at i level cut as An AnL , AnR , i P ~ i i ~ Step 4.Compute the earliest starting fuzzy time for each activity ( ES ) Fuzzy PERT usually uses forward method in network to compute the earliest starting fuzzy time for each activity. In proposed modified fuzzy PERT, the computing procedures of earliest starting fuzzy time for each activity are as bellow: (1). Suppose that there are items in total project network, the starting item is and completed item is , the earliest starting time with trapezoid fuzzy ~ number for item is： ES (0,0,0,0) (2). We can compute the earliest starting fuzzy time for each activity n with the ~ ~ ~ equation: ES n max ( ES m Am ) m pred n It means that the earliest starting fuzzy time for activity n is the maximum fuzzy time of all precedence activities m completing fuzzy time. In this step, ~ ~ ~ in order to get result of ES n max ( ES m Am ) , have to compute the fuzzy m pred n max ( ES m Am ) . Here, we propose the Max i level method to number ~ ~ m pred n ~ ~ calculate the membership of max ( ES m Am ) . The computing procedures can m pred n be stated as following with trapezoid fuzzy number. (3). Decision maker sets numbers of i level cut p , value is decided, we can get the set, i 0 ,1 , 2 , 3 ,...., p , of i level cut. ~ ~ ~ (4). Suppose that two fuzzy numbers are A and B , the membership values of A and B at i level cut are A a i , a i , B i bL i , bRi , i P 0,1,2,3,..., p . ~ ~ i L R ~ ~ ~ (5). Comparing all i level cut values of A and B fuzzy numbers at 0 ,1 , 2 , 3 ,...., p level and take the maximum value at each level cut. The set of maximum value is C i max A i , B i . The computing result of ~ ~ ~ i p ~ ~ Max i level for A and B can be illustrated as: 4 ~ A 1.0 ~ 0.75 B ~ B 0.5 0.25 x Figure 1. Computation result of C i max Ai , B i ~ ~ ~ i p From above computing procedure of Max i level cut, in this paper, we use it to compute the earliest starting fuzzy time for each activity and get the results: ~ i i i i ES n [m predn ( ES mL AmL ), m predn ( ES mR AmR )] , i P max max ~ Step 5. Compute the earliest completing fuzzy time for each activity ( E C ) To compute the earliest completing fuzzy time for n activity, refer to Slyeptsov ~ ~ ~ et al. [2], applying the equation: ECn ESn An , n R; is the operator symbol of fuzzy number addition. Applying Max i level method, the earliest completing fuzzy time for n activity at i level cut can be written as follow: ~ i i i i ECn [ ( ES nL AnL ) , ( ES nR AnR ) ], i P ~ Step 6. Compute the overall completing fuzzy time of total project ( T F ) ~ Suppose that the overall completing fuzzy time of total project is T F . From the ~ ~ ~ equation ECn ESn An , if the last item of project activity is , we know that overall ~ ~ ~ ~ ~ ~ completing fuzzy time of total project T F EC . To get T F EC = ES A , Using Max i level method, we can get: ~ i i i i T F [ ( ESL AL ) , ( ESR AR ) ], i P i i =[ ECL , ECR ], i P ~ Step 7. Compute the latest completing fuzzy time for each activity ( L C ) To avoid the fuzzy number extending and unreasonable negative value after fuzzy number substraction operator, propose fuzzy algebra method instead of fuzzy substraction method to compute the fuzzy latest dates of each activity. ~ i i ~ L C [ LCL , LC R ] , i P EC ~ ~ ~ ~ LCn L S n An min L S m mSucc i i = [ min LS mL , min LS mL ] , i P mSuccn mSuccn 5 = [ min LCmL AmL , min LCmR AmR ], i P i i i i mSuccn mSuccn ~ Step 8. Compute the latest starting fuzzy time for each activity ( L S ) Here, also use fuzzy algebra method to compute the latest starting fuzzy time for each activity. The computing procedures are these: (1). From step 6, we have computed the overall completing fuzzy time of total ~ ~ project T F EC . The earliest and latest completing fuzzy time is same for last item activity in project. There, the latest completing fuzzy time last item ~ ~ ~ ~ activity is L C , L C = T F EC . ~ ~ ~ (2). Based on forward method, calculate: L S n An min L Sm mSucc (3). Base on i level cut method, use fuzzy algebra method to compute the latest ~ starting fuzzy time for each activity L S n . The