A New Mathematical Model to Fully Fuzzy Project Crashing Problems

A New Mathematical Model to Fully Fuzzy Project Crashing Problems

M. Evangeline Jebaseeli1

PG and Research Department of Mathematics, Bishop Heber College,

Tiruchirappalli – 620 017, Tamil Nadu, India.

1. Paul Dhayabaran2

   Principal, Bishop Heber College, Tiruchirappalli – 620 017 Tamil Nadu, India.

   Abstract: Optimization problems in project management have been traditionally solved by two distinctive approaches: heuristic methods and optimization techniques. Although heuristic methods can handle large-size projects; they do not guarantee optimal solutions. In this paper, a linear mathematical optimization model for fuzzy project time cost trade off problem is developed. An illustration is provided to demonstrate the efficiency of the proposed method.

   Keywords: Fuzzy Project crashing – Triangular fuzzy number – Linear Programming – cut Interval.

   1. INTRODUCTION

      Project Management is a very important field employed for scheduling activities and monitoring the progress, in competitive and fluctuating environments. The feasible duration required to perform a specific project is determined using critical path method. However, because of competitive priorities, time is important and the completion time of a project determined using critical path method should be reduced to meet a deadline requested.

      In scheduling a project, it is generally considered to expedite the duration of some activities through expanding extra budget in order to compress the project completion time. This procedure can be considered under either some fixed available budget or a threshold of project completion time. This problem is known as time cost trade off problem or project crashing problem in the project management literature.

      The main objective of these kinds of problem is to determine the optimum duration and cost should be assigned to the activities such that the overall cost is minimized.

      The project duration can be shortened by the acceleration of the critical activity times. The acceleration of the activity times can be achieved using more resources (using more productive equipment, material or hiring more workers) which means higher costs. Project crashing problem analyzes how to modify project activities so as to achieve the trade off between the project cost and the completion time.

      A. LITERATURE REVIEW

      By reviewing the literature, it is observed that there are several studies investigated and analyzed the project management problems. Mathematical and heuristic methods are the two major approaches used to solve the time cost trade off problems in project scheduling. Mathematical methods convert the project time cost trade

      off problems to mathematical models and utilize linear programming, integer programming, dynamic programming, goal programming or multi-objective linear programming to solve the problems. However, formulating the objective function as well as the required constraints is time-consuming and prone to errors. Heuristic methods provide a way to obtain good solutions but do not guarantee optimality. However, they require less computational effort than mathematical methods.

      The problem of project time cost trade off was first introduced by Kelly [9]. By assuming that direct cost of an activity changes with time, mathematical programming models were developed to minimize the projects direct cost [9]. Thereafter, many researchers have developed mathematical programming model for these kinds of problems. The time cost trade off problems have been extensively investigated. Many models have been proposed and they can be categorized into two types: Deterministic scheduling and non-deterministic scheduling. Recently, Yang [19] proposed a chance-constrained programming model to analyze the time cost trade off problem, where funding variability is considered. Yang [19] took budget uncertainty into account on project time cost trade off in a chance-constrained programming model. A hybrid intelligent algorithm integrating simulation and genetic algorithm was designed for solving the proposed models.

      The above mentioned time cost trade off models mainly based on probability theory. As generally known, it requires a prior predictable regularity or a posterior frequency distribution to construct the probability distribution of activity times. However, in real world applications some activity times must be forecasted subjectively; for example, we have to use human judgment instead of stochastic assumptions to determine activity times. An alternative way to deal with imprecise data is to employ the concept of fuzziness, whereby the vague activity times can be represented by fuzzy sets. The main advantages of methodologies based on fuzzy theory are that they do not require prior predictable regularities or posterior frequency distributions, and they can deal with imprecise input information containing feelings and emotions quantified based on the decision-makers subjective judgment.

