Grey-fuzzy solution for multi-objective linear ...

Grey Systems: Theory and Application Grey-fuzzy solution for multi-objective linear programming with interval coefficients Amin Mahmoudi, Mohammad Reza Feylizadeh, Davood Darvishi, Sifeng Liu,

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Article information: To cite this document: Amin Mahmoudi, Mohammad Reza Feylizadeh, Davood Darvishi, Sifeng Liu, (2018) "Grey-fuzzy solution for multi-objective linear programming with interval coefficients", Grey Systems: Theory and Application, https://doi.org/10.1108/GS-01-2018-0007 Permanent link to this document: https://doi.org/10.1108/GS-01-2018-0007 Downloaded on: 25 May 2018, At: 08:12 (PT) References: this document contains references to 35 other documents. To copy this document: [email protected] Access to this document was granted through an Emerald subscription provided by emeraldsrm:277069 []

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Grey-fuzzy solution for multi-objective linear programming with interval coefficients

Grey-fuzzy solution for MOLP

Amin Mahmoudi and Mohammad Reza Feylizadeh

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Department of Industrial Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran

Davood Darvishi

Received 20 January 2018 Revised 31 March 2018 Accepted 1 April 2018

Department of Basic Sciences-Mathematics, Payame Noor University, Tehran, Iran, and

Sifeng Liu Institute for Grey Systems Studies, Nanjing University of Aeronautics and Astronautics, Nanjing, China Abstract Purpose – The purpose of this paper is to propose a method for solving multi-objective linear programming (MOLP) with interval coefficients using positioned programming and interactive fuzzy programming approaches. Design/methodology/approach – In the proposed algorithm, first, lower and upper bounds of each objective function in its feasible region will be determined. Afterwards using fuzzy approach, considering a membership function for each objective function and finally using grey linear programming, the solution for this problem will be obtained. Findings – According to the presented example, in this paper, the proposed method is both simple in use and suitable for solving different problems. In the numerical example mentioned in this paper, the proposed method provides an acceptable solution for such problems. Practical implications – As in most real-world situations, the coefficients of decision models are not known and exact. In this paper, the authors consider the model of MOLP with interval data, since one of the solutions to cover uncertainty is using interval theory. Originality/value – Based on using grey theory and interactive fuzzy programming approaches, an appropriate method has been presented for solving MOLP problems with interval coefficients. The proposed method, against the complex methods, has less effort and offers acceptable solutions. Keywords Grey linear programming, Grey system, Interactive fuzzy programming approach, Interval coefficient, Multi-objective linear programming Paper type Research paper

1. Introduction In real world, a mathematical model that includes only one objective function does not represent the requirements of the decision maker, which reduces the utility of the model’s results. In addition, the data of the problems are not always accurately expressed. One of the solutions to cover uncertainty is using interval theory. In an interval programing, the coefficients of variables are considered as interval variables but in stochastic programming, coefficients of variables are considered as random variables, and their probability distribution is assumed to be known (Wallace and Fleten, 2003; Cho, 2005; Birge and Louveaux, 2011; Liu, 2013). There are common methods to solve interval programing problem, but sometimes the models derived from these methods are very complex, so that solving them is a serious challenge. For the first time, Ishibuchi and Tanaka (1990) have studied multi-objective linear programming (MOLP) problems with interval coefficients in the objective function. In these cases, only the objective function with interval coefficients was investigated. To solve these problems, the objective functions with interval coefficients are converted into

Grey Systems: Theory and Application © Emerald Publishing Limited 2043-9377 DOI 10.1108/GS-01-2018-0007

