application of fuzzy sets to multi objective project management decisions
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International Journal of General Systems
ISSN: 0308-1079 (Print) 1563-5104 (Online) Journal homepage: http://www.tandfonline.com/loi/ggen20
Application of fuzzy sets to multi-objective projectmanagement decisions
Tien-Fu Liang
To cite this article: Tien-Fu Liang (2009) Application of fuzzy sets to multi-objective project
management decisions, International Journal of General Systems, 38:3, 311-330, DOI:10.1080/03081070701785833
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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 toincomplete or unavailable information, and the decision maker (DM) generally faces afuzzy multi-objective PM decision problem in uncertain environments. This work focuses on the application of fuzzy sets to solve fuzzy multi-objective PM decisionproblems. The proposed possibilistic linear programming (PLP) approach attempts tosimultaneously minimise total project costs and completion time with reference todirect costs, indirect costs, relevant activities times and costs, and budget constraints.An industrial case illustrates the feasibility of applying the proposed PLP approach topractical PM decisions. The main advantage of the proposed approach is that the DMmay adjust the search direction during the solution procedure, until the efficientsolution satisfies the DM’s preferences and is considered to be the preferredsatisfactory solution. In particular, computational methodology developed in this work can easily be extended to any other situations and can handle the realistic PM decisionproblems 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 relevantoperating 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
q 2009 Taylor & Francis
DOI: 10.1080/03081070701785833
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*Email: [email protected]
International Journal of General Systems
Vol. 38, No. 3, April 2009, 311–330
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introduced fuzzy set theory into ordinary single-goal LP problems. That study consideredLP 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 combinationrules 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 andHwang (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 planningproblems 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,
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Buckley 1990). Alternatively, the proposed possibilistic programming provides a moreefficient 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 canbe 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.
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Assumption 1, relates to the fuzziness of the objective functions in real-world PMproblems 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)
d ij ¼ minimum crashed time for activity (i, j; days)
C Dij ¼ normal (direct) cost for activity (i, j; $)
C d ij ¼ minimum crashed (direct) cost for activity (i, j; $)~k ij ¼ incremental crashing costs for activity (i, j; $/day)
t ij ¼ crashed duration time for activity (i, j; days)
Y ij ¼ crash time for activity (i, j; days)
E i ¼ earliest time for event i (days)
E 1 ¼ project start time (days)
E n ¼ project completion time (days)T o ¼ project completion time under normal conditions (days)~T ¼ specified project completion time (days)C I ¼ fixed indirect costs under normal conditions ($)
~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,
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by increasing direct expenses to accelerate project progress. Generally, most practicaldecisions 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 ¼X
i
X j
C Dij þX
i
X j
~k ijY ij þ ½C I þ ~mðE n 2 T oÞ ð1Þ
where ~k ij 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,
Xi
X j
C Dij þX
i
X j
~k ijY ij;
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 ½C I þ ~mðE n 2 T oÞ 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 thevariablecostsper unit time are thesame regardless of project completion time.
. Minimise total completion time
Min z2 ø E n 2 E 1 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.
