Application 1 – Analysis and Synthesis of Prior Research

Application of fuzzy sets to multi-objective project management decisions

Tien-Fu Liang*

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

( Received 1 May 2007; final version received 16 August 2007 )

In real-world project management (PM) decision problems, input data and/or related parameters are frequently imprecise/fuzzy over the planning horizon owing to incomplete or unavailable information, and the decision maker (DM) generally faces a fuzzy multi-objective PM decision problem in uncertain environments. This work focuses on the application of fuzzy sets to solve fuzzy multi-objective PM decision problems. The proposed possibilistic linear programming (PLP) approach attempts to simultaneously minimise total project costs and completion time with reference to direct costs, indirect costs, relevant activities times and costs, and budget constraints. An industrial case illustrates the feasibility of applying the proposed PLP approach to practical PM decisions. The main advantage of the proposed approach is that the DM may adjust the search direction during the solution procedure, until the efficient solution satisfies the DM’s preferences and is considered to be the preferred satisfactory solution. In particular, computational methodology developed in this work can easily be extended to any other situations and can handle the realistic PM decision problems with simplified triangular possibility distributions.

Keywords: project management; fuzzy sets; possibilistic linear programming; triangular distribution

1. Introduction

Project management (PM) decisions have attracted considerable interest from both

practitioners and academics. Since the program evaluation and review technique (PERT)

and the critical path method (CPM) were both developed in the 1950s, numerous models

including mathematical programming techniques and heuristics have been developed for

solving PM problems, each with its own advantages and disadvantages. However, when

any of the conventional CPM, linear programming (LP) and heuristics was used to solve

PM decision problems, the goals and model inputs are generally assumed to be

deterministic/crisp (Davis and Patterson 1975, Elsayed 1982, DePorter and Ellis 1990).

In real-world PM decision problems, input data or related parameters, such as relevant

operating costs, activities times, available resources and costs budget, are frequently

imprecise/fuzzy owing to incomplete or unobtainable information. Conventional

mathematical programming techniques and heuristics clearly do not solve all fuzzy PM

programming problems. Fuzzy set theory, was presented by Zadeh (1965), has been found

extensive applications in various fields (Rommelfanger 1996). Zimmermann (1976) first

ISSN 0308-1079 print/ISSN 1563-5104 online

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DOI: 10.1080/03081070701785833

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*Email: [email protected]

International Journal of General Systems

Vol. 38, No. 3, April 2009, 311–330

introduced fuzzy set theory into ordinary single-goal LP problems. That study considered

LP problems with fuzzy goal and constraints. Subsequently, Zimmermann’s fuzzy linear

programming (FLP) has developed into several fuzzy optimisation methods for solving the

PM problems (Chanas and Kamburowsi 1981, Buckley 1989, DePorter and Ellis 1990,

Wang and Fu 1998, Liang 2006, Wang and Liang 2006).

However, in practical situations, the project managers must generally handle

conflicting goals that govern the use of the resources within organisations. These

conflicting goals are required to be optimised simultaneously by the project managers,

often in the framework of fuzzy aspiration levels. Particularly, solutions to fuzzy

multi-objective optimisation problems benefit from assessing the imprecision of the

decision maker’s (DM’s) judgments, such as ‘the objective function of project duration

should be substantially less than or equal to 120 days’, and ‘total project cost should be

substantially less than or equal to five million’. Zimmermann (1978) first extended his FLP

approach to a conventional multi-objective linear programming (MOLP) problem.

Moreover, Arikan and Gungor (2001) employed fuzzy goals programming (FGP) method

developed by Zimmermann (1978) to solve PM problems with two fuzzy objectives,

minimising both completion time and crashing costs. Wang and Liang (2004a) recently

developed an interactive multiple fuzzy goals programming (MFGP) model using the

linear membership function for solving the fuzzy multi-objective PM problems.

Furthermore, Zadeh (1978) presented the theory of possibility, which is related to the

theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy

restriction, which acts as an elastic constraint on the values that can be assigned to a

variable. Moreover, Zadeh (1978) showed that the importance of the theory of possibility

is based on the fact that much of the information on which human decisions is possibilistic

rather than probabilistic in nature. Since the expression of a possibility distribution can be

viewed as a fuzzy set, possibility distribution may be manipulated by the combination

rules of fuzzy sets and more particular of fuzzy restrictions (Dubois and Prade 1980).

Buckley (1988) formulated a mathematical programming problem in which all parameters

may be fuzzy variables specified by their possibility distribution; he also illustrated this

problem using the possibilistic linear programming (PLP) approach. Lai and Hwang

(1992a) designed an auxiliary MOLP model for solving a PLP problem with imprecise

objective and/or constraint coefficients. Tang et al. (2001) established two types of PLP

with general possibilistic distribution, including LP problems with general possibilistic

resources and general possibilistic objective coefficients. Moreover, Hsu and Wang (2001)

developed a possibilistic programming model integrating the PLP method of Lai and

Hwang (1992a) and the fuzzy programming method of Zimmermann (1978) for managing

production planning decision problems involving ambiguous cost goal and uncertain

demand in an assemble-to-order environment. Wang and Liang (2005) more recently

formulated a possibilistic programming model to solve single-goal production planning

problems with imprecise objective and constraints. Related works on PLP problems

include, Inuiguchi and Sakawa (1996), Hussein (1998) and Tanaka et al. (2000).

Possibilistic programming approach may provide an important aspect in handling

practical multi-objective PM decisions in uncertain environments. The possibilistic

programming provides more computational efficiency and flexibility of fuzzy arithmetic

operations than the stochastic programming model. The critical problems of applying

stochastic programming to solve PM problems 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 probability

distribution function (Chanas and Kamburowsi 1981, Mjelde 1986, Yazenin 1987,

T.-F. Liang312

Buckley 1990). Alternatively, the proposed possibilistic programming provides a more

efficient way of solving imprecise PM problems and additionally, preserves the original

linear model for all imprecise objectives and constraints with the proposed simplified

weighted average, and fuzzy ranking techniques (Zadeh 1978, Buckley 1988, Lai and

Hwang 1992a). Additionally, the possibilistic programming model differs from general

FLP problems in terms of its meaning. The FLP is based on the subjective preferred

concept for establishing membership functions with fuzzy data, while the possibilistic

programming is based on the objective degree of event occurrence required to obtain

possibilistic distributions with imprecise data (Kaufmann and Gupta 1991, Lai and Hwang

1992b, Klir and Yuan 1995, Inuiguchi and Sakawa 1996).