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GS

two objective functions with crisp coefficients and then resolved. In this method, the coefficients of constraints were considered as crisp numbers. Chanas and Kuchta (1996) used the same approach as Ishibuchi and Tanaka (1990), with the difference that the objective function was considered as a linear combination of the parameter t, which could change in [0,1]. So, for different values of t, the model could be solved. In this method, the constraints were considered as crisp numbers. Chinneck and Ramadan (2000) provided a solution to solve linear programming problems with interval coefficients. They believed that the only method to consider the uncertainty is to conduct sensitivity analysis. In their research, they calculated the best and worst solutions for the interval model. In their research, only problems with one objective were investigated. Sengupta et al. (2001) provided a satisfactory solution to solve the linear programming problems. In this research, problems were also addressed with a single objective. Allahdadi and Mishmast Nehi (2013) also proposed a solution for a linear programming in which the best and worst solutions of the method were calculated for the model. Then, taking into account the linear combinations of the worst solution, they performed the sensitivity analysis towards the best solution. Oliveira and Antunes (2007) examined the multi-objective problems in which objective function and constraints had interval coefficients. One of the disadvantages of this method was a great deal of computational effort, which has led to many steps being taken. In addition, an interval approach was used to solve fuzzy linear programming problems (Allahdadi and Nehi, 2011; Dubey et al., 2012; Allahviranloo and Ghanbari, 2012). Razavi Hajiagha et al. (2013) used a fuzzy approach to solve multi-objective problems. This method was less complicated than other methods presented. Nevertheless, there are serious shortcomings in this method. After solving the model, the upper limit values will be less than the lower limit values for some variables. In the present article, Razavi Hajiagha et al.’s (2013) method has been investigated and then an appropriate method has been presented for solving MOLP problems with interval coefficients. The proposed method, against the complex methods, has less effort and offers acceptable solutions. In the following, we first discuss on the shortcomings of the method proposed by Razavi Hajiagha et al. (2013), and then we will propose a new method. 2. Uncertainty theories In mathematical programming, most problems are assumed with crisp coefficients. In the real world, it is not possible to express the crisp coefficients because there is uncertainty in real problem. The specifications of uncertain systems are the incompleteness and insufficiency of their information (Liu et al., 2017). There are several theories to consider uncertainty as follows: •

fuzzy approach;

•

grey theory; and

•

interval theory.

Fuzzy sets theory was introduced by Zadeh in 1965 (Kaviani et al., 2014). Fuzzy theory is a reliable tool to model uncertainty base of human judgments (Parsaei et al., 2012; Zare et al., 2018). Deng established grey theory in 1982. The focus of grey theory is on the study of problems in which there is a lack of information ( Javed and Liu, 2017; Kaviani and Abbasi, 2014). Today, the grey approach is used in domains such as appraisal, forecast modelling, decision-making, control, etc. (Liu and Forrest, 2010; Javed and Liu, 2018). In the interval uncertainty, just the information about the lower bound and upper bound of the interval is known (Soltani, 2014). In linear programming problems, the coefficients of the objective function and constraints can be considered as intervals. Interval number has upper and L U lower bounds. Let x1 ¼ ½xL1 ; xU 1  and x2 ¼ ½x2 ; x2 , then we have the following operations

(Chakrabortty et al., 2010):
 U x1 þx2 ¼ xL1 þxL2 ; xU 1 þx2

(1)


 U L x1 x2 ¼ x1 þ ðx2 Þ ¼ xL1 xU 2 ; x1 x2

(2)


   L L U U L U U L  U L U U L x1  x2 ¼ min xL1 xL2 ; xU 1 x2 ; x1 x2 ; x1 x2 ; max x1 x2 ; x1 x2 ; x1 x2 ; x1 x2

(3)

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"

! !# U U xL1 xL1 xU xL1 xL1 xU 1 x1 1 x1 x1 Cx2 ¼ min L ; U ; L ; U ; max L ; U ; L ; U x2 x2 x2 x2 x2 x2 x2 x2

(4)

3. Shortcomings of an existing method The study by Razavi Hajiagha et al. (2013) conducted on multi-objective linear problems with interval coefficients has some shortcomings. In their study, the multi-objective interval model was transformed into the following equations, and the sum of membership functions of the objectives was maximised: max

k X

m~ l ðxÞ

(5)

l¼1

0

n X j¼1

m~ l ðxÞp 1 1

p B C a~ ij x~ j @ ¼ Ab; X X~ j X0;

i ¼ 1; 2; . . .; m

j ¼ 1; 2; . . .; n

m~ l ðxÞ: unrestricted;

l ¼ 1; 2; . . .; k

(6)

(7)

(8) (9)

In constraint (6), it has been mentioned that the value of m~ l ðxÞ is equal to or less than 1, while in Equation (9), it has been mentioned that m~ l ðxÞ is unrestricted in sign. Therefore, considering constraint (6), constraint (9) is redundant. Now, the following counterexample adopted from the paper by Razavi Hajiagha et al. (2013) is explored: maxZ 1 ¼ ½1; 3~x 1 þ ½1; 1:5~x 2

(10)

maxZ 2 ¼ ½0:5; 2~x 1 þ ½1:5; 1~x 2

(11)

½1; 2~x 1 þ ½1:5; 3~x 2 p ½4; 6

(12)

½1; 3~x 1 þ ½2:5; 3:5~x 2 p12

(13)

x~ 1 ; x~ 2 X0

(14)

s.t.:

According to the paper by Razavi Hajiagha et al. (2013), the problem stated in Equations (10)-(14) needs to be transformed into two single-objective problems and solved.