computing procedure is as following. An AnL , AnR , i P ~ i i min L Sm = , i P ~ i i min LS mL , min LS mR mSucc mSuccn mSuccn = L C n LC nL , LC nR , i P ~ i i ~ ~ L S n An min LS mL , min LS mR , i P i mSuccn i mSuccn ~ LSn min ( LS mL AnL ) , min ( LS mR AnR ) , i P mSuccn i i i i mSuccn LC nL AnL , LC nR AnR i i i i , i P Step 9. Compute project contract time risk level index: The definition of project contract time risk level index is the possibility which overall project completing fuzzy time is longer than project contract time. The relationship between the overall project completing fuzzy time and project contract time is shown in figure 2. In step 6, we have gotten the overall project completing fuzzy time. If project contract time is a crisp value Tcon , we can compute the project contract time risk level index . TFnRi 1.0 TFnRi i i 0 x Tcon Figure 2. The overall completion fuzzy time vs. project contract time Tcon 6 If TFnRi Tcon ， DTF i Tcon 0 ， DTF i Tcon 1 ~ ~ If TFnLi Tcon ， DTF i Tcon 1 ， DTF i Tcon 0 ~ ~ If TFnLi Tcon TFnRi ， DT F i Tcon TFi Tconi i ~ TF nR nR TFnL Project risk level index DT F Tcon i0 ~ m ~ D T F i Tcon ， m . m ~ ~ D TF Tcon D TF Tcon 1 m ：number ofα-level cut, the accuracy of project risk level index is relative to m . If = m 2m 1% ，to accept computation result of project risk level index . m Step 10. Output the computation results. SIMULATION EXPERIMENTS AND PERFORMANCE EVALUATIONS In order to compare the quantity of computing results and CPU time for overall completing fuzzy time and project risk level index, which were computed by Max i level and defuzzifying method respectively. Simulation experiment are conducted to simulate the varied construction project situations and process the Hypothesis testing to confirm whether the results are significant. 3.1 The setting of experiment environment For the setting of experiment environment, consider the project activity items and time to express the construction scale and complicated degree. In experiment, set that the activity items are large and small, fuzzy time of activities are short and long. This is an experiment with two factors and two levels. Two factors are activity items (7 vs. 21) and fuzzy time of activities. Two levels are large and small, short and long. We will combine different factor and level to simulate the overall completing fuzzy time and project risk level index by MAX i level and defuzzifying method separately, compare the quantity of computing results and CPU time. Suppose that the fuzzy time for each activity is An an , bn , cn , d n , n is the number of ~ items. The values of an , bn , cn , d n can be random generated during fuzzy interval by simulation program. In this experiment, we controlled the short and long activity fuzzy times by C value and random value in interval. In simulation experiment, there are two variable factors, which are activities and operation fuzzy time. Each variable factor has two levels. In order to process the simulation experiment for different conditions, to assemble the different variable factors and levels. Four groups experiment, shown as table 4 and figure 3, will be simulated. 7 Table 1 Simulation conditions of plant construction project Activity Activity Working Capacity of working Resource Precedence Fuzzy Operation Time item description Volume resource (days) quantity item a b c d 1 Piling 480 24 25 26 27 1 None 18 18 19 20 2 Foundation 4,800 30 30 32 33 3 1 48 50 53 53 3 Steel structure Fabrication 3,200 26 28 30 32 4 None 25 27 29 31 4 Steel structure Install 3,200 30 32 33 34 2 2- 3 47 48 50 53 5 Equipment type 1 200 2 3 3 4 2 2 29 33 33 50 6 Equipment type 2 400 4 4 5 5 2 4 40 40 50 50 7 Equipment type 3 1,000 8 10 11 14 2 4 36 45 50 63 8 Piping Fabricated 20,000 50 52 55 57 8 None 44 45 48 50 9 Piping Installation 16,000 16 18 19 22 10 5-8 73 84 89 100 10 Control room 10,000 20 21 23 25 2 1 200 217 238 250 11 Equipment Flushing 16 0.26 0.34 0.34 0.42 1 9 38 47 47 61.5 12 Piping Flushing 3,600 60 70 80 85 1 9 42 45 51 60 13 Equipment Insulation 8,500 75 85 90 120 2 9 35 47 50 57 14 Piping