PROJECT RELIABILITY: PROBABILITY OF A PROJECT MEETS … · 220 International Journal of Electronic Business Management, Vol. 9, No. 3, pp. 220-230 (2011) PROJECT RELIABILITY: PROBABILITY

220 International Journal of Electronic Business Management, Vol. 9, No. 3, pp. 220-230 (2011)

PROJECT RELIABILITY: PROBABILITY OF A PROJECT MEETS ITS QUALITY-COST-TIME TARGET UNDER

UNCERTAINTY

Yudha Andrian Saputra1* and Ode Siti Andini Ladamay2 1Department of Industrial Engineering

Institut Teknologi Sepuluh Nopember (ITS) Surabaya (60111), East Java, Indonesia

2The 10th BPBPJN Ministry of Public Works

Jayapura (12110), Papua, Indonesia

ABSTRACT

For so long, Quality, Cost, and Time become an Iron Triangle for measuring project success. Many researchers have investigated about Quality-Cost-Time in a Project Network under uncertainty, particularly it focuses on single parameter (Quality/Cost/Time only) or combination between parameters (trade off or relation between parameters). Knowing the probability to meet these three targets is important, especially during the project planning. Management can gain information insights about the confidence level to meet the target and potential risk in the future. However, none of the previous research has combined these three parameters under uncertainty. If we define quality as: 1) a standard or requirement to be met with failed and successful probability; 2) it has consequences or impact on additional cost and/or additional time for nonconformity; and 3) the time and cost have their own relationship and form of uncertainty, here our proposed method to evaluate probability of a project whether it met its Quality-Cost-Time target under uncertainty. We call this as Project Reliability. A Monte Carlo Simulation and Generalized Activity Network are utilized. A numerical example is provided to support and validating the propose method.

Keywords: Quality-cost-time, Generalized Activity Network, Monte Carlo Simulation, Uncertainty

1. INTRODUCTION *

Quality, cost and time have become indicators for measuring project success. These three parameters are called as The Iron Triangle [1]. An assessment before running the project is important for measuring capability to meet these three constraints. A set of action plans can be set up by the management in the planning stage as preventive or response based on the assessment result.

In details, several authors have investigated the Iron Triangle, particularly focus on single criteria (Quality/Cost/time only) or for combination, e.g.: relation between criteria. In the field of Project duration, we found a lot of discussion to overcome PERT weakness and oversimplified in Williams [21], Lu and Rizk [15], Lee and Arditi [14], and Huang and Wang [12]. In terms of Project Cost, Elkjaer [11] and Chou, J.S. [6] proposed different methodology to assess project cost and uncertainties. Chan [5] discusses about time and cost relationship in Malaysia *Corresponding author: [email protected]

case studies based on Bromilow equation. In Project Quality, Burati et al. [4] conducts a research in 9 industrial fast track and record the amount of cost that associated with non-conformities.

The idea of comprehensive evaluation of Quality-Cost-Duration under uncertainty was stated by Kidd [13] Kidd suggested the usability of VERT not only for estimating duration, but also for evaluating quality/performance and cost integrative using an indicator called as Reliability. Unfortunately, the discussion was only stopped at single or interaction time-cost parameters. The most nearby result can be found in Babu and Suresh [2] and Tareghian and Taheri [18] which discusses optimization and trade off of Quality-Cost-Duration evaluation under deterministic assumption. However, we can’t find any scholars proposed Quality-Cost-Duration evaluation under uncertainty.

This research can be applied in the Project Planning stages. After an organization; e.g.: a contractor, awarded a contract to perform a project within specific quality-cost-duration; and a planning already established; then the question would be how

Y. A. Saputra and O. S. A. Ladamay: Project Reliability 221

confidence (probability) we are with the performance of this project to meet the target of The Iron Triangle. To answer this question, we must be able to model the Quality-Cost-Time in a Project Network.

Now if: 1) The environment under uncertainty; 2) The Project Reliability defines as the probability to meet the total cost and total time/duration target, and pass all the Quality/Standard required by the Project; 3) The Quality/Standard has some uncertainty about successful and failed. If failed, there are some additional process/consequences with some additional time and cost; thus, this research proposes a comprehensive methodology on how to evaluate a project planning under a network rules and uncertainty to find its probability to meet the Quality-Cost-Time target. The reliability indicator will become an important consideration for management to set a preventive action to protect the achievement of the targets as the basis of System Safety and further Risk Analysis.