Grey-fuzzy solution for MOLP

GS

Problem (1): maxZ 1 ¼ ½1; 3~x 1 þ ½1; 1:5~x 2

(15)

½1; 2~x 1 þ ½1:5; 3~x 2 p ½4; 6

(16)

½1; 3~x 1 þ ½2:5; 3:5~x 2 p12

(17)

x~ 1 ; x~ 2 X 0

(18)

maxZ 2 ¼ ½0:5; 2~x 1 þ ½1:5; 1~x 2

(19)

½1; 2~x 1 þ ½1:5; 3~x 2 p ½4; 6

(20)

½1; 3~x 1 þ ½2:5; 3:5~x 2 p12

(21)

x~ 1 ; x~ 2 X 0

(22)

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s.t.:

Problem (2):

s.t.:

Problem (1) will be transformed into the following form according to the method presented by Razavi Hajiagha et al. (2013): maxZ L ¼ x 1 x 2

(23)

1 1 3 3 maxZ C ¼ x 1  x 2 þ x1 þ x2 2 2 2 4

(24)

2x1 þ3x2 p6

(25)

1 3 3 x þ x þx1 þ x2 p5 2 1 4 2 2

(26)

7 3x1 þ x2 p12 2

(27)

1 5 3 7 x þ x þ x1 þ x2 p12 2 1 4 2 2 4

(28)

x 1 ; x1 ; x 2 ; x2 X0

(29)

s.t.:

If problem (1) is correctly solved using software LINGO 11, and considering the same weight for objective functions as 1/2(ZL+ZC), the solution is shown as: x~ n1 ¼ ½10; 0

(30)

x~ n2 ¼ ½0;0

(31)

n Z~ 1 ¼ ½0;30

(32)

Grey-fuzzy solution for MOLP

But the solution presented by Razavi Hajiagha et al. (2013) for problem (1) is as given below which is not correct:

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n Z~ 1 ¼ ½3;9

(33)

It is seen that the solution obtained in Equation (33) is completely different from the solution presented by Razavi Hajiagha et al. (2013). Also, in the solution obtained (Equation (30)), the lower bound for the variable x~ n1 is higher than the upper bound and this is not logical. In addition, problem (1) was also solved using LP Metric method and the solutions obtained are presented in Table I. As seen, again, the method does not have a good performance, and in the solutions obtained, for the variable x~ n1 , the lower bound values are always greater than the upper bound values and this result indicates that the example solved in the paper by Razavi Hajiagha et al. (2013) is itself a counterexample for their method. It should be noted that Razavi Hajiagha et al. (2012) had solved problem (1) and provided solutions presented in the following equations and these values were not correct and were outside the feasible region of the problem (Mahmoudi et al., 2018): x~ n1 ¼ ½10;10

(34)

x~ n2 ¼ ½0;0

(35)

n Z~ 1 ¼ ½0;6

(36)

Now, the second problem which was in Euations (19)-(22) is explored. The solution presented by Razavi Hajiagha et al. (2013) for the second problem is seen in the following equations: n Z~ 2 ¼ ½1:5;6

(37)

After solving problem (2) precisely using LINGO 11, and considering equal weight for objective functions in the form of 1/2 (ZL+ZC), we will have the following solutions:

Variables n

x~ 1 x~ 2 n n Z~

x~ n1 ¼ ½10;0

(38)

x~ n2 ¼ ½0;0

(39)

n Z~ 1 ¼ ½0;20

(40)

LP ¼ 1

LP ¼ 2

LP ¼ 30

LP ¼ ∞

[10,0] [0,0] [0,30]

[9.22,0.38] [0,0] [0.38,27.66]

[8.76,0.61] [0,0] [0.61,26.28]