      In the literature, there are several studies that have investigated the project management problems with fuzzy parameters. Liu et al. [14] proposed a fuzzy optimal construction time cost trade off method. In their study, the

      activity duration is accepted as fuzzy number. An acceptable risk level is defined as the minimum concept of the fuzzy set theory; fuzzy durations are transformed into crisp sets. Then the genetic algorithm techniques are used to find the optimal or near optimal solutions. Arikan and Gungor [2] applied fuzzy goal programming to the time cost trade off problem with two objectives which are minimum completion time and crashing costs. The aspiration levels of the objective are defined as fuzzy numbers. The goal programming is solved using max-min approach. Wang and Liang [15] solved project management decision problem with multiple fuzzy goals in their study. The goals of the problem are defined using linear membership functions, and the multiple fuzzy goal programming problem is solved after transforming into its crisp equivalent using Bellman and Zadehs fuzzy decision concept. Eshtehardian et al. [4] presented a new approach for the solution of time cost trade off problems with uncertain costs. An appropriate genetic algorithm is used to find the solution of the multi objective fuzzy time cost problem. Lin [12] proposed an approach to solve project crashing problems with uncertain activity times and crash costs. The confidence-interval estimates and the previous statistical data are used to solve the fuzzy project crashing

      problem. In this study, level (1- ) fuzzy numbers were

      derived from (1 ) 100% confidence interval

      levels are obtained. Tolunay Gocken [18] proposed a solution process for the fuzzy multi objective project crashing problem.

      In this paper, a mathematical optimization model for solving time cost trade off problem is presented. The model is formulated in the form of linear programming that produces optimal solution. The proposed model minimizes the total cost of the project and this model concerns both direct cost and indirect cost.

   2. PRELIMINARIES

      In this section, some basic definitions of fuzzy theory defined by Kaufmann, Gupta and Zimmermann, are presented.

      Definition 2.1

      A

      The characteristic function A of a crisp set A X assigns a value either 0 or 1 to each member in X. This function can be generalized to a function ~ such that the

      A

      value assigned to the element of the universal set X fall within a specified range i.e. ~ : X [0,1] . The

      assigned values indicate the membership grade f the element in the set A.

      The function ~ is called the membership function and

      estimates of the statistical data with a distance ranking A

      which is used to define the fuzzy ordering. The activities ~

      execution times and costs, the daily costs are accepted as triangular fuzzy numbers. The proposed approach explicitly embeds the fuzzy set theory into the optimization procedure and then a multi objective genetic algorithm is used to solve the discontinuous and multi objective fuzzy time cost model. Maa and Ke [10] solved time cost trade off problem with fuzzy activity duration times. The fuzzy activity duration times defined as fuzzy variables based on self-dual credibility measure. Then, the obtained time cost trade off problem is solved with a hybrid intelligent algorithm integrating fuzzy simulation and genetic algorithm. Ghazanfari et al. [6] proposed a mathematical model to deal with fuzzy time cost trade off problem. The normal and crash durations of activities are considered as triangular fuzzy numbers. For the solution of the fuzzy problem, a ranking fuzzy numbers method used. Liang [16] proposed a possibilistic linear programming approach for the solution of fuzzy multi objective project management decision problem. The fuzzy parameters are defined using the triangular possibility distribution. In the proposed possibilistic linear programming approach, the fuzzy objectives and the fuzzy constraints are transformed into their crisp equivalents. Then, the obtained multi objective linear programming problem transformed into an equivalent linear programming problem using Zimmermans fuzzy decision concept and the minimum

      operator. Chen and Tsai [3] proposed a new approach to

      the set A {(A, A (x) : x X }

      A

      defined by ~ (x) for each x X is called a fuzzy set.

      Definition 2.2

      ~

      A fuzzy set A defined on the set of real numbers R is said

      to be a fuzzy number if its membership function has the following characteristics:

      A

      1. ~ (x) : R [0,1] is continuous.

      A

      2. ~ (x) 0 for all (, a] [c, ).

      A

      1. ~ (x) is strictly increasing on [a, b] and strictly decreasing on [b, c].

         A

      2. ~ (x) 1 for all x b where a b c.

         Definition 2.3

         Triangular fuzzy number is a fuzzy number represented with three points as follows: A = (a1, a2, a3) this representation is interpreted as membership functions.

         0

         1 3

         if x a and x a

         a

         A 1 2

         ( x) x a1 if a x a

         2 a1

         solve time cost trade off problems. This approach described

         a3 x

         if a

         x a

         the minimum total crash cost of a project network via a membership function which completely conserves all the fuzziness of parameters, and the corresponding optimal activity time for each activity under different possibility

         2 3

         a

         • a

         3 2

         cut interval of TFN

         [0,1]

         A [a ( ) , a ( ) ]

         A shall be obtained as follows

         programming models and proposed the associated solution methods.