2. THE PROPOSED IDEA

Our Proposed idea for calculating the Project Reliability is simple. The idea is shown in the Figure 1. The basic idea is similar with calculating the Probability to meet duration in PERT methodology. We just need to transform the Quality, Cost and Time and its relationship from beginning activities until end activities and to aggregate the result based on each parameter rule. Then, each aggregate is compared with its target.

If all aggregate can meet the target, we can say that the project is success (Index = 1). On other hand, if at least one target can’t be met, the project is fail (Index=0). By repeating this experiment for X trials we can collect how many numbers of successes and how many numbers of fails. The portion of success comparing to the number of trials is the value of Project Reliability. For this purpose, we can easily utilize Monte Carlo Simulation approach.

The problem here, we have to deal with 3 parameters in each activity. Those 3 parameters have relationship each other and form an uncertainty. For Cost and Time relationship maybe it’s not a big problem since many references already discussed about this issue [5,6,19]. For Quality-Time and Quality-Cost, the relationship forms depend on how we define the quality on this model. If quality is defined as an index as discussed in Babu and Suresh [2], the problem maybe in the practical point of view and how to translate each Quality definition in each activity into an index that ‘apple to apple’. Rather than using this definition, we would like to use another definition of quality that was already defined by Burati etal. [4]. This definition later on drive us to use and did little bit modification on the Project Network proposed by Dawson and Dawson [10]

called Generalized Activity Network. By this modification, we can transform each Quality-Cost-Time from beginning to the end activities of the project easily. The detail development of our proposed idea is shown in subsection 2.1 and 2.2. 2.1 Quality-Cost-Time (Iron Triangle) Relationship

This section is trying to explain about the basic concept of cost and time relationship, the quality definition, and the quality association with the additional cost and/or time due to failed to meet the quality level. Based on those concepts, we can develop a conceptual model of Quality-Cost-Time relationship in section 2.2. 2.1.1 Cost and Time Relationship

Bromilow [3] propose an idea on how to model total time and total cost relationship. They developed an equation based on historical data aimed to predict the construction time in form of the formula: T=KCB. T is the duration of construction; C is the final cost of building in millions of dollars adjusted to constant labour and material prices; K is a constant describing the general level of time performance for an Australian $1 million project and B is a constant describing how the time performance is affected by project size as measured by cost. This equation then tested by Chan [5] for Malaysia case studies.

Although it was developed for the total cost and total time in a project, this method can be used to develop activity cost and time relationship if there is any similar experience in the past (historical data). How if there is no historical data? In this situation, then we must rely on the subjective estimation by an experienced Project Manager. A good approach was provided by Taylor and Davis [19]. Taylor and Davis [19] assumed that the activity duration followed the beta/constant distribution and the cost is deterministic which consists of set up cost and variable cost. The set up cost is independent to the time, but the variable cost is dependent on how many days spent in each activity (where the time is random variable). Sometimes each activity has only one of either set up or variable cost, or both. It’s conditional situation, depend on the cost perception in each activity.

The illustration given by Taylor and Davies [19] maybe is the most probable methods during the planning stages, which is the time and cost both are random variable or the cost is a function of time (dependent on time). About the form of distribution, the time and cost actually have free form, depend on the case. Most of the case assumes it followed Beta Distribution due to the PERT logic. Actually, a planner may substitute the time distribution with any continuous distribution form i.e.: normal, exponential, triangular, or a discrete distribution form i.e.: Poisson, discrete uniform, etc. [13]. The concept of setup and

222 International Journal of Electronic Business Management, Vol. 9, No. 3 (2011)

variable cost can also be improved. Let’s say an activity consists of procurement and installation, then we can separate the cost of the activity into 2 part, fixed cost for procurement with certain days (due to contractual issue) and variable cost for installation

process with uncertain days. Another improvement form that might occur that we can imagine can describe as Figure 2.

Quality

Cost

Time UNCERTAIN ENVIRONMENT

Project Reliability: Pr { Project Perform over or equal to the target }

Target (and form)

Duration <= x days

Cost <= y US$

Meet QualityTarget

How to transforming Time, Cost and Quality value from beginning to end of project network and

aggregating each target

A

B

C

DE

Do the Quality-Cost-Time

relationship requires special form of Project

Network?

