- Email: [email protected]
Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method
Computers & Industrial Engineering xxx (2013) xxx–xxx
Contents lists available at ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method Ming-Feng Yang a,⇑, Yi Lin b a b
Department of Transportation Science, National Taiwan Ocean University, 2 Pei-Ning Road, Keelung 20224, Taiwan, ROC College of Management, National Taipei University of Technology, No. 1, Sec. 3, Chung-Shiao E. Rd., Taipei 10643, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 13 March 2013 Received in revised form 20 July 2013 Accepted 14 August 2013 Available online xxxx Keywords: Project management Possibilistic programming approach Two-phase approach Fuzzy multi-objective linear programming
a b s t r a c t The aim of this paper is to develop an interactive two-phase method that can help the Project Manager (PM) with solving the fuzzy multi-objective decision problems. Therefore, in this paper, we first revisit the related papers and focus on how to develop an interactive two-phase method. Next, we establish to consider the imprecise nature of the data by fulfilling the possibilistic programming model, and we also assume that each objective work has a fuzzy goal. Finally, for reaching our objective, the detailed numerical example is presented to illustrate the feasibility of applying the proposed approach to PM decision problems at the end of this paper. Results show that our model can be applied as an effective tool. Furthermore, we believe that this approach can be applied to solve other multi-objective decision making problems. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, the project managers have faced the competitive environment such as the product’s life cycle is becoming short and customers want more-customized services. It means when the project managers face the complicated situations, it is difficult for them to use resources and take decisions in a perfect way. With today’s projects, much of the uncertainty surrounding information management simply can’t be eliminated. In this case, we apply the fuzziness to improve the chances of success in project management. In addition, the degree of fuzziness not only deals with the lack of information but also supports the project managers that can make the wrong decisions in lower possibility. In another word, the experiences of the project managers on the project appropriate application reduces errors due to poor decisions may lead to opportunities for project failure. Recently, both practitioners and academicians have been more interested in considering the issues of the relationship between project management decisions and possible problems. Numerous mathematical programming techniques and heuristics for considering the fuzzy theory have been developed for solving PM
⇑ Corresponding author. Tel.: +886 931179007. E-mail addresses: [email protected] (M.-F. Yang), iwc3706@yahoo. com.tw (Y. Lin).
problems, each with its own advantages and disadvantages. Okuhara, Shibata, and Ishii (2007) utilized the genetic algorithm to the adaptive assignment of worker and workload control in PM decision problems. After that, Lin (2008) utilized statistical confidence-interval estimates and level (1 a) fuzzy numbers to solve project time–cost tradeoff problems. Arikan and Güngör (2001) utilized fuzzy goal programming (FGP) approach to solve PM decision problems with two objectives—minimizing both completion time and crashing cost. After that, Wang and Fu’s work (1998) applied fuzzy mathematical programming to solve PM decision problems. The aim of these models was to minimize complete project cost and whole crashing cost simultaneously. In addition, Wang, Liang, Li and other scholars have developed and researched an interactive multiple fuzzy goal programming (MFGP) model to solve PM decision problems in a fuzzy environment. It aimed to minimize total costs, whole completion time, and complete crashing costs simultaneously (Li, Huang, & Xiao, 2008; Liu, Liang, Yeh, & Chen, 2009; Lv et al., 2010; Suo, Li, & Huang, 2012; Wang & Fu, 1998; Wang & Liang, 2004). According to some related studies, the decision-making process is closer to the possibilistic than probability. Besides, Zadeh (1978) presented the theory of possibility, which was related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction. It acts as an elastic constraint on the values that can be assigned to a variable. Since the expression of a possibility distribution could be viewed as a fuzzy set, possibility distribution might be manipulated by the combination rules of fuzzy sets and
0360-8352/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2013.08.009
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
2
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
more particular about fuzzy restrictions. Buckley (1988) formulated a mathematical programming problem in which all parameters might be fuzzy variables by their possible distribution, and moreover, illustrated this problem using the possibilistic linear programming (PLP) approach. After that, Liang (2009) formulated a possibilistic programming (PLP) model to solve fuzzy multi-objective PM problems with imprecise objectives and constraints. Some related works such as Inuiguchi and Sakawa (1996), Hussein (1998) and Tanaka and Guo (2000) applied possibilistic programming linear method to address the decision-maker problems. In addition, some researchers extended the related research scope such as Kwong, Chen, Chan, and Wong (2008) proposed a hybrid fuzzy least-squares regression (HFLSR) approach to modeling manufacturing processes which features the capability of dealing with the two types of uncertainty and addresses the consideration of replication of responses in experiments. After that, Kwong, Chen, Chan, and Luo (2010) addressed a generalized FLSR approach to modeling relationships in QFD is described that can be used to develop models of the relationships based on fuzzy observations and/or crisp observations. And, Chan, Kwong, and Hu (2012) proposed a new methodology to perform market segmentation based on consumers’ customer requirements with fuzziness consideration. Besides, we found some inadequacies based on above-mentioned literatures: 1. The PLP approach for an optimization problem with fuzzy parameters is possibilistic, which lead to the increase of the number of objectives function and constraints of the model. In addition, the computing efficiency of the solutions obtained by the max– minimum operators has not been considered. Since the results obtained by the max–minimum operator cannot be compensated by other members. As a result, the efficiency of the optimal solution yielded by the minimum-max operators can be improved as it obtains multiple ideal solutions. 2. Different membership functions are formulated from decision-maker preferences and experiences, but the decision-makers have the difficulties in making tradeoffs between the alternatives because of their inexperience and incomplete information. Although several methods have been proposed to treat this problem in above, the two-phase approach has the value described as: if the decision maker is seeking an efficient solution which can ameliorate the max–minimum operators’ solution so that each membership degree should be ameliorated, then the two-phase method automatically obtains this desire if there is room for improvement (Guu & Wu, 1999). Finally, in this paper, we consider the imprecise nature of the input data by implementing the interactive two-phase operators; and we also assume that each objective function has a fuzzy goal. The result can be obtained by determining the suitable membership function and seek an efficient solution.
2. Project management in fuzzy environment The fuzziness comes from incomplete information and uncertainty. For example, to execute a project, the project manager will consider how to lead, how to organize, how the employer, how to plan and control the process of thinking in which ideas will be generated with incomplete and uncertainty. In addition, the higher incomplete and uncertain information is controlled; the least degree of fuzziness of the message becomes. Therefore, if the project
managers have more complete and usage information, they can reduce the degree of fuzziness.
2.1. Notations and assumptions Assume that a project has n interrelated activities that must be executed in a certain order before the entire task can be completed in the fuzzy environment. In general, environmental coefficients and related parameters are incomplete and/or vague about the planning horizon. Therefore, the incremental crashing cost for all activities, variable indirect cost per unit time, specified project completion time, and total budget is imprecise or/ and fuzzy. This problem focuses on the development of the multiple objective possibilistic linear programming (MOPLP) model to the optimum duration of each activity in the project, given a specified project completion time, the crash time tolerance for each activity and allocated total budget, and the optimal solution obtained by two-phase operator approach. Aims of this PM decision are to minimize simultaneously whole project cost, total crashing cost and total completion time. The original multi-objective linear programming (MOLP) model proposed in this work is based on the following assumptions: 1. All the objective functions and constraints are linear. 2. Direct costs increase linearly as the duration of an activity is reduced from its normal value to its crash value. 3. The common time and the shortest possible time for each activity and the cost of completing the activity in the regular time and crash time are certain to the planning horizon. 4. Indirect cost can be divided into two categories, i.e., fixed cost and variable cost, and the variable cost per unit time is the same regardless of project completion time. 5. The decision-makers adopted the pattern of triangular possibility distribution to represent the estimated objectives and related imprecise numbers. 6. Two-phase operator is used to aggregate all fuzzy sets. Assumptions 1, 2 and 3 imply that both the linearity and certainty properties must be technically satisfied in order to represent an optimization problem as a LP problem. For the sake of model facilitation, Assumption 4 represents that the indirect costs can be divided into fixed costs and variable costs. Fixed cost represents the indirect cost under regular condition and remains constant regardless of project duration. Meanwhile, variable cost, which is used to measure savings or increases in variable indirect cost, varies directly with the difference between actual completion and normal duration of the project. Assumption 4 concerns the simplicity and flexibility of the model formulation and the fuzzy arithmetic operations. Assumption 5 addresses the effectiveness of applying triangular possibility distribution to represent imprecise objectives and related imprecise numbers. In general, 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 of its central value. Assumptions 5 and 6 convert the original MOLP problem into an equivalent ordinary single-objective LP form that can be solved efficiently by the standard simplex method.
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
The following notation is used.
The crash time for activity (i, j):
Y ij 6 Dij dij 8i; 8j
ði; jÞ e1 Z
activity between event i and event j total project cost ($)
e2 Z e3 Z
total completion time (days)
Dij dij CDij Cdij ~ k
Ei E1 En To e T
normal time for activity (i, j) (days) minimum crashed time for activity (i, j) normal (direct) cost for activity (i, j) ($) minimum crashed (direct) cost for activity (i, j) ($) incremental crashing costs for activity (i, j)(representing the cost–time slopes) ($/day) duration time for activity (i, j) (days)(difference between normal time and crash time) crash time for activity (i, j) (days)(difference between normal time and duration time) earliest time for event i (days) project start time (days) project completion time (days) project completion time under normal conditions (days) specified project completion time (days)
CI ~ m ~ b
fixed indirect cost under normal conditions ($) variable indirect cost per unit time ($/day) total allocated budget ($)
e
cut level code of defuzzy objective functions by Lai and Hwang’s (1992) fashion decisive factor in weight
ij
tij Yij
k r
total crashing cost ($)
XX XX ~ Y þ C I þ mðE ~ n T o Þ: k C Dij þ ij ij j
i
ð7Þ ð8Þ
The total budget:
~ e 1 6 b: Z
ð9Þ
Non-negativity constraints on decision variables:
tij ; Y ij ; Ei ; Ej P 0 8i; 8j
ð10Þ
In real-world situations, the incremental crashing costs for all activities in Eq. (1), (3) and (9), the specified completion time for the project in Eqs. (2) and (8) are often imprecise because some relevant information, such as the skills of the workers, law and regulations, available resources, and other factors, is incomplete or unavailable.
This work assumes that the DM has already adopted the pattern of triangular possibility distribution to represent the crashing cost, ~ij , the specified project completion time, T e , variable indirect cost k ~ in the original fuzzy ~ and total allocated budget, b, per unit time, m, linear programming problem. The primary advantages of the triangular fuzzy number are the simplicity and flexibility of the fuzzy arithmetic operations. For instance, Fig. 1 shows the distribution ~ij . of the triangular fuzzy number k In practical situations, the DM can construct the triangular disij in objective (1) based on the following three promtribution of k p inent data: 1. the most pessimistic value kij that has a very low likelihood of belonging to the set of available values (possibility de m gree 0 if normalized); 2. the most likely value kij that definitely
1. Minimize total project cost:
i
E0 ¼ 0: Em 6 Te
3. Model development
Three objective functions with minimizing total project cost, total crashing cost, and total completion time are simultaneously considered to development of the proposed multiple objective’s linear programming (MOLP) model as follows:
e1 ¼ Min Z
ð6Þ
The project start time and total completion time is as follows:
2.2. Basic model
2.