         Infact, there are many different types of FLP models, Lai

         1

         Where

         3

         a ( ) (a

         3

         1 2 1 1

         a ( ) (a a ) a

         3

         a2 ) a3

         and

         and Hwang provided a systematical classification in detail. Instead of describing all types of FLP models, here the FLP model with fuzzy profit coefficients of the objective function and total resources available is exemplified as

         Thus

         A [(a2 a1 ) a1, (a3 a2 ) a3 ]

         follows:

         Max ~ x

         T

         1. OPERATIONS ON cut INTERVAL

      subject to

      Ax ~

      If A [a ( ) , a ( ) ] and B [b ( ) , b ( ) ] are two

      b

      C

      1 3

      1 3

      x 0

      cut intervals of triangular fuzzy numbers A and B, ~ ~

      then the arithmetic operations between follows:

      1 1 3 3

      A + B = [a ( ) b ( ) , a ( ) b ( ) ]

      A – B = [a ( ) b ( ) , a ( ) b ( ) ]

      A and B is

      Where C and b are the vectors of profit coefficients and resources available being fuzzy numbers, respectively. Clearly, the assumption of certainty of the crisp LP is

      relaxed to fit real-world situations. Moreover, the operations of addition and multiplication are operations of

      1

      3 3 1

      fuzzy arithmetic, and denotes the ordering of fuzzy

      numbers.

   3. FUZZY LINEAR ROGRAMMING

      On the basis of LP formulation, this paper proposes an approach to the fuzzy time cost trade off problem. Here crisp LP and one of fuzzy LP models are briefly introduced, together with discussion on relaxing the assumptions of LP into fuzzy LP for solving the case.

      LP has been demonstrated to be one of the most frequently applied Operational Research/Management Science techniques in practical problems. LP is concerned with the effective and efficient allocation of limited resources to interrelated activities with the objective of achieving a pre- specific goal such as maximizing profit. The general LP model can be described in the canonical firm using matrix notation as follows:

      Max CT x

      subject to Ax b

      x 0

      Where C is the n-vector of profit coefficients of the objective function, A is the m n constraint matrix, x is the n-vector of decision variables and b is the m-vector of

      total resources available. As generally known, four assumptions of LP are proportionality, additivity, divisibility and deterministic. In general, the first two assumptions are also called linearity. Note that in this model, all coefficients of A, b, and C are crisp numbers, and each constraint must be satisfied strictly. In practical situations, these input data for this model are usually imprecise because of incomplete information; in addition, vagueness in these input data may not be of a probabilistic type. As Rommelfanger pointed out that LP requires much well-defined and precise data which involves high information costs, these four assumptions are often too strict to limit the applications. To deal quantitatively with imprecise information in making decisions, Bellman and Zadeh introduced the concept of fuzzy set theory. Since then, much research has developed specific fuzzy linear

      1. PROBLEM DESCRIPTION

         With the progress of the project, project managers always need to make tradeoff between the cost and the completion time. Sometimes managers may make decision in order to finish the project sooner with project cost augment by accelerating the project schedule, which is also named as project crashing in project management. In other cases, motivated by reducing the project cost, managers may be conscripted to sacrifice with prolonging the project completion time. Therefore, it is naturally desirable for managers to find a schedule to complete a project with the balance of the cost and completion time.

         The total cost function of a project has two components: direct and indirect costs. Direct costs are incurred because of the performance of project activities, while indirect costs include those items that are not directly related to individual project activities and thus can be assessed for the entire project. In general, indirect cost increases almost linearly with the increase of project duration and usually assumed as a percentage of project direct cost. The project time cost trade off problem, thus, is reduced to determine project cost against project duration. A possible way to solve time cost trade off problem is to use a mathematical programming model whose objective function is constructed so that project direct cost is minimized and the imposed constraints guarantee a desired project deadline, while the precedence requirements of the network are maintained.