[8.33,0.83] [0,0] [0.83,24.99]

Table I. Solution to problem (1) using LP metric method

GS

As can be seen, the solution obtained by the authors of the present study is different from the solution by Razavi Hajiagha et al. (2013). In addition, the lower bounds obtained for the variable x~ 1 n are greater than the upper bounds of this variable, which indicates that the method by Razavi Hajiagha et al. (2013) is unacceptable. The following counterexample too is adopted from the study by Razavi Hajiagha et al. (2013):     5 17 1 1 5 x~ 1 þ  ; x~ 2  (41) max ; 18 18 2 36 6

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s.t.: ½1; 3~x 1 þ ½1; 1:5~x 2 X3

(42)

½0:5; 2~x 1 þ ½1:5; 1~x 2 X1:5

(43)

½1; 2~x 1 þ ½1:5; 3~x 2 X ½4; 6

(44)

½1; 3~x 1 þ ½2:5; 3:5~x 2 p12

(45)

x~ 1 ; x~ 2 X 0

(46)

In order to solve the problem stated in Equations (41)-(46), it is transformed into following multi-objective model: 5 1 5 x1 x2 18 2 6

(47)

5 1 17 1 5 x 1  x 2 þ x1 þ x2  36 4 36 72 6

(48)

maxZ L ¼

maxZ C ¼ s.t.:

x1 þx2 p3

(49)

3 3 1 1  x 1  x 2  x1 þ x2 p3 2 4 2 2

(50)

1 3 3  x1 þ x2 p 2 2 2

(51)

1 1 3 3 x 1 þ x 2  x1 þ x2 p 2 4 4 2

(52)

2x1 þ3x2 p6

(53)

1 3 3 x þ x þx1 þ x2 p5 2 1 4 2 2

(54)

7 3x1 þ x2 p 12 2

(55)

1 5 3 7 x þ x þ x1 þ x2 p12 2 1 4 2 2 4

(56)

x 1 ; x1 ; x 2 ; x2 X0

(57)

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After solving precisely, the problem expressed in Equations (47)-(57) using LINGO 11, the solutions obtained are as: x~ n1 ¼ ½4;3

(58)

x~ n2 ¼ ½0;0

(59)

The solution presented for the problem expressed by Razavi Hajiagha et al. (2013) that was expressed in Equations (41)-(46) is as follows: x~ n1 ¼ ½3;3

(60)

x~ n2 ¼ ½0;0

(61)

n

As seen, the solution obtained for the variable x~ 1 by the authors of the present paper is different from the solution presented by Razavi Hajiagha et al. (2013). In addition, in the solution obtained by the authors of the present study, the lower bound for the variable x~ n1 is greater than the upper bound which indicates the inappropriate performance of the method presented by Razavi Hajiagha et al. (2013). It will be shown below that in another point of the feasible region of Equations (25)-(29) of the method by Razavi Hajiagha et al. (2013), again the lower bound becomes greater than the upper bound. As Razavi Hajiagha et al. (2013) used the approach in the study by Razavi Hajiagha et al. (2012) that was presented for solving grey linear models, the definition of interval grey number is presented here: Definition 1. Liu et al. (2012) and Chatterjee and Chatterjee (2012) have defined interval grey number as:
 (62) A a; a ; a oa Now, the following steps are followed for the examination the problems of method by Razavi Hajiagha et al. (2013): (1) First, it is supposed that: x1 ¼ x 1 þe1

(63)

x2 ¼ x 2 þe2

(64)

e1 ; e2 40

(65)

(2) The hypotheses presented in Equations (63)-(65) are inserted in inequalities (26) and (28):  3 1 3 x 1 þ x 2 þ x 1 þe1 þ x 2 þe2 p5 (66) 2 4 2 7 1 5 3 (67) x 1 þ x 2 þ x 1 þe1 þ x 2 þe2 p12 2 4 2 4

Grey-fuzzy solution for MOLP

GS

(3) Then, the constraints mentioned in Equations (66) and (67) are turned into the following equations using slack variables:  3 1 3 x 1 þ x 2 þ x 1 þe1 þ x 2 þe2 þs1 ¼ 5 2 4 2  1 5 3 7 x 1 þ x 2 þ x 1 þe1 þ x 2 þe2 þs2 ¼ 12 2 4 2 4