3
ð1Þ
j
where the terms PP e PP~ (1) i j C Dij þ i j kij Y ij : total direct cost including to total normal cost and total crashing cost, obtained using additional direct resources such as overtime, personnel and equipment. ~ n T o Þ: indirect cost, including those of administra(2) C I þ mðE tion, depreciation, financial and other variable overhead cost that can be avoided by reducing total project time. ~ij ¼ ð g (3) k C dij C Dij Þ=ðDij dij Þ: the analysis in this problem depends primarily on the cost–time slopes for the various activities. 3. Minimize total completion time:
e 2 ’ En E0 Min Z
belongs to the set of available values (membership degree = 1 if o normalized); and 3. the most optimistic value kij that has a very low likelihood of belonging to the set of available values (membership degree = 0 if normalized). e , in constraints ~ in objective (1), T Similarly, the fuzzy data, m, ~ (8), and b, in objective (9) thus can be modeled using the distribu ; m, tion of the triangular fuzzy number. Hence, the fuzzy data for k ij ~ ~ e b and T can be symbolized as follows.
ð2Þ
4. Minimize total crashing cost:
e3 ¼ Min Z
XX ~ Y k ij ij i
ð3Þ
j
The time between event i and event j:
Ei þ t ij Ej 6 0 8i; 8j
ð4Þ
t ij ¼ Dij Y ij 8i; 8j
ð5Þ
~ij . Fig. 1. Triangular membership function of k
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
4
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
~ij ¼ ko ; km ; kp k ij;e ij;e ij;e p ~ e ¼ moe ; mm m e ; me ~ ¼ bo ; bm ; bp b e e e p T~ ¼ T oe ; T m e ; Te
3.2. Solving the fuzzy constraints e is fuzzy and Recalling Eq. (8) from the original MOLP model; T has triangular distribution. It can be substituted by:
~ Te e : En6
3.1. Solving the fuzzy objective functions The imprecise objective of minimizing the total project cost can be minimized by moving the three prominent points toward the left. Using Lai and Hwang’s (1992) approach, the proposed approach substitutes simultaneously minimizing Z m 1 , maximizing p 0 m o p minimizing Z Z Z for minimizing Zm Zm 1 1 1 1 ; Z 1 and Z 1 . 1 The imprecise objective of minimizing total crashing cost is the same method. The results show there are six new objective functions still guarantee the declaration of moving the triangular distribution of the left. Fig. 2 illustrates the strategy for minimizing the imprecise objective function; that is, the auxiliary MOLP problem generated by this proposed approach comprises simultaneously minimizing the most likely value of imprecise total costs Z m 1 , maximizing the possibility of obtaining lower total costs (region o I of the possibility distribution in Fig. 2) Z m 1 Z 1 , and minimizing the risk of obtaining higher total cost (region II of the possibil ity distribution in Fig. 2) Z p1 Z m 1 . As indicated in Fig. 2, possibility distribution is preferred to possibility distribution. Expressions (11)–(13) list is the results for the three new objective functions of total cost in Eq. (1). Expressions (14)–(16) list is the results for the three new objective functions of total crashing cost in Eq. (3).
XX XX m C Dij þ kij Y ij þ ½C I þ mm ðEn T 0 Þ ð11Þ
Min Z 11 ¼ Z m 1 ¼
i
Max Z 12 ¼
Zm 1
j
Z o1
i
i
Min Z 13
j
þ ½ðmm mo ÞðEn T 0 Þ XX p m ¼ ðZ p Z m Þ ¼ kij kij Y ij i p
Min Z 31
j
XX m o kij kij Y ij ¼
þ ½ðm m ÞðEn T 0 Þ XX m ¼ Zm K ij Y ij 3 ¼ i
Max Z 32 ¼
Min Z 33 ¼
Zm 3
Z p3
j
Z o3
Zm 3
¼
XX m o K ij kij Y ij i
j
i
j
XX p kij K m ¼ ij Y ij
e1 . Fig. 2. The strategy to minimize the imprecise objective of Z
This work applies the signed distance method of Yao and Wu e into a crisp number. If the minimum accept(2000) to convert T able membership degree, e-cut, is given, the auxiliary crisp inequality constraints can be presented as follows:
P ~ T oe þ 2T m En6 e þ T e =4:
i XX Xh o m p C Dij þ kij;e þ 2kij;e þ kij;e =4 Y ij þ C I i
o
i
p þ moe þ 2mm e þ me =4 ðEn T o Þ o m p 6 be þ 2be þ be =4
ð14Þ ð15Þ ð16Þ
ð19Þ
3.3. Two-phase for solving the possibilistic linear programming problem The original MOLP model designed above can be converted into an equivalent ordinary LP form based on the fuzzy linear programming method of Zimmermann (1978), to represent imprecise goals of the decision-maker. The two-phase operator is used to aggregate all fuzzy sets to be solved efficiently (Li & Li, 2006). Based on Bellman and Zadeh’s concept (Bellman & Zadeh, 1970), fuzzy goals (G), fuzzy constraints (C), and fuzzy decisions (D) of the fuzzy decision are defined as follows:
ð20Þ
Next, this problem is characterized by the following membership function:
lD ðxÞ ¼ lG ðxÞ ^ lc ðxÞ ¼ MinðlG ðxÞ; lc ðxÞÞ ð13Þ
ð18Þ
~ m; ~ are fuzzy ~ b Recalling Eq. (9) from the original MOLP model, k; and have triangular distribution. This work applies the signed dis~ m; ~ into crisp number. If the minimum ~ b tance method to convert k; acceptable membership degree, e, is given, the auxiliary crisp inequality constraints can be presented as follows:
D¼G\C ð12Þ
j
o
ð17Þ
ð21Þ
Furthermore, the corresponding linear membership functions of the fuzzy objective functions of the auxiliary MOLP problem are defined by
8 1; if Z 11 < Z PIS > 11 > > > ZNIS Z ðxÞ > < 11 11 ; if Z PIS 6 Z ðxÞ 6 Z NIS 11 NIS PIS 11 11 l11 ðZ 11 ðxÞÞ ¼ Z11 Z11 > Nis > > 0; if Z 11 > Z 11 > > : 1
ð22Þ
8 1; if Z 12 < Z PIS > 12 > > > NIS > < Z12 Z12 ðxÞ ; if Z PIS 6 Z ðxÞ 6 Z NIS 12 NIS 12 12 12 Z 12 l12 ðZ 12 ðxÞÞ ¼ ZPIS > NIS > > 0; if Z > Z 12 > 12 > : 1
ð23Þ
The linear membership functions l2(Z2(x)) and l13(Z13(x)) is similar to l11(Z11(x)). And the linear membership functions l31(Z31(x)),l32(Z32(x)) l33(Z33(x)) and are similar to l11(Z11 (x)),l12(Z12(x)) and l13(Z13(x)). Accordingly, positive ideal solutions (PIS) and negative ideal solutions (NIS) of the seven objective functions of the auxiliary MOLP problem can be specified as follows, respectively. And, a payoff table (see Table 1) is constructed by using the solutions of single
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
5
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx Table 1 The corresponding PIS and NIS for the fuzzy objective functions. Objective functions
MinZ 11 ¼ Z m 1
o Max Z 12 ¼ Z m 1 Z1
Min Z 13 ¼ Z p1 Z m 1
Min Z2
Min Z 31 ¼ Z m 3
o Max Z 32 ¼ Z m 3 Z3
Min Z 33 ¼ Z p3 Z m 3
Z11
Z 111
Z 211
Z311
Z411
–
–
–
Z12
Z 112
Z13
Z 113
Z 212
Z 312
Z 412
–
–
–
Z 213
Z 313
Z 413
–
–
–
Z2
Z 12
Z 22
Z 32
Z 42
Z 52
Z 62
Z 72
Z31
–
–
–
Z 131
Z 231
Z 331
Z 431
Z32
–
–
–
Z 132
Z 232
Z 332
Z 432
Z33
–
–
–
Z 133
Z 233
Z 333
Z 433
objective where Zi is the original MOLP objective function i; Z kij is the value of six new objective function ij at a solution by the model that has objective function k, i = 1, and 3; j = 1, 2, and 3; k = 1, 2, . . . and 7; is the value of objective function of total completion time at a solution by the model that has objective function k; k = 1, 2, . . . and 7. These results are as follows: NIS Z PIS ij 6 Z ij 6 Z ij ; k Z PIS ij ¼ Min Z ij ;
Z NIS ij
¼ Max
k ¼ 1; . . . ; 7
Z PIS i2 Z NIS i2
ð24Þ
k ¼ 1; . . . ; 7
PIS Z NIS i2 6 Z i2 6 Z i2 ;
i ¼ 1; 3
¼ Max
Z ki2 ;
k ¼ 1; . . . ; 7
¼ Min
Z ki2 ;
k ¼ 1; . . . ; 7
ð25Þ
k ¼ 1; . . . :; 7
Z NIS 2
k ¼ 1; . . . :; 7
¼ Min
Z k2 ;
p ~ T oe þ 2T m En6 e þ T e =4 h i X XX o m p C Dij þ kij;e þ 2kij;e þ kij;e =4 Y ij þ C I i
j
i
p þ ððmoe þ 2mm e þ me Þ=4Þ ðEn T o Þ o
m
p
6 ððbe þ 2be þ be Þ=4Þ After getting the optimal solution from phase1, the biggest disadvantage of phase1 is that the results obtained by the max–min operator is not efficient solution and cannot be compensated for by other members, which according to the optimal objective function values, the phase 1 satisfaction degree can be used in phase 2. lþr 1 X Max k ¼ ki l þ r i¼1
s:t:
NIS Z PIS 2 6 Z2 6 Z2 k Z PIS 2 ¼ Max Z 2 ;
NIS Z PIS 11 ; Z 11 NIS Z PIS 12 ; Z 12 NIS Z PIS 13 ; Z 13 NIS Z PIS 2 ; Z2 NIS Z PIS 31 ; Z 31 NIS Z PIS 32 ; Z 32 NIS Z PIS 33 ; Z 33
k 2 ½0; 1; t ij ; Y ij ; Ei ; Ej P 0 8i; 8j
i ¼ 1; 3 j ¼ 1; 3
Z kij ;
(PIS, NIS)
ð26Þ
l1;s ðZ 1;s ðx ÞÞ 6 ki 6 l1;s ðZ 1;s ðxÞÞ; s ¼ 1; 2; 3 l2 ðZ 2 ðx ÞÞ 6 ki 6 l2 ðZ 2 ðxÞÞ l3;s ðZ 3;s ðx ÞÞ 6 ki 6 l3;s ðZ 3;s ðxÞÞ; s ¼ 1; 2; 3
ð28Þ
Ei þ tij Ej 6 0 8i; 8j
Finally, the complete FMOLP model for solving PM decision problems can be formulated as follows:
tij ¼ Dij dij
Max k
E0 ¼ 0
s:t: k 6 l1i ðZ 1i ðxÞÞi ¼ 1; 2; 3 k 6 l2 ðZ 2 ðxÞÞ
Y ij 6 Dij dij
8i; 8j 8i; 8j
p ~ T oe þ 2T m En6 e þ T e =4 i XX Xh o m p C Dij þ kij;e þ 2kij;e þ kij;e =4 Y ij þ C I
k 6 l3i ðZ 1i ðxÞÞi ¼ 1; 2; 3
i
Eqs. (4)–(7), (18), and (19)
t ij ; Y ij ; Ei ; Ej P 0 8i; 8j The auxiliary variable k represents the overall degree of DM satisfaction with determined goal values. In the phase 1, based on the above analysis and Zimmermann’s fuzzy programming model, the model can be converted to a single objective linear programming model by using max–minimum operator.
