Application 1 – Analysis and Synthesis of Prior Research

Expert Systems with Applications 37 (2010) 8499–8507

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Expert Systems with Applications

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e s w a

Applying fuzzy goal programming to project management decisions with multiple goals in uncertain environments

Tien-Fu Liang *

Department of Industrial Engineering and Management, Hsiuping Institute of Technology, 11 Gungye Road, Dali City, Taichung 412, Taiwan

a r t i c l e i n f o

Keywords: Project management Multi-objective linear programming Fuzzy sets Fuzzy goal programming

0957-4174/$ - see front matter � 2010 Elsevier Ltd. A doi:10.1016/j.eswa.2010.05.026

* Tel.: +886 4 24961123x1409; fax: +886 4 249611 E-mail address: [email protected]

a b s t r a c t

In real-life situations, the project manager must handle multiple conflicting goals and these conflicting goals are normally fuzzy owing to information is incomplete and unavailable. This study develops a two-phase fuzzy goal programming (FGP) method for solving the project management (PM) decision problems with multiple goals in uncertain environments. The original multi-objective linear program- ming (MOLP) model designed here attempts to simultaneously minimize total project costs, total com- pletion time and total crashing costs with reference to direct costs, indirect and contractual penalty costs, duration of activities and the constraint of available budget. An industrial case is implemented to demonstrate the feasibility of applying the proposed two-phase FGP method to practical PM decisions. The contribution of this study lies in presenting a fuzzy mathematical programming methodology to fuzzy multi-objective PM decisions, and provides a systematic decision-making framework that facili- tates the decision maker to interactively adjust the search direction until the preferred efficient solution is obtained.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Since the program evaluation and review technique (PERT) and the critical path method (CPM) were both developed in the 1950s, relevant project management (PM) decision issues have long at- tracted interest from both practitioners and academics. Numerous techniques including mathematical programming, algorithms and heuristics have also been presented to PM decisions. When any of the conventional models was used to solve PM decision prob- lems, however, the goals and related parameters were often as- sumed to be deterministic/crisp (Al-Fanzine & Haouari, 2005; Davis & Patterson, 1975; Deckor & Hebert, 1989; DePorter & Ellis, 1990; Elsayed, 1982; Kotiah & Wallace, 1973; Kurtulus & Davis, 1982; Lin & Gen, 2007; MacCrimmon & Ryavec, 1964; Rabbani, Ghomi, Jolai, & Lahiji, 2007; Russell, 1986; Wiley, Deckro, & Jack- son, 1998). In real-life PM decisions, model inputs and environ- mental coefficients, such as operating costs, activities duration, available resources and total cost budget, are typically fuzzy/ imprecise owing to incomplete and unobtainable information over the project planning horizon. Conventional deterministic tech- niques described above obviously cannot solve practical PM deci- sion problems in an uncertain environment.

Moreover, the existing PM decision models consider only direct costs (including labor, materials, equipment and other costs

ll rights reserved.

87.

directly related to projected activities), neglecting relevant indirect costs (including interest, administration, depreciation, contractual penalty and other variable overhead costs). In practical situations, a project’s total costs are the sum of direct costs and indirect costs over the project planning horizon. Generally, the real PM decisions focus on the minimization of project completion time, and/or the minimization of total project costs through crashing or shortening duration of particular activities. The aim of evaluating time-cost trade-offs is to develop a suitable PM plan that will minimize the total project costs. Thus, a project decision maker (DM) may be able to shorten project completion time, realizing savings on indi- rect costs, by increasing direct expenses to accelerate the project.

Additionally, although various PM decision techniques have been developed to minimize project duration, most do not also minimize the total costs (Karshenas & Haber, 1990; Li, 1995; Rus- sell, 1986). In practice, the project DM must frequently handles conflicting goals in term of the use of organizational resources, and these conflicting goals are required to be optimized simulta- neously by the DM. These goals are to minimize total costs, crashing cost, completion time, contractual penalties, and/or max- imizing profits and the utilization of equipment (Al-Fanzine and Haouari, 2005; Arikan and Gungor, 2001; DePorter and Ellis, 1990; Liang, 2004, 2009; Lin and Gen, 2007; Viana and Sousa, 2000; Yin and Wang, 2008). Particularly, it is critical that the satisfying goal values should normally be uncertain due to unit cost/time coefficients and related parameters are fuzzy/imprecise in nature. Solutions to fuzzy multi-objective PM optimization

8500 T.-F. Liang / Expert Systems with Applications 37 (2010) 8499–8507

problems benefit from assessing the imprecision of the DM’s judg- ments, such as ‘‘the objective function of project duration should be substantially less than or equal to 200 days,” and ‘‘total costs should be substantially less than or equal to 2 millions”. Conven- tional deterministic PM decision techniques cannot clearly solve the fuzzy multi-objective PM programming problems.

This study aims to develop a two-phase fuzzy goal program- ming (FGP) method for solving the PM decision problems with multiple fuzzy goals in uncertain environment. The original mul- ti-objective linear programming (MOLP) model designed here at- tempts to simultaneously minimize total project costs, total completion time and total crashing costs with reference to direct costs, indirect and contractual penalty costs, duration of activities and the constraint of available budget. The remainder of this study is organized as follows. Section 2 dedicates to a review of the rel- evant literature. Section 3 describes the problem, details the assumptions and formulates the fuzzy multi-objective PM decision model. Subsequently, Section 4 develops the FGP method for solv- ing the fuzzy multi-objective PM decision problems. Next, a real industrial case is used to implement the feasibility of applying the proposed method in Section 5. Finally, conclusions are drawn in Section 6.