This work presents a PLP approach for solving fuzzy multi-objective PM decision

problems in uncertain environments. The proposed approach attempts to simultaneously

minimise total project costs and completion time with reference to direct costs, indirect

and contractual penalties costs, and budget constraints. The remainder of this work is

organised as follows. Section 2 describes the problem, details the assumptions and

formulates the problem. Section 3, then develops the interactive PLP approach and

procedure for solving PM problems. Subsequently, Section 4 presents an industrial case

for implementing the feasibility of applying the proposed approach to real PM decisions.

Next, Section 5 discusses the findings for the practical application of the proposed PLP

approach. Finally, conclusions are drawn in Section 6.

2. Problem formulation

2.1 Problem description, assumptions and notation

This section describes the PM decision problem examined here. Assume a project involves

n interrelated activities that must be executed in a certain order before the entire task can

be completed. Generally, the environmental coefficients and related parameters are

uncertain over the planning horizon. Consequently, the incremental crashing costs,

variable indirect cost per unit time, specified project completion time and total budget are

imprecise in nature. Assigning a set of crisp values for the environmental coefficients and

related parameters is inappropriate for dealing with such ambiguous PM decision

problems. Hence, the proposed PM decision focuses on developing a PLP approach to the

optimum duration of each activity in the project, given a specified project completion time

T, crash time tolerances for each activity and budget constraints. The developed approach

attempts to simultaneously minimise total project costs and completion time in uncertain

environments.

The mathematical programming model formulated here is based on the following

assumptions.

(1) Objective functions are fuzzy/imprecise and have imprecise aspiration levels.

(2) The pattern of triangular possibility distribution is adopted to represent the

imprecise objective function and related imprecise numbers.

(3) The linear membership functions are specified for all fuzzy objectives involved

and the minimum operator is used to aggregate all fuzzy sets.

(4) Direct costs increase linearly as the duration of activity is reduced from its normal

time to its crash value.

(5) Indirect costs can be divided into two categories – fixed costs and variable costs –

and the variable cost per unit time is the same regardless of the project completion

time.

International Journal of General Systems 313

Assumption 1, relates to the fuzziness of the objective functions in real-world PM

problems and incorporates the variations in the DM judgments regarding the solutions of

fuzzy/imprecise multiple goals optimisation problems in a framework of fuzzy aspiration

levels. Assumption 2, addresses the effectiveness of applying triangular possibility

distribution to represent imprecise objectives and related imprecise numbers. Generally,

the project managers are familiar with estimating optimistic, pessimistic and most likely

parameters from the use of the Beta distributions specified by the class PERT. The pattern

of triangular distribution is commonly adopted due to ease in defining the maximum and

minimum limit of deviation of the fuzzy number from its central value (Yang et al. 1991).

Hershauer and Nabielsky (1972) recommended employing triangular distribution, when

only the mode (most likely value) and range (limit of optimistic and pessimistic values) of

a fuzzy number are known. Additionally, when knowledge of the distribution is limited,

triangular distribution is appropriate for representing a fuzzy number (MacCrimmon and

Ryavec 1964, Kotiah and Wallace 1973, Chanas and Kamburowsi 1981, Buckley 1989).

Assumption 3, is made to convert the fuzzy MOLP problem into an equivalent ordinary

single-goal LP form that can be solved efficiently by the ordinary simplex method

(Zimmermann 1976). Assumption 4, implies that direct costs increase linearly with

reducing project duration. Assumption 5, 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 duration of the project.

The following notation is used.

(i, j) ¼ activity between events i and j

~z1 ¼ total project costs ($)

z2 ¼ total completion time (days)

Dij ¼ normal time for activity (i, j; days)

dij ¼ minimum crashed time for activity (i, j; days)

CDij ¼ normal (direct) cost for activity (i, j; $)

Cdij ¼ minimum crashed (direct) cost for activity (i, j; $) ~kij ¼ incremental crashing costs for activity (i, j; $/day)

tij ¼ crashed duration time for activity (i, j; days)

Yij ¼ crash time for activity (i, j; days)

Ei ¼ earliest time for event i (days)

E1 ¼ project start time (days)

En ¼ project completion time (days)

To ¼ project completion time under normal conditions (days) ~T ¼ specified project completion time (days)

CI ¼ fixed indirect costs under normal conditions ($)

~m ¼ variable indirect costs per unit time ($/day) ~B ¼ total budget ($)

2.2 Original multi-objective PLP model

2.2.1 Objective functions

The proposed PLP model selected multiple imprecise goals for solving the PM decision

problems based on a literature review and by considering industrial situations. In practice,

project managers can shorten project completion time, realising savings on indirect costs,

T.-F. Liang314

by increasing direct expenses to accelerate project progress. Generally, most practical

decisions for solving PM problems usually consider total costs and completion time

(DePorter and Ellis 1990, Wang and Fu 1998, Arikan and Gungor 2001, Wang and Liang

2004a, Liang 2006, Wang and Liang 2006). Notably, the related cost coefficients

frequently are imprecise owing to some information being incomplete or unobtainable

over the planning horizon. The major goal of PM decisions is to determine just which

time–cost trade-offs should be made for each activity to meet the specified project

completion time with the minimum total costs. In practice, a project DM may be able to

shorten project completion time, realising savings on indirect costs, by increasing direct

expenses to accelerate the project. If a DM faces costly penalties for failing to complete a

project on time, then using extra resources to complete the project may be economical.

Consequently, two objective functions are considered simultaneously in designing the

original multi-objective PLP model, as follows:

. Minimise total project costs

Min ~z1 ¼ X i

X j

CDij þ X i

X j

~kijYij þ ½CI þ ~mðEn 2 ToÞ� ð1Þ

where ~kij and ~m are imprecise coefficients with triangular possibility distributions. The

total project costs are imprecise and are the sum of the direct costs and the indirect costs

over the planning horizon. The terms,

X i

X j

CDij þ X i

X j

~kijYij;

are used to calculated 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. Total direct costs increase with decreasing project

duration. The terms ½CI þ ~mðEn 2 ToÞ� denote total indirect costs, including adminis-

tration, contractual penalties, depreciation, financial and other variable overhead costs that

can be avoided by reducing total completion time. For facilitating the model, this work

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.