j
i
p þ moe þ 2mm e þ me =4 ðEn T o Þ o m p 6 be þ 2be þ be =4
k 2 ½0; 1;
tij ; Y ij ; Ei ; Ej P 0 8i; 8j
Max kð1Þ s:t: k 6 l1;s ðZ 1;s ðxÞÞ;
s ¼ 1; 2; 3
k 6 l2 ðZ 2 ðxÞÞ k 6 l3;s ðZ 3;s ðxÞÞ;
s ¼ 1; 2; 3
Ei þ t ij Ej 6 0 8i; 8j t ij ¼ Dij dij Y ij 6 Dij dij E0 ¼ 0
ð27Þ
8i; 8j 8i; 8j Fig. 3. The strategy to modify of membership functions.
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
6
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
The auxiliary variable k represents the satisfaction of decisionmakers by using determined goal values. If the solution is k = 1, subsequently each goal is fully satisfied; if 0 < k < 1, next all goals are satisfied at the level of k, and if k = 0, then none of the goals are satisfied. A two-phase approach is used to overcome the above-mentioned disadvantage of max–minimum approach. On the other hand, if k(1) in compromise approach is improperly given, it will make the interaction process more complicated. The steps for defining k(1) properly are given by Li and Li (2006) as: (1) Solve the model to get an optimal solution x⁄, then calculate the relative membership lk(x⁄) of each objective value’s satisfaction degree. (2) Solve the model to get an optimal solution of phase2 for solving MOLP. 3.4. An interactive satisfying method After solving the PM problem to yield a compromise solution by the two-phase approach, the decision-maker that is not satisfied with the initial solution can change the model parameters until a set of preferred satisfactory solution is found. However, this research only addresses the decision-makers that can change the PIS and NIS of model parameters until a set of preferable satisfactory solution is found, but there may be no feasible solution for the arbitrary modification of membership functions (Hu, Shen, & Li, 2009; Liang, 2009; Wang & Liang, 2004) (Fig. 3.). Accordingly, this approach presents a new modification in solution procedures to reconstruct the membership functions step by step until a set of preferred satisfactory solution is found. The algorithm procedures for modification of objective functions are shown as follows:
1. Let the optimal solution of objective functions of Z be Z ij . Comij pare each Z ij with the existing Z NIS ij and apply the rule 1 and 2 to update the aspiration levels. If rule 1 and 2 are both not fitted, rule 3 is applied. Furthermore, by considering these rules, the membership values and aspiration levels are updated to generate another optimal solution. NIS Rule 1: If Z ij < Z NIS by Z ij . ij , then replace Z ij NIS Rule 2: If Z ij ¼ Z ij , then keep these aspiration levels. PIS Rule 3: If Z ij < Z PIS ij , then replace Z ij by Z ij and keep it/them until the solution procedure is terminated. 2. Let the optimal solution of objective function of Z2 be Z 2 . Compare Z 2 with the existing Z NIS 2 and apply the rule 4 and 5 to update the aspiration levels. If rule 4 and 5 are both not fitted, rule 6 is applied. To increase the effectiveness of obtaining the preferred solutions, we move the prominent point toward the left. NIS Rule 4: If Z 2 < Z NIS by Z 2 . 2 , then replace Z 2 Rule 5: If Z 2 ¼ Z NIS , then keep these aspiration levels as they are. 2 PIS Rule 6: If Z 2 < Z PIS 2 , then replace Z 2 by Z 2 and keep it/them until the solution procedure is terminated. e ij ; i ¼ 1, and 3. Let the optimal solution of objective functions of Z 3; j = 2 be Z ij . Compare each Z ij with the existing Z PIS ij and apply the rules 7 and 8 to update the aspiration levels. If rules 7 and 8 are both not fitted, rule 9 is applied. To increase the effectiveness of obtaining the preferred solutions, we move the prominent point () toward the left. Fig. 4 illustrates the strategy for minimizing the imprecise objective function. PIS Rule 7: If Z ij < Z PIS ij , then replace Z ij by Z ij (such as Fig. 4). PIS Rule 8: If Z ij ¼ Z ij , then keep these aspiration levels as they are. NIS Rule 9: If Z ij ¼ Z NIS by Z ij and keep it/them ij , then replace Z ij until the solution procedure is terminated. The solution process terminates whenever one of the following criteria is satisfied:
Denotes critical path Denotes an activity Denotes an dummy activity
Fig. 4. The strategy to modify the optimal solution of Zij, k = 1, 3; q = 2.
1. The decision-maker accepts the modified solution and considers it as the preferred compromise solution. 2. There is no significant improvement in the objective function values after additional modifications. 3. The modification of the Z NIS or Z PIS ij ij leads to infeasible solution. 3.5. Solution procedure The solution procedure of the proposed interactive two-phase approach for solving PM problems is as follows. Step 1: Formulate the original MOLP model for solving PM decision problems Step 2: Specify the fuzzy data using triangular fuzzy numbers. Step 3: Develop the crisp new objective functions of the auxiliary MOLP problem for the imprecise goal, and specify the inequality for the fuzzy constraints. Step 4: Formulate the PIS and NIS of all objective functions of the auxiliary MOLP problem according to payoff Table 1. Step 5: Solve two-phase approach according to Eqs. (27) and (28), and get the optimal solution. Step 6: If the DM is dissatisfied with the optimal solutions, the model must be modified until a set of preferred satisfactory solutions is obtained. 4. Numerical example The example focuses on the development of an interactive twophase model to solve the new facility layout problem for a middlesized metal fabricating firm. The purpose of this PM decision is to minimize simultaneously whole project cost, complete completion time and total crashing cost, concerning direct cost, indirect cost,
Table 2 Summarised data n the Daya case (in US dollar). (i, j)
Dij (days)
dij (days)
CDij ($)
Cdij ($)
kij ($/days)
1–2 1–5 2–3 2–4 4–7 4–10 5–6 5–8 6–7 7–9 8–9 9–10 10–11
14 18 19 15 8 19 22 24 27 20 22 18 20
10 15 19 13 8 16 20 24 24 16 18 18 18
1000 4000 1200 200 600 2100 4000 1200 5000 2000 1400 700 1000
1600 4540 1200 440 600 2490 4600 1200 5450 2200 1900 1150 1200
(132, 150, 164) (164, 180, 198) – (112, 120, 128) – (112, 130, 140) (280, 300, 324) – (136, 150, 166) (34, 50, 58) (111, 125, 139) (120, 150, 160) (80, 100, 108)
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
7
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
Fig. 5. The project network of the Daya case.
Max kð1Þ
Table 3 Compared results for numerical example. Item Objective function e 1 ($) Z
e 2 (days) Z
LP-1
LP-2
Liang [23]
s:t: The proposed approach
Min Z1
Min Z2
Max L
Max k
35900
38020
(36153.50, 36364.81, 36446.37) 115.48
(35763.48, 35941.23, 36038.46) 111.62
116
108
k 6 ð39670 Z 11 Þ=3770 k 6 ðZ 12 68Þ=438 k 6 ð362 Z 13 Þ=338 k 6 ð119 Z 2 Þ=11 k 6 ð4170 Z 31 Þ=3770 k 6 ð386 Z 33 Þ=338 Ei þ t ij Ej 6 0 8i; 8j t ij ¼ Dij dij
duration of activities and the budget constraint. Table 2 summarizes the basic data of the example (in US thousand dollars) (Liang, 2009). 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 starts time (E1) is set to zero. The e-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 allocation and economic considerations, and related factors. Fig. 5 shows the activity on-arrow network. The critical path is 1–5-6–7–9–10–11. The solution procedure using the proposed IPLP and two-phase approach for the Daya case is described as follows. First, formulate the original multi-objective model for the PM decision problem according to Eqs. (1)–(10). Second, develop six new objective functions of the auxiliary MOLP problem for the imprecise objective function (1), and (3) using Eqs. (11)–(16), and the complete equivalent single-goal LP model can be formulated as follows.
ð29Þ
k 6 ðZ 32 104Þ=438
Y ij 6 Dij dij
8i; 8j 8i; 8j
E0 ¼ 0 ~ E11 6119 24; 400 þ 149:5Y 12 þ 180:25Y 15 þ 118:75Y 24 þ 129Y 410 þ300:5Y 56 þ 150:25Y 67 þ 49Y 79 þ125Y 89 þ 147:5Y 910 þ 98:5Y 1011 þ 12000 þ 149:5E11 18687:5 6 45; 250 k 2 ½0; 1; t ij ; Y ij ; Ei ; Ej P 0 8i; 8j According to the optimal objective function values, the phase 1 satisfaction degree can be used in phase 2. Consequently, the equivalent common LP model for solving the PM decision problem can be formulated using the two-phase operators from (27) and (28) to aggregate fuzzy sets. Run this regular LP model by using e 1 ¼ ð3658 Lingo computer software. The initial solutions are Z e 2 ¼ 113:34 and Z e 3 ¼ ð1866:62; 222 7:51; 36880:90; 37033:58Þ, Z 9:98; 2429:30Þ, and the overall DM satisfaction with the given
Table 4 Solutions comparison. Item
Initial compromise solutions (phase I)
DM satisfaction Goal values
L(1) = 0.5146 Z11 = 36880.90, Z12 = 293.39, Z13 = 152.68, Z2 = 113.34 Z31 = 2229.98, Z32 = 363.36,
e 1 ¼ ð36587:51; 36880:90; 37033:58Þ Z e 2 ¼ 113:34 (days) Z e 3 ¼ ð1866:62; 2229:98; 2429:30Þ Z
Z33 = 199.32 Improved compromise solutions (phase II) DM satisfaction Goal values
L(2) = 0.5807 Z11 = 36854.10,Z12 = 291.4797, Z13 = 152.1304,Z2 = 113.176 Z31 = 2227.613,Z32 = 362.42,
e 1 ¼ ð36562:6; 36854:1; 37006:2Þ Z e 2 ¼ 113:176 days Z e 3 ¼ ð1865:19; 2227:613; 2427:03Þ Z
Z33 = 199.424
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
8
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
Table 5 Modification of membership functions for the example. Initial solutions
Phase II
(PIS, NIS)
Objective values
Z11 = (35,900, 39,670), Z12 = (506, 68), Z13 = (24, 362), Z2 = (108,119) Z31 = (400, 4170), Z32 = (542, 104), Z33 = (48, 386)
e 1 =(36562.6, 36854.1, 37006.2) Z
k = 0.5807
e 2 ¼ 113:176 (days) Z e 3 ¼ ð1865:19; 2227:613; 2427:03Þ Z
Improved solutions
Phase II
(PIS, NIS)
Objective values
Z11 = (35,900, 36880.90),
e 1 ¼ (35763.48, 35941.23, Z 36038.46) e 2 ¼ 111:62 days Z
Z12 = (293.39, 68), Z13 = (24, 152.68), Z2 = (108, 113.34), Z31 = (400, 2229.98), Z32 = (363.36, 104), Z33 = (48, 199.32)
5. Experimental results
k = 0.4978
e 3 ¼ ð1289:37; 1547:36; 1698:09Þ Z
e 3 ¼ ðZ 31 Z 32 ; Z 31 ; Z 31 þ Z 33 Þ. e 1 ¼ ðZ 11 Z 12 ; Z 11 ; Z 11 þ Z 13 Þ; Z Note: Z
Fig. 6. The triangular distribution of the total project costs.