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2.2.2 Constraints
. Constraints on the time between events i and j
E i þ t ij 2 E j # 0 ;i;; j ð3Þ
t ij ¼ Dij 2 Y ij ;i;; j ð4Þ
. Constraints on the crash time for activity (i, j)
Y ij # Dij 2 d ij ;i;; j ð5Þ
. Constraints on project start time and total completion time
E 1 ¼ 0 ð6Þ
E n # ~T ð7Þ
. Constraint on the total budget
Xi
X j
C Dij þX
i
X j
~k ijY ij þ ½C I þ ~mðE n 2 T oÞ # ~ B ð8Þ
. Non-negativity constraints on decision variables
t ij; Y ij; E i; E j $ 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 thecompletion 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 withimprecise 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), ~k ij, based on the three prominent data: (1) themost optimistic value (k oij) 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 (k mij ) that
definitely belongs to the set of available values (possibility degree ¼ 1 if normalised) and
(3) the most pessimistic value (k pij ) that has a very low likelihood of belonging to the set
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of available values (possibility degree ¼ 0 if normalised), where the base is on theinterval ½k oij ; k
pij and vertex at x ¼ k
mij . Figure 1 presents the triangular possibility
distribution of ~k ij ¼ ðk oij ; k
mij ; k
pij Þ. 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 ð zo1 ; 0Þ, ð zm1 ; 1Þ and ð z
p1 ; 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 z m
1
, maximising
ð z m1 2 zo1 Þ and minimising ð z
p1 2 z
m1 Þ for minimising z
m1 ; z
o1 and z
p1 . 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 ð z m1 Þ,
maximising the possibility of obtaining lower total costs (region I of the possibility
distribution in Figure 2) ð z m1 2 zo1 Þ, and minimising the risk of obtaining higher total costs
(region II of the possibility distribution in Figure 2) ð z p1 2 z
m1 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 zm
1 ¼X
i
X j
C Dij þX
i
X j
k m
ij Y ij þ ½C I þ mm
ðE n 2 T oÞ ð10Þ
Max z12 ¼ zm1 2 z
o1
¼
Xi
X j
k mij 2 k oij
Y ij þ ½ðm
m2m oÞðE n 2 T oÞ ð11Þ
Min z13 ¼ z p2 z mð Þ ¼
Xi
X j
k pij 2 k
mij
Y ij þ ½ðm
p2 m mÞðE n 2 T oÞ ð12Þ
Figure 1. The triangular possibility distribution of ~k ij.
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3.2.2 Strategy for solving the imprecise constraintsRecalling 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.
E n # w1T oa þ w2T
ma þ w3T
pa 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 valuesproposed 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, T oa and T pa 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.
Xi
X j
C Dij þX
i
X j
k mij;aY ij þ C I þ mma ðE n 2 T oÞ
# Bma ð14Þ
Xi
X j
C Dij þX
i
X j
k oij;aY ij þ C I þ moaðE n 2 T oÞ
# Boa ð15Þ
Xi
X j
C Dij þX
i
X j
k pij;aY ij þ C I þ m
paðE n 2 T oÞ
# B pa ð16Þ
Figure 2. The strategy to minimise the imprecise objective function.
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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 zm1 ; z
NIS11 ¼ Max z
m1 ð17aÞ
zPIS12 ¼ Max zm1 2 z
o1
; zNIS12 ¼ Min z
m1 2 z
o1
ð17bÞ
zPIS13 ¼ Min z p1 2 z
m1
; zNIS13 ¼ Max z
p1 2 z
m1
ð17cÞ
zPIS2 ¼ Min z2; zNIS2 ¼ Max z2 ð18Þ
Furthermore, the corresponding linear membership functions of the fuzzy objective
functions of the auxiliary MOLP problem are defined by
f 11ð z11Þ ¼
1 if z11 , zPIS11
zNIS11 2 z11
zNIS11 2 zPIS11 if zPIS11 # z11 # z
NIS11
0 if z11 . zNIS11
8
>>>>>:ð19Þ
f 12ð z12Þ ¼
1 if z12 . zPIS12
z122 zNIS12
zPIS12 2 zNIS
12
if zNIS12 # z12 # zPIS12
0 if z12 , zNIS12
8>>><>>>:
ð20Þ
The linear membership functions f 13( z13) and f 2( z2) is similar to f 11( 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 1gð z1gÞ g ¼ 1; 2; 3
L # f 2ð z2Þ
Equations (3)–(6), (13)–(16)
t ij; Y ij; E i; E j $ 0 ;i;; j;
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where the auxiliary variable L represents the overall degree of DM satisfaction withdetermined 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 inan 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 projectcompletion time under normal conditions 125 days. The project start time (E 1) 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
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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 ; zNIS11
¼ ð$30; 000; $100; 000Þ;
zPIS12 ; zNIS12
¼ ð$200; $30Þ;
zPIS13 ; zNIS13
¼ ð0; $200Þ;
and
zPIS2 ; zNIS2
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 canbe 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; $36; 029:57Þ;
z2 ¼ 111.83 days, and overall degree of DM satisfaction is up to 0.8817. Table 2 lists
Table 1. Summarised data in the Daya case (in US dollar).