. Minimise total completion time

Min z2 ø En 2 E1 ð2Þ

where the symbol ‘ ø ’ is the fuzzified version of ‘ ¼ ’ and refers to the fuzzification of the aspiration levels. In practical situations, substantial amounts of information for the inputs

required to solve a PM decision problem are often fuzzy in nature. This may be true for

objectives as well as parameters. This work assumes that the DM has such imprecise goals,

such as ‘the project total completion time should essentially equal some value’. In real-

world PM problems, total completion time is usually fuzzy with imprecise aspiration

levels, incorporating variations in the DM’s judgments concerning solutions for fuzzy

multi-objective PM optimisation problems, and the project start time is often set to zero.

International Journal of General Systems 315

2.2.2 Constraints

. Constraints on the time between events i and j

Ei þ tij 2 Ej # 0 ;i; ;j ð3Þ

tij ¼ Dij 2 Yij ;i; ;j ð4Þ

. Constraints on the crash time for activity (i, j)

Yij # Dij 2 dij ;i; ;j ð5Þ

. Constraints on project start time and total completion time

E1 ¼ 0 ð6Þ

En # ~T ð7Þ

. Constraint on the total budget

X i

X j

CDij þ X i

X j

~kijYij þ ½CI þ ~mðEn 2 ToÞ� # ~B ð8Þ

. Non-negativity constraints on decision variables

tij; Yij; Ei; Ej $ 0 ;i; ;j ð9Þ

In constraint (7) this work assumes that a specific deadline T has been fixed (perhaps

by contract, resource allocation and economic considerations, and/or other factors) for the

completion of the project. In real-world situations, the specified completion time T for the

project in Equation (7) and the total budget in Equation (8) is never obtained precisely in a

dynamic environment, because some relevant information, such as contractual

information, the skills of the workers, public policy, law and regulations, available

resources and other factors, is incomplete or unavailable. Therefore, Equations (7) and (8)

are normally imprecise constraints.

3. Model development

3.1 Model the imprecise data with triangular possibility distribution

The possibility distribution can be stated as the degree of occurrence of an event with

imprecise data. This work assumes the DM to have already adopted the pattern of

triangular possibility distribution for all imprecise numbers. To simplify experts’

evaluation, similarity as in the class PERT, employing the most likely, optimistic and

pessimistic parameters are appropriate (Chanas and Kamburowsi 1981, Buckley 1989).

According to experts, the most likely parameter is the most appropriate value of activity

performance, and optimistic and pessimistic values determine the limit of toleration.

In practice, for example, the DM can establish the triangular distribution of the

incremental crashing costs for activity (i, j), ~kij, based on the three prominent data: (1) the

most optimistic value (koij) that has a very low likelihood of belonging to the set of

available values (possibility degree ¼ 0 if normalised); (2) the most likely value (kmij ) that

definitely belongs to the set of available values (possibility degree ¼ 1 if normalised) and

(3) the most pessimistic value (k p ij) that has a very low likelihood of belonging to the set

T.-F. Liang316

of available values (possibility degree ¼ 0 if normalised), where the base is on the

interval ½koij; k p ij� and vertex at x ¼ k

m ij . Figure 1 presents the triangular possibility

distribution of ~kij ¼ ðk o ij; k

m ij ; k

p ijÞ. Similarly, the related imprecise data of the original PLP

model thus can be modelled using triangular possibility distributions.

3.2 Developing the auxiliary MOLP model

3.2.1 Strategy for solving the imprecise objective function

The objective function (1) in the original PLP model formulated above has triangular

possibility distribution. Geometrically, this imprecise objective is fully defined by three

prominent points ðzo1; 0Þ, ðz m 1 ; 1Þ and ðz

p 1 ; 0Þ. The imprecise objective can be minimised by

moving the three prominent points toward the left. Using Lai and Hwang’s (1992a)

approach, the proposed approach substitute simultaneously minimising zm1 , maximising

ðzm1 2 z o 1Þ and minimising ðz

p 1 2 z

m 1 Þ for minimising z

m 1 ; z

o 1 and z

p 1 . The resulting three new

objective functions still guarantee the declaration of moving the triangular distribution

toward the left. Figure 2 illustrates the strategy for minimising the imprecise objective

function; that is, the auxiliary MOLP problem generated by this proposed approach

comprises simultaneously minimising the most likely value of imprecise total costs ðzm1 Þ,

maximising the possibility of obtaining lower total costs (region I of the possibility

distribution in Figure 2) ðzm1 2 z o 1 Þ, and minimising the risk of obtaining higher total costs

(region II of the possibility distribution in Figure 2) ðz p 1 2 z

m 1 Þ.

As indicated in Figure 2, possibility distribution ~A2 is preferred to possibility

distribution ~A1. Expressions (10)–(12) list the results for the three new objective functions

of total costs in Equation (1).

Min z11 ¼ z m 1 ¼

X i

X j

CDij þ X i

X j

kmij Yij þ ½CI þ m m ðEn 2 ToÞ� ð10Þ

Max z12 ¼ z m 1 2 z

o 1

� � ¼

X i

X j

kmij 2 k o ij

� � Yij þ ½ðm

m 2 moÞðEn 2 ToÞ� ð11Þ

Min z13 ¼ z p 2 zmð Þ ¼

X i

X j

k p ij 2 k

m ij

� � Yij þ ½ðm

p 2 mmÞðEn 2 ToÞ� ð12Þ

Figure 1. The triangular possibility distribution of ~kij.

International Journal of General Systems 317

3.2.2 Strategy for solving the imprecise constraints

Recalling Equation (7) from the original PLP model; consider the situations in which the

specified project completion time (the right-hand side), ~T, are imprecise and have

triangular possibility distribution with the most and least possible values. This work

applies the weighted average method to convert ~T into a crisp number (Lai and Hwang

1992a, Wang and Liang 2005). If the a-cut level (minimum acceptable possibility level) is

given, the auxiliary crisp equality constraints can be presented as follows.