         A project can be representedby an activity-on-arc network G = (V, A), where V = {1, 2,n} is the set of nodes representing the milestones and A is the set of arcs representing the activities. In the network, node 1 and n represent the start and end of the project respectively. In this paper, the normal activity durations are assumed to be uncertain variables.

      2. FORMULATION OF THE DEVELOPED MODEL

      To formulate the time cost trade off problem, one of the important issues is to access the value of that time, total cost of the project. Clearly, project completion time can be calculated by determining the total completion time for the critical path. A critical path in a project is the path with longest duration. Moreover the total cost is equal to the sum of cost used for all activities. The main goal in this problem is to minimize the total cost while the total time is minimized.

      The primary information obtained from traditional scheduling is basically activities start and finish timings and floats. The duration and the corresponding cost for an activity are selected optimally form their utility data to satisfy the objective function and the imposed constraints. If the start time of an activity is determined, the finish time can be specified by adding the selected activity duration, and vice versa.

      Parameters and decision variables of model are as follows:

      n Number of nodes

      ~

      the equation, in which T is the desired deadline of the project.

      ~ ~

      Tn T

      The upper and lower bounds on T are the normal project duration and crash project duration respectively.

      • Bounded constraint of duration:

        The planned duration of activity i j (i.e. xij ) is always lies between normal duration and crash duration of that activity. The mathematical form of this bounded constraint is

        CTij xij NTij

      • Cost constraint:

        ~ ~ ~ ~ ~

        The cost constraint involves the normal cost, slope cost, normal time and planned duration of activity ij. Thus the project cost of each activity is written in the form of

        Cij NCij sij (NTij xij )

      • Objective function:

      Our main objective of this model is to minimize the cost of the project. The linear programming objective will be

      Ti Starting time of event i

      Min

      ~

      ~ ~ ~

      ~

      Cij

      Planned cost of activity i

      j (ie. Normal

      Cij I (Tn T1 )

      i j

      ~

      sij

      cost + crash cost)

      Slope cost of activity i j

      Thus the complete fuzzy mathematical model can be summarized as follows:

      ~ ~ ~ ~

      I

      ~

      ~ Indirect cost per day

      Min

      Z Cij I * (Tn T1 )

      i j

      xij

      Planned duration of activity i j

      Subject to

      T

      ~

      ~ Project completion time

      ~

      T1 0

      NCij Normal cost of activity i j

      ~ ~ ~

      ~ T j Ti xij 0

      NTij Normal time of activity i j ~ ~

      ~

      CTij

      Crash time of activity i j

      Tn T

      ~ ~ ~ ~ ~

      • Precedence relationship constraint:

        The completion time of project could be constrained by one of two methods. The first approach is to allow for a precedence constraint for each immediate preceding relationship in the project network. This approach was used

        in almost all existing optimization techniques. The second

        Cij NCij sij (NTij xij )

        Tij ij Tij

        C ~ ~x N ~ (i, j) P

        Theorem 4.1

        If the optimal solution of the fully fuzzy linear programming problem (P2) lies in the interval

        is to allow for one constraint for each path from the first

        [x ( ) , x ( ) ] , then

        x ( )

        is the optimal solution of the

        1 3 1

        activity to the last one in the project network. In the present model, the first approach will be adopted.

        lower linear programming model (P4) and

        ( )

        x

        3

        is the

        The logical relationship between any two consecutive optimal solution of the upper linear programming model

        activities i and its immediate predecessor j, is expressed

        (P ) where

        [x ( ) , x( ) ]

        is the

        cut interval of

        ~ ~ ~

        mathematically as

        5 1 3

        ~

        Tj Ti xij 0

      • Project completion constraint:

      Project completion is controlled by the latest finish time of ending activities. If the number of ending activities is denoted by n, the project completion constraint is given by

      triangular fuzzy number x .

   4. ALGORITHM TO SOLVE THE FULLY FUZZY PROJECT CRASHING PROBLEMS

      In this section, we have presented a new algorithm for solving fuzzy time cost trade off problems through fuzzy linear programming problems.

      1. Formulate the chosen fuzzy time cost trade off problem (P) into the following fuzzy linear programming problem (P1):

         Min

         Z Cij I * (Tn T1 )

         ~ ~ ~ ~

         i j

         namely lower linear programming model (P4) and upper linear programming model (P5).