objective values is 0.5146. Furthermore, entering the phase II, initial solutions in phase I forced to improve by adding satisfaction degrees as a constraint, and the weighted average compensatory operator is used to obtain the overall DM satisfactory degree. Cone 1 ¼ ð36562:60; 36 sequently, the improved efficient solutions are Z e 2 ¼ 113:17 and Z e 3 ¼ ð1865:19; 2227:61; 24 854:10; 37006:2Þ, Z 27:03Þ, and the overall degree of DM satisfaction increases sharply to 0.5807, so the optimal solution of the max–minimum operator approach is not efficient. Obviously, the result of the two-phase approach is better than that of max–minimum operator approach. From Table 4, the optimal results by the proposed approach are an efficient solution, because of the solutions obtained using the proposed two-phase fuzzy programming is obviously better than that of max–minimum operator approach. The related investigation, Table 3 compares the results obtained by the ordinary single-objective LP with the proposed interactive two-phase approach based on the original problem. Guu and Wu (1999) and Li and Li (2006) also had proven why the output result by the two-phase fuzzy programming approach is always an efficient solution. Besides, if the DM is dissatisfied with the initial solutions, he may try to modify the results by adjusting the related parameters (PIS, NIS) until a set of preferred satisfactory solution is found. e 1 ¼ ð35763:48; 35941:23; Hence, the improved solutions are Z e 2 ¼ 111:62 and Z e 3 ¼ ð1289:37; 1547:36; 16 98:09Þ. Ta36038:46Þ, Z ble 5 lists are initial and improved PM plans for the example with the proposed approach based on current information (Chen & Chou, 1996; Lee & Li, 1993).
The decision-making is closer to the possibilistic rather than the probability in real life. Besides, in recent years, regarding the project management, Liang formulated a Possibilistic Programming (PLP) model to solve fuzzy multi-objective problems. With consideration of imprecise objectives and constraints, the PLP approach attempts to minimize the costs and completion time. Table 3 compares the results obtained by the ordinary single-objective LP with the proposed interactive two-phase approach based on current information from the Daya case. To evaluate the performance of the proposed approach, let us consider the solution of the illustrative example. From Table 3 applying LP-1 to minimize the complete project cost, the optimal value of the whole project cost, crashing cost, and completion time was $35,900 and 116 days. Applying LP-2 to minimize the total completion time, the optimal value of the total project cost, crashing cost, and completion time was $38020 and 108 days. In fact, the project managers must handle conflicting goals of project total costs, total completion time, and total crashing costs in real-world situations, so in this research most important advantage of the proposed approach is to address a more systematic procedure for modifying the initial solutions until a set of preferred satisfactory solutions is obtained. Fig. 6 shows the change in triangular possibility distributions of total project costs (Z1) for the Daya case. As indicated in Fig. 6, improved solutions are preferred to initial solutions. To summarize, several significant characteristics distinguish the proposed model from the other models. Firstly, the proposed model meets the requirements for actual application because it simultaneously minimizes the total project cost, crashing cost and total completion time. Secondly, the proposed approach yields an efficient preferable solution and the DM’s overall levels of satisfaction. Thirdly, the proposed model sets up a systematic framework that facilitates the decision-making process, enabling the DM interactively to modify the membership grades of the objectives until a set of preferred satisfactory solutions is obtained. Finally, proposed model yield’s more wide-ranging decision information than other models. It provides more information on alternative crashing strategies in terms of direct cost, indirect cost, specified project completion time and allocated the total budget. From these results of comparison, the reasons for the good performance of the proposed approach are explained as follows: 1. The decision-maker is seeking an efficient solution which can improve the max–minimum operators’ Solution. Also avoiding the potential infeasibility in the fuzzy compromise approach. 2. This method presents an easy way to determine suitable membership function instead of relying on the DMs experience which may lead to an inaccurate solution and increase the solution time. 3. Based on our proper parameter settings, all the results in this study were performed on a PC with IntelÒ Core™ i7-3770 3.4 GHz processor. Furthermore, programs were performed in Lingo software and Matlab software to solve the algorithms. As shown in results, the processing time is less than 0.01 s. 6. Conclusion To sum up, in this paper, we have discussed the issue of the interactive two-phase method with the limit owing consideration. The project managers may not have enough information to h i NIS estimate the possible interval Z PIS for imprecise objective ij ; Z ij values. Based on the results of examples, the proposed integrating interactive PLP model and two-phase approach can improve DM
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
satisfaction degree and attempts to minimize total project costs, total completion time and total crashing costs. Also, we will focus on interactive satisfying method based on rebuilding the membership functions as our future research. This method can get the satisfying result according to DM without knowing the specific marginal rates of substitution. In comparison with the conventional model, our model has taken on the customer’s satisfaction into consideration. With other constraints, in reality, our model can handle more flexible and quality of the PM. The most important advantage of the proposed approach is to address a more systematic procedure, enabling a decision-maker to interactively modify the imprecise data and parameters of a set of satisfactory compromise solution is obtained. Based on our results, this article provides a practical example of a project which we get the data in the computer processing operations in less than 0.01 s to prove the validity of this method, and this method can be applied in more complex project issues. References Arikan, F., & Güngör, Z. (2001). An application of fuzzy goal programming to a multiobjective project network problem. Fuzzy Sets and Systems, 119(1), 49–85. Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management Science, 17, 141–164. Buckley, J. J. (1988). Possibilistic linear programming with triangular fuzzy numbers. Fuzzy Sets and Systems, 26, 135–138. Chan, K. Y., Kwong, C. K., & Hu, B. Q. (2012). Market segmentation and ideal point identification for new product design using fuzzy data compression and fuzzy clustering methods. Applied Soft Computing, 12(4), 1371–1378. Chen, H. K., & Chou, H. W. (1996). Solving multi-objective linear programming problems – A generic approach. Fuzzy Sets and Systems, 82, 35–38. Guu, S. M., & Wu, Y. K. (1999). Two-phase approach for solving the fuzzy linear programming problems. Fuzzy Sets and Systems, 107, 191–195. Hu, C. F., Shen, Y., & Li, S. (2009). An interactive satisficing method based on alternative tolerance for fuzzy multiple objective optimization. Applied Mathematical Modelling, 1886–1893. Hussein, M. L. (1998). Complete solutions of multiple objective transportation problems with possibilistic coefficients. Fuzzy Sets and Systems, 93, 293–299. Inuiguchi, M., & Sakawa, M. (1996). Possible and necessary efficiency in possibilistic multi-objective linear programming problems and possible efficiency test. Fuzzy Sets and Systems, 78, 231–241.
9
Kwong, C. K., Chen, Y., Chan, K. Y., & Luo, X. (2010). A generalised fuzzy least-squares regression approach to modelling relationships in QFD. Journal of Engineering Design, 21(5), 601–613. Kwong, C. K., Chen, Y., Chan, K. Y., & Wong, H. (2008). The hybrid fuzzy least-squares regression approach to modeling manufacturing processes. IEEE Transactions on Fuzzy Systems, 16(3), 644–651. Lai, Y. J., & Hwang, C. L. (1992). A new approach to some possibilistic linear programming problems. Fuzzy Sets and Systems, 49, 121–133. Lee, S., & Li, R. J. (1993). Fuzzy multiple objective programming and compromise programming with Pareto optimum. Fuzzy Sets and Systems, 53, 275–288. Liang, T. F. (2009). Application of fuzzy sets to multi-objective project management decisions. International Journal of General Systems, 38, 311–330. Li, Y. P., Huang, G. H., & Xiao, H. N. (2008). Municipal solid waste management under uncertainty: An interval-fuzzy two-stage stochastic programming approach. Journal of Environmental Informatics, 12(2), 96–104. Li, X., & Li, B. (2006). Computing efficient solutions to fuzzy multiple objective linear programming problems. Fuzzy Sets and Systems, 157, 1328–1332. Lin, F. T. (2008). Time–cost tradeoff problem based on confidence-interval estimates and level (1 a) fuzzy numbers. International Journal of Innovative Computing, Information and Control, 4(10), 2551–2564. Liu, K. F. R., Liang, H. H., Yeh, K., & Chen, C. W. (2009). A qualitative decision support for environmental impact assessment using fuzzy logic. Journal of Environmental Informatics, 13(2), 93–103. Lv, Y., Huang, G. H., Li, Y. P., Yang, Z. F., Liu, Y., & Cheng, G. H. (2010). Planning regional water resources system using an interval fuzzy Bi-Level programming method. Journal of Environmental Informatics, 16(2), 43–56. Okuhara, K., Shibata, J., & Ishii, H. (2007). Adaptive worker’s arrangement and workload control for project management by genetic algorithm. International Journal of Innovative Computing, Information and Control, 3(1), 175–178. Suo, M. Q., Li, Y. P., & Huang, G. H. (2012). Multicriteria decision making under uncertainty: An advanced ordered weighted averaging operator for planning electric power systems. Engineering Applications of Artificial Intelligence, 25(1), 72–81. Tanaka, H., & Guo, P. (2000). Portfolio selection based on fuzzy probabilities and possibility distributions. Fuzzy Sets and Systems, 111, 387–397. Wang, H. F., & Fu, C. C. (1998). Fuzzy resource allocations in project management. International Journal of Operations and Quantitative Management, 4(3), 187–197. Wang, R. C., & Liang, T. F. (2004). Project management decisions with multiple fuzzy goals. Construction Management and Economics, 22(10), 1047–1056. Yao, J. S., & Wu, K. (2000). Ranking fuzzy numbers based on decomposition principle and signed distance. Fuzzy Sets and Systems, 116, 275–288. Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1(1), 3–28. Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems, 1, 45–55.
