Construction Project Management Approach by riv11333

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									    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 Ani  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 Ai , 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 predn ( ES mL  AmL ), m predn ( 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  [ ( ESL  AL ) , ( ESR  AR ) ], i  P
                                                              i     i
                                                         =[ ECL , ECR ], 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  [ LCL , LC R ] , i  P  EC
                                                ~     ~       ~        ~
                                                LCn  L S n  An  min L S m
                                                                               mSucc



                                                     i                    i
                                          = [ min LS mL ,           min LS mL ] , i  P
                                            mSuccn             mSuccn 




                                                              5
                     = [ min LCmL  AmL  , min LCmR  AmR  ], i  P
                                   i i               i i
                        mSuccn           mSuccn 

                                                                   ~
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
                                                                                  mSucc

(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
                               mSucc                mSuccn           mSuccn 


                                        = L C n   LC nL , LC nR  , i  P
                                          ~            i      i


                              ~       ~
                              L S n  An             min LS mL , min LS mR , i  P
                                                              i
                                                     mSuccn 
                                                                          i
                                                                         mSuccn 


                       ~
                       LSn            min ( LS mL  AnL ) , min ( LS mR  AnR )  , i  P
                                    mSuccn 
                                                 i    i              i    i
                                                                    mSuccn 


                                                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 , DTF i  Tcon   0 , DTF i  Tcon   1
                                        ~                      ~

                    If TFnLi  Tcon , DTF i  Tcon   1 , DTF i  Tcon   0
                                       ~                      ~

                     If TFnLi  Tcon  TFnRi , DT F i  Tcon   TFi  Tconi
                                                                      i
                                               ~
                                                                        TF               
                                                                     nR
                                                                             
                                                                            nR    TFnL

         Project risk level index   DT F  Tcon   i0
                                        ~
                                                            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
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