         ~

         Subject to

         Min

         C

         ( ) I ( ) (T ( ) T ( ) )

         T1 0

         ij1

         i j

         1 n1 11

         ~ ~ ~

         (P4)

         T j Ti xij 0

         Subject to

         ~ ~ ( )

         Tn T

         ~ ~ ~ ~ ~

         T11

         ( )

         0

         ( )

         ( )

         Cij NCij sij (NTij xij )

         T j1

         • Ti1

         • xij1 0

           ~ ~ ~

           ( )

           ( )

           CTij xij NTij (i, j) P

           (P1)

      2. Change the representation of fuzzy number (i.e.

         Tn1 Cij1

         T1

         ( ) NC

         ij1

         ( ) s

         ij1

         ( )

         (NT

         ij1

         ( ) x

         ij1

         ( ) )

         triangular fuzzy number into

         cut

         interval

         CTij1

         ( ) x

         ij1

         ( )

         NT

         ij1

         ( )

         (i, j) P

         fuzzy number) in the above fuzzy linear

         Min

         C ( ) I ( ) * (T ( ) T ( ) )

         programming problem (P1). The representation of fuzzy linear programming problem (P2) where the

         datas are in the form of cut interval of

         ij3

         i j

         Subject to

         3 n3 13

         fuzzy number is the following:

         3.

         T ( ) 0

         ~ ~ ~ ~ 13

         Min

         C

         I * (T T )

         ( )

         ( )

         ( )

         ij n 1

         i j

         Subject to

         T j 3

         Tn3

         • Ti 3

         3

         ( ) T ( )

         • xij3 0

         T1

         ~ 0

         Cij3

         ( ) NC

         ij3

         ( ) s

         ij3

         ( )

         (NT

         ij3

         ( ) x

         ij3

         ( ) )

         ~ ~ ~x 0

         (P )

         CT ( ) x

         ( ) NT

         ( )

         (i, j) P

         T j Ti ij

         T

         Tn

         ~ ~

         2 ij3

         ij3

         ij3

         (P5)

         ~

         Cij

         ~

         NCij

         ~

         • s

         • x

         ~

         )

         ~

         ij (NTij ij

         4. Solve the above lower linear programming problem (P ) and upper linear programming

         x

         CT

         ~ ~

         ij ij

         ~

         NTij

         (i, j) P

         4

         problem (P5) using LINGO software package for

         1. The above mathematical formation of fuzzy linear

            programming problem (P2) can be written in the following form (P3):

            different values of , where [0,1]

         2. Substituting the obtained solutions of (P4) and (P5) in the problem (P1), the solution of (P) for

         [C ( ) , C

         ( ) ] [I ( ) , I ( ) ] *

         different values of , where [0,1]

         can be

         Min

         ij1

         ij3

         ( )

         1

         ( )

         3

         ( )

         ( )

         obtained.

         i j

         Subject to

         ([Tn1

         , Tn3

         ] [T11

         , T13 ])

   5. NUMERICAL ILLUSTRATION

      1. Example 5.

         [T11

         ( )

         , T

         13

         ( ) ] 0

         Consider an AOA project network in fuzzy environments whose corresponding network is given in Fig. 1. In this

         [T j1

         ( ) , T

         ( ) ] [T

         ( )

         ,T

         i3

         ( ) ] [x

         ij1

         ( ) , x

         ij 3

         ( ) ] 0

         section, first, parameter and decision variables are considered in triangular fuzzy number form and then

         i1

         j 3

         [T ( ) ,T

         ( ) ] [T ( ) ,T ( ) ]

         reconsidered in cut interval form of triangular fuzzy

         5. n1

         [Cij1

         n3

         , C

         ( )

         ij 3

         1

         ( ) ] [NC

         ij1

         3

         ( ) , NC

         ij 3

         ( ) ] [s

         ij1

         ( ) , s

         ij 3

         ( ) ]

         number. Activities information is given in Table I.