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
Contents lists available at ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method Ming-Feng Yang a,⇑, Yi Lin b a b
Department of Transportation Science, National Taiwan Ocean University, 2 Pei-Ning Road, Keelung 20224, Taiwan, ROC College of Management, National Taipei University of Technology, No. 1, Sec. 3, Chung-Shiao E. Rd., Taipei 10643, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 13 March 2013 Received in revised form 20 July 2013 Accepted 14 August 2013 Available online xxxx Keywords: Project management Possibilistic programming approach Two-phase approach Fuzzy multi-objective linear programming
a b s t r a c t The aim of this paper is to develop an interactive two-phase method that can help the Project Manager (PM) with solving the fuzzy multi-objective decision problems. Therefore, in this paper, we first revisit the related papers and focus on how to develop an interactive two-phase method. Next, we establish to consider the imprecise nature of the data by fulfilling the possibilistic programming model, and we also assume that each objective work has a fuzzy goal. Finally, for reaching our objective, the detailed numerical example is presented to illustrate the feasibility of applying the proposed approach to PM decision problems at the end of this paper. Results show that our model can be applied as an effective tool. Furthermore, we believe that this approach can be applied to solve other multi-objective decision making problems. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, the project managers have faced the competitive environment such as the product’s life cycle is becoming short and customers want more-customized services. It means when the project managers face the complicated situations, it is difficult for them to use resources and take decisions in a perfect way. With today’s projects, much of the uncertainty surrounding information management simply can’t be eliminated. In this case, we apply the fuzziness to improve the chances of success in project management. In addition, the degree of fuzziness not only deals with the lack of information but also supports the project managers that can make the wrong decisions in lower possibility. In another word, the experiences of the project managers on the project appropriate application reduces errors due to poor decisions may lead to opportunities for project failure. Recently, both practitioners and academicians have been more interested in considering the issues of the relationship between project management decisions and possible problems. Numerous mathematical programming techniques and heuristics for considering the fuzzy theory have been developed for solving PM
⇑ Corresponding author. Tel.: +886 931179007. E-mail addresses: [email protected] (M.-F. Yang), iwc3706@yahoo. com.tw (Y. Lin).
problems, each with its own advantages and disadvantages. Okuhara, Shibata, and Ishii (2007) utilized the genetic algorithm to the adaptive assignment of worker and workload control in PM decision problems. After that, Lin (2008) utilized statistical confidence-interval estimates and level (1 a) fuzzy numbers to solve project time–cost tradeoff problems. Arikan and Güngör (2001) utilized fuzzy goal programming (FGP) approach to solve PM decision problems with two objectives—minimizing both completion time and crashing cost. After that, Wang and Fu’s work (1998) applied fuzzy mathematical programming to solve PM decision problems. The aim of these models was to minimize complete project cost and whole crashing cost simultaneously. In addition, Wang, Liang, Li and other scholars have developed and researched an interactive multiple fuzzy goal programming (MFGP) model to solve PM decision problems in a fuzzy environment. It aimed to minimize total costs, whole completion time, and complete crashing costs simultaneously (Li, Huang, & Xiao, 2008; Liu, Liang, Yeh, & Chen, 2009; Lv et al., 2010; Suo, Li, & Huang, 2012; Wang & Fu, 1998; Wang & Liang, 2004). According to some related studies, the decision-making process is closer to the possibilistic than probability. Besides, Zadeh (1978) presented the theory of possibility, which was related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction. It acts as an elastic constraint on the values that can be assigned to a variable. Since the expression of a possibility distribution could be viewed as a fuzzy set, possibility distribution might be manipulated by the combination rules of fuzzy sets and
0360-8352/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2013.08.009
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
2
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
more particular about fuzzy restrictions. Buckley (1988) formulated a mathematical programming problem in which all parameters might be fuzzy variables by their possible distribution, and moreover, illustrated this problem using the possibilistic linear programming (PLP) approach. After that, Liang (2009) formulated a possibilistic programming (PLP) model to solve fuzzy multi-objective PM problems with imprecise objectives and constraints. Some related works such as Inuiguchi and Sakawa (1996), Hussein (1998) and Tanaka and Guo (2000) applied possibilistic programming linear method to address the decision-maker problems. In addition, some researchers extended the related research scope such as Kwong, Chen, Chan, and Wong (2008) proposed a hybrid fuzzy least-squares regression (HFLSR) approach to modeling manufacturing processes which features the capability of dealing with the two types of uncertainty and addresses the consideration of replication of responses in experiments. After that, Kwong, Chen, Chan, and Luo (2010) addressed a generalized FLSR approach to modeling relationships in QFD is described that can be used to develop models of the relationships based on fuzzy observations and/or crisp observations. And, Chan, Kwong, and Hu (2012) proposed a new methodology to perform market segmentation based on consumers’ customer requirements with fuzziness consideration. Besides, we found some inadequacies based on above-mentioned literatures: 1. The PLP approach for an optimization problem with fuzzy parameters is possibilistic, which lead to the increase of the number of objectives function and constraints of the model. In addition, the computing efficiency of the solutions obtained by the max– minimum operators has not been considered. Since the results obtained by the max–minimum operator cannot be compensated by other members. As a result, the efficiency of the optimal solution yielded by the minimum-max operators can be improved as it obtains multiple ideal solutions. 2. Different membership functions are formulated from decision-maker preferences and experiences, but the decision-makers have the difficulties in making tradeoffs between the alternatives because of their inexperience and incomplete information. Although several methods have been proposed to treat this problem in above, the two-phase approach has the value described as: if the decision maker is seeking an efficient solution which can ameliorate the max–minimum operators’ solution so that each membership degree should be ameliorated, then the two-phase method automatically obtains this desire if there is room for improvement (Guu & Wu, 1999). Finally, in this paper, we consider the imprecise nature of the input data by implementing the interactive two-phase operators; and we also assume that each objective function has a fuzzy goal. The result can be obtained by determining the suitable membership function and seek an efficient solution.
2. Project management in fuzzy environment The fuzziness comes from incomplete information and uncertainty. For example, to execute a project, the project manager will consider how to lead, how to organize, how the employer, how to plan and control the process of thinking in which ideas will be generated with incomplete and uncertainty. In addition, the higher incomplete and uncertain information is controlled; the least degree of fuzziness of the message becomes. Therefore, if the project
managers have more complete and usage information, they can reduce the degree of fuzziness.
2.1. Notations and assumptions Assume that a project has n interrelated activities that must be executed in a certain order before the entire task can be completed in the fuzzy environment. In general, environmental coefficients and related parameters are incomplete and/or vague about the planning horizon. Therefore, the incremental crashing cost for all activities, variable indirect cost per unit time, specified project completion time, and total budget is imprecise or/ and fuzzy. This problem focuses on the development of the multiple objective possibilistic linear programming (MOPLP) model to the optimum duration of each activity in the project, given a specified project completion time, the crash time tolerance for each activity and allocated total budget, and the optimal solution obtained by two-phase operator approach. Aims of this PM decision are to minimize simultaneously whole project cost, total crashing cost and total completion time. The original multi-objective linear programming (MOLP) model proposed in this work is based on the following assumptions: 1. All the objective functions and constraints are linear. 2. Direct costs increase linearly as the duration of an activity is reduced from its normal value to its crash value. 3. The common time and the shortest possible time for each activity and the cost of completing the activity in the regular time and crash time are certain to the planning horizon. 4. Indirect cost can be divided into two categories, i.e., fixed cost and variable cost, and the variable cost per unit time is the same regardless of project completion time. 5. The decision-makers adopted the pattern of triangular possibility distribution to represent the estimated objectives and related imprecise numbers. 6. Two-phase operator is used to aggregate all fuzzy sets. Assumptions 1, 2 and 3 imply that both the linearity and certainty properties must be technically satisfied in order to represent an optimization problem as a LP problem. For the sake of model facilitation, Assumption 4 represents that the indirect costs can be divided into fixed costs and variable costs. Fixed cost represents the indirect cost under regular condition and remains constant regardless of project duration. Meanwhile, variable cost, which is used to measure savings or increases in variable indirect cost, varies directly with the difference between actual completion and normal duration of the project. Assumption 4 concerns the simplicity and flexibility of the model formulation and the fuzzy arithmetic operations. Assumption 5 addresses the effectiveness of applying triangular possibility distribution to represent imprecise objectives and related imprecise numbers. In general, 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 of its central value. Assumptions 5 and 6 convert the original MOLP problem into an equivalent ordinary single-objective LP form that can be solved efficiently by the standard simplex method.
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
The following notation is used.
The crash time for activity (i, j):
Y ij 6 Dij dij 8i; 8j
ði; jÞ e1 Z
activity between event i and event j total project cost ($)
e2 Z e3 Z
total completion time (days)
Dij dij CDij Cdij ~ k
Ei E1 En To e T
normal time for activity (i, j) (days) minimum crashed time for activity (i, j) normal (direct) cost for activity (i, j) ($) minimum crashed (direct) cost for activity (i, j) ($) incremental crashing costs for activity (i, j)(representing the cost–time slopes) ($/day) duration time for activity (i, j) (days)(difference between normal time and crash time) crash time for activity (i, j) (days)(difference between normal time and duration time) earliest time for event i (days) project start time (days) project completion time (days) project completion time under normal conditions (days) specified project completion time (days)
CI ~ m ~ b
fixed indirect cost under normal conditions ($) variable indirect cost per unit time ($/day) total allocated budget ($)
e
cut level code of defuzzy objective functions by Lai and Hwang’s (1992) fashion decisive factor in weight
ij
tij Yij
k r
total crashing cost ($)
XX XX ~ Y þ C I þ mðE ~ n T o Þ: k C Dij þ ij ij j
i
ð7Þ ð8Þ
The total budget:
~ e 1 6 b: Z
ð9Þ
Non-negativity constraints on decision variables:
tij ; Y ij ; Ei ; Ej P 0 8i; 8j
ð10Þ
In real-world situations, the incremental crashing costs for all activities in Eq. (1), (3) and (9), the specified completion time for the project in Eqs. (2) and (8) are often imprecise because some relevant information, such as the skills of the workers, law and regulations, available resources, and other factors, is incomplete or unavailable.
This work assumes that the DM has already adopted the pattern of triangular possibility distribution to represent the crashing cost, ~ij , the specified project completion time, T e , variable indirect cost k ~ in the original fuzzy ~ and total allocated budget, b, per unit time, m, linear programming problem. The primary advantages of the triangular fuzzy number are the simplicity and flexibility of the fuzzy arithmetic operations. For instance, Fig. 1 shows the distribution ~ij . of the triangular fuzzy number k In practical situations, the DM can construct the triangular disij in objective (1) based on the following three promtribution of k p inent data: 1. the most pessimistic value kij that has a very low likelihood of belonging to the set of available values (possibility de m gree 0 if normalized); 2. the most likely value kij that definitely
1. Minimize total project cost:
i
E0 ¼ 0: Em 6 Te
3. Model development
Three objective functions with minimizing total project cost, total crashing cost, and total completion time are simultaneously considered to development of the proposed multiple objective’s linear programming (MOLP) model as follows:
e1 ¼ Min Z
ð6Þ
The project start time and total completion time is as follows:
2.2. Basic model
2.