En # w1T o a þ w2T

m a þ w3T

p a w1; w2; w3 $ 0 ð13Þ

where, w1 þ w2 þ w3 ¼ 1, w1, w2 and w3 represent the weights of the most optimistic,

most possible and most pessimistic values of the imprecise completion time, respectively.

Detailed investigation of the effects of various weighting methods should be based on a

DM’s experience and knowledge. This work applies the concept of the most likely values

proposed by the approach of Lai and Hwang (1992a), assuming w2 ¼ 4/6 and

w1 ¼ w3 ¼ 1/6. The reason is that the most likely value generally is the most important

ones and thus should be assigned greater weights. However, Toa and T p a which provided the

boundary solutions of the imprecise available resource is respectively too optimistic and

pessimistic, and thus should be assigned smaller weights. Changes to the weights of the

three critical points of the triangular possibility distribution influence solutions.

Furthermore, to solve Equation (8) with imprecise technological coefficient and

available resource, the presented approach converted these imprecise inequality

constraints into a crisp one using the fuzzy ranking concept (Tanaka et al. 1984,

Ramik and Rimanek 1985, Lai and Hwang 1992a). Accordingly, the auxiliary inequality

constraints in Equation (8) can be presented as follows.

X i

X j

CDij þ X i

X j

kmij;aYij þ CI þ m m a ðEn 2 ToÞ

� � # Bma ð14Þ

X i

X j

CDij þ X i

X j

koij;aYij þ CI þ m o aðEn 2 ToÞ

� � # Boa ð15Þ

X i

X j

CDij þ X i

X j

k p ij;aYij þ CI þ m

p aðEn 2 ToÞ

� � # Bpa ð16Þ

Figure 2. The strategy to minimise the imprecise objective function.

T.-F. Liang318

3.3 Solving the auxiliary MOLP problem

The auxiliary MOLP problem developed above can be converted into an equivalent

ordinary LP problem using Zimmermann’s (1978) linear membership function to

represent the imprecise goals of the DM, together with the minimum operator of the fuzzy

decision-making of Bellman and Zadeh (1970) to aggregate all fuzzy sets, and can be

solved efficiently using the standard simplex method. First, the positive ideal solutions

(PIS) and negative ideal solutions (NIS) of the three objective functions of the auxiliary

MOLP problem and the fuzzy objective function (2) can be specified as follows,

respectively.

zPIS11 ¼ Min z m 1 ; z

NIS 11 ¼ Max z

m 1 ð17aÞ

zPIS12 ¼ Max z m 1 2 z

o 1

� � ; zNIS12 ¼ Min z

m 1 2 z

o 1

� � ð17bÞ

zPIS13 ¼ Min z p 1 2 z

m 1

� � ; zNIS13 ¼ Max z

p 1 2 z

m 1

� � ð17cÞ

zPIS2 ¼ Min z2; z NIS 2 ¼ Max z2 ð18Þ

Furthermore, the corresponding linear membership functions of the fuzzy objective

functions of the auxiliary MOLP problem are defined by

f 11ðz11Þ ¼

1 if z11 , z PIS 11

zNIS 11

2z11

zNIS 11

2zPIS 11

if zPIS11 # z11 # z NIS 11

0 if z11 . z NIS 11

8 >>>< >>>:

ð19Þ

f 12ðz12Þ ¼

1 if z12 . z PIS 12

z122z NIS 12

zPIS 12

2zNIS 12

if zNIS12 # z12 # z PIS 12

0 if z12 , z NIS 12

8 >>>< >>>:

ð20Þ

The linear membership functions f13(z13) and f2(z2) is similar to f11(z11). Finally, using

the minimum operator of the fuzzy decision-making of Bellman and Zadeh (1970) to

aggregate all fuzzy sets, the complete equivalent ordinary LP model for solving the PM

decision problems can be formulated as follows.

Max L

s:t: L # f 1gðz1gÞ g ¼ 1; 2; 3

L # f 2ðz2Þ

Equations (3)–(6), (13)–(16)

tij; Yij; Ei; Ej $ 0 ;i; ;j;

International Journal of General Systems 319

where the auxiliary variable L represents the overall degree of DM satisfaction with

determined goal values. The concepts of fuzzy sets and fuzzy decision-making of Bellman

and Zadeh (1970) are presented in the Appendix.

To summarise, the solution procedure of the proposed PLP approach for solving the

PM decision problems is as follows.

Step 1. Formulate the original multi-objective PLP model for the PM decision problems

according to Equations (1)–(9).

Step 2. Model the imprecise coefficients and right-hand sides using the triangular

possibility distributions.

Step 3. Develop the three new crisp objective functions of the auxiliary MOLP problem

for the imprecise goal using Equations (10)–(12).

Step 4. Given the a-cut level, then convert the imprecise constraints into crisp ones

using the weighted average method and/or the fuzzy ranking concept.

Step 5. Specify the linear membership functions for the three new objective functions,

and then convert the auxiliary MOLP problem into an equivalent LP model using the

minimum operator to aggregate fuzzy sets.

Step 6. Solve the ordinary LP model to delivery a set of compromise solutions. If the

DM is dissatisfied with the initial solutions, the model must be modified until a set of

preferred satisfactory solutions is obtained.

4. Model implementation

4.1 Case description

Daya Technology Corporation was used as a case study demonstrating the practicality of

the proposed methodology (Wang and Liang 2004a). Daya is the leading producer of

precision machinery and transmission components in Taiwan. The products of Daya are

primarily distributed throughout Asia, North America and Europe and recently have been

in high demand. The PM decision examined here, involves expanding a metal finishing

plant owned by Daya. Currently, the deterministic CPM approach used by Daya suffers

from the limitation owing to the fact that a DM does not have sufficient information related

to the model inputs and related parameters. Alternatively, the proposed possibilistic

programming approach introduced by Daya can effectively handle vagueness and

imprecision in the statement of the objectives and related parameters by using simplified

triangular distributions to model imprecise data. It is critical that the satisfying objective

values should often be imprecise as the cost coefficients and parameters are imprecise and

such imprecision always exists in real-world PM decision problems. The case study

focuses on developing a possibilistic programming approach to solve the PM problem in

an uncertain environment. Incremental crashing costs for all activities, variable indirect

cost per unit time and budget are imprecise and have triangular possibility distributions

over the planning horizon. The PM decision of Daya aims to simultaneously minimise

total project costs and completion time in terms of direct costs, indirect costs, activity

duration, and budget constraints. Table 1 lists the basic data of the real industrial case.