         * ([NTij1

         ( )

         , NT

         ij 3

         ( ) ] [x

         ij1

         ( ) , x

         ij3

         ( ) ])

         [CTij1

         ( )

         , CT

         ij 3

         ( ) ] [x

         ij1

         ( ) , x

         ij 3

         ( ) ] 2 5

         1 6

         [NTij1

         ( )

         , NT

         ij 3

         ( ) ]

         (i, j) P

         3 4

         The above fuzzy linear programming model (P3) can be decompose into two crisp linear programming model,

         Table I. THE FUZZY DATA OF EXAMPLE 5.1

         Activity	Normal time ~ ( NTij )	Crash time ~ ( CTij )
         (1, 2)	(13, 14, 15)	(4, 6, 6)
         (1, 3)	(10, 12, 13)	(7, 8, 9)
         (2, 5)	(16, 18, 19)	(10, 14, 14)
         (2, 4)	(5, 6, 8)	(3, 4, 5)
         (3, 4)	(3, 4, 5)	(2, 2, 3)
         (4, 5)	(7, 8, 10)	(4, 6, 7)
         (5, 6)	(9, 12, 14)	(6, 8, 8)
         Normal cost ~ ( Cij )		Slope cost ( ~s ) ij
         (1200, 1400, 1600)		(80, 100, 120)
         (1100, 1200, 1400)		(190, 200, 220)
         (1600, 1700, 1800)		(90, 100, 130)
         (700, 800, 900)		(160, 200, 210)
         (400, 500, 600)		(180, 200, 210)
         (600, 800, 1000)		(80, 100, 110)
         (1000, 1100, 1200)		(90, 100, 120)

         Using step 1, the complete fuzzy linear mathematical model of the example project has been formulated. The input data are dependency relationships (predecessors), activities utility data.

         Procedure to solve fuzzy time cost trade off problems presented in section should be used in order to solve this problem. As mentioned in step 4, problem (P3) can be decomposed in to (P4) and (P5). (i.e lower linear programming model and upper linear programming model) The values of minimum total cost and planned duration of the project have been determined using LINGO solver. A computer package called LINGO (LINGO 2000) is used on a personal computer to solve the mathematical model of the example project. LINGO is a commercial package using the power of linear and non-linear optimization to formulate large problems concisely, solve them, and analyze the solution. In all tested runs, the linear mathematical model of the example project requires less than one second on LINGO to obtain the optimal solution. The Fuzzy Mathematical model of the example project is solved for different values , where [0,1] . Table III

         shows the lower and upper bounds of the cost values of the example project.

         TABLE III. LOWER AND UPPER BOUNDS OF COST VALUES OF EXAMPLE 5.1.

         	Lower bound	Upper bound
         0.0	11490	16160
         0.1	11691.9	15928.6
         0.2	11893.6	15698.4
         0.3	12095.1	15469.4
         0.4	12309.2	15241.6
         0.5	12532.5	15015
         0.6	12759.2	14806.4
         0.7	12989.3	14565.4
         0.8	13282.8	14342.4
         0.9	13589.7	14120.6
         1.0	13900	13900

         5 6 ~ ~ ~

         Min

         Z Cij I * T6

         i1

         ~

         Subject

         T1 0

         j2

         to

         (P1)

         ~ ~ ~

         T j Ti xij 0

         i, j 1,2…6

         ~ ~

         T6 T

         ~ ~ ~ ~ ~

         Cij NCij sij (NTij xij )

         i, j 1,2…6

         ~ ~ ~

         CTij xij NTij

         (i, j) P

         Now, change the form of triangular fuzzy number in the problem (P1) into cut interval as given in step 2, in