3
ð1Þ
j
where the terms PP e PP~ (1) i j C Dij þ i j kij Y ij : total direct cost including to total normal cost and total crashing cost, obtained using additional direct resources such as overtime, personnel and equipment. ~ n T o Þ: indirect cost, including those of administra(2) C I þ mðE tion, depreciation, financial and other variable overhead cost that can be avoided by reducing total project time. ~ij ¼ ð g (3) k C dij C Dij Þ=ðDij dij Þ: the analysis in this problem depends primarily on the cost–time slopes for the various activities. 3. Minimize total completion time:
e 2 ’ En E0 Min Z
belongs to the set of available values (membership degree = 1 if o normalized); and 3. the most optimistic value kij that has a very low likelihood of belonging to the set of available values (membership degree = 0 if normalized). e , in constraints ~ in objective (1), T Similarly, the fuzzy data, m, ~ (8), and b, in objective (9) thus can be modeled using the distribu ; m, tion of the triangular fuzzy number. Hence, the fuzzy data for k ij ~ ~ e b and T can be symbolized as follows.
ð2Þ
4. Minimize total crashing cost:
e3 ¼ Min Z
XX ~ Y k ij ij i
ð3Þ
j
The time between event i and event j:
Ei þ t ij Ej 6 0 8i; 8j
ð4Þ
t ij ¼ Dij Y ij 8i; 8j
ð5Þ
~ij . Fig. 1. Triangular membership function of k
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
4
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
~ij ¼ ko ; km ; kp k ij;e ij;e ij;e p ~ e ¼ moe ; mm m e ; me ~ ¼ bo ; bm ; bp b e e e p T~ ¼ T oe ; T m e ; Te
3.2. Solving the fuzzy constraints e is fuzzy and Recalling Eq. (8) from the original MOLP model; T has triangular distribution. It can be substituted by:
~ Te e : En6
3.1. Solving the fuzzy objective functions The imprecise objective of minimizing the total project cost can be minimized by moving the three prominent points toward the left. Using Lai and Hwang’s (1992) approach, the proposed approach substitutes simultaneously minimizing Z m 1 , maximizing p 0 m o p minimizing Z Z Z for minimizing Zm Zm 1 1 1 1 ; Z 1 and Z 1 . 1 The imprecise objective of minimizing total crashing cost is the same method. The results show there are six new objective functions still guarantee the declaration of moving the triangular distribution of the left. Fig. 2 illustrates the strategy for minimizing the imprecise objective function; that is, the auxiliary MOLP problem generated by this proposed approach comprises simultaneously minimizing the most likely value of imprecise total costs Z m 1 , maximizing the possibility of obtaining lower total costs (region o I of the possibility distribution in Fig. 2) Z m 1 Z 1 , and minimizing the risk of obtaining higher total cost (region II of the possibil ity distribution in Fig. 2) Z p1 Z m 1 . As indicated in Fig. 2, possibility distribution is preferred to possibility distribution. Expressions (11)–(13) list is the results for the three new objective functions of total cost in Eq. (1). Expressions (14)–(16) list is the results for the three new objective functions of total crashing cost in Eq. (3).
XX XX m C Dij þ kij Y ij þ ½C I þ mm ðEn T 0 Þ ð11Þ
Min Z 11 ¼ Z m 1 ¼
i
Max Z 12 ¼
Zm 1
j
Z o1
i
i
Min Z 13
j
þ ½ðmm mo ÞðEn T 0 Þ XX p m ¼ ðZ p Z m Þ ¼ kij kij Y ij i p
Min Z 31
j
XX m o kij kij Y ij ¼
þ ½ðm m ÞðEn T 0 Þ XX m ¼ Zm K ij Y ij 3 ¼ i
Max Z 32 ¼
Min Z 33 ¼
Zm 3
Z p3
j
Z o3
Zm 3
¼
XX m o K ij kij Y ij i
j
i
j
XX p kij K m ¼ ij Y ij
e1 . Fig. 2. The strategy to minimize the imprecise objective of Z
This work applies the signed distance method of Yao and Wu e into a crisp number. If the minimum accept(2000) to convert T able membership degree, e-cut, is given, the auxiliary crisp inequality constraints can be presented as follows:
P ~ T oe þ 2T m En6 e þ T e =4:
i XX Xh o m p C Dij þ kij;e þ 2kij;e þ kij;e =4 Y ij þ C I i
o
i
p þ moe þ 2mm e þ me =4 ðEn T o Þ o m p 6 be þ 2be þ be =4
ð14Þ ð15Þ ð16Þ
ð19Þ
3.3. Two-phase for solving the possibilistic linear programming problem The original MOLP model designed above can be converted into an equivalent ordinary LP form based on the fuzzy linear programming method of Zimmermann (1978), to represent imprecise goals of the decision-maker. The two-phase operator is used to aggregate all fuzzy sets to be solved efficiently (Li & Li, 2006). Based on Bellman and Zadeh’s concept (Bellman & Zadeh, 1970), fuzzy goals (G), fuzzy constraints (C), and fuzzy decisions (D) of the fuzzy decision are defined as follows:
ð20Þ
Next, this problem is characterized by the following membership function:
lD ðxÞ ¼ lG ðxÞ ^ lc ðxÞ ¼ MinðlG ðxÞ; lc ðxÞÞ ð13Þ
ð18Þ
~ m; ~ are fuzzy ~ b Recalling Eq. (9) from the original MOLP model, k; and have triangular distribution. This work applies the signed dis~ m; ~ into crisp number. If the minimum ~ b tance method to convert k; acceptable membership degree, e, is given, the auxiliary crisp inequality constraints can be presented as follows:
D¼G\C ð12Þ
j
o
ð17Þ
ð21Þ
Furthermore, the corresponding linear membership functions of the fuzzy objective functions of the auxiliary MOLP problem are defined by
8 1; if Z 11 < Z PIS > 11 > > > ZNIS Z ðxÞ > < 11 11 ; if Z PIS 6 Z ðxÞ 6 Z NIS 11 NIS PIS 11 11 l11 ðZ 11 ðxÞÞ ¼ Z11 Z11 > Nis > > 0; if Z 11 > Z 11 > > : 1
ð22Þ
8 1; if Z 12 < Z PIS > 12 > > > NIS > < Z12 Z12 ðxÞ ; if Z PIS 6 Z ðxÞ 6 Z NIS 12 NIS 12 12 12 Z 12 l12 ðZ 12 ðxÞÞ ¼ ZPIS > NIS > > 0; if Z > Z 12 > 12 > : 1
ð23Þ
The linear membership functions l2(Z2(x)) and l13(Z13(x)) is similar to l11(Z11(x)). And the linear membership functions l31(Z31(x)),l32(Z32(x)) l33(Z33(x)) and are similar to l11(Z11 (x)),l12(Z12(x)) and l13(Z13(x)). Accordingly, positive ideal solutions (PIS) and negative ideal solutions (NIS) of the seven objective functions of the auxiliary MOLP problem can be specified as follows, respectively. And, a payoff table (see Table 1) is constructed by using the solutions of single
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
5
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx Table 1 The corresponding PIS and NIS for the fuzzy objective functions. Objective functions
MinZ 11 ¼ Z m 1
o Max Z 12 ¼ Z m 1 Z1
Min Z 13 ¼ Z p1 Z m 1
Min Z2
Min Z 31 ¼ Z m 3
o Max Z 32 ¼ Z m 3 Z3
Min Z 33 ¼ Z p3 Z m 3
Z11
Z 111
Z 211
Z311
Z411
–
–
–
Z12
Z 112
Z13
Z 113
Z 212
Z 312
Z 412
–
–
–
Z 213
Z 313
Z 413
–
–
–
Z2
Z 12
Z 22
Z 32
Z 42
Z 52
Z 62
Z 72
Z31
–
–
–
Z 131
Z 231
Z 331
Z 431
Z32
–
–
–
Z 132
Z 232
Z 332
Z 432
Z33
–
–
–
Z 133
Z 233
Z 333
Z 433
objective where Zi is the original MOLP objective function i; Z kij is the value of six new objective function ij at a solution by the model that has objective function k, i = 1, and 3; j = 1, 2, and 3; k = 1, 2, . . . and 7; is the value of objective function of total completion time at a solution by the model that has objective function k; k = 1, 2, . . . and 7. These results are as follows: NIS Z PIS ij 6 Z ij 6 Z ij ; k Z PIS ij ¼ Min Z ij ;
Z NIS ij
¼ Max
k ¼ 1; . . . ; 7
Z PIS i2 Z NIS i2
ð24Þ
k ¼ 1; . . . ; 7
PIS Z NIS i2 6 Z i2 6 Z i2 ;
i ¼ 1; 3
¼ Max
Z ki2 ;
k ¼ 1; . . . ; 7
¼ Min
Z ki2 ;
k ¼ 1; . . . ; 7
ð25Þ
k ¼ 1; . . . :; 7
Z NIS 2
k ¼ 1; . . . :; 7
¼ Min
Z k2 ;
p ~ T oe þ 2T m En6 e þ T e =4 h i X XX o m p C Dij þ kij;e þ 2kij;e þ kij;e =4 Y ij þ C I i
j
i
p þ ððmoe þ 2mm e þ me Þ=4Þ ðEn T o Þ o
m
p
6 ððbe þ 2be þ be Þ=4Þ After getting the optimal solution from phase1, the biggest disadvantage of phase1 is that the results obtained by the max–min operator is not efficient solution and cannot be compensated for by other members, which according to the optimal objective function values, the phase 1 satisfaction degree can be used in phase 2. lþr 1 X Max k ¼ ki l þ r i¼1
s:t:
NIS Z PIS 2 6 Z2 6 Z2 k Z PIS 2 ¼ Max Z 2 ;
NIS Z PIS 11 ; Z 11 NIS Z PIS 12 ; Z 12 NIS Z PIS 13 ; Z 13 NIS Z PIS 2 ; Z2 NIS Z PIS 31 ; Z 31 NIS Z PIS 32 ; Z 32 NIS Z PIS 33 ; Z 33
k 2 ½0; 1; t ij ; Y ij ; Ei ; Ej P 0 8i; 8j
i ¼ 1; 3 j ¼ 1; 3
Z kij ;
(PIS, NIS)
ð26Þ
l1;s ðZ 1;s ðx ÞÞ 6 ki 6 l1;s ðZ 1;s ðxÞÞ; s ¼ 1; 2; 3 l2 ðZ 2 ðx ÞÞ 6 ki 6 l2 ðZ 2 ðxÞÞ l3;s ðZ 3;s ðx ÞÞ 6 ki 6 l3;s ðZ 3;s ðxÞÞ; s ¼ 1; 2; 3
ð28Þ
Ei þ tij Ej 6 0 8i; 8j
Finally, the complete FMOLP model for solving PM decision problems can be formulated as follows:
tij ¼ Dij dij
Max k
E0 ¼ 0
s:t: k 6 l1i ðZ 1i ðxÞÞi ¼ 1; 2; 3 k 6 l2 ðZ 2 ðxÞÞ
Y ij 6 Dij dij
8i; 8j 8i; 8j
p ~ T oe þ 2T m En6 e þ T e =4 i XX Xh o m p C Dij þ kij;e þ 2kij;e þ kij;e =4 Y ij þ C I
k 6 l3i ðZ 1i ðxÞÞi ¼ 1; 2; 3
i
Eqs. (4)–(7), (18), and (19)
t ij ; Y ij ; Ei ; Ej P 0 8i; 8j The auxiliary variable k represents the overall degree of DM satisfaction with determined goal values. In the phase 1, based on the above analysis and Zimmermann’s fuzzy programming model, the model can be converted to a single objective linear programming model by using max–minimum operator.
j
i
p þ moe þ 2mm e þ me =4 ðEn T o Þ o m p 6 be þ 2be þ be =4
k 2 ½0; 1;
tij ; Y ij ; Ei ; Ej P 0 8i; 8j
Max kð1Þ s:t: k 6 l1;s ðZ 1;s ðxÞÞ;
s ¼ 1; 2; 3
k 6 l2 ðZ 2 ðxÞÞ k 6 l3;s ðZ 3;s ðxÞÞ;
s ¼ 1; 2; 3
Ei þ t ij Ej 6 0 8i; 8j t ij ¼ Dij dij Y ij 6 Dij dij E0 ¼ 0
ð27Þ
8i; 8j 8i; 8j Fig. 3. The strategy to modify of membership functions.