2. Literature reviews

In practical situations, the goals and model inputs for the PM decisions are normally imprecise/vague owing to relevant informa- tion is incomplete and unavailable over the project planning hori- zon. To deal with imprecision, Lukaszewicz (1965) and Parks and Ramsing (1969) first introduced the techniques of randomness the- ory to solve PM decision problems for minimizing the expected project duration, in which each activity duration for a PERT type project is defined as a random variable with known probabilistic density function. Later works on stochastic PM decisions included Diaz and Hadipriono (1993), Golenko-Ginzburz and Goink (1997, 1998), Kotiah and Wallace (1973), Rabbani et al. (2007) and Touran and Edward (1992). The critical drawback of applying probability theory to PM decisions are lack of computational efficiency and inflexible probabilistic doctrines which might not be able to model the real imprecise meaning of DM because they can only take the limited form of a given non-linear distribution function, such as normal, Beta, poisson and compound poisson (Chanas & Kambu- rowsi, 1981; Lai & Hwang, 1992; Lootsma, 1989; Yazenin, 1987).

Fuzzy set theory, was presented by Zadeh (1965), has been found extensive applications in various fields (Klir & Yuan, 1995; Rommelfanger, 1996; Slowinski, 1986; Zimmermann, 1996). Alter- natively, fuzzy set theory has also provided an appropriate meth- odology to deal quantitatively with decision problems that are formulated as mathematical programming models with imprecise parameters. Zimmermann (1976) first introduced fuzzy set theory into an ordinary linear programming (LP) problem with fuzzy goal and constraints. Following the fuzzy decision-making concept of Bellman and Zadeh (1970), it confirmed that an equivalent crisp LP problem exists. Subsequently, fuzzy set theory and Zimmer- mann’s fuzzy programming technique have been developed into several fuzzy optimization methods to solve imprecise PM decision problems and avoiding unrealistic modeling in an uncertain environment.

Chanas and Kamburowsi (1981) presented a fuzzy PERT (FPERT) method that can derive the possibility distribution of the project completion time in the situation when particular activity duration times were given in the form of fuzzy variables on the time space. Mjelde (1986) originated the special structure of fuzzy resource allocation problems and defined a dedicated algorithm for their solution which is based upon a LP formulation in terms of the re-

source allocation variables and a single additional variable describ- ing the aspiration level of resource consumptions and activity returns. Chang, Tsujimuta, Gen, and Tozawa (1995) adopted the fuzzy Delphi method to estimate a reliable time interval of each activity in the project network analysis and an efficient methodol- ogy for calculating the fuzzy project completion time and the de- gree of criticality for each path in a project was proposed based on these time estimates. Yao and Lin (2000) introduced a signed distance ranking method for fuzzy numbers in a CPM of activity- on-edge (AOE) networks, and used them to obtain fuzzy critical path. Liang (2006) designed an interactive fuzzy linear program- ming (FLP) approach for solving the single-goal PM decision prob- lems with fuzzy goal and fuzzy constraints. Long and Ohsato (2008) developed a fuzzy critical approach for imprecise project scheduling problems under resource constraints which consisted of developing a desirable deterministic schedule and adding a pro- ject buffer to the end of the schedule to deal with uncertainty. Additional investigations in which fuzzy set theory was applied to PM decisions included Buckley (1989), Chanas and Zieliński (2001), Hapke and Slowinski (1996), Hussein and Abo-Sinna (1995), Okada and Soper (2000) and Wang and Fu (1998).

In practical PM decisions, however, the project DM must simul- taneously handle multiple conflicting goals that govern the use of the constrained resources within organizations, and these conflict- ing goals are generally fuzzy in nature and are required to be solved simultaneously by the DM in the framework of imprecise aspiration levels. Conventional solution techniques clearly cannot solve fuzzy multi-objective PM optimization problems. Zimmer- mann (1978) extended his fuzzy programming technique (1976) to an ordinary MOLP problem. For each of the objective functions of this MOLP problem, the DM is assumed to have a fuzzy goal such as, ‘‘the objective functions should be substantially less than or equal to some value.” Then, the corresponding linear membership function of each fuzzy objective functions were defined and the minimum operator is applied to aggregate all fuzzy sets to trans- form this MOLP problem into an equivalent ordinary LP problem. Subsequent studies on FGP included those of Chen and Tsai (2001), Hannan (1981), Kuwano (1996), Leberling (1981), Luhandj- ula (1982), Sakawa (1988, 1993) and Vasant et al. (2002).