(i, j) Dij (days) d ij (days) C Dij ($) C d ij ($) k ij ($/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)
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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.
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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
Y ij (days) Y 12 ¼ 0, Y 15 ¼ 0, Y 23 ¼ 0, Y 24 ¼ 0.98,Y 34 ¼ 0, Y 47 ¼ 0, Y 410 ¼ 0, Y 56 ¼ 0,Y 58 ¼ 0, Y 67 ¼ 0, Y 79 ¼ 0, Y 89 ¼ 0,Y 910 ¼ 3, Y 1011 ¼ 2.
Y 12 ¼ 0, Y 15 ¼ 1.17, Y 23 ¼ 0, Y 24 ¼ 0,Y 34 ¼ 0, Y 47 ¼ 0, Y 410 ¼ 0, Y 56 ¼ 0,Y 58 ¼ 0, Y 67 ¼ 3, Y 79 ¼ 4, Y 89 ¼ 0,Y 910 ¼ 3, Y 1011 ¼ 2.
t ij (days) t 12 ¼ 14, t 15 ¼ 18, t 23 ¼ 19, t 24 ¼ 14.02t 34 ¼ 0, t 47 ¼ 8, t 410 ¼ 19, t 56 ¼ 22,t 58 ¼ 24, t 67 ¼ 27, t 79 ¼ 16, t 89 ¼ 22,t 910 ¼ 15, t 1011 ¼ 18.
t 12 ¼ 14, t 15 ¼ 16.83, t 23 ¼ 19, t 24 ¼ 15t 34 ¼ 0, t 47 ¼ 8, t 410 ¼ 19, t 56 ¼ 22,t 58 ¼ 24, t 67 ¼ 24, t 79 ¼ 16, t 89 ¼ 22,t 910 ¼ 15, t 1011 ¼ 18.
E i (days) E 1 ¼ 0, E 2 ¼ 14, E 3 ¼ 33, E 4 ¼ 33E 5 ¼ 18, E 6 ¼ 40, E 7 ¼ 67, E 8 ¼ 42,E 9 ¼ 83, E 10 ¼ 98, E 11 ¼ 116.
E 1 ¼ 0, E 2 ¼ 14, E 3 ¼ 33, E 4 ¼ 33,E 5 ¼ 16.83, E 6 ¼ 38.83, E 7 ¼ 62.83,E 8 ¼ 40.83, E 9 ¼ 78.83, E 10 ¼ 93.83,
E 11 ¼ 111.83.Objectivevalues
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,$36,029.57)*, z2 ¼ 111.83 days.
Note: ~ z1 ¼ ð z11 2 z12; z11; z11 þ z13Þ
Table 3. Results of sensitivity analysis for varying the project duration.
Item Run 1 Run 2 Run 3 Run 4 Run 5
E n(days)
107 113 119 125 131
L 0.8700 0.8100 0.7500 0.6900~ z1($) 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
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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 sufficientinformation 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 DMmay 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, $36,029.57) and 111.83 days, respectively. These analytical
Table 4. Comparison of solutions.
Item LP-1 LP-2Wang and Liang
(2004a)The proposed PLP
approach
Objectivefunction
Min z1 Min z2 Max L Max L
L 100% 100% 92% 88.17%~ z1($) 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.