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s1 ; s2 X0

(68) (69) (70)

After solving the system formed by Equations (65) and (70) using software MATLAB 2016, we will have the following results: 7 3 9 37 (71) e1 ¼ s1 3s2  x 1  x 2 þ 2 4 8 2 1 3 e2 ¼ 3s1 þ2s2  x 1  x 2 9 (72) 2 4 It is clear that the variables x 1 , x 2 , s1 and s2 can all take values greater than or equal to 0. If we consider for all of them the value of 0 in Equation (72) as a point of feasible region, we will have: (73) e2 ¼ 9 The result obtained in Equation (73) indicates that, for the variable x~ n2 , the lower bound has become greater than the upper bound and the hypothesis expressed in Equation (65) is violated. 4. Grey-fuzzy solution for interval LP models In this section, we present a proposed method for solving interval multi-objective problems using positioned programming and fuzzy MOLP approaches. Linear programming with interval coefficients is expressed in the form of the following model Gabrel et al. (2008): n h i X cLj ; cU (74) maxZ ¼ j xj j¼1

s.t.: n h i h i X L U aLij ; aU ij xj p bi ; bi j¼1

xj X0 j ¼ 1; . . .; n: i ¼ 1; . . .; m: To find the solution for model (74), we calculate the best and worst solutions. In this case, surely the solution will be in this interval. To find the lower bound of solution, we need to solve the following model: n X maxZ ¼ cLj xj (75) j¼1

s.t.: n X

L aU ij xj pbi

j¼1

xj X0 j ¼ 1; . . .; n:

i ¼ 1; . . .; m:

To achieve the upper bound of the solution, we also need to solve the following model: n X maxZ ¼ cU (76) j xj j¼1

s.t.: n X

aLij xj pbU i

j¼1

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xj X0; n

j ¼ 1; . . .; n; nL

nU

i ¼ 1; . . .; m:

n

The result obtained is Z ¼ [Z , Z ] and xj ¼ ½xnj L ; xnj U ; problems are expressed in the form of the following model: n h i X maxðminÞZ 1 ¼ cLj ; cU j xj

j ¼ 1; . . .; n. The MOLP (77)

j¼1

maxðminÞZ 2 ¼

n h X

i cLj ; cU j xj

j¼1

⋮ maxðminÞZ S ¼

n h i X cLj ; cU j xj j¼1

s.t.: n h i h i X L U aLij ; aU ij xj p ð X Þ bi ; bi j¼1

xj X0;

j ¼ 1; . . .; n;

i ¼ 1; . . .; m:

To solve the aforementioned multi-objective problems, we first need to solve the problem separately with each objective. As an interactive and appropriate fuzzy approach to handle practical MOLP models, we use Alavidoost et al. (2016) approach. Then, we will define the proper membership functions for all objective functions. If the objective function has the form of maximum, the proper membership function is obtained using the following equation (Bagherpour et al., 2012): 8 > 1 if Z X XZ U > X > < L U ZX  ZXL if Z pZ (78) mðZ X Þ ¼ Z X U Z X L X pZ X X > > > L :0 if Z X pZ X If the objective function is the minimum type, the proper membership function is obtained using the following equation (Bagherpour et al., 2012): 8 1 if Z X pZ LX > > > < Z U Z L U X (79) mðZ X Þ ¼ Z XU Z L if Z X pZ X pZ X X X > > > :0 if Z X XZ U X

Grey-fuzzy solution for MOLP

GS

In Equations (78) and (79), the parameters are defined as: •

ZX: the objective function.

•

ZU X : upper bounds of optimal solution for objective function.

•

Z LX : lower bounds of optimal solution for objective function.

After defining the proper membership functions, we need to combine the objectives. The model used to aggregate the objectives is shown as (Alavidoost et al., 2016): K h i X WLk ; WU (80) maxlðxÞ ¼ l0 þd k lk Downloaded by UNIVERSITY OF TOLEDO LIBRARIES At 08:12 25 May 2018 (PT)

k¼1

s.t.:

h i h i L U ; WLk ; WU k l0 þlk p mk ðxÞ ; mk ðxÞ lk ; l0 A ½0; 1;