Several studies about the use of FGP methods to solve PM deci- sion problems had been presented. DePorter and Ellis (1990) devel- oped a FGP technique for a multiple imprecise goal optimization problem in a way that compromised among the goals, and was solvable using ordinary LP computer software. Arikan and Gungor (2001) introduced a practical application of FGP technique in a pro- ject network problem with two fuzzy goals as minimize project duration and minimize total crashing costs wanted to be optimized simultaneously, and further comparisons between solutions of FGP, FLP and lexicographic maximization method (LMM) were also presented. Wang and Liang (2004) developed a multiple fuzzy goals programming (MFGP) approach to PM decisions that can yield the DM’s overall degree of satisfaction with the given goal values. Chen and Huang (2006) proposed a fuzzy programming model by combining fuzzy set theory with PERT to calculate the to- tal cycle time of a supply chain system. That model adopted trian- gular fuzzy numbers to describe these uncertain variables and the promise delivery possibility index is defined to indicate the order fulfillment degree of a supply chain system based on the fuzzy completion time and fuzzy due date. Wang and Liang (2006) devel- oped an interactive FGP model with piecewise membership func- tions to solve multi-objective PM decision problems in a fuzzy environment. More recently, Liang (2009) proposed a possibilistic linear programming (PLP) approach attempts to simultaneously minimize total project costs and completion time with reference to direct costs, indirect costs, relevant activities times and costs, and considers constraint on the available budget.

T.-F. Liang / Expert Systems with Applications 37 (2010) 8499–8507 8501

However, these simplified models described above neglect indi- rect costs, contractual penalties, available budget constraint and conditions of insufficient resources; thus, they are unrealistic in practical applications. Particularly, although it had been indicated that the minimum operator adopted extensively by the above fuz- zy programming methods has some good properties, the optimal solution yielded by the minimum operator may not be an efficient solution (Dubois & Fortemps, 1999; Guu & Wu, 1999; Lee & Li, 1993; Li, Zhang, & Li, 2006; Özgen, Önut, Gülsün, Tuzkaya, & Tuz- kaya, 2008).

3. Problem formulation

3.1. Problem description, assumptions and notation

The fuzzy multi-objective PM decision problem examined here can be described as follows. Assume a project involves n interre- lated activities that must be executed in a certain order before the entire task can be completed. The objective functions of the PM decision cannot be precisely measured because some infor- mation regarding the environmental coefficients and parameters is incomplete and unobtainable over the project planning hori- zon. This study focuses on developing a two-phase FGP technique to optimize the duration time and the crash time tolerance of each activity in the project with reference to direct costs, indirect costs and the constraints of available budget. The original fuzzy MOLP model designed in this study aims to simultaneously min- imize total project costs, total completion time and total crashing costs.

The proposed fuzzy mathematical programming model is based on the following assumptions:

(1) All of the objective functions are fuzzy with imprecise aspi- ration levels.

(2) All of the objective functions and constraints are linear equations.

(3) Direct costs increase linearly as the duration of activity is reduced from its normal time to its crash value.

(4) The normal time and shortest possible time for each activity and the cost of completing the activity in the normal time and crash time are certain.

(5) The available total budget is known over the planning horizons.

(6) The linear membership functions are adopted to specify fuzzy goals, and the minimum operator and the average operator are sequentially used to aggregate fuzzy sets.

(7) The total indirect costs can be divided into fixed costs and variable costs, and the variable costs per unit time are the same regardless of project completion time.

Assumption 1 relates to the fuzziness of the objective functions in practical PM decisions, and incorporates the variations in the DM judgments regarding the solutions of fuzzy optimization prob- lems in a framework of imprecise aspiration levels. Assumptions 2–5 indicate that the linearity, proportionality and certainty prop- erties must be technically satisfied as a standard LP form. Assump- tion 6 is made to specify the fuzzy objective functions with simplified linear membership functions and to convert the original fuzzy MOLP problem into an equivalent ordinary LP form. Assump- tion 7 represents that the indirect costs can be divided into fixed costs and variable costs. Fixed costs represent the indirect costs under normal conditions and remain constant regardless of project duration. Meanwhile, variable costs, which are used to measure savings or increases in variable indirect costs, vary directly with the difference between actual completion and normal.

The following notation is used.

(i, j) = activity between events i and j, g = index for objective function, for all g = 1, 2, . . . , K, z1 = total project costs, z2 = total completion time, z3 = total crashing costs, Dij = normal time for activity (i, j), dij = minimum crashed time for activity (i, j), CDij = normal (direct) cost for activity (i, j), Cdij = minimum crashed (direct) cost for activity (i, j), kij = incremental crashing costs for activity (i, j), tij = crashed duration time for activity (i, j), Yij = crash time for activity (i, j), Ei = earliest time for event i, E1 = project start time, En = project completion time, Tnc = project completion time under normal conditions, T = specified project completion time, CI = fixed indirect costs under normal conditions, m = variable indirect costs per unit time, B = available total budget.

3.2. Fuzzy MOLP model

3.2.1. Objective functions This study chose multiple goals for the PM decisions by review-

ing the literature and considering practical situations. In real-world situations, the project DM must generally handle conflicting goals that govern the use of the resources within organizations. Most practical PM decisions must consider minimizing total project costs, completion duration, crashing costs and contractual penal- ties, and/or maximizing profits and the utilization of equipment (Al-Fanzine & Haouari, 2005; Arikan & Gungor, 2001; DePorter & Ellis, 1990; Lin & Gen, 2007; Viana & Sousa, 2000; Wang & Liang, 2004, 2006; Yin & Wang, 2008). Notably, these goals are frequently fuzzy owing to incomplete and unavailable information over the planning horizon. In practice, a DM may be able to shorten project completion time, realizing savings on indirect costs, by increasing direct expenses to accelerate the project. The major goal of PM decisions is to determine just which time-cost trade-offs should be made for each activity to meet the minimum total costs with the optimal completion duration and crashing time. Accordingly, three fuzzy objective functions are simultaneously considered dur- ing the formulation of the multi-objective PM decision model, as follows.