Insulation 4,000 80 85 95 120 1 9 33 42 47 50 15 Instrument 2,000 30 50 52 60 1 9 33 38 40 67 16 Electricity 4,000 35 43 44 55 2 9 36 45 47 57 17 Test run 16 0.5 0.6 0.6 0.7 1 10-16 23 27 27 32 Start 0 Piling 1 Steel structure fabrication Foundation 2 3 Steel structure 4 installation Equip. type 1 Equip. type 2 Equip. type 3 5 6 7 8 Piping fabrication Piping installation 9 Equip. Insulation 10 11 12 15 16 13 14 Piping Instrument. Equip. Piping. Control Flushing Insulation Insulation room Electricity 17 Test run 18 Finish Figure 3. Project network of plant construction 8 3.2 Simulation results and analysis To simulate four groups experiment, we can get the results of overall project time, project contract time, project risk level index and CPU time by Max i level method and defuzzifying method individually. The simulation results show as in table 3, the P value are all less than the level of significance 0.05, the value of test statistics are all in the rejection region for four experiment items. Mean of project risk level index of Max i level is higher than defuzzifying method in four experiments items. For fuzzy number ranking method, the operator of MAX belongs to composite method. The benefit of composite method is: the composite result of MAX operator can fully express the character of two or more fuzzy numbers, but using this method to compute the project time is complicated in real world. For defuzzifying method, the result is the maximum value of fuzzy numbers which participating in comparison. It can’t fully express the character of two or more fuzzy numbers. In this studying, we propose Max i level method, it is modified operator of composite method. Using this method to compute the project time, it is more suitable than defuzzifying method Also, the testing results are shown as table 4, P value are all less than the level of significance 0.05, the value of test statistics are all in the rejection region for four experiment items. CPU time mean of Max i level is shorter than defuzzifying method in four experiments item. Table 3 Mean of project risk level index test result for each experiment item. Mean of project risk level index Testing result Exp. item MAX i level method Defuzzifying method P value H 0 : 0 = 1 Mean 0 Std dev. Mean 1 Std dev. 1 0.1614 0.0250 0.1384 0.0349 0.000 Rejection 2 0.1614 0.0250 0.1384 0.0349 0.000 Rejection 3 0.1462 0.0142 0. 1239 0.0256 0.000 Rejection 4 0.1462 0.0142 0. 1219 0.0273 0.000 Rejection Table 4 Testing result of CPU time mean for each experiment item. CPU time mean Test results Exp. item MAX i level method Defuzzifying method P value Mean 2 Std dev. Mean 3 Std dev. 1 0.3469 0.0652 0.4329 0.0875 0.000 2 0.2282 0.0612 0.2666 0.0939 0.004 3 0.3913 0.0634 0.9885 0.1659 0.000 4 0.3746 0.0585 0.9543 0.1646 0.000 9 CONCLUSIONS AND DISCUSSIONS For the purpose of decreasing the risk of managing the petroleum chemical plant construction project, we presented an extended fuzzy PERT approach to resolve the difficulties of traditional fuzzy PERT and the major achievements are as follows: (1). Activity operation durations in project network are computed from task volumes and capacity of resources. The computing model is feasible and was proofed by simulation experiment. (2). The proposed Max i level cut method outperformed the defuzzifying method to rank fuzzy number for determining the reasonable earliest startting date of each activity. (3). The proposed procedure to calculate the project time risk level index is convenient for decision maker for bidding a potential project. REFERENCES Wang, J., 2002, A fuzzy project scheduling approach to minimize schedule risk for product development, Fuzzy Sets and Systems, 127, 99-116. Slyeptsov, A.I. & Tyshchuk, T.A., 2003, Fuzzy temporal characteristics of operations for project management on the network models basis, European Journal of Operational Research, 147, 253-265. Chanas, S. & Zielinski, P. 2001, Critical patch analysis in the network with fuzzy activity times, Fuzzy Sets and Systems, 122, 195-204. 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