k ¼ 1; 2; . . .; K

k ¼ 1; 2; . . .; K

x AF ðxÞ The parameters and variables used in Equation (80) are presented in Table II. In order to solve the proposed model in model (80), the positioned programming approach should be used. Now, we review the positioned programming approach (Mahmoudi and Feylizadeh, 2018). If the general form of the interval linear programming model is expressed in model (74), we will have the following definition: Definition 2. Assume that the values of ρj, βj and δij for i ¼ 1, …, m and j ¼ 1, …, n are in the closed interval [0,1], so the following linear combinations are (Liu et al., 2009):  L (81) cj ¼ rj cU j þ 1rj cj ; j ¼ 1; 2; . . .; n   L U (82) bj ¼ bj bj þ 1bj bj ; i ¼ 1; 2; . . .; m   L aij ¼ dij aU ij þ 1dij aij ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n

(83)

If we substitute Equations (81)-(83) into Equation (80), we will have the following model: maxlðxÞ ¼ l0 þd

K  X

  L rj WU k þ 1rj Wk lk

(84)

k¼1

Variables and parameters

Table II. Variables and parameter of proposed model

λ0 δ ϑk λk μk(x) F(x)

The minimum level of satisfaction from the objective function (between 0 and 1) A small positive number, usually 0.01 The weight of the kth objective function, which can be calculated using a variety of methods The satisfaction level variable of the kth objective function (between 0 and 1) The membership function for the kth objective function A feasible region that includes all constraints of the original problem

s.t.: dij WU k l0 þ

    1dij WLk l0 þlk pbj mk ðxÞU þ 1bj mk ðxÞL

Grey-fuzzy solution for MOLP

k ¼ 1; 2; . . .; K

lk ; l0 ; rj ; dij ; bj A ½0; 1 x A F ðxÞ

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Theorem 1. In model (84), for each ρ, β and δ in the closed interval [0,1], we have the following equation (Liu et al., 2009): maxlL p maxlðr; b; dÞp maxlU

(85)

For ( ρ, β, δ) ¼ (1, 1, 0) and ( ρ, β, δ) ¼ (0, 0, 1), maxλU and maxλL are obtained, respectively, which are ideal, and critical values of are the desired problem (Liu and Lin, 2006). The solution to this problem will be taken into account with all uncertainties in this interval. According to the management’s view, crisp solutions of the problem can also be obtained for different values of ( ρ, β, δ), hence we use positioned programming of grey theory. 5. Algorithm for solving the proposed method Based on the previous sections, the following algorithm is proposed for solving the interval MOLP: (1) Based on Equations (75) and (76), solve the problem separately for each objective function. The upper and lower bounds are obtained for each problem. (2) Provide the proper membership function for each objective function. (3) Calculate the weight of each objective function according to its importance. (4) Based on the data obtained, write model (84). (5) Solve the problem in modes ( ρ, β, δ) ¼ (1, 1, 0) and ( ρ, β, δ) ¼ (0, 0, 1). According to the management’s view, crisp solutions of the problem can also be obtained for different values of ( ρ, β, δ). 6. Numerical example Consider the problem expressed in the following equations: maxZ 1 ¼ ½1;3x1 þ ½1;1:5x2

(86)

maxZ 2 ¼ ½0:5;2x1 þ ½1:5; 1x2

(87)

½1; 2x1 þ ½1:5; 3x2 p ½4;6

(88)

½1; 3x1 þ ½2:5; 3:5x2 p 12

(89)

x1 ; x2 X0

(90)

s.t.:

First, we solve problem (1) based on Equations (75) and (76). The solution obtained for the problem (1) is as follows: xn1 ¼ ½2;6

(91)

xn2 ¼ ½0;0

(92)

Z n1 ¼ ½2;18

GS

(93)

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For the second problem, we will have the following solution: xn1 ¼ ½2;6

(94)

xn2 ¼ ½0;0

(95)

Z n2 ¼ ½1;12

(96)