� Minimize total project costs

Min z1 ffi X

i

X j

CDij þ X

i

X j

kij Y ij þ½CI þ mðEn � T ncÞ� ð1Þ

where the terms P

i

P j CDij þ

P i

P jkij Y ij are used to calculate total

direct costs. Total direct costs include total normal cost and total crashing cost, obtained using additional direct resources such as overtime, personnel and equipment. Generally, the major direct costs such as overtime, personnel and equipment, depend either on activity times or on project completion time, although materials costs are fixed during the planning horizon. The terms [CI + m(En � Tnc)] denote total indirect costs, including administra- tion, financial, contractual penalties, depreciation and other variable overhead costs that can be avoided by reducing total com- pletion time. For facilitating the model, this study assumes that the total indirect costs are divided into two categories, fixed costs and variable costs, and the variable costs per unit time are the same regardless of project completion time.

8502 T.-F. Liang / Expert Systems with Applications 37 (2010) 8499–8507

� Minimize total completion time

(gf

1

Fig.

Min z2 ffi En � E1 ð2Þ

� Minimize total crashing time

Min z3 ffi X

i

X j

kijY ij ð3Þ

The symbol ‘ ffi ’ is the fuzzified version of ‘=’ and refers to the fuzzification of the aspiration levels. In practical situations, Eqs. (1)–(3) are normally fuzzy and incorporate the variations in the DM’s judgments relating to the solutions of the fuzzy PM optimiza- tion problem and these conflicting goals are required to be simul- taneously optimized by the DM in the framework of imprecise aspiration levels.

3.2.2. Constraints

� Constraints on the time between events i and j

Ei þ tij � Ej 6 0 8i; 8j ð4Þ tij ¼ Dij � Y ij 8i; 8j ð5Þ

� Constraints on the crashing time for activity (i, j)

Y ij 6 Dij � dij 8i; 8j ð6Þ

� Constraint on the total budget

z1 6 B 8i; 8j ð7Þ

� Non-negativity constraints on decision variables

tij; Y ij; Ei P 0 8i; 8j ð8Þ

4. Model development

4.1. Solving the fuzzy multi-objective PM decision problems

4.1.1. Phase I: the minimum operator method In phase I, the original fuzzy multi-objective PM decision model

designed above can be solved using the fuzzy decision-making concept of Bellman and Zadeh (1970), together with the Zimmer- mann’s fuzzy programming technique. First, the positive ideal solution (PIS) and negative ideal solution (NIS) for each of the fuzzy objective functions can be specified as follows:

zPISg ¼ Min zg; z NIS g ¼ Max zg g ¼ 1; 2; . . . ; K ð9Þ

Moreover, the linear membership functions are defined for rep- resenting fuzzy objective functions involved. The linear member- ship functions can be specified by requiring the DM to select the

)gz

gz 0 PIS

gz NIS gz

1. The non-increasing continuous linear membership functions of fg(zg).

goal value interval zPISg ; z NIS g

h i . Accordingly, the corresponding

non-increasing continuous linear membership functions for the fuzzy objective functions can be expressed by

fgðzgÞ¼

1 zg 6 zPISg zNISg �zg

zNISg �z PIS g

zPISg < zg < z NIS g

0 zg P zNISg

8 >>>< >>>:

g ¼ 1; 2; . . . ; K ð10Þ

In practice, the corresponding possible value interval for a fuzzy objective function can be estimated based on the DM’s experience and knowledge, and the equivalent membership grade of the DM are in the interval [0, 1]. Fig. 1 shows the graph of the non-increas- ing continuous linear membership functions.

Additionally, the minimum operator is used to aggregate all fuz- zy sets. By introducing the auxiliary variable L(1), the original fuzzy MOLP problem can be converted into an equivalent ordinary LP model. Consequently, the complete equivalent single-goal LP mod- el is as follows. The fuzzy decision-making concept using the min- imum operator is presented in the Appendix A.

Max Lð1Þ

s:t 0 6 Lð1Þ 6 fgðzgÞ 8g Eqs: ð4Þ—ð8Þ

ð11Þ

where the auxiliary variable L(1) represents overall DM satisfaction with the determined goal values. Notably, Guu and Wu (1999) and Li et al. (2006) indicated that the optimal solution yielded by phase I may not be an efficient solution, and the computational efficiency of the solutions obtained using the minimum operator is not been as- sured. Hence, this work presents a further improvement technique to overcome this main drawback of the minimum operator method used in phase I.

4.1.2. Phase II: the weighted average operator method In phase II, the initial solution obtained via the minimum

operator method in phase I is improved by adding the lower bound of satisfaction degrees for each fuzzy objective function, Llgðg ¼ 1; 2; . . . ; KÞ as a constraint; and the compensatory weighted average operator is then adopted to aggregate the fuzzy set. By introducing the auxiliary variable L(2), moreover, the fuzzy MOLP problem can be converted into an equivalent ordinary LP model by the weighted average operator method, as follows:

Max Lð2Þ ¼ XK g¼1

wg Lg

s:t: Llg 6 Lg 6 fgðzgÞ 8g XK g¼1

wg ¼ 1

Eqs: ð4Þ—ð8Þ 0 6 Lð2Þ 6 1

0 6 Llg 6 1 8g 0 6 wg 6 1 8g

ð12Þ

where wg (g = 1, 2, . . . , K) is the corresponding weight of the gth fuz- zy objective function chosen by DM. In real-life PM decisions, the changes in the weight and the lower bound of satisfaction degree for each fuzzy objective function in model (12) influence the com- putational results. The value of the relative weights among of multi- ple goals can be adjusted subjectively based on the DM’s experience and knowledge. The derived degree of membership of each fuzzy objective function is taken as its initial lower bound of satisfaction degree in model (12). Notably, when the lower bound of satisfaction

T.-F. Liang / Expert Systems with Applications 37 (2010) 8499–8507 8503

degree for each of the objective functions is improperly given, the complexity of the solution procedure will increase. If the lower bound of satisfaction degree specified by a DM is too high, model (12) may not generate a solution (Li et al., 2006). Generally, when a DM increases the minimum satisfaction degree of one fuzzy objec- tive function, it implies that the value of this fuzzy objective func- tion is then closer to the optimal value; however, this may make other fuzzy objective values far from their optimal values. In prac- tice, the ordinary single-goal LP optimal solution is typically utilized as a starting point of the PIS for specifying the interval of member- ship degree for each of the objective functions in model (11).

4.2. Solution procedure

Step 1. Formulate the original fuzzy MOLP model for solving the fuzzy multi-objective PM decision problems according to Eqs. (1)–(8).

Step 2. Specify the PIS and NIS for each fuzzy objective function zg (g = 1, 2, . . . , K), and then define the corresponding linear membership functions, as Eqs. (9) and (10).

Step 3. Introduce the auxiliary variable L(1), thus enabling aggre- gation of the original fuzzy MOLP problem into an equiva- lent ordinary LP form with the minimum operator, as in model (11).

Step 4. Solve the model (11) to obtain an initial compromise solution.

Step 5. Specify the lower bound of satisfaction degree Llg and the corresponding weight wg for each fuzzy objective function based on the initial solution in model (11).

Step 6. Reformulates model (11) into an equivalent ordinary LP model using the compensatory weighted average operator, as in model (12).

Table 1 Summarized data in the Daya case (in US dollar).

(i, j) Dij (days) dij (days) CDij ($) Cdij ($) kij ($/day)

1–2 14 10 1000 1600 150 1–5 18 15 4000 4540 180 2–3 19 19 1200 1200 – 2–4 15 13 200 440 120 4–7 8 8 600 600 – 4–10 19 16 2100 2490 130 5–6 22 20 4000 4600 300 5–8 24 24 1200 1200 – 6–7 27 24 5000 5450 150 7–9 20 16 2000 2200 50 8–9 22 18 1400 1900 125 9–10 18 15 700 1150 150 10–11 20 18 1000 1200 100

27

67 14

15 14

33

8

0

18

18 22

24

42

22

19

33

0

40 1

2

3

4

5

6

7

8

Fig. 2. The project netwo

Step 7. Solve the model (12) to generate improved compromise solutions. If the DM is dissatisfied with the improved com- promise solutions, then the model (12) should be adjusted interactively until a preferred efficient solutions is obtained.

5. Implementation

5.1. Case description

Daya Technology is used as a case study to demonstrate the practicality of the proposed method. The Daya Technology is the leading producer of precision machinery and transmission compo- nents in Taiwan, and is the main manufacturer producing the super precision ballscrew, linear stage, linear bearing, guideways, and aerospace parts. Its products are distributed throughout Asia, North America and Europe and have been in high demand for sev- eral years. The real-life PM decision examined here involves expanding a metal finishing plant owned by Daya. The determinis- tic CPM technique currently used by Daya suffers from the limita- tion owing to the fact that the project manager does not have sufficient information over the planning horizon. The case study focuses on developing a fuzzy programming method to develop a suitable PM plan for the metal finishing plant in an uncertain envi- ronment. The PM decision of Daya aims to simultaneously mini- mize total project costs, total completion time and total crashing costs in terms of direct costs, indirect costs, activity and crash durations, and the constraint of available budget. Table 1 lists the basic data of the case.

Other relevant data are as follows: fixed indirect costs $12,000, saved daily variable indirect costs $150, total budget $38,500 and project completion duration under normal conditions 125 days. The project start time is set to zero. The critical path is 1–5–6–7– 9–10–11. Fig. 2 shows the activity-on-arrow network diagram.

5.2. Solution procedure for the Daya case

The solution procedure with the proposed two-phase FGP method to solve fuzzy PM decision problem for the Daya case is demonstrated as follows. In phase I, the original fuzzy MOLP model is first formulated according to Eqs. (1)–(8). Moreover, the fuzzy MOLP problem is respectively solved using the ordinary single-goal LP model, and the PIS and NIS for each of the fuzzy objective func- tion can be specified with Eq. (9). The results are zPIS1 ; z

NIS 1

� � ¼

ð35;900;36;400Þ; zPIS2 ; zNIS2 � �

¼ð108;125Þ, and zPIS3 ; zNIS3 � �

¼ð0;2440Þ. Table 2 lists the optimal solutions obtained by the ordinary LP model and the corresponding interval values of (PIS, NIS) for each of the fuzzy objective functions.

denotes critical path

denotes an activity

denotes an dummy activity

105

19

12587 18 20 20

10 119

rk of the Daya case.

Table 2 The PIS and NIS for the fuzzy objective functions.