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
6
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
The auxiliary variable k represents the satisfaction of decisionmakers by using determined goal values. If the solution is k = 1, subsequently each goal is fully satisfied; if 0 < k < 1, next all goals are satisfied at the level of k, and if k = 0, then none of the goals are satisfied. A two-phase approach is used to overcome the above-mentioned disadvantage of max–minimum approach. On the other hand, if k(1) in compromise approach is improperly given, it will make the interaction process more complicated. The steps for defining k(1) properly are given by Li and Li (2006) as: (1) Solve the model to get an optimal solution x⁄, then calculate the relative membership lk(x⁄) of each objective value’s satisfaction degree. (2) Solve the model to get an optimal solution of phase2 for solving MOLP. 3.4. An interactive satisfying method After solving the PM problem to yield a compromise solution by the two-phase approach, the decision-maker that is not satisfied with the initial solution can change the model parameters until a set of preferred satisfactory solution is found. However, this research only addresses the decision-makers that can change the PIS and NIS of model parameters until a set of preferable satisfactory solution is found, but there may be no feasible solution for the arbitrary modification of membership functions (Hu, Shen, & Li, 2009; Liang, 2009; Wang & Liang, 2004) (Fig. 3.). Accordingly, this approach presents a new modification in solution procedures to reconstruct the membership functions step by step until a set of preferred satisfactory solution is found. The algorithm procedures for modification of objective functions are shown as follows:
1. Let the optimal solution of objective functions of Z be Z ij . Comij pare each Z ij with the existing Z NIS ij and apply the rule 1 and 2 to update the aspiration levels. If rule 1 and 2 are both not fitted, rule 3 is applied. Furthermore, by considering these rules, the membership values and aspiration levels are updated to generate another optimal solution. NIS Rule 1: If Z ij < Z NIS by Z ij . ij , then replace Z ij NIS Rule 2: If Z ij ¼ Z ij , then keep these aspiration levels. PIS Rule 3: If Z ij < Z PIS ij , then replace Z ij by Z ij and keep it/them until the solution procedure is terminated. 2. Let the optimal solution of objective function of Z2 be Z 2 . Compare Z 2 with the existing Z NIS 2 and apply the rule 4 and 5 to update the aspiration levels. If rule 4 and 5 are both not fitted, rule 6 is applied. To increase the effectiveness of obtaining the preferred solutions, we move the prominent point toward the left. NIS Rule 4: If Z 2 < Z NIS by Z 2 . 2 , then replace Z 2 Rule 5: If Z 2 ¼ Z NIS , then keep these aspiration levels as they are. 2 PIS Rule 6: If Z 2 < Z PIS 2 , then replace Z 2 by Z 2 and keep it/them until the solution procedure is terminated. e ij ; i ¼ 1, and 3. Let the optimal solution of objective functions of Z 3; j = 2 be Z ij . Compare each Z ij with the existing Z PIS ij and apply the rules 7 and 8 to update the aspiration levels. If rules 7 and 8 are both not fitted, rule 9 is applied. To increase the effectiveness of obtaining the preferred solutions, we move the prominent point () toward the left. Fig. 4 illustrates the strategy for minimizing the imprecise objective function. PIS Rule 7: If Z ij < Z PIS ij , then replace Z ij by Z ij (such as Fig. 4). PIS Rule 8: If Z ij ¼ Z ij , then keep these aspiration levels as they are. NIS Rule 9: If Z ij ¼ Z NIS by Z ij and keep it/them ij , then replace Z ij until the solution procedure is terminated. The solution process terminates whenever one of the following criteria is satisfied:
Denotes critical path Denotes an activity Denotes an dummy activity
Fig. 4. The strategy to modify the optimal solution of Zij, k = 1, 3; q = 2.
1. The decision-maker accepts the modified solution and considers it as the preferred compromise solution. 2. There is no significant improvement in the objective function values after additional modifications. 3. The modification of the Z NIS or Z PIS ij ij leads to infeasible solution. 3.5. Solution procedure The solution procedure of the proposed interactive two-phase approach for solving PM problems is as follows. Step 1: Formulate the original MOLP model for solving PM decision problems Step 2: Specify the fuzzy data using triangular fuzzy numbers. Step 3: Develop the crisp new objective functions of the auxiliary MOLP problem for the imprecise goal, and specify the inequality for the fuzzy constraints. Step 4: Formulate the PIS and NIS of all objective functions of the auxiliary MOLP problem according to payoff Table 1. Step 5: Solve two-phase approach according to Eqs. (27) and (28), and get the optimal solution. Step 6: If the DM is dissatisfied with the optimal solutions, the model must be modified until a set of preferred satisfactory solutions is obtained. 4. Numerical example The example focuses on the development of an interactive twophase model to solve the new facility layout problem for a middlesized metal fabricating firm. The purpose of this PM decision is to minimize simultaneously whole project cost, complete completion time and total crashing cost, concerning direct cost, indirect cost,
Table 2 Summarised data n the Daya case (in US dollar). (i, j)
Dij (days)
dij (days)
CDij ($)
Cdij ($)
kij ($/days)
1–2 1–5 2–3 2–4 4–7 4–10 5–6 5–8 6–7 7–9 8–9 9–10 10–11
14 18 19 15 8 19 22 24 27 20 22 18 20
10 15 19 13 8 16 20 24 24 16 18 18 18
1000 4000 1200 200 600 2100 4000 1200 5000 2000 1400 700 1000
1600 4540 1200 440 600 2490 4600 1200 5450 2200 1900 1150 1200
(132, 150, 164) (164, 180, 198) – (112, 120, 128) – (112, 130, 140) (280, 300, 324) – (136, 150, 166) (34, 50, 58) (111, 125, 139) (120, 150, 160) (80, 100, 108)
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
7
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
Fig. 5. The project network of the Daya case.
Max kð1Þ
Table 3 Compared results for numerical example. Item Objective function e 1 ($) Z
e 2 (days) Z
LP-1
LP-2
Liang [23]
s:t: The proposed approach
Min Z1
Min Z2
Max L
Max k
35900
38020
(36153.50, 36364.81, 36446.37) 115.48
(35763.48, 35941.23, 36038.46) 111.62
116
108
k 6 ð39670 Z 11 Þ=3770 k 6 ðZ 12 68Þ=438 k 6 ð362 Z 13 Þ=338 k 6 ð119 Z 2 Þ=11 k 6 ð4170 Z 31 Þ=3770 k 6 ð386 Z 33 Þ=338 Ei þ t ij Ej 6 0 8i; 8j t ij ¼ Dij dij
duration of activities and the budget constraint. Table 2 summarizes the basic data of the example (in US thousand dollars) (Liang, 2009). 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 starts time (E1) is set to zero. The e-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 allocation and economic considerations, and related factors. Fig. 5 shows the activity on-arrow network. The critical path is 1–5-6–7–9–10–11. The solution procedure using the proposed IPLP and two-phase approach for the Daya case is described as follows. First, formulate the original multi-objective model for the PM decision problem according to Eqs. (1)–(10). Second, develop six new objective functions of the auxiliary MOLP problem for the imprecise objective function (1), and (3) using Eqs. (11)–(16), and the complete equivalent single-goal LP model can be formulated as follows.
ð29Þ
k 6 ðZ 32 104Þ=438
Y ij 6 Dij dij
8i; 8j 8i; 8j
E0 ¼ 0 ~ E11 6119 24; 400 þ 149:5Y 12 þ 180:25Y 15 þ 118:75Y 24 þ 129Y 410 þ300:5Y 56 þ 150:25Y 67 þ 49Y 79 þ125Y 89 þ 147:5Y 910 þ 98:5Y 1011 þ 12000 þ 149:5E11 18687:5 6 45; 250 k 2 ½0; 1; t ij ; Y ij ; Ei ; Ej P 0 8i; 8j According to the optimal objective function values, the phase 1 satisfaction degree can be used in phase 2. Consequently, the equivalent common LP model for solving the PM decision problem can be formulated using the two-phase operators from (27) and (28) to aggregate fuzzy sets. Run this regular LP model by using e 1 ¼ ð3658 Lingo computer software. The initial solutions are Z e 2 ¼ 113:34 and Z e 3 ¼ ð1866:62; 222 7:51; 36880:90; 37033:58Þ, Z 9:98; 2429:30Þ, and the overall DM satisfaction with the given
Table 4 Solutions comparison. Item
Initial compromise solutions (phase I)
DM satisfaction Goal values
L(1) = 0.5146 Z11 = 36880.90, Z12 = 293.39, Z13 = 152.68, Z2 = 113.34 Z31 = 2229.98, Z32 = 363.36,
e 1 ¼ ð36587:51; 36880:90; 37033:58Þ Z e 2 ¼ 113:34 (days) Z e 3 ¼ ð1866:62; 2229:98; 2429:30Þ Z
Z33 = 199.32 Improved compromise solutions (phase II) DM satisfaction Goal values
L(2) = 0.5807 Z11 = 36854.10,Z12 = 291.4797, Z13 = 152.1304,Z2 = 113.176 Z31 = 2227.613,Z32 = 362.42,
e 1 ¼ ð36562:6; 36854:1; 37006:2Þ Z e 2 ¼ 113:176 days Z e 3 ¼ ð1865:19; 2227:613; 2427:03Þ Z
Z33 = 199.424
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
8
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
Table 5 Modification of membership functions for the example. Initial solutions
Phase II
(PIS, NIS)
Objective values
Z11 = (35,900, 39,670), Z12 = (506, 68), Z13 = (24, 362), Z2 = (108,119) Z31 = (400, 4170), Z32 = (542, 104), Z33 = (48, 386)
e 1 =(36562.6, 36854.1, 37006.2) Z
k = 0.5807
e 2 ¼ 113:176 (days) Z e 3 ¼ ð1865:19; 2227:613; 2427:03Þ Z
Improved solutions
Phase II
(PIS, NIS)
Objective values
Z11 = (35,900, 36880.90),
e 1 ¼ (35763.48, 35941.23, Z 36038.46) e 2 ¼ 111:62 days Z
Z12 = (293.39, 68), Z13 = (24, 152.68), Z2 = (108, 113.34), Z31 = (400, 2229.98), Z32 = (363.36, 104), Z33 = (48, 199.32)
5. Experimental results
k = 0.4978
e 3 ¼ ð1289:37; 1547:36; 1698:09Þ Z
e 3 ¼ ðZ 31 Z 32 ; Z 31 ; Z 31 þ Z 33 Þ. e 1 ¼ ðZ 11 Z 12 ; Z 11 ; Z 11 þ Z 13 Þ; Z Note: Z
Fig. 6. The triangular distribution of the total project costs.
objective values is 0.5146. Furthermore, entering the phase II, initial solutions in phase I forced to improve by adding satisfaction degrees as a constraint, and the weighted average compensatory operator is used to obtain the overall DM satisfactory degree. Cone 1 ¼ ð36562:60; 36 sequently, the improved efficient solutions are Z e 2 ¼ 113:17 and Z e 3 ¼ ð1865:19; 2227:61; 24 854:10; 37006:2Þ, Z 27:03Þ, and the overall degree of DM satisfaction increases sharply to 0.5807, so the optimal solution of the max–minimum operator approach is not efficient. Obviously, the result of the two-phase approach is better than that of max–minimum operator approach. From Table 4, the optimal results by the proposed approach are an efficient solution, because of the solutions obtained using the proposed two-phase fuzzy programming is obviously better than that of max–minimum operator approach. The related investigation, Table 3 compares the results obtained by the ordinary single-objective LP with the proposed interactive two-phase approach based on the original problem. Guu and Wu (1999) and Li and Li (2006) also had proven why the output result by the two-phase fuzzy programming approach is always an efficient solution. Besides, if the DM is dissatisfied with the initial solutions, he may try to modify the results by adjusting the related parameters (PIS, NIS) until a set of preferred satisfactory solution is found. e 1 ¼ ð35763:48; 35941:23; Hence, the improved solutions are Z e 2 ¼ 111:62 and Z e 3 ¼ ð1289:37; 1547:36; 16 98:09Þ. Ta36038:46Þ, Z ble 5 lists are initial and improved PM plans for the example with the proposed approach based on current information (Chen & Chou, 1996; Lee & Li, 1993).
The decision-making is closer to the possibilistic rather than the probability in real life. Besides, in recent years, regarding the project management, Liang formulated a Possibilistic Programming (PLP) model to solve fuzzy multi-objective problems. With consideration of imprecise objectives and constraints, the PLP approach attempts to minimize the costs and completion time. Table 3 compares the results obtained by the ordinary single-objective LP with the proposed interactive two-phase approach based on current information from the Daya case. To evaluate the performance of the proposed approach, let us consider the solution of the illustrative example. From Table 3 applying LP-1 to minimize the complete project cost, the optimal value of the whole project cost, crashing cost, and completion time was $35,900 and 116 days. Applying LP-2 to minimize the total completion time, the optimal value of the total project cost, crashing cost, and completion time was $38020 and 108 days. In fact, the project managers must handle conflicting goals of project total costs, total completion time, and total crashing costs in real-world situations, so in this research most important advantage of the proposed approach is to address a more systematic procedure for modifying the initial solutions until a set of preferred satisfactory solutions is obtained. Fig. 6 shows the change in triangular possibility distributions of total project costs (Z1) for the Daya case. As indicated in Fig. 6, improved solutions are preferred to initial solutions. To summarize, several significant characteristics distinguish the proposed model from the other models. Firstly, the proposed model meets the requirements for actual application because it simultaneously minimizes the total project cost, crashing cost and total completion time. Secondly, the proposed approach yields an efficient preferable solution and the DM’s overall levels of satisfaction. Thirdly, the proposed model sets up a systematic framework that facilitates the decision-making process, enabling the DM interactively to modify the membership grades of the objectives until a set of preferred satisfactory solutions is obtained. Finally, proposed model yield’s more wide-ranging decision information than other models. It provides more information on alternative crashing strategies in terms of direct cost, indirect cost, specified project completion time and allocated the total budget. From these results of comparison, the reasons for the good performance of the proposed approach are explained as follows: 1. The decision-maker is seeking an efficient solution which can improve the max–minimum operators’ Solution. Also avoiding the potential infeasibility in the fuzzy compromise approach. 2. This method presents an easy way to determine suitable membership function instead of relying on the DMs experience which may lead to an inaccurate solution and increase the solution time. 3. Based on our proper parameter settings, all the results in this study were performed on a PC with IntelÒ Core™ i7-3770 3.4 GHz processor. Furthermore, programs were performed in Lingo software and Matlab software to solve the algorithms. As shown in results, the processing time is less than 0.01 s. 6. Conclusion To sum up, in this paper, we have discussed the issue of the interactive two-phase method with the limit owing consideration. The project managers may not have enough information to h i NIS estimate the possible interval Z PIS for imprecise objective ij ; Z ij values. Based on the results of examples, the proposed integrating interactive PLP model and two-phase approach can improve DM
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009
M.-F. Yang, Y. Lin / Computers & Industrial Engineering xxx (2013) xxx–xxx
satisfaction degree and attempts to minimize total project costs, total completion time and total crashing costs. Also, we will focus on interactive satisfying method based on rebuilding the membership functions as our future research. This method can get the satisfying result according to DM without knowing the specific marginal rates of substitution. In comparison with the conventional model, our model has taken on the customer’s satisfaction into consideration. With other constraints, in reality, our model can handle more flexible and quality of the PM. The most important advantage of the proposed approach is to address a more systematic procedure, enabling a decision-maker to interactively modify the imprecise data and parameters of a set of satisfactory compromise solution is obtained. Based on our results, this article provides a practical example of a project which we get the data in the computer processing operations in less than 0.01 s to prove the validity of this method, and this method can be applied in more complex project issues. References Arikan, F., & Güngör, Z. (2001). An application of fuzzy goal programming to a multiobjective project network problem. Fuzzy Sets and Systems, 119(1), 49–85. Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management Science, 17, 141–164. Buckley, J. J. (1988). Possibilistic linear programming with triangular fuzzy numbers. Fuzzy Sets and Systems, 26, 135–138. Chan, K. Y., Kwong, C. K., & Hu, B. Q. (2012). Market segmentation and ideal point identification for new product design using fuzzy data compression and fuzzy clustering methods. Applied Soft Computing, 12(4), 1371–1378. Chen, H. K., & Chou, H. W. (1996). Solving multi-objective linear programming problems – A generic approach. Fuzzy Sets and Systems, 82, 35–38. Guu, S. M., & Wu, Y. K. (1999). Two-phase approach for solving the fuzzy linear programming problems. Fuzzy Sets and Systems, 107, 191–195. Hu, C. F., Shen, Y., & Li, S. (2009). An interactive satisficing method based on alternative tolerance for fuzzy multiple objective optimization. Applied Mathematical Modelling, 1886–1893. Hussein, M. L. (1998). Complete solutions of multiple objective transportation problems with possibilistic coefficients. Fuzzy Sets and Systems, 93, 293–299. Inuiguchi, M., & Sakawa, M. (1996). Possible and necessary efficiency in possibilistic multi-objective linear programming problems and possible efficiency test. Fuzzy Sets and Systems, 78, 231–241.
9
Kwong, C. K., Chen, Y., Chan, K. Y., & Luo, X. (2010). A generalised fuzzy least-squares regression approach to modelling relationships in QFD. Journal of Engineering Design, 21(5), 601–613. Kwong, C. K., Chen, Y., Chan, K. Y., & Wong, H. (2008). The hybrid fuzzy least-squares regression approach to modeling manufacturing processes. IEEE Transactions on Fuzzy Systems, 16(3), 644–651. Lai, Y. J., & Hwang, C. L. (1992). A new approach to some possibilistic linear programming problems. Fuzzy Sets and Systems, 49, 121–133. Lee, S., & Li, R. J. (1993). Fuzzy multiple objective programming and compromise programming with Pareto optimum. Fuzzy Sets and Systems, 53, 275–288. Liang, T. F. (2009). Application of fuzzy sets to multi-objective project management decisions. International Journal of General Systems, 38, 311–330. Li, Y. P., Huang, G. H., & Xiao, H. N. (2008). Municipal solid waste management under uncertainty: An interval-fuzzy two-stage stochastic programming approach. Journal of Environmental Informatics, 12(2), 96–104. Li, X., & Li, B. (2006). Computing efficient solutions to fuzzy multiple objective linear programming problems. Fuzzy Sets and Systems, 157, 1328–1332. Lin, F. T. (2008). Time–cost tradeoff problem based on confidence-interval estimates and level (1 a) fuzzy numbers. International Journal of Innovative Computing, Information and Control, 4(10), 2551–2564. Liu, K. F. R., Liang, H. H., Yeh, K., & Chen, C. W. (2009). A qualitative decision support for environmental impact assessment using fuzzy logic. Journal of Environmental Informatics, 13(2), 93–103. Lv, Y., Huang, G. H., Li, Y. P., Yang, Z. F., Liu, Y., & Cheng, G. H. (2010). Planning regional water resources system using an interval fuzzy Bi-Level programming method. Journal of Environmental Informatics, 16(2), 43–56. Okuhara, K., Shibata, J., & Ishii, H. (2007). Adaptive worker’s arrangement and workload control for project management by genetic algorithm. International Journal of Innovative Computing, Information and Control, 3(1), 175–178. Suo, M. Q., Li, Y. P., & Huang, G. H. (2012). Multicriteria decision making under uncertainty: An advanced ordered weighted averaging operator for planning electric power systems. Engineering Applications of Artificial Intelligence, 25(1), 72–81. Tanaka, H., & Guo, P. (2000). Portfolio selection based on fuzzy probabilities and possibility distributions. Fuzzy Sets and Systems, 111, 387–397. Wang, H. F., & Fu, C. C. (1998). Fuzzy resource allocations in project management. International Journal of Operations and Quantitative Management, 4(3), 187–197. Wang, R. C., & Liang, T. F. (2004). Project management decisions with multiple fuzzy goals. Construction Management and Economics, 22(10), 1047–1056. Yao, J. S., & Wu, K. (2000). Ranking fuzzy numbers based on decomposition principle and signed distance. Fuzzy Sets and Systems, 116, 275–288. Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1(1), 3–28. Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems, 1, 45–55.
Please cite this article in press as: Yang, M.-F., & Lin, Y. Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.009