Figure 1: The framework of the proposed Idea

A c t iv it ie s P D F G r a p h

A / D u ra t io n P o is s o n ( 2 8 , a n d T ru n c a te ( 2 6 ,3 1 ) )

B / D u ra t io nD is c r e te ( { 1 4 ,1 5 ,1 7 ,1 8 ,1 9 ,2 0 ,2 1 ,1 6} ,{ 0 .1 ,0 .1 ,0 .1 5 ,0 .2 ,0 .1 ,0 .0 1 ,0 .0 5 ,0.2 }

C / D u ra t io n P o is s o n ( 2 7 , a n d T ru n c a te ( 2 0 ,3 4 ) )

D / D u ra t io n D is c r e te U n if o rm ( { 8 ,9 ,1 0 ,1 1 ,1 2 } )

E / D u ra t io n D is c r e te ( { 1 7 ,2 0 ,2 5 } ,{ 0 .2 ,0 .5 ,0 .3 } )

TIME/ DURATION

TIME/ TIME/ DURATIONDURATION

Name

Activity A

Triangular(900,1000,1300) for 1st 20 days. Cost of fabrication/procurement of the items of A. The duration is fixed, but the cost is uncertain

E3

After fabrication/procurement, the rest activity A is installation. It required some days, uncertain. The cost required per day is following Normal Distributionl(200,40)

Activity BMaterial cost will follow normal distribution (2400,200)

E4For each installation days the cost will be charged following Normal distribution (100,15) / days

Activity CThe material cost would follow uniform distribution (900, 1200) independent to the time/duration

E5

Activity D 0Each day activity D will be charged in cost following Normal Distribution (180,40)

Activity EThe 1st 15 days the cost for fabrication/procurement will somehow fall in Triangular (400,620,900)

E7the next after 15 days, the project cost will be charged following uniform distribution (300,350) / day

Additional Cost / day Package (Fabrication, procurement, delivery, etc.) Charged

COSTCOSTCOST

Figure 2: Various time and cost relationship form

2.1.2 Quality Definition and its Association with Time and Cost

In this paper, we define Quality as "conformance to established requirements." This definition provides a basis for measurement, i.e.: the requirements are either met or not met [4]. For non-conformities, it is called as Deviation. Deviation includes changes to the requirements that results in rework, as well as products or results that do not

conform to all specification requirements, but do not require rework [7].

In Burati et al. [4] the quality is a representative of the scope of project, which is defined in earlier and it’s completed with the associated cost. Absolutely, we can see this situation as a Bernouli trials with the probability to meet a quality level as a successful process (with value of p) and a probability to fail meet quality level as failure process (1-p=q). Now, how if


there is any failure process (non-conformities)? From the literature review, there are some corrective actions, i.e.: • Reprocessing (loop the process again). It might

charge us additional cost and additional time to the original/baseline process, due to we must repeat the process [10].

• Rework or repair. It might charge us also some additional cost and time. In most situations, the amount is lower than original process, but in some case, it probable to have additional time and cost higher than the original process [4,17]

When we try to enrich another possibility about what happen if we fail to meet the quality, we found another situation, i.e.: • In some project, the extreme consequences of

non-conformities sometimes made the project fail [19]. If this happen, it might charge us cost but not time.

• In product design, failed to meet a standard have means activating another scenario or alternative pathways to develop a product with lower level quality [20]. Some pharmaceutical industry experiencing with this situation when they develop their product. This situation might charge us some additional cost and time.

• In practical, we also found there is a situation that non-conformity still might be accepted by the project owner but it is charged by some penalty (In some management or IT project, Burati et al. [4]).

2.1.3 The Conceptual Model of Quality-Cost-Time Relationship under Uncertainty

Therefore, the relationship between the Quality-Cost-Time can be described in Figure 3. We start the discussion with a standard process which consist of baseline time and baseline cost (as described in 2.1.1). If the process have quality requirement, then a Quality Checking (Meet or not meet with the requirement) mechanism is required (as described in 2.1.2). If the result can meet the requirement (success=p) means we can go thorughly the next process. If the result fails (q=1-p), then the next question is what kind of corrective actions is needed or any consequences for this situation (one from five possibilities as discussed in 2.1.2). Whatever the corrective actions/consequences, it would charge us some additional time and/or cost.

Moreover, the Figure 3 is representing our conceptual model for Quality-Cost-Time relationship. The time is represented by the baseline time and additional time if the QC process indicates failure to meet quality requirement. Cost is represented by cost function and additional cost if QC process indicates failure to meet quality requirement. And the quality is represented by a QC mechanism with success and failure probability and a set of options for corrective actions/consequences for failure situation.

