Solving Multi-Level Programming Problem with Interval ... - IJIREM

International Journal of Innovative Research in Engineering & Management (IJIREM) ISSN: 2350-0557, Volume-3, Issue-4, July-2016

Solving Multi-Level Programming Problem with Interval Coefficients by Using Fuzzy Programming Samir A. Abass Atomic Energy Authority, Nuclear Research Center, Cairo, Egypt. P.O. Box 13759, Egypt.

Asmaa S. Abdallah Atomic Energy Authority, Nuclear Research Center, Cairo, Egypt. P.O. Box 13759, Egypt.

[email protected]

ABSTRACT This paper proposes a solution method for three level programming (TLP) problem with interval coefficients in both of objective functions and constraints. This method uses the concepts of tolerance membership function at each level to develop a fuzzy Max–Min decision model for generating Pareto optimal (satisfactory) solution. The first level decision maker (FLDM) specifies his/her objective functions and decisions with possible tolerances. Then, the second level decision-maker (SLDM) specifies his/her objective functions and decisions, in the view of the FLDM, with possible. Finally, the third level decision-maker (TLDM) uses the preference information for the FLDM and SLDM to solve his/her problem subject to the two upper level decision-makers restrictions. An illustrative numerical example is provided to clarify the proposed approach.

Keywords:

Multi-level programming, interval coefficients, fuzzy programming.

1. INTRODUCTION Three-level programming (TLP) problem is a class of multilevel programming (MLP) problem in which there are three independent decision makers [3, 5, 14, 16]. Each DM attempts to optimize its objective function and is affected by the actions of the others decision makers. TLP problem, whether from the stand point of the three-planner Stackelberg behavior or from the interactive organizational behavior, is a very practical problem and encountered frequently in actual practice. Several approaches have been used widely in searching for the optimal solutions of three level programming problems. The hybrid extreme-point search algorithm [4, 7], mixed-integer problem with complementary slackness [7], and the penalty function approach [2, 7]. Apart from these approaches, fuzzy sets have been employed to formulate and solve three-level non-linear multi-objective decision-making [12]. A multi-level linear programming problem with random rough coefficients in objective functions was presented by Emam et al [6]. In traditional mathematical programming, the coefficients of the problems are always treated as deterministic values. However in practice, it is very common for the coefficients values to be only approximately known. Hence, in order to develop good Operations Research methodology, fuzzy and stochastic approaches are frequently used to describe and treat uncertain elements present in a real decision problem. In fuzzy

programming, the constraints and goals are viewed as fuzzy sets and it is assumed that their membership functions are known. On the other hand, in stochastic programming, the coefficients are viewed as random variables and it is also assumed that their probability distributions are known. These membership functions and probability distributions play important roles in their corresponