Grey Systems: Theory and Application A note on “a multi-objective programming approach to solve grey linear programming” Amin Mahmoudi, Mohammad Reza Feylizadeh, Davood Darvishi,
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Article information: To cite this document: Amin Mahmoudi, Mohammad Reza Feylizadeh, Davood Darvishi, (2018) "A note on “a multi-objective programming approach to solve grey linear programming”", Grey Systems: Theory and Application, Vol. 8 Issue: 1, pp.35-45, https://doi.org/10.1108/GS-08-2017-0027 Permanent link to this document: https://doi.org/10.1108/GS-08-2017-0027 Downloaded on: 05 February 2018, At: 08:42 (PT) References: this document contains references to 8 other documents. To copy this document:
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A note on “a multi-objective programming approach to solve grey linear programming”
Grey linear programming
35
Amin Mahmoudi and Mohammad Reza Feylizadeh Department of Industrial Engineering, Islamic Azad University Shiraz, Shiraz, Iran, and
Received 8 August 2017 Revised 5 October 2017 Accepted 6 October 2017
Davood Darvishi
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Department of Mathematics, Payame Noor University, Tehran, Iran Abstract Purpose – The purpose of this paper is to examine the shortcomings and problems associated with the method proposed by Razavi Hajiagha et al. (2012). Design/methodology/approach – A multi-objective approach is proposed to solve the grey linear programming problems. In this method, the grey linear problem is converted into a multi-objective problem and then solved. Findings – According to the numerical example presented in the study by Razavi Hajiagha et al. (2012), this method does not have a correct solution because the solution does not satisfy the constraints and the upper bounds of the variables are equal or less than their lower bound. Originality/value – In recent years, various methods have been proposed for solving grey linear programming problems. Razavi Hajiagha et al. (2012) proposed a multi-objective approach to solve grey linear programming problems, but this method does not have a correct solution and using this method in other researches studies can reduce the value of the grey system theory. Keywords Grey systems, Grey linear programming, Multi-objective approach Paper type Technical paper
1. Introduction The grey system theory was introduced in 1982 by a Chinese researcher named Deng (Liu and Lin, 2010). The grey system theory is a very suitable method for dealing with unknown and incomplete information. When information is completely clear and transparent in a system, it forms a system with white colour, while it forms a black system if the information is completely unknown. In the real world, information is not always white or black, but is a mixture of the two, i.e. the grey colour. These systems are called grey systems (Liu and Lin, 2010). Linear programming problems with grey parameters are defined in the following model (Liu et al., 2009): M ax S ¼ C ðÞX
(1)
s.t.: AðÞX pbðÞ X X0 This problem is defined as a linear programming problem with grey parameters (LPGP) in which C(⊗) is the grey cost index, A(⊗) is grey consumption index matrix, b(⊗) is grey constraint index for resource consumption, and X is the decision-making variable of the LPGP problem.
Grey Systems: Theory and Application Vol. 8 No. 1, 2018 pp. 35-45 © Emerald Publishing Limited 2043-9377 DOI 10.1108/GS-08-2017-0027
GS 8,1
36
There are several approaches for solving grey linear programming problems. Liu et al. (2009) presented a method called positioned programming for solving grey problems. In this method, an LPGP was considered as a set of common linear programming problems (Liu et al., 2009). Razavi Hajiagha et al. (2012) presented a method for solving grey linear programming problems using a multi-objective approach. In this method, the grey linear problem is converted into the following model and then solved: (2) M in Z ðxÞ; Z C ðxÞ f s.t.:
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LHS i p bi ðLH S i ÞC p bC x 1 ; x1 ; x 2 ; x2 X 0 Li et al. (2014) in a study presented a general formula for calculating the inverse of the grey matrix and used the results to solve grey linear programming problem. Nasseri et al. (2016) presented a new approach to solve linear programming problem with grey cost coefficients based on the simplex algorithm. In the next section, we intend to explore the shortcomings of Razavi Hajiagha et al.’s (2012) method. 2. Errors in “a multi-objective programming approach to solve grey linear programming” The example in the following equations is taken from the paper by Razavi Hajiagha et al. (2012): Max ½1; 3 x1 þ ½1; 1:5 x2
(3)
½1; 2 x1 þ ½1:5; 3 x2 p ½4; 6
(4)
½1; 3 x1 þ ½2:5; 3:5 x2 p 12
(5)
x1 ; x2 X 0
(6)
s.t.:
According to the paper by Razavi Hajiagha et al. (2012), for solving the problem, it is converted into the following equations: h i (7) M ax x 1 x 2 ; 3x1 þ1:5x2 LH S 1 : 2x1 þ 3x2 p 6
(8)
1 3 ðLH S 1 ÞC : x 1 þ x 2 þx1 þx2 p 5 2 4
(9)
7 LHS 2 : 3x1 þ x2 p 6 2
(10)
1 5 3 7 ðLHS 2 ÞC : x 1 þ x 2 þ x1 þ x2 p 5 2 4 2 4
(11)
x 1 ; x1 ; x 2 ; x2 X 0
(12)
Grey linear programming
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Then, based on the proposed approach by Razavi Hajiagha et al. (2012), the above problem is turned into a multi-objective problem, as shown in the following equations: M ax Z L ¼ x 1 x 2
(13)
1 1 3 3 M ax Z C ¼ x 1 x 2 þ x1 þ x2 2 2 2 4
(14)
s.t.:
x 1 ; x 2 ; x1 ; x2 AX
(15)
Equations (13)-(15) were solved by Razavi Hajiagha et al. (2012) and the following solution (Equations (16)-(18)) was obtained with the combination of objective functions ZL and ZC i.e. 1/2(ZL + ZC): xn1 ¼ ½10; 10
(16)
xn2 ¼ ½0; 0
(17)
Z n ¼ ½0; 6
(18)
If the values xn1 and xn2 are inserted in Equations (8)-(11), it is seen that the obtained optimal solution is not in the feasible region and does not satisfy constraints of the problem. Also, the values xn1 and xn2 are inserted in Equation (3) to calculate the value of the objective function, the value Z* ¼ [10, 30] is obtained which is completely different from the solution presented by Razavi Hajiagha et al. (2012). These results are presented in Table I. If the values xn1 and xn2 are inserted in Equations (3)-(6), it is seen that the optimal solution obtained is out of the feasible region and does not satisfy constraints of the problem either. These results are presented in Table II.
Razavi Hajiagha et al. (2012) model
Razavi Hajiagha et al. (2012) Substituting the optimal solution in optimal solution Razavi Hajiagha et al. (2012)
xn1 ¼ ½10; 10 Max [1, 3]⊗x1+[−1, 1.5]⊗x2 n LH S 1 : 2x1 þ 3x p 6 x 2 2 ¼ ½0; 0 LH S 2 : 3x1þ 7=2 x p 6 2 ðLH S 1 ÞC : 1=2x 1 þ 3=4x 2 þ x1 þ x2 p 5 ðLH S2 ÞC : 1=2 x 1 þ 5=4 x 2 þ 3=2 x1 þ 7=4 x2 p 5
Max [1, 3] × [10, 10]+[−1, 1.5] × [0, 0] ≠ [0, 6] LH S 1 : 2 10 þ 3 0 Ü 6 LH S 2 : 3 10 þ 7=2 0 Ü 6 ðLH S 1 ÞC : 1=2 10 þ 3=4 0 þ 10 þ 0 Ü 5 ðLH S2 ÞC : 1=2 10 þ 5=4 0 þ 3=2 10 þ 7=4 0 Ü 5
Table I. Substitution of the solution in the proposed approach by Razavi Hajiagha et al. (2012)
GS 8,1
It can be seen that the solution presented in Equations (16)-(18) was incorrect. We solved the multi-objective model presented in Equations (13)-(15) using the proposed approach by Razavi Hajiagha et al. (2012) and the obtained solution is presented in the following equations:
38
xn1 ¼ ½10; 0
(19)
xn2 ¼ ½0; 0
(20)
Z n ¼ ½0; 30
(21)
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As can be seen, the proposed solution is not logical because the lower bound of xn1 is greater than the upper bound value. After further investigation, it is observed that Equations (9)-(11) are false and should be corrected as the following equations: h i M ax x 1 x 2 ; 3x1 þ1:5x2 (22) LH S 1 : 2x1 þ 3x2 p 6
(23)
7 LHS 2 : 3x1 þ x2 p 12 2
(24)
1 3 3 ðLH S 1 ÞC : x 1 þ x 2 þx1 þ x2 p 5 2 4 2
(25)
1 5 3 7 ðLH S 2 ÞC : x 1 þ x 2 þ x1 þ x2 p 12 2 4 2 4
(26)
x 1 ; x1 ; x 2 ; x2 X 0
(27)
We solved the problem stated in Equations (22)-(27) using the proposed approach by Razavi Hajiagha et al. (2012) by combining the objective functions 1/2(ZL + ZC) and the solution is obtained in the following equations:
Table II. Substitution of the solution in Example (1) of Razavi Hajiagha et al. (2012)
Example (1) of Razavi Hajiagha et al. (2012) model Max [1, 3] ⊗ x1+[−1, 1.5] ⊗ x2 [1, 2]⊗x1+[1.5, 3]⊗x2 ⩽ [4, 6] [1, 3]⊗x1+[2.5, 3.5]⊗x2 ⩽ 12
xn1 ¼ ½10; 0
(28)
xn2 ¼ ½0; 0
(29)
Z n ¼ ½0; 30
(30)
Example (1) of Razavi Hajiagha Substituting the optimal solution in et al. (2012) optimal solution Example (1) of Razavi Hajiagha et al. (2012) xn1 ¼ ½10; 10 xn2 ¼ ½0; 0
Max [1, 3] × [10, 10] + [−1, 1.5] × [0,0] ≠ [0, 6] [1, 2] × [10, 10] + [1.5, 3] × [0, 0] ≮ [4, 6] [1, 3] × [10, 10] + [2.5, 3.5] × [0, 0] ≮ 12
Again the interval for x1 is not correct. For further investigation, the authors of the present paper solved the problem presented in Equations (22)-(27) using the LP-Metric method. The solution is summarised in Table III. The obtained solutions for the various values of p are not still correct since the intervals are incorrect (the lower bound of xn1 has become greater than its upper bound) and, thus, the Razavi Hajiagha et al.’s (2012) method does not have an appropriate performance. 3. Causes of errors and the appropriate solutions In this section, the reasons responsible for generating errors in the method proposed by Razavi Hajiagha et al. (2012) are presented. In the study by Razavi Hajiagha et al. (2012), the variables are divided into three different subcategories:
Grey linear programming
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(1) k+ subcategory includes variables whose coefficients of upper and lower bounds are both positive; (2) k− subcategory includes variables whose coefficients of upper and lower bounds are both negative; and (3) k0 subcategory includes variables whose coefficients of upper and lower bounds have different signs, or, in other words, the interval includes zero. In addition, in the research study of Razavi Hajiagha et al. (2012), the grey linear objective function is considered in the following form: CU X ¼
n X
cj U xj ¼
X jAk
j¼1
þ
cjþ U xjþ þ
X
c j U xj þ
jAk
X
c0j U x0j (31)
0
jAk
Razavi Hajiagha et al. (2012), using grey multiplication, have presented the following equations: h i (32) if jA k þ - cjþ U xjþ ¼ c jþ x jþ ; cjþ xjþ h i if jA k - c j U xj ¼ c j xj ; cj x j
(33)
h i if j A k0 - c0j U x0j ¼ c 0j x 0j ; c0j x0j
(34)
Razavi Hajiagha et al. (2012), by substituting Equations (32)-(34) into Equation (31), have presented the objective function as shown in the following equation: 2 P þ þ P P 0 0 3 cj xj þ c j xj þ cj xj ; j A k j A k0 6 jAkþ 7 7 (35) Z ðxÞ ¼ CU X ¼ 6 4 P c þ x þ þ P c x þ P c0 x0 5 jAkþ
Variables n
x1 xn2 ⊗Z *
j
j
j A k
j
j
j A k0
j j
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 III. The solution of Example (1) using LP-Metric method
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GS 8,1
It is observed that Equation (34) is written incorrectly. Therefore, we consider the interval [−1, 1] ⊗ xj in which the signs of the upper and lower bounds are different and as a result the following equation is obtained: h i ½1; 1 xj ¼ x j ; xj (36)
40
Equation (36) is not correct and its correct form is shown in the following equation: ½1; 1 xj ¼ xj ; xj
(37)
The proof of the aforementioned subject is shown in the following equation: h i h i c j ; cj x j ; xj c j o0; cj 40; x j X 0; xj X 0
(38)
h
i h i h n o n oi c j ; cj x j ; xj ¼ Min c j x j ; c j xj ; cj x j ; cj xj ; Max c j x j ; c j xj ; cj x j ; cj xj h i ¼ c j xj ; cj xj
(39)
Therefore, the objective function proposed by Razavi Hajiagha et al. (2012) in Equation (35) needs to be corrected and its correct form is shown in the following equation: 2 P þ þ P P 0 0 3 cj xj þ c j xj þ c j xj ; j A k 6 jAkþ 7 j A k0 7 Z ðxÞ ¼ CU X ¼ 6 (40) 4 P c þ x þ þ P c x þ P c0 x 0 5 jAkþ
j
j
j A k
j
j
j A k0
j j
In the proposed objective function and constraints, Razavi Hajiagha et al. (2012) have used the following definitions: h i Definition 1. If x ¼ x; x and y ¼ y; y are two grey numbers, the order relation ⩽ LC is defined as shown in the following equations: x p LC y x o LC y
if x p y and xc p yc
(41)
if x p LC y and xa y
(42)
h i Definition 2. If x ¼ x; x and y ¼ y; y are two grey numbers, the order relation p nRC is defined as shown in the following equations: x p nRC y
if x p y and xc pyc
x p nRC y if x p nRC y and x a y
(43) (44)
The order relations expressed in Equations (41)-(44) that are used by Razavi Hajiagha et al. (2012) do not have an appropriate performance and cannot be used as a general order relation. Consider the following grey numbers for illustrating (Karmakar and Bhunia, 2012): A ¼ ½0; 10 ¼ o5; 54
(45)
B ¼ ½4; 8 ¼ o6; 2 4
(46)
Grey linear programming
According to the order relation expressed in Equation (43), the following equation is obtained: B p A and Ac p Bc - Ap nRC B or Bp nRC A
(47)
According to Equation (47), it is observed that the order relation expressed in Equation (43) fails to find the preferred grey number from ⊗A and ⊗B (see Figure 1). It is better to use another order relation for comparing grey numbers which are explored below:
41
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Definition 3. The greyness degree for the grey number ⊗ is calculated through the following equation (Liu et al., 2017): g 0 ðÞ ¼ mðÞ=mðOÞ
(48)
In Equation (48), the symbol Ω represents background of grey numbers and the symbol μ represents measurement for grey numbers. h i Definition 4. The kernel of grey number ⊗A∈ A; A is calculated through the following equation (Liu et al., 2017): ^ ¼ ⊗A
1 AþA 2
(49)
Definition 5. If ⊗A and ⊗B are two grey numbers, the order relations for these numbers will be as follows (Liu et al., 2017): ^ < ⊗B^ thus ⊗A< G ⊗B If ⊗A
(50)
^ ¼ ⊗B^ thus: If ⊗A
(51)
ð1Þ g o ð⊗AÞ ¼ g o ð⊗BÞ thus ⊗A ¼ G ⊗B
(52)
ð2Þ g o ð⊗AÞ < g o ð⊗BÞ thus ⊗A>G ⊗B
(53)
ð3Þ g o ð⊗AÞ > g o ð⊗BÞ thus ⊗A< G ⊗B
(54)
B
AC
0
BC
10 8
4
A
Figure 1. The ⊗A and ⊗B numbers and their comparison
GS 8,1
Based on the kernel of grey number expressed in Equation (49) for numbers ⊗A = [0, 10] and ⊗B = [4, 8] while they are all in the field of Ω∈[−1, 14], the following equation is obtained: ^ < ⊗B←5 ^ ⊗A