Other relevant data are as follows: fixed indirect costs $12,000, saved daily variable

indirect costs ($144, $150, $154), total budget ($40,000, $45,000, $51,000), and project

completion time under normal conditions 125 days. The project start time (E1) is set to

zero. The a-cut level for all imprecise numbers is specified as 0.5. The specified project

completion time is set to (116, 119, 122) days based on contractual information, resource

T.-F. Liang320

allocation and economic considerations, and related factors. Figure 3 shows the activity-

on-arrow network. The critical path is 1–5–6–7–9–10–11.

4.2 Solution procedure for the Daya case

The solution procedure using the proposed PLP approach for the Daya case is described as

follows. First, formulate the original multi-objective PLP model for the PM decision

problem according to Equations (1)–(9). Second, develop the three new objective

functions of the auxiliary MOLP problem for the imprecise objective function (1) using

Equations (10)–(12). Third, formulate the auxiliary crisp constraints using Equations

(13)–(16) at a ¼ 0.5. Additionally, specify the PIS and NIS of the imprecise/fuzzy

objective functions in the auxiliary MOLP problem with Equations (17a)–(18). The results

are

zPIS11 ; z NIS 11

� � ¼ ð$30; 000; $100; 000Þ;

zPIS12 ; z NIS 12

� � ¼ ð$200; $30Þ;

zPIS13 ; z NIS 13

� � ¼ ð0; $200Þ;

and

zPIS2 ; z NIS 2

� � ¼ ð$100; $500Þ;

respectively. The corresponding linear membership functions of the three new objective

functions can be defined according to Equations (19) and (20). Consequently, the

equivalent ordinary LP model for solving the PM decision problem for the Daya case can

be formulated using the minimum operator to aggregate fuzzy sets.

LINGO computer software is used to run this ordinary LP model. The initial solutions

are ~z1 ¼ ð$35; 859:94; $36; 017:58; $36; 067:42Þ; z2 ¼ 116 days, and overall degree of DM satisfaction with determined goal values is 0.7508. Furthermore, the DM may attempt to

modify the results by adjusting and related parameters to obtain a satisfactory solution.

Consequently, the improved solutions are ~z1 ¼ ð$35; 759:44; $35; 935:16; $36; 029:57Þ; z2 ¼ 111.83 days, and overall degree of DM satisfaction is up to 0.8817. Table 2 lists

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

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

1–2 14 10 1000 1600 (132, 150, 164) 1–5 18 15 4000 4540 (164, 180, 198) 2–3 19 19 1200 1200 – 2–4 15 13 200 440 (102, 120, 128) 4–7 8 8 600 600 – 4–10 19 16 2100 2490 (112, 130, 140) 5–6 22 20 4000 4600 (280, 300, 324) 5–8 24 24 1200 1200 – 6–7 27 24 5000 5450 (136, 150, 166) 7–9 20 16 2000 2200 (34, 50, 58) 8–9 22 18 1400 1900 (111, 125, 139) 9–10 18 15 700 1150 (120, 150, 160) 10–11 20 18 1000 1200 (80, 100, 108)

International Journal of General Systems 321

initial and improved PM plans for the Daya case with the proposed PLP approach based on

current information. Figure 4 shows the change in triangular possibility distributions of

total project costs (z1) for the Daya case.

Furthermore, sensitivity analysis results for varying project duration indicate that

minimising completion time conflicts with minimising total project costs. From Table 3, as

project duration increases, total costs increase significantly because the indirect and

penalty costs increase with project duration. Thus, if a project DM faces costly indirect and

penalties for completing a project late, using additional resources to reduce project

duration is likely worthwhile. In particular, chosen PIS and NIS of fuzzy objective

functions and weights in inequality constraints affect decision results.

5. Computational analysis

Several significant management implications regarding the practical application of the

proposed approach are as follows. First, the proposed PLP approach yields an efficient

solution. The proposed approach is based on Zimmermann’s fuzzy programming method,

which assumes that the minimum operator is the proper representation of the human DM

who aggregates fuzzy sets using logical ‘and’ operations. It follows that maximisation of

two or more membership functions is best accomplished by maximising the minimum

Figure 3. The project network of the Daya case.

Figure 4. The triangular distribution of the total project costs.

T.-F. Liang322

membership values. Zimmermann (1976, 1978) explained why the ‘maximising solution’

is always an efficient solution for the minimum operator. Table 4 compares the results

using the ordinary single-goal LP model with the proposed PLP approach. This work

assumed that the DM specified the most likely value of the possibility distribution of each

imprecise data as the precise numbers. From Table 4, applying LP-1 to minimise the total

costs (z1), the optimal value and total completion time were $35,900 and 113 days,

respectively. Applying LP-2 to minimise the completion time (z2), the optimal value and

total costs were 108 days and $36,290, respectively. Alternatively, using the multiple

fuzzy goals programming method developed by Wang and Liang (2004a) with linear

membership function to simultaneously minimise total project costs and completion time

obtains z1 ¼ $37,030, z2 ¼ 108 days, and the overall degree of DM satisfaction is 0.9200.

These figures indicate that the results obtained using the proposed PLP method are a set of

efficient solutions, compared to the solutions obtained by the ordinary single-goal LP and

Wang and Liang (2004a).

Second, project managers generally face a planning problem with multiple imprecise

goals, when making a PM decision. The comparison as shown in Table 4 reveals that the

interaction of trade-offs and conflicts exists among dependent objective functions. Hence,

a project DM may be able to shorten project completion time, realising savings on indirect

costs, by increasing direct expenses to accelerate the project. If a project DM faces costly

Table 2. PLP solutions for the Daya case.

Initial solutions Improved solutions

Yij (days) Y12 ¼ 0, Y15 ¼ 0, Y23 ¼ 0, Y24 ¼ 0.98, Y34 ¼ 0, Y47 ¼ 0, Y410 ¼ 0, Y56 ¼ 0, Y58 ¼ 0, Y67 ¼ 0, Y79 ¼ 0, Y89 ¼ 0, Y910 ¼ 3, Y1011 ¼ 2.

