An efficient approach for large scale project planning ... - Science Direct

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sets and systems Fuzzy Sets and Systems 76 (1995)277-288

An efficient approach for large scale project planning based on fuzzy Delphi method In Seong Chang a'*, Yasuhiro Tsujimura b, Mitsuo Gen b, Tatsumi Tozawa a aDepartment of Production and Information Science, Utsunomiya University, Utsunomiya 321, Japan bDepartment of Industrial and Systems Engineering. Ashikaga Institute of Technology, Ashikaga 326, Japan

Received December 1993;revisedJuly 1994

Abstract The goal of this paper is to replace probabilistic or deterministic considerations in the project network analysis by possibilistic ones and to reduce the difficulty arising from the inexact and insufficient information of activity times. The activity times are considered as fuzzy numbers (fuzzy intervals or time intervals) and the fuzzy Delphi method is used to estimate a reliable time interval of each activity. Based on these time estimates, we then propose an efficient methodology for calculating the fuzzy project completion time and the degree of criticality for each path in a project. Keywords: Composite method; Comparison method; Fuzzy activity times; Fuzzy project completion time; Degree of

criticality; Fuzzy Delphi method; Combination of composite and comparison methods

1. Introduction Project network analysis represents an effective tool frequently used whenever complex and structured technological process are to be planned. C P M (Critical Path Method) and P E R T (Project Evaluation and Review Technique) are typical techniques for project network analysis. A project defines a combination of interrelated activities that must be executed in a certain order before the entire task can be completed. A network is a picture of a project, a map of requirements tracing the work from a departure point to the final completion objective. Let G ( N , A) be a directed network with node set N and arc set A. The nodes stand for the events, while the arcs stand for the activities of the project. Each activity is defined as a task requiring time for its completion, and each event represents a point in time that signifies the completion of some activities and the beginning of new ones. The beginning and end points of an activity are thus described by two events usually known as the tail and head events. The application of both P E R T and C P M should ultimately find a critical path that represents the shortest duration needed to complete the project. To identify the critical

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path, three parameters of each event are determined: the earliest occurrence time of each event, the latest occurrence time of each event, and the slack (float) time of each event. The critical path is the path from a start event to an end event where the slack times are all zeros. The constructed network represents the first step toward achieving the above goal. The time to conduct each activity is not required for network logic, but times are necessary to determine the critical path. An estimate of the elapsed time required to accomplish the objectives stated in an activity's description is called the activity time. The time estimate for each activity is largely subjective evaluation, which is influenced by the estimator and the operating environment. CPM and PERT require the availability of deterministic time estimate for each activity. However, in real world problems, it is very difficult to forecast the precise estimates of activity times. Although PERT can be used to represent the uncertain activity times by modeling them as random variables with Beta distributions [9], this approach is theoretically valid only when there is some reason to assume an underlying Beta distribution as prior experience or information. Therefore, if the uncertainty is due to a lack of them, then each activity time should be estimated as being within a certain interval without even any knowledge of a probability distribution within the interval. This time interval can be naturally represented by a fuzzy number which is described using the concept of an interval of confidence as in Kaufmann and Gupta [12]. In this paper, we present an efficient methodology for solving project planning problem when activity times are uncertain due to a lack of information. The activity times are considered as fuzzy numbers, and the fuzzy Delphi method [11] is used to estimate a reliable time interval (fuzzy interval) of each activity. If the activity times are fuzzy numbers and the ordinary fuzzy operations on fuzzy numbers are used, then the minimum required project time is a fuzzy number. Furthermore, a unique critical path is not identified. Thus, the naive approach for solving fuzzy project planning problems is estimating the fuzzy project completion time and the degree of criticality for each path in a project network. The calculations can be carried out by using the fuzzy max-min operation (composite method) and discrete max-min operation (comparison method) [5, 6, 17, 19]. The comparison method is much easier to use than the composite method. The increase in computational effort by the composite method is tremendous. However, the composite method is a more realistic model. Accordingly, this paper discusses a method which can suffice for the weaknesses of the two methods. The paper shows that the fuzzy project completion time and the degree of criticality for each path can be efficiently determined using the time estimate of each activity gained by the fuzzy Delphi method and the combination of the composite and comparison methods.

2. Fuzzy Delphi method for forecasting of fuzzy activity times The traditional Delphi method is one of the effective methods which enables forecasting by converging a possibility value through the feedback mechanism of the results of questionnaires, based on experts' judgments. For estimating a reliable time interval of each activity, the fuzzified Delphi method can be effectively used. Occasionally, a manager is challenged by large scale projects that have never been attempted before, such as research and development (R&D) projects to develop new materials, energy, processes, and devices. In such cases, highly qualified experts are interviewed to give their opinions for the possibility of each activity time in the projects. In such projects it is very difficult to forecast the precise estimates of activity times because there is a lack of human intelligence and suitable equipment at present (e.g. Metasynchrotron for high energy nuclear studies). The difficulty concerning time estimations can be relieved by using fuzzy numbers. In this work, triangular fuzzy numbers (TFNs) which represent the pessimistic, moderate and optimistic estimate are used to represent the opinions of experts for each activity time. A TFN ,4 can be defined by a triplet (a, b, c) [12]. The membership function is defined

LS. Chang et aL / Fuzzy Sets and Systems 76 (1995) 277 288

279

as

0,

x < a,

x--a

b --a'

~(x) =

O--X

a

5152.

61

64

Fig. 4. Illustration of the calculation Cp(9).

X Fig. 5. Network considered paths 5 and 9.

LS. Chang et al. / Fuzzy Sets and Systems 76 (1995) 277-288

287

Table 5 Forward calculations using the composite method Activity (p)

E'-Sp(? ~(p))

E"Fp(? ~p) + alp)

A B E G L

0 0 (4,8, 13) (3,8,8)_. ~ max[EFE,EFc]

(4,8, 13) (3, 8, 8) (14,20,26) (11,21,23)

=

x g

14 6'

14~