Figure 3: Quality-cost-time relationship model


2.2 Modification on Generalized Activity Network as the Project Network to Accommodate the Quality-Cost-Time Relationship

Our next question is how we select an appropriate project network to accommodate the transferring process of Quality-Cost-Time from beginning to the end activity in the project. Standard AoA or AoN can’t accommodate our relationship model in Figure 3. We need a probabilistic branching from the arch/node. For this purpose, we try to use the Generalized Activity Network proposed by Dawson and Dawson [9,10].

Initially, GAN is developed by introducing probabilistic branching between activities in AoA type, which is explicitly identified the uncertainties and possibilities within a project's scope. There are 2 applications of GAN have been created over the year. First is Graphical Evaluation and Review Technique (GERT) in 1966 and second is Venture Evaluation Review Technique (VERT) in 1967.

After so long, Dawson and Dawson [8] criticized and made some clarification on previous GAN. They showed some ambiguity and complexity of previous GAN and identified it as the reason why GAN never been established as famous project-management techniques. By the provision of a clearer definition of the GAN structure, they stated that the generalized activity network should be prove as the next logical step that managers can make in planning ever more complex development projects. In their clarification, they still use Activity on Arch (AoA) as the form of network.

In 1995, Dawson and Dawson tried to propose a modification of GAN (Activity on Arch based) to Generalized Activity on Node Network. They claimed that since the weakness of PERT to handle complex project planning and managing uncertainties, GAN have the ability to deal with that situation. Unfortunately, GAN did not become familiar for some Project Managers due to its limitation with Activity on Arch (AoA) form. Dependency logic in GAN is hard to understand in AoA form. On the other hand, most of Project Managers are familiar with Activity on Node (AoN) form as basis for famous PERT. Thus, Dawson and Dawson tried to modify their previous

clarification on GAN in the form of Generalized Activity on Node Network.

Again, in 1998, Dawson and Dawson proposed extensions of GAN to existing software tools to specify and manage such uncertainties that would be easy to learn and use. A survey has shown that if these extensions were available, commercial and government organizations would regularly use them. Now, how was exactly their proposed idea on the reform of GAN? Here, some of their proposed idea and why their proposed idea matches with our requirement to modeling Quality-Cost-Time relationship as we shown in Figure 3.

In two next figures (Figure 4 and 5), Dawson and Dawson proposed a concept of looping. Some activities may be performed more than once. Dawson and Dawson [10] defined this type of activity can occur if either alternatives input are specified or if the task exists in a task loop. The probability may change in each looping process. The simple example that was used in Dawson and Dawson is “test and fix example”. The test and fix example is a type of rework or repair activity, which is an activity performed when the test is failed (Figure 4). Another type of loop is re-processing, which is if a test is failed, the project must repeat some process before the test with same duration and cost estimation as in the first looping (Figure 5).

Figure 4: Example of rework/repair (Test and fix

looping process)

Develop SoftwareTriangular (4,8,12)

ImplementBeta (5,20,80)

TestFixed (1)

MaintenanceTriangular(0,5,200)

0.4, 0.6, 0.8, 1

0.6, 0.4, 0.2, 0 Figure 5: Example of re-processing (Test process, if fail it back again to develop software process)

From figure 4 and 5, clearly we can get an

example on how rework/repair and reprocessing can be accommodated in GAN. But how about the other 3 failure meet quality situation as we discussed in

subsection 2.1.2? How about the extensions form of rework/repair and reprocessing above? We try to make some modification of GAN proposed by


Dawson and Dawson [9,10] to fit and adjust the concept with our problem. 2.2.1 Rework Process

The Figure 6 shows a simple project network consists of 2 processes, A and B. Activity A have quality requirement, while B does not. Activity A and B have basic time and cost function. For activity A, a QC mechanism is performed (1B) to check does the process (1A) have met the quality requirement. In first attempt of QC, it has probability of success 0.8 and probability to failure 0.2. If the 1A result passes the QC test (1B), the Project can go to the next process (Process B). If fail, the Project must be performed an additional process. In the example, the additional process type is rework (1C). An additional cost and time due to the rework is charged to the total cost and total duration.

Next, after performing the rework, the looping process goes back to the QC test. In second attempt, the process would never be failed; because during the rework process there is a certainty to guarantee the process to meet the standard of quality. 2.2.2 Stop the Project and the Extension the Number of Looping in Rework Process

If the planner does not have certainty in number of looping as shown in Figure 6, the planner can modify this process. The looping process in Dawson and Dawson [10] just assumed it allowed repeating until X times, with probability of success at X-th trials is certain (p=1). There is some potential improvement about the concept of looping. Our propose concept of looping is presented in Figure 7.