Y12 ¼ 0, Y15 ¼ 1.17, Y23 ¼ 0, Y24 ¼ 0, Y34 ¼ 0, Y47 ¼ 0, Y410 ¼ 0, Y56 ¼ 0, Y58 ¼ 0, Y67 ¼ 3, Y79 ¼ 4, Y89 ¼ 0, Y910 ¼ 3, Y1011 ¼ 2.

tij (days) t12 ¼ 14, t15 ¼ 18, t23 ¼ 19, t24 ¼ 14.02 t34 ¼ 0, t47 ¼ 8, t410 ¼ 19, t56 ¼ 22, t58 ¼ 24, t67 ¼ 27, t79 ¼ 16, t89 ¼ 22, t910 ¼ 15, t1011 ¼ 18.

t12 ¼ 14, t15 ¼ 16.83, t23 ¼ 19, t24 ¼ 15 t34 ¼ 0, t47 ¼ 8, t410 ¼ 19, t56 ¼ 22, t58 ¼ 24, t67 ¼ 24, t79 ¼ 16, t89 ¼ 22, t910 ¼ 15, t1011 ¼ 18.

Ei (days) E1 ¼ 0, E2 ¼ 14, E3 ¼ 33, E4 ¼ 33 E5 ¼ 18, E6 ¼ 40, E7 ¼ 67, E8 ¼ 42, E9 ¼ 83, E10 ¼ 98, E11 ¼ 116.

E1 ¼ 0, E2 ¼ 14, E3 ¼ 33, E4 ¼ 33, E5 ¼ 16.83, E6 ¼ 38.83, E7 ¼ 62.83, E8 ¼ 40.83, E9 ¼ 78.83, E10 ¼ 93.83, E11 ¼ 111.83.

Objective values

L ¼ 0.7508, z11 ¼ $36,017.58, z12 ¼ $157.46, z13 ¼ $49.84, ~z1 ¼ ($35,859.94, $36,018.58, $36,067.42)*, z2 ¼ 116.00 days.

L ¼ 0.8817, z11 ¼ $35,935.16, z12 ¼ $175.72, z13 ¼ $94.41, ~z1 ¼ ($35,759.44, $35,935.16, $36,029.57)*, z2 ¼ 111.83 days.

Note: ~z1 ¼ ðz11 2 z12; z11; z11 þ z13Þ

Table 3. Results of sensitivity analysis for varying the project duration.

Item Run 1 Run 2 Run 3 Run 4 Run 5

En (days)

107 113 119 125 131

L 0.8700 0.8100 0.7500 0.6900 ~z1($) Infeasible (36,019.40,

36,183.40, 36,283.20)

(36,078.40, 36,208.00, 36,252.53)

(36,580.00, 37,000.00, 37,042.67)

(37,561.60, 37,672.00, 37,723.47)

z2 (days)

113 119 125 131

International Journal of General Systems 323

penalties for failing to complete a project on time, then using extra resources to complete

the project may be economical. Currently, the deterministic CPM method used by Daya

suffers from the limitation owing to the fact that a project manager does not have sufficient

information related to the model inputs and related parameters. In real-world PM

problems, these data are often imprecise/fuzzy in nature. Moreover, due to conflicting

nature of the multiple goals and vagueness in the information relating to the cost

coefficients over the planning horizon, the deterministic CPM method is unsuitable to

obtain an effective solution. The results obtained from the CPM may not comply with the

actual aims of modelling PM problems. Alternatively, the proposed PLP approach

introduced by Daya can effectively handle vagueness and imprecision in the statement of

the goals and related parameters by using simplified triangular distributions to model

imprecise data. Analytical results obtained by implementing indicate that the proposed

approach satisfies the requirement for the practical application since it simultaneously

minimises total project costs and completion time in uncertain environments.

Third, the proposed PLP approach determines the overall degree of DM satisfaction

under the proposed strategy of minimising the most possible values and the risk of

obtaining higher values, and maximising the possibility of obtaining lower values for all

imprecise objective functions. If the solution is L ¼ 1, then each goal is fully satisfied; if

0 , L , 1, then all of the goals are satisfied at the level of L, and if L ¼ 0, then none of

the goals are satisfied. Moreover, the proposed possibilistic programming approach

comprises a rational fuzzy decision-making process for solving PM problems with

multiple goals. For instance, the overall degree of DM satisfaction with determined goal

values for the Daya case, ~z1 ¼ ð$35; 859:94; $36; 017:58; $36; 067:42Þ; z2 ¼ 116 days, and overall degree of DM satisfaction with determined goal values is 0.7508. Furthermore, the

L value was adjusted to seek a set of better compromise solutions as the DM was not

satisfied with this value. Consequently, the improved results are ~z1 ¼

ð$35; 759:44; $35; 935:16; $36; 029:57Þ and z2 ¼ 111.83 days, with an overall degree of DM satisfaction of 0.8817. The main advantage of the proposed approach is that the DM

may adjust the search direction during the solution procedure, until the efficient solution

satisfies the DM’s preferences and is considered to be the preferred solution.

Fourth, comparisons of initial and improved solutions reveal that the changes in the

PIS and NIS of the fuzzy objective functions of the auxiliary MOLP problem influence

both objective and L values. The L value rapidly increased from 0.7508 to 0.8817 when the

PIS and NIS of the four fuzzy objective functions changed from ($30,000, $100,000),

($200, $30), (0, $200) and ($100, $500) to ($33,000, $100,000), ($160, $30), ($40, $200)

and ($100, $200), respectively (Tables 4 and 5). Conversely, total project costs and

completion time reduce from ($35,859.94, $36,017.58, $36,067.42) and 116 days to

($35,759.44, $35,935.16, $36,029.57) and 111.83 days, respectively. These analytical

Table 4. Comparison of solutions.

Item LP-1 LP-2 Wang and Liang

(2004a) The proposed PLP

approach

Objective function

Min z1 Min z2 Max L Max L

L 100% 100% 92% 88.17% ~z1($) 35,900.00

* 36,290.00 37,030.00 (35,759.44, 35,935.16, 36,029.57) z2 (days) 116.00 108.00

* 108.00 111.83

Note: *denotes optimal value by the ordinary single-goal LP model.