Item LP-1 LP-2 LP-3 (PIS, NIS)

Objective function Min z1 Min z2 Min z3 – z1 ($) 35,900

a 36,290 36,400 (35,900, 36,400) z2 (days) 113 108

a 125 (108, 125) z3 ($) 1300 2440 0

a (0, 2440)

a Denotes the optimal value by ordinary single-goal LP model.

Table 4 Results of sensitivity analysis for varying the completion time.

Item Run 1 Run 2 Run 3 Run 4 Run 5 Run 6 Run 7

En (days) < 108.00 108.00 116.50 125.00 133.50 139.00 >139.00 z1($) 36,290 35,900 36,400 37,675 38,500

z2 (days) Infeasible 108.00 116.50 125.00 133.50 139.00 Infeasible z3($) 2440.00 775.00 0 0 0

1.0

z1 (z2) (z3)

40,000(150)(2500)

8504 T.-F. Liang / Expert Systems with Applications 37 (2010) 8499–8507

Accordingly, the corresponding non-increasing continuous lin- ear membership functions for each of the fuzzy objective functions can be defined via Eq. (10), as follows:

Completion time

0.0

0.2

0.4

0.6

0.8

108 116.5 125 133.5 139

z1

z2

z3

32,000(120)(2000)

24,000( 90)(1500)

16,000( 60)(1000)

8000( 30)( 500)

0 ( 0)( 0)

Fig. 3. Goal values of analyzing sensitivity for the completion time.

f1ðz1Þ¼ 1 z1 6 35; 900 36;400�z1

500 35; 900 < z1 < 36; 400 0 z1 P 36; 400

8 >< >:

ð13Þ

f2ðz2Þ¼ 1 z2 6 108 125�z2

17 108 < z2 < 125 0 z2 P 125

8 >< >:

ð14Þ

f3ðz3Þ¼ 1 z3 6 0 2440�z3

2440 0 < z3 < 2440 0 z3 P 2440

8 >< >:

ð15Þ

By introducing the auxiliary variable L(1), the resulting equiva- lent ordinary LP model for solving the fuzzy multi-objective PM decision problem for the Daya case can be formulated according to model (11). LINDO computer software is used to run this ordin- ary LP model. Using the proposed FGP method in phase I to simul- taneously minimize the total project costs, total completion time and total crashing costs, the resulting the goal values of the initial solutions are z1 = $35,900, z2 = 114.98 days, z3 = $1002.41, and the overall DM satisfaction degree is 0.5892. Moreover, the project manager specifies the lower bound of satisfaction degree ðLl1; L

l 2; L

l 3Þ¼ ð0:84; 0:86; 0:27Þ and the corresponding weights

(w1, w2, w3) = (0.4, 0.4, 0.2) for three fuzzy objective functions. The equivalent ordinary LP model can be formulated via the model (12). Consequently, the improved efficient solutions are z1 = $35,978.60, z2 = 110.38 days, z3 = $1771.60, and overall degree of DM satisfaction is up to 0.7359. Table 3 lists initial and improved PM plans for the Daya case with the proposed method based on current information.

Table 3 Initial and improved PM plans for the Daya case.

Item Initial solutions (phase I) Improved solutions (phase II)

Goal values L(1) = 0.5892, L(2) = 0.7359, z1 = $35,900.00, z1 = $35,978.60, z2 = 114.98 days, z2 = 110.38 days, z3 = $1002.41. z3 = $1771.60.

Yij (days) Y12 = 0, Y15 = 0, Y23 = 0, Y24 = 0,

Y12 = 0, Y15 = 2.62, Y23 = 0, Y24 = 0,

Y34 = 0, Y47 = 0, Y410 = 0, Y56 = 0, Y58 = 0,

Y34 = 0, Y47 = 0, Y410 = 0, Y56 = 0, Y58 = 0,

Y67 = 1.02, Y79 = 4, Y89 = 0, Y910 = 3,

Y67 = 3, Y79 = 4, Y89 = 0, Y910 = 3,

Y1011 = 2. Y1011 = 2.

tij (days) t12 = 14, t15 = 18, t23 = 19, t24 = 15,

t12 = 14, t15 = 15.38, t23 = 19, t24 = 15,

t47 = 8, t410 = 19, t56 = 22, t58 = 24,

t47 = 8, t410 = 19, t56 = 22, t58 = 24,

t67 = 25.98, t79 = 16, t89 = 22, t910 = 15,

t67 = 24, t79 = 16, t89 = 22, t910 = 15,

t1011 = 18. t1011 = 18.

Additionally, the sensitivity of the DM’s expected completion time is analyzed under the decision conditions of the Daya case. Table 4 presents the results of the implementation. The results of analyzing sensitivity for varying project duration indicate that minimizing completion time conflicts with minimizing the total project costs and the total crashing costs, as depicted in Fig. 3. Notably, the solution is infeasible when the duration of the project is far below 108 days, because the cumulative crashing time for all activities on the critical path exceeds the allowed upper limit (17 days). Conversely, if the project duration is extended beyond 139 days, the project becomes infeasible due to the total costs ex- ceed the available budget. Thus, a project DM may be able to short- en project duration for realizing savings on indirect costs, by increasing direct expenses to accelerate the project. If the DM faces costly indirect and contractual penalties for being late in complet- ing a project, the use of additional resources to reduce the project duration may be worthwhile.