From figure 7 in process A, the QC test is conducted with probability of success 0.9. The maximum loop is unlimited. So in this type of activity, the looping will continue until the first success occurs. From the same figure in process B, the looping is limited only for 3 times. In each QC test, the successful probability is similar (0.9). If the process can’t pass the QC test, then the project is terminated (failed).

Figure 6: Simple project network with rework process

Figure 7: Extension of the looping process in GAN

2.2.3 Accepted at Below Quality Standard with Some Penalties

Figure 8a shows a condition of accepted the non-conforming with penalty cost as the consequences. After the process 3a completed, the QC test will be performed. If the QC test indicate a grade = 100 (p=0.25), then the process succeeds. If the QC test indicates a grade < 100 (q=0.75, with grade value somewhere between 90-99.99%), the result is accepted but penalty cost must be added by following a specific function. The Additional cost due to penalty must also be added to the total of project cost.

Figure 8: GAN for acceptance non- conforming by

add penalty cost (a) and reprocessing (b) 2.2.4 Re-processing

Figure 8b show the figure of re-processing. After 4a completed, QC test is performed (4b). If successful, it can go to the fifth process. If fails, the process will loop back repeat the 4a process with same time and cost probability distribution. The number of looping in re-processing can be extended in

(a)

(b)


term of limited and unlimited looping, similar as rework process in Figure 7. 2.2.5 Activating new path/scenario

The last model is activating new path/scenario as shown in Figure 9. In the 9(a), the basic/main process actually only process 1 (A) and process 3 (Assembly). Due to QC Checking mechanism on process A, the process can’t go directly to process 3. If fails the QC test at 1B, it will activate process 2 (Process B). Additional time and cost is required. The new path will be: 1-2-3. The probability of success passes the QC test at 1B is 0.7 and the failure

probability is 0.3. It’s also possible that the path will have different ending point due to the difference of quality output. This situation is possible in product development project case. The situation is described at Figure 9(b).

The alternative modification shown from Figure 6 until 9 are the most possible situation in the project. Definitely there are any other improvement possibilities to enrich the scenario based on each case or PM’s creativity. However, we have given the basic concept.

Figure 9: Activating new path or scenario for GAN

3. NUMERICAL EXAMPLE

A manufacturer awarded a contract to do a project fabrication spare part called X-123. The scope of job was shown in the Figure 10. The contract agreed at 75 days and $575,000.00. 4 of 5 activities above have QC mechanism. The activities with QC mechanism are: Engineering Design, Process A, Process B, and Process C. Engineering Design and Process A have reworking additional process, Process B has penalty additional process, and process C has re-processing. The actual duration, success and failure in QC test, and the cost of each activity is uncertain. If the target passes all the process and its required quality target, completion time <= 75 days and spending budget <=$520,000.00 ($55,000.00 for profit allocation), how to the probability meet these targets?

Engineering Design

Engineering Engineering DesignDesign

Process AProcess AProcess A

Process BProcess BProcess B Process CProcess CProcess C

AssemblyAssemblyAssembly

Figure 10: Activity on node the fabrication of x-123

A small team is formed to develop a project plan. The team consists of some experience planner/PM. After conducting a detail assessment based on past performance data and makes some subjective probability, the form of uncertainty in this project is shown in Figure 11. The management is confidence to start the project if the reliability is greater than 90%. The penalty cost will occur if the completion time of the project is over than 75 days. The penalty is 1/1000 to the total contract ($575,000.00) and must be added to the total project cost. In the example, time is discretized. The process B applies penalty cost which is followed the quality function.

For conducting the experiment, we use @Risk software from Palisade Decision Tools [16]. The simulation is conducted for 15 replications, each consist of 5000 iterations. After conducting the experiment by applying a Monte Carlo Simulation, here are some important statistics.

We found that the Probability of the Project will complete less than equal to 75 days and under US$ 520,000.00 is around 0.847770 ≤ µPr ≤ 0.853880 (95% Confidence Interval). This parameter indicates the Project reliability is below the target (90%).


Figure 11: The project network planning (modified of GAN) for x-123

-0.1

0

-0.0

5

0.00

0.05

0.10

0.15

0.20

Figure 12: Tornado diagram for the risk factors of the x-123 Project

Another important statistics i.e.: 1) Probability

to complete the project <= 75 days itself is 0.868, while perform below the budget target is 0.938. 2) The Project cost will follow normal distribution with Φ US$(453,206; 44,097); 3) 15.41% of total cost and 16.66% the total duration is happened due to non-conformity (Quality issue).