Now we write a proper membership for each objective function. For the objective function Z1, we have the following membership function: 8 1 if ½1;3x1 þ ½1; 1:5x2 X18 > > < ½1; 3x þ ½1; 1:5x 2 1 2 if 2 p ½1; 3x1 þ ½1; 1:5x2 p18 (97) mðZ 1 Þ ¼ 182 > > : 0 if ½1;3x1 þ ½1;1:5x2 p 2 For the objective function Z2, we have the following membership function: 8 1 if ½0:5; 2x1 þ½1:5; 1x2 X 12 > > < ½0:5; 2~x þ ½1:5; 1~x 1 1 2 if 1 p ½0:5; 2x1 þ ½1:5; 1x2 p12 mðZ 2 Þ ¼ > 121 > : 0 if ½0:5; 2x1 þ ½1:5; 1x2 p1 Finally, the following model will be obtained using Equation (84):     maxlðxÞ ¼ l0 þ ð0:01Þ  r1  0:5 þ 1r1  0:5 l1      þ r2  0:5 þ 1r2  0:5 l2

(98)

(99)

s.t.:



   3 1:5 2 1 1 2 þ 1b1  0:5d11 l0 þ ð1d11 Þ0:5l0 þl1 pb1  x1 þ x2  x1  x2  16 16 16 16 16 16

0:5d21 l0 þ ð1d21 Þ0:5l0 þl2 pb2 



   2 1 1 0:5 1:5 1 x1  x2  þ 1b2  x1  x2  11 11 11 11 11 11

  2d31 x1 þ ð1d31 Þx1 þ3d32 x2 þ1:5ð1d32 Þx2 p6b3 þ4 1b3   3d41 x1 þ ð1d41 Þx1 þ3:5d42 x2 þ2:5ð1d42 Þx2 p12b4 þ12 1b4 lk ; l0 ; rj ; dij ; bj A ½0; 1 x AF ðxÞ The solution of model (99) is presented in Table III.

7. Practical example In this part, we investigate a practical example in project management. Consider the project network in Figure 1. The information of activities in desired arrow network of project is shown in Table IV. We want to crash the duration of the project to [10, 16]. The interval model shown in the following equation expresses this issue (Mahmoudi and Feylizadeh, 2017):

Grey-fuzzy solution for MOLP

minZ 1 ¼ ½200; 250Y 12 þ ½450; 460Y 13 þ ½150; 210Y 24 þ ½190; 195Y 25

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þ ½205; 220Y 35 þ ½280; 300Y 46 þ½310; 325Y 56

(100)

minZ 2 ¼ X 6 X 1 s.t.: Y 12 p ½1; 2 Y 13 p ½2; 3 Variables λ0 λ1 λ2 x1 x2

Lower bounds

Upper bounds

0.0000 0.0000 0.0000 2.0000 0.0000

1.0000 0.5000 0.5000 6.0000 0.0000

2

4

6

1

1-2 1-3 2-4 2-5 3-5 4-6 5-6

Figure 1. Arrow network of the practical example

5

3

Activity

Table III. The solution of numerical example of model (99)

Original duration

Crashed duration

Direct cost per unit time for activity

[2,3] [5,7] [4,6] [6,7] [3,4] [5,5.5] [7,10]

[1,1] [3,4] [3,3.25] [4,5] [1,2] [4,4] [5,6]

[200,250] [450,460] [150,210] [190,195] [205,220] [280,300] [310,325]

Table IV. The information of activities in project arrow network of Figure 1

Y 24 p ½1;2:75

GS

Y 25 p ½2;2 Y 35 p ½1;2 Y 46 p ½1;1:5

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Y 56 p ½2;4 X 2 XX 1 þ ½2;3Y 12 X 3 XX 1 þ ½5;7Y 13 X 4 XX 2 þ ½4;6Y24 X 5 XX 2 þ ½6;7Y 25 X 5 XX 3 þ½3;4Y 35 X 6 XX 4 þ ½5;5:5Y 46 X 6 XX 5 þ ½7;10Y56 X 6 p ½10;16 Y 12 ; Y 13 ; Y 24 ; Y 25 ; Y 35 ; Y 46 ; Y 56 X0 X 1 ; X 2 ; X 3 ; X 4 ; X 5 ; X 6 X0 After solving the interval model (100) using the proposed approach of this paper, the results are presented in Table V. Variable

Table V. Optimal solution of the practical example using proposed model (100)