5.3. Computational analysis and comparisons

Several significant characteristics when practically applying the proposed FGP method to PM decisions are as follows. First, the pro- posed method yields an efficient compromise solution. The Zim- mermann’s fuzzy programming technique which uses the linear membership function and the minimum operator can not generate an efficient solution to fuzzy programming problems (Dubois, Far- gier, & Prade, 1996; Dubois & Fortemps, 1999; Guu & Wu, 1999; Lee & Li, 1993). Li et al. (2006) verified why the output results ob- tained by the two-phase FGP technique are always efficient for using the minimum and the compensatory weighted average oper- ator sequentially to aggregate fuzzy sets in the decision-making process. As listed in Table 3, the solutions obtained with the pro- posed method are obviously better than the results using the one-stage minimum operator method. As a result, an improved PM plan can be provided by the proposed method under an accept- able degree of DM satisfaction.

Second, the proposed method comprises a rational fuzzy deci- sion-making process for solving the PM decision problems with multiple fuzzy goals. The two-phase FGP method developed here presents the overall DM satisfaction with the known goal values for fuzzy multi-objective PM decision problem. If the overall DM

T.-F. Liang / Expert Systems with Applications 37 (2010) 8499–8507 8505

satisfaction is L = 1, then each goal is fully satisfied; if it is 0 < L < 1, then all of the goals are satisfied at the level of L, and if it is L = 0, then none of the goals are satisfied. Generally, the L value may be adjusted to identify a better PM results if the project DM did not accept the initial overall degree of this satisfaction value. For in- stance, the overall degree of DM satisfaction of the initial solutions in phase I for the Daya case was 0.5892. Furthermore, the obtained L value was adjusted by the project DM to seek a better solu- tion, and the improved efficient results are z1 = $35,978.60, z2 = 110.38 days, z3 = $1771.60, with an overall degree of DM satis- faction of 0.7359.

Third, the project DM generally must make fuzzy multi-objec- tive PM decisions owing to some information being incomplete and unobtainable during the planning horizon, and these conflict- ing goals must be optimized simultaneously by the DM in the framework of imprecise aspiration levels. Due to conflicting and vagueness nature of the multiple goals, the conventional CPM and deterministic techniques may not comply with the actual aims of modeling PM decisions and are unsuitable to yield an effective solution. The comparisons from Table 4 and Fig. 3 showed that the interaction of trade-offs and conflicts among dependent multi- ple goals. Analytical results obtained by implementing Daya case indicate that the proposed method satisfies the requirement for the practical application since it attempts to simultaneously mini- mize the total project costs, total completion time and total crash- ing costs in fuzzy environments.

Additionally, the optimal solution yielded by the minimum operator method may not be an efficient solution, and the compu- tational efficiency of the solution is not been assured (Guu & Wu, 1999; Li et al., 2006). The minimum operator is preferable when a DM wants to make values of the optimal membership functions

Table 5 Comparisons of common aggregation operators.

Operator Example Brief

Intersection (t-norms) � Minimum � Algebraic product � Bounded sum � Drastic intersection

� An ‘and � The � The

Union (t-conorms) � Maximum � Algebraic sum � Bounded difference � Drastic union

� An ‘or’ � The sma

Averaging (Compensative) � Mean � Weighted � c � OWA (The ordered weighted averaging)

� Hav � Con � The � OW fuzz

Table 6 Comparisons of the major PM decision models.

Factor Conventional crisp CPM/LP Stochastic programming (Rabbani et al., 2007)

Fuz (W

Objective function Single, linear Single, nonlinear Sin Main consideration Cost or time Cost or time Cos Objective property Crisp Probabilistic Fuz Degree of satisfaction Not presented Not presented Pre Main consideration Time or cost Time or cost Tim Aggregate operator – – Min Output solution efficient Not quaranteed Not Decision parameter crisp Probabilistic Fuz Revised flexibility – Low Me Activity time crisp Probabilistic Fuz Indirect cost Not included Not included Not Budget limit Not included Not included Not

approximately equal or when a DM believes that the minimum operator is an approximate representation. The critical drawback of the minimum operator is its lack of discriminatory power be- tween solutions that strongly differ with respect to the fulfillment of membership to the various constraints (Dubois et al., 1996; Weners, 1987). For some practical situations, the application of the aggregation operator to draws maps above the maximum oper- ator and below the minimum operator is important. As listed in Ta- ble 5, average operators consider the relative importance of fuzzy sets and have the compensative property so that the result of com- bination will be medium (Klir & Yuan, 1995; Zimmermann, 1996). The proposed two-phase FGP technique can overcome the disad- vantage of using the minimum operator by adding phase I satisfac- tion degrees to phase II as a constraint, and the compensatory weighted average operator is employed for to obtain overall DM satisfaction degree. Zimmermann (1996) pointed out that the fol- lowing eight criteria must be applied selecting an adequate aggre- gation operator-axiomatic strength, empirical fit, compensation, numerical efficiency, range of compensation, adaptability, aggre- gating behavior and required scale level of membership function.