The next important information for management is which factor has the big contribution

to the value of Project Reliability. For this purpose, we develop a sensitivity analysis by conducting a Tornado Diagram from each uncertainty variable. The result is showed in Figure 12. Tornado diagram from a sensitivity analysis display a ranking of the input distributions (the risks events) which impact an output (Project Reliability). Inputs that have the largest impact on the distribution of the output will have the longest bars in the diagram and located at


the top. From the Figure 12, we can understand the Risk Profile from each variable in the project to the comprehensive Quality-Cost-Time target which is represented by Project Reliability. Further Risk Analysis may be required as Post Project Reliability Analysis.

4. CONCLUSION

This research has shown about how and the importance of comprehensive evaluation on the Iron Triangle (Quality-Cost-Duration) in a project under uncertainty situation. We introduce an index called as Project Reliability. By modeling the quality as a success/failure process, Project Reliability becomes an index of measurement to evaluate the probability of project completion within the time and cost target and passes the entire QC requirements.

Even maybe looks so complex and lead the managers have a little base on their estimate due to a vague value of probability, all the proposed method above still become important and worth to conduct. Understanding the nature of all possibilities and accommodating the risks in decision is worth in managing the project. Finally, we have developed a comprehensive overview about measuring of success probability regarded with Iron Triangle. Moreover, numerical example is provided to show how the implementation of the proposed concept above. Is this approach worth to be applied? The answer is similar as Dawson and Dawson [10]. In many high cost and risk project, which is a comprehensive evaluation required (i.e.: Product Development Feasibility Project Proposal), this approach is important to be applied as input for decision making, i.e.: running the project normally, run with some further decision/consideration, or stop the project.

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ABOUT THE AUTHORS Yudha Andrian Saputra is a Lecturer at Department of Industrial Engineering at Institut Teknologi Sepuluh Nopember (ITS) Surabaya, Indonesia. He received his Bachelor Degree (in Industrial Engineering) from ITS Surabaya and MBA from Department of Industrial Management of National Taiwan University of Science and Technology. He has practical experience and involved on many project management and consultancy in several private and government institutions in Indonesia. His research interests are: Simulation, Decision Analysis, Risk Analysis, and Strategic Management. Ode Siti Andini Ladamay is a staff at The 10th BPBPJN, Ministry of Public Works Republic of Indonesia. She has responsibility to prepare bidding documents and monitoring several road projects in Papua Island, East of Indonesia. She received her Bachelor Degree (in Civil Engineering) and Master (in Industrial Engineering) from ITS Surabaya. (Received May 2011, revised August 2011, accepted September 2011)


工程可靠度：在不確定性下工程達到品質－成本－工時目標的機率

Yudha Andrian Saputra1*、Ode Siti Andini Ladamay2

1 工業工程系 Institut Teknologi Sepuluh Nopember (ITS)

印尼東爪哇泗水市 2 印尼公共工程部

印尼巴布亞新幾內亞查亞普拉

摘要

品質、成本、與工時這三者長久以來一直是測量工程是否成功的鐵三角。許多研究探討

這三者在不確定性下的工程網路圖，大都著重探討單一指標(質量/成本/工期)或是結合多

重指標 (折衷指標間的關係)。但在工程計劃階段，滿足這三樣指標的要求是重要的。在

管理上可以得知工程計劃在一定信賴水準下，達到目標、了解潛在的不確定的因素的相

關資訊是十分有用的。然而，過去的研究並未將這三項指標結合在不確定性下作探討。

如果品質的定義如下：1) 為一可能達到的標準或條件； 2) 對於額外成本或額外工時會

有不一致性的結果及影響；3) 此外，工時和成本本身也有其關聯及其不確定性。我們提

出的驗證方法在不確定性下，檢測工程是否能夠達到其品質-成本-工期目標的機率，此即

工程可靠性。並使用蒙特卡羅模擬及一般化活動網路圖檢測。同時本文利用多個樣本支

持及驗證本文所建議的方法。

關鍵詞：品質－成本－工期、一般化活動網路圖、蒙特卡羅模擬、不確定性（*聯絡人：[email protected]）

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PROJECT RELIABILITY: PROBABILITY OF A PROJECT MEETS … · 220 International Journal of Electronic Business Management, Vol. 9, No. 3, pp. 220-230 (2011) PROJECT RELIABILITY: PROBABILITY