λ0 λ1 λ2 Y12 Y13 Y24 Y25 Y35 Y46 Y56

Critical optimal value

Ideal optimal value

0.0000 0.3582 0.0000 0.0000 2.0000 0.0000 1.0000 1.0000 0.0000 2.0000

1.0000 0.8731 0.1666 1.0000 0.0000 0.0000 0.0000 1.0000 0.0000 4.0000

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8. Conclusion In this paper, we proposed an algorithm for solving MOLP problem with interval coefficients using positioned programming and fuzzy MOLP. In this method, we modified the shortcomings of a solving method, in which some variables solutions have the interval with lower bounds greater than their upper bounds. In addition, in our method, one can determine different weights for each objective function using interval linguistic variables. Moreover, using interval coefficients in the original problem, interval solution for the variables will be obtained. This method has the capability of producing crisp solution if decision maker wants this type of solution for the variables. According to the mentioned example, in this paper, the proposed method is both simple in use and suitable for solving different problems. References Alavidoost, M.H., Babazadeh, H. and Sayyari, S.T. (2016), “An interactive fuzzy programming approach for bi-objective straight and U-shaped assembly line balancing problem”, Applied Soft Computing, Vol. 40, pp. 221-235. Allahdadi, M. and Nehi, H.M. (2011), “Fuzzy linear programming with interval linear programming approach”, Advanced Modeling and Optimization, Vol. 13 No. 1, pp. 1-12. Allahdadi, M. and Mishmast Nehi, H. (2013), “The optimal solution set of the interval linear programming problems”, Optimization Letters, Vol. 7 No. 8, pp. 1893-1911. Allahviranloo, T. and Ghanbari, M. (2012), “On the algebraic solution of fuzzy linear systems based on interval theory”, Applied Mathematical Modelling, Vol. 36 No. 11, pp. 5360-5379. Bagherpour, M., Feylizadeh, M.R. and Cioffi, D.F. (2012), “Time, cost, and quality trade-offs in material requirements planning using fuzzy multi-objective programming”, Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, Vol. 226 No. 3, pp. 560-564. Birge, J.R. and Louveaux, F. (2011), Introduction to Stochastic Programming, Springer Science & Business Media. Chakrabortty, S., Pal, M. and Nayak, P.K. (2010), “Multi section technique to solve interval-valued purchasing inventory models without shortages”, Journal of Information and Computing Science, Vol. 5 No. 3, pp. 173-182. Chanas, S. and Kuchta, D. (1996), “Multiobjective programming in optimization of interval objective functions – a generalized approach”, European Journal of Operational Research, Vol. 94 No. 3, pp. 594-598. Chatterjee, P. and Chatterjee, R. (2012), “Supplier evaluation in manufacturing environment using compromise ranking method with grey interval numbers”, International Journal of Industrial Engineering Computations, Vol. 3 No. 3, pp. 393-402. Chinneck, J.W. and Ramadan, K. (2000), “Linear programming with interval coefficients”, Journal of the Operational Research Society, Vol. 51 No. 2, pp. 209-220. Cho, G.M. (2005), “Log-barrier method for two-stage quadratic stochastic programming”, Applied Mathematics and Computation, Vol. 164 No. 1, pp. 45-69. Dubey, D., Chandra, S. and Mehra, A. (2012), “Fuzzy linear programming under interval uncertainty based on IFS representation”, Fuzzy Sets and Systems, Vol. 188 No. 1, pp. 68-87. Gabrel, V., Murat, C. and Remli, N. (2008), “Best and worst optimum for linear programs with interval right hand sides”, Modelling, Computation and Optimization in Information Systems and Management Sciences, Springer, Berlin and Heidelberg. Ishibuchi, H. and Tanaka, H. (1990), “Multiobjective programming in optimization of the interval objective function”, European Journal of Operational Research, Vol. 48 No. 2, pp. 219-225. Javed, S.A. and Liu, S. (2017), “Evaluation of project management knowledge areas using grey incidence model and AHP”, 2017 International Conference on Grey Systems and Intelligent Services, IEEE, August, pp. 120-120. Javed, S.A. and Liu, S. (2018), “Predicting the research output/growth of selected countries: application of even GM (1, 1) and NDGM models”, Scientometrics, Vol. 115 No. 1, pp. 395-413.

Grey-fuzzy solution for MOLP

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GS

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1. MahmoudiAmin, Amin Mahmoudi, FeylizadehMohammad Reza, Mohammad Reza Feylizadeh. A grey mathematical model for crashing of projects by considering time, cost, quality, risk and law of diminishing returns. Grey Systems: Theory and Application, ahead of print. [Abstract] [Full Text] [PDF]