T.-F. Liang324

findings demonstrate that the DM must specify an appropriate set of PIS and NIS values of

the new objective functions to generate PM decisions in effectively seeking the

corresponding linear membership function for each fuzzy objective function. In practice,

single-goal LP solutions were often used as a starting point for both the PIS and NIS and,

furthermore, both intervals must cover the LP solutions. Table 5 presents the

corresponding PIS and NIS values of the initial and improved solutions.

Additionally, the proposed PLP approach uses the simplified pattern of triangular

possibility distribution for representing all imprecise numbers. MacCrimmon and Ryavec

(1964) proposed the triangular distribution which has the desirable ability to compute

exactly the mean and the variance while generating findings comparable to those obtained

by a Beta model. Generally, the possibility distribution provides an effective method for

dealing with ambiguities in determining environmental coefficients and parameters

(Zadeh 1978, Buckley 1988, Lai and Hwang 1992b). To summarise, differently shaped

fuzzy numbers fields can be divided into several patterns, for example triangular,

trapezoid, bell-shaped, exponential, hyperbolic and so on. Among the various types of

possibility distributions, the triangular distribution is used most often for representing

imprecise data for solving possibilistic mathematical problems, through other patterns

may be preferable in some applications. The main advantages of the triangular distribution

are the simplicity and flexibility of the fuzzy arithmetic operations (MacCrimmon and

Ryavec 1964, Hershauer and Nabielsky 1972, Kotiah and Wallace 1973, Chanas and

Kamburowsi 1981, Inuiguchi and Sakawa 1996, Zimmermann 1997).

The minimum operator used in this work is preferable when the DM wishes to make

the optimal membership function values approximately equal or when the DM feels that

the minimum operator is an approximate representation. However, for some practical

situations, the application of the aggregation operator to draws maps above the maximum

operator and below the minimum operator is important. Alternatively, as shown in Table 6,

averaging operators consider the relative importance of fuzzy sets and have the

compensative property so that the result of combination will be medium (Klir and Yuan

1995, Zimmermann 1996, 1997, Wang and Liang 2004b). The primary drawback of the

minimum operator is its lack of discriminatory power between solutions that strongly

differ with respect to the fulfillment of membership to the various constraints (Werner

1987, Dubois et al. 1996). Dubois et al. (1995, 1996) noted that the maximin method for

fuzzy optimisation, which incorporates the fuzzy decision-making concept of Bellman and

Zadeh (1970) in multiple criteria decision-making in fact models flexible constraints

rather than objective functions. Additionally, two refinements of the ordering of solutions

– discrimin partial ordering and the leximin complete preordering – were developed to

compute improved optimal solutions obtained by the minimum operator for maximin

flexible constraint satisfaction problems (Dubois et al. 1996, Dubois and Fortemps 1999).

To summarise, various features distinguish the proposed PLP approach from other PM

models. First, the proposed approach outputs more diverse PM decision information than

other decision methods. It provides more information on alternative crashing strategies

with reference to direct costs, indirect and contractual penalty costs and budget

constraints. Moreover, the proposed approach exhibits greater computational efficiency

and flexibility of the fuzzy arithmetic operations by employing the linear membership

functions to represent fuzzy goals, and then the original fuzzy multi-objective PM problem

formulated here can be converted into an equivalent ordinary LP form by the minimum

operator to aggregate fuzzy sets, and is easily solved by the simplex method. In particular,

computational time using LINGO to deliver the optimal solution in the Daya case is very

shortly. The proposed model has the advantage that commercially available software, such

International Journal of General Systems 325

as LINGO and related mathematical programming packages, can be easily used to solve it.

Additionally, this work does not restrict the goal values and decision variables to be an

integer because the lack of such a restriction avoids the need to use an inefficient integer

programming method. Since the solution to LP is only the basis for the next planning

horizon, the non-integer value of the total completion time can be rounded to the next

integer. Computational methodology developed here can easily be extended to any other

situations and can handle the realistic PM decision problems. Although, it only involves

about 250 decision variables and parameters, the industrial case illustrated here lays a

strong foundation upon which the DM can formulate additional applications for the

proposed approach in solving large-scale PM problems.

6. Conclusions

In real-world PM decision problems, input data or related parameters are frequently

imprecise/fuzzy owing to incomplete and/or unavailable information over the planning

horizon. This work presents a PLP approach for solving PM problems with multiple

imprecise goals having triangular possibility distribution. The proposed approach attempts

to simultaneously minimise total project costs and completion time with reference to direct

costs, indirect costs, relevant activities times and costs, and budget constraints.

An industrial case demonstrates the feasibility of applying the proposed approach to real

PM decisions. Consequently, the proposed PLP approach yields a set of efficient

compromise solutions and the overall degree of DM satisfaction with determined goal

values. The proposed PLP approach provides a systematic framework that facilitates

decision- making, enabling a DM to interactively modify the imprecise data and

parameters until a set of satisfactory compromise solution is obtained.

The main contribution of this work lies in presenting a possibilistic programming

methodology for fuzzy multi-objective PM decisions. It is critical that the satisfying

objective values should often be imprecise as the cost coefficients and parameters are

imprecise and such imprecision always exists in real-world PM decisions. Computational

methodology developed here can easily be extended to any other situations and can handle

the realistic PM decisions. Additionally, the proposed approach is based on the fuzzy

programming of Zimmermann, which implicitly assumes the minimum operator to proper

represent the human DM that aggregates fuzzy sets by ‘and’ (intersection). Future

investigations may apply the discrimin partial ordering and leximin complete preordering

methods to the refinement of improved optimal solutions determined by the minimum

operator for maximin flexible constraint satisfaction problems, and may also adopt union,

averaging and other compensative operators to solve fuzzy multi-objective PM problems.

Finally, the project indirect costs are practically charged in terms of percentage of direct

costs, and contractor bonus and penalties must also be considered.

Table 5. The PIS and NIS for the fuzzy objective functions.

LP-11 LP-12 LP-13 LP-2 (PIS, NIS)

(Initial solution) (PIS, NIS)

(Improved solution)

Objective function

Min z11 Max z12 Min z13 Min z2 – –

z11 ($) 35,900.00 – – – (30,000, 100,000) (33,000, 100,000) z12 ($) – 506.00 – – (200, 30) (160, 0) z13 ($) – – 24.00 – (0, 200) (40, 500) z2 (days) – – – 108.00 (100, 500) (100, 200)

T.-F. Liang326

Notes on contributor

Tien-Fu Liang is currently an associate professor in the Department of

Industrial Engineering and Management, Hsiuping Institute of

Technology, Taiwan. He received his MS and PhD degree in Department

of Industrial Management from National Taiwan Universisty of Science

and Technology. His research interests include project management,

logistics and supply chain management, aggregate production planning,

and fuzzy optimisation. He has published in the Asia-Pacific Journal of

Operational Research, Computers and Industrial Engineering, Construction

Management and Economics, Fuzzy sets and Systems, International

Journal of Production Economics, International Journal of Production

Research, International Journal of Systems Science, Journal of the Chinese Institute and Industrial

Engineers, Production Planning and Control, and International Journal of General Systems.

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Table 6. Comparisons of common aggregation operators.

Operator Example Brief description

Intersection (t-norms)

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Algebraic product The result of combination is high if and only if all values are high

Bounded sum The minimum operator is a greatest t-norm Drastic intersection

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Bounded difference The minimum operator is a smallest t-conorm Drastic union

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T.-F. Liang328

Appendix

A.1 A brief introduction to fuzzy sets (Dubois and Prade 1980, Sakawa 1988, Kaufmann and Gupta 1991, Klir and Yuan 1995, Zimmermann 1996)

. Fuzzy sets. Let X denote a universal set, then a fuzzy subset ~A in X is defined by its membership function

m~A : X ! ½0; 1� ðA1Þ

which assigns to each element x [ X a real number m~AðxÞ in the interval [0, 1], where the value of m~AðxÞ at x represents the grade of membership of x in A. A fuzzy subset

~A can be characterised as a set of ordered pairs of element x and grade m~AðxÞ and is normally written

~A ¼ {ðx; m~AðxÞÞjx [ X} ðA2Þ

. Intersection. The membership function of the intersection of two fuzzy sets ~A and ~B in X is defined by

~A > ~B , m~A>~BðxÞ ¼ Min{m~AðxÞ; m~BðxÞ} ¼ m~AðxÞ ^ m~BðxÞ ðA3Þ

. Union. The membership function of the union of two fuzzy sets ~A and ~B in X is defined by

~A < ~B , m~A<~BðxÞ ¼ Max{m~AðxÞ; m~BðxÞ} ¼ m~AðxÞ _ m~BðxÞ ðA4Þ

. a-cuts (a-level set). The a-cuts of a fuzzy set ~A is defined by

Aa ¼ {x [ Xjm~A $ a}; a [ ½0; 1� ðA5Þ

a-cuts Aa is an ordinary (crisp) set for which the degree of its membership function exceeds the level a.

. Convex fuzzy set. A fuzzy set ~A is convex if and only if

m~A{lx1 þ ð1 2 lÞx2} $ Min{m~Aðx1Þ; m~Aðx2Þ}; x1; x2 [ U; l [ ½0; 1� ðA6Þ

. Fuzzy numbers. A fuzzy number ~N is a convex normalised fuzzy set of the real line R such that it exists exactly ones x0 [ R with m ~Mðx0Þ ¼ 1 and m ~MðxÞ is piecewise continuous.

. Triangular fuzzy numbers. A fuzzy number ~N may by characterised by triangular distribution function parameterised by a triplet (a, b, c), where the base is on the interval [a, c] and vertex at x ¼ b. The membership function of the triangular fuzzy number ~N is defined by

m ~NðxÞ ¼

0 if x , a

x2a b2a

if a # x # b

c2x c2b

if b # x # c

0 if x . c

8 >>>>>< >>>>>:

ðA7Þ

. Possibility distribution. Let ~A is a fuzzy set that acts as a fuzzy restriction on the possible value of v. Then ~A induces a possibility distribution m~A that is equal to on the value of v and is defined by

Y ðv ¼ xÞ ¼ pðxÞ ¼ m~AðxÞ ðA8Þ

Since the expression of a possibility distribution can be viewed as a fuzzy set, possibility distributions may be manipulated by the combination rules of fuzzy sets, and more particular of fuzzy restrictions.

. Possibilistic programming versus stochastic programming. Possibilistic and stochastic programming are both suitable techniques for an overall analysis of the effects of imprecision

International Journal of General Systems 329

in decision parameters. In stochastic programming problems, the parameters can be random variables, but in possibilistic programming problems, they are fuzzy variables defined by their possibility distribution. The stochastic programming approach handles situations where related parameters are imprecise and described by random variables that are normally defined by non-linear probability distribution functions. Alternatively, the possibilistic programming provides a more efficient technique, and also preserves the original linear model for all of the imprecise goals and constraints.

A.2 Fuzzy decision-making of Bellman and Zadeh (1970)

Let X be a given set of all possible solutions to a decision problem. A fuzzy goal G is a fuzzy set on X characterised by its membership function

mG : X ! ½0; 1� ðA9Þ

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

mC : X ! ½0; 1� ðA10Þ

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 characterised by its membership function

L ¼ mDðxÞ ¼ mGðxÞ ^ mCðxÞ ¼ MinðmGðxÞ; mCðxÞÞ ðA11Þ

and the corresponding maximising decision is defined by

Max L ¼ Max mDðxÞ ¼ Max MinðmGðxÞ; mCðxÞÞ ðA12Þ

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

L ¼ mDðxÞ ¼ mG1ðxÞ ^ mG2ðxÞ ^ · · · ^ mGk ^ mC1 ^ mC2 ^ · · · ^ mCm

¼ MinðmG1ðxÞ; mG2ðxÞ; · · ·; mGk ðxÞ; mC1ðxÞ; mC2ðxÞ; · · ·; mCmðxÞÞ ðA13Þ

and the corresponding maximising decision is defined by

Max L ¼ Max mDðxÞ ¼ Max Min mG1ðxÞ; mG2 ðxÞ; · · ·; mGk ðxÞ; mC1ðxÞ; · · ·; mCmðxÞ � �

ðA14Þ

T.-F. Liang330