Finally, Table 6 presents the comparisons among the proposed two-phase FGP method with those of the representative PM tech- niques including deterministic CPM/LP, stochastic programming (Rabbani et al., 2007), FLP (Wang & Fu, 1998) and FGP (Arikan & Gungor, 2001) models. To summarize, several main advantages of the proposed method are presented that distinguish it from other PM decision techniques. First, The proposed method satisfies the practical application requirements because it simultaneously min- imize total project costs, total completion time and total crashing costs with reference to direct costs, indirect costs, contractual pen- alty costs, duration of activities and the constraint on total budget.

description

aggregation scheme is implemented where fuzzy sets are connected by a logical ’

result of combination is high if and only if all values are high minimum operator is a greatest t-norm

aggregation scheme is implemented where fuzzy sets are connected by a logical

result of combination is high if some values are high. The minimum operator is a llest t-conorm

e the compensative property so that the result of combination will be medium sider the relative importance of the fuzzy sets c-operator is the convex combination of the min-operator and the max-operator A enables a DM to specify linguistically his agenda for aggregating a collection of

y sets

zy linear programming ang & Fu, 1998)

FGP (Arikan & Gungor, 2001) The proposed method

gle, linear Multiple, linear Multiple, linear t or time Cost and time Cost and time zy Fuzzy Fuzzy sented Presented Presented e or cost Time & cost Time & cost imum Minimum Minimum & average quaranteed Not quaranteed efficient

zy crisp crisp dium Medium High zy crisp crisp included Not included Included included Not included Included

8506 T.-F. Liang / Expert Systems with Applications 37 (2010) 8499–8507

Second, the proposed method provides a systematic decision-mak- ing framework that the DM adjusts interactively the search direc- tion during the solution procedure, until the efficient solution satisfies the DM’s preferences and is considered to be the preferred satisfactory solution. Third, the proposed model exhibits greater computational flexibility of the fuzzy arithmetic operations by employing the simplified linear membership functions to repre- sent fuzzy goals, and the original fuzzy MOLP model can be con- verted into an equivalent ordinary LP form that is easily solved by the simplex method. Finally, computational methodology developed here can easily be extended to any other situations and can handle the realistic PM decisions. The industrial case illus- trated here lays a strong foundation upon which the project DM can formulate additional applications of the proposed method to large-scale PM decisions in uncertain environments.

6. Conclusions

In practical PM decision problems, the project DM must simul- taneously handle multiple conflicting goals that govern the use of the constrained resources, and these conflicting objectives are of- ten fuzzy because information is incomplete and unavailable over the project planning horizon. This work aims to develop a two- phase FGP technique for solving the multi-objective PM decision problems in uncertain environments. The proposed fuzzy multi- objective PM decision model attempts to minimize total project costs, total completion time and total crashing costs with reference to direct costs, indirect and penalty costs, duration of activities and the constraint of available budget. The main advantage of the pro- posed method is that it provides a systematic framework that facil- itates the decision-making process, enabling a DM to interactively modify the fuzzy data until a satisfactory efficient solution is ob- tained. An industrial case is used to demonstrate the feasibility of applying the proposed method to real PM decisions. Sensitivity analysis results for varying project duration indicate that minimiz- ing completion time conflicts with minimizing the total costs. Overall, the main contribution of this study lies in presenting a fuz- zy mathematical programming methodology to fuzzy multi-objec- tive PM decisions, and provides a systematic decision-making framework that facilitates the decision maker to interactively ad- just the search direction until the preferred efficient solution is obtained.

The major limitations of the proposed method concern the cer- tain assumptions made for each of the unit cost/time coefficients in the fuzzy objective functions and related available resources in the constraints. Hence, the proposed method must be modified make it better suited to the practical application. Furthermore, the pro- posed method is based on Zimmermann’s fuzzy programming technique, which implicitly assumes that the linear membership function is the proper representative fuzzy goals of the human DM for the PM decision problems. Future researchers may also ap- ply the piecewise linear, non-linear and related membership func- tions to construct fuzzy multi-objective PM decision models.

Appendix A. The fuzzy decision-making concept using the minimum operator (Bellman & Zadeh, 1970)

Let X be a given set of all possible solutions to a decision prob- lem. A fuzzy goal G is a fuzzy set on X characterized by its member- ship function

lG : X !½0; 1� ðA1Þ

A fuzzy constraint C is a fuzzy set on X characterized by its membership function

lC : X !½0; 1� ðA2Þ

Then, G and C combine to generate a fuzzy decision D on X, which is a fuzzy set resulting from intersection of G and C, and is characterized by its membership function

L ¼ lDðxÞ¼ lGðxÞ^ lCðxÞ¼ Min ðlGðxÞ; lCðxÞÞ ðA3Þ

and the corresponding maximizing decision is defined by

Max L ¼ Max lDðxÞ¼ Max Min ðlGðxÞ; lCðxÞÞ ðA4Þ

More generally, suppose the fuzzy decision D results from k fuz- zy goals G1, . . . , Gk and m constraints C1, . . . , Cm. Then the fuzzy deci- sion D is the intersection of G1, . . . , Gk and C1, . . . , Cm, and is characterized by its membership function

L ¼ lDðxÞ ¼ lG1ðxÞ^ lG2ðxÞ^ �� �^ lGk ^ lC1 ^ lC2 ^�� �^ lCm ¼ Min lG1ðxÞ; lG2ðxÞ; . . . ; lGkðxÞ; lC1ðxÞ; lC2ðxÞ; . . . ; lCmðxÞ

� �

ðA5Þ

and the corresponding maximizing decision is defined by

Max L ¼ Max lDðxÞ

¼ Max Min lG1ðxÞ; lG2ðxÞ; . . . ; lGkðxÞ; lC1ðxÞ; . . . ; lCmðxÞ � �

ðA6Þ

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