         TABLE IV. THE LOWER AND UPPER BOUNDS OF THE ACTIVITY TIMES OF EXAMPLE 5.1

         	xij	x12	x13	x25	x24	x34	x45	x56
         1.0	U L	6 6	12 12	14 14	6 6	2 2	6 6	8 8
         0.9	U L	6 5.8	12 11.8	14.2 14.4	6.2 5.9	2.1 2.6	6.1 5.8	8 7.8
         0.8	U L	6 5.6	12 11.6	14.4 14.8	6.4 5.8	2.2 3.2	6.2 5.6	8 7.6
         0.7	U L	6 5.4	12 11.4	14.6 15.2	6.6 5.7	2.3 3.7	6.3 5.5	8.0 7.4
         0.6	U L	6 5.2	12 11.2	14.8 15.6	6.8 5.6	2.4 3.6	6.4 6	8 7.2
         0.5	U L	6 5	12 11	15 16	7 5.5	2.5 3.5	6.5 6.5	8 7
         0.4	U L	6 4.8	12 10.8	15.2 16.4	7.2 5.4	2.6 3.4	6.6 7	8 6.8
         0.3	U L	6 4.6	12 10.6	15.4 16.6	7.4 5.3	2.7 3.3	6.7 7.3	8 6.6
         0.2	U L	6 4.4	12 10.4	15.6 16.4	7.6 5.2	2.8 3.2	6.8 7.2	8 6.4
         0.1	U L	6 4.2	12 10.2	15.8 16.2	7.8 5.1	2.9 3.1	6.9 7.1	8 6.2
         0.0	U L	6 4	12 10	16 16	8 5	3 3	7 7	8 6

         the procedure. The following is the cut interval form

         of the given information.

         ~ ~ ~ ~

         Table II.

         cut

         INTERVAL FORM OF NTij , CTij , Cij , sij

         Normal Time NT ( ) ij	Crash Time CT ( ) ij
         ( +13, – +15)	(2 +4, 6)
         (2 +10, – +13)	( +7, – +9)
         (2 +16, – +19)	(4 +10, 14)
         ( +5, -2 +8)	( +3, – +5)
         ( +3, – +5)	(2, – +3)
         ( +7, -2 +10)	(2 +4, – +7)
         (3 +9, -2 +14)	(2 +6, 8)
         Slope Cost s ( ) ij	Normal Cost C ( ) ij
         (20 +80, -20 +120)	(200 +1200, -200 +1600)
         (10 +190, -20 +220)	(100 +1100, -200 +1400)
         (10 +90, -30 +130)	(100 +1600, -100 +1800)
         (40 +160, -10 +210)	(100 +700, -100 +900)
         (20 +180, -10 +210)	(100 +400, -100 +600)
         (20 +80, -10 +110)	(200 +600, -200 +1000)
         (10 +90, -20 +120)	(100 +1000, -100 +1200)

      2. RESULT ANALYSIS

      C

      C

      As generally known, when the parameters are fuzzy numbers, the minimal total cost ~ is a fuzzy number as well. The cut of ~ represents the possibility that the

      total crashing time of this time cost trade off problem in

      this project network will appear in the associated range. Specially, the 1.0cut shows the crashing time that is most likely, and the cut that approaches 0 shows the

      range in which the crashing cost could appear.

      Table IV lists the optimal activity times corresponding to the lower and upper bounds of the minimum total cost at these 11 values of , 0, 0.1, 0.21.0. Notably, the total

      cost associated with smallest possible values of parameters need not be lowest, even if they are cost coefficients in the objective function, such as CTij .

   6. CONCLUSION

A fuzzy mathematical model has been developed which links the CPM with least cost optimization, mathematical programming in order to optimize the traditional time cost trade off problem. The developed model is a stand-alone piece of generic technique which may well be applied to projects of any kind; provided the projects can be defined within the boundaries of the techniques used (i.e. project is being divided into precedence related activities, each with normal and crash time and cost data.)

This paper investigated the time cost trade off problem in the project network with several fuzzy parameters which makes this problem being complicated. The underlying idea is based on linear programming formulation and the

concept of cut to transform the fuzzy time cost trade

off problem to pair of mathematical programming. By enumerating different values of the possibility level , the lower and upper bound of the cuts of the fuzzy

minimum total cost are calculated to approximate the membership function; and the corresponding optimal activity times are also obtained. A numerical example solved by other researchers is being taken into consideration to illustrate the validity of the proposed model and approach.

It is clear that if the obtained minimum total cost is a crisp value, then it may lose some useful information. Compared with other studies, the proposed model and approach can obtain more reasonable solutions suitable for all cases ranging from the pessimistic case to the optimistic case, and so more information is provided for making project management decisions.

The proposed model is also applicable to more complicated project networks in real world. Clearly, the proposed approach is not confined to the fuzzy parameters of triangular type. This is illustrated by successfully solving an example with fuzzy parameters. Other types such as

trapezoidal type and interval type are also applicable. The proposed model suits well for the fuzzy time cost trade off problem involving both direct costs and indirect costs.

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