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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 PMmodels. 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
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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 integerprogramming 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 multipleimprecise 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. Computationalmethodology 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)
Objectivefunction
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)
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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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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 itsmembership function
m ~ A : X ! ½0; 1 ðA1Þ
which assigns to each element x [ X a real number m ~ Að xÞ in the interval [0, 1], where thevalue of m ~ Að xÞ at x represents the grade of membership of x in A. A fuzzy subset
~ A can becharacterised as a set of ordered pairs of element x and grade m ~ Að xÞ and is normally written
~ A ¼ {ð x;m ~ Að xÞÞj x [ X } ðA2Þ
. Intersection. The membership function of the intersection of two fuzzy sets ~ A and ~ B in X isdefined by
~ A> ~ B , m ~ A> ~ Bð xÞ ¼ Min{m ~ Að xÞ;m ~ Bð xÞ} ¼ m ~ Að xÞ ^ m ~ Bð xÞ ðA3Þ
. Union. The membership function of the union of two fuzzy sets ~ A and ~ B in X is defined by
~ A< ~ B , m ~ A< ~ Bð xÞ ¼ Max{m ~ Að xÞ;m ~ Bð xÞ} ¼ m ~ Að xÞ _m ~ Bð xÞ ðA4Þ
. a-cuts (a-level set ). The a-cuts of a fuzzy set ~ A is defined by
Aa ¼ { x [ X jm ~ 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{l x1 þ ð12 lÞ x2} $ Min{m ~ Að x1Þ;m ~ Að x2Þ}; 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 thatit exists exactly ones x0[ R with m ~ M ð x0Þ ¼ 1 and m ~ M ð xÞ is piecewise continuous.
. Triangular fuzzy numbers. A fuzzy number ~ N may by characterised by triangular distributionfunction parameterised by a triplet (a, b, c), where the base is on the interval [a, c ] and vertexat x ¼ b. The membership function of the triangular fuzzy number ~ N is defined by
m ~ N ð xÞ ¼
0 if x , a
x2ab2a
if a # x # b
c2 xc2b
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 possiblevalue of v. Then ~ A induces a possibility distribution m ~ A that is equal to on the value of v and isdefined by
Yðv ¼ xÞ ¼ p ð xÞ ¼ m ~ Að xÞ ðA8Þ
Since the expression of a possibility distribution can be viewed as a fuzzy set, possibilitydistributions may be manipulated by the combination rules of fuzzy sets, and more particularof fuzzy restrictions.
. Possibilistic programming versus stochastic programming. Possibilistic and stochasticprogramming are both suitable techniques for an overall analysis of the effects of imprecision
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in decision parameters. In stochastic programming problems, the parameters can be randomvariables, but in possibilistic programming problems, they are fuzzy variables defined bytheir possibility distribution. The stochastic programming approach handles situations whererelated parameters are imprecise and described by random variables that are normally definedby non-linear probability distribution functions. Alternatively, the possibilistic programmingprovides a more efficient technique, and also preserves the original linear model for all of theimprecise 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 fromintersection of G and C , and is characterised by its membership function
L ¼ m Dð xÞ ¼ mGð xÞ ^mC ð xÞ ¼ MinðmGð xÞ;mC ð xÞÞ ðA11Þ
and the corresponding maximising decision is defined by
Max L ¼ Maxm Dð xÞ ¼ MaxMinðmGð xÞ;mC ð xÞÞ ðA12Þ
More generally, suppose the fuzzy decision D results from k fuzzy goals G1, . . . , Gk and m
constraints C 1, . . . , C m. Then the fuzzy decision D is the intersection of G1, . . . , Gk and C 1, . . . , C m,and is characterised by its membership function
L ¼ m Dð xÞ ¼ mG1 ð xÞ ^ mG2 ð xÞ ^ · · · ^ mGk ^mC 1 ^ mC 2 ^ · · · ^ mC m
¼ MinðmG1 ð xÞ;mG2 ð xÞ; · · ·;mGk ð xÞ;mC 1 ð xÞ;mC 2 ð xÞ; · · ·;mC m ð xÞÞ ðA13Þ
and the corresponding maximising decision is defined by
Max L ¼ Maxm Dð xÞ ¼ MaxMin mG1 ð xÞ;mG2 ð xÞ; · · ·;mGk ð xÞ;mC 1 ð xÞ; · · ·;mC m ð xÞ
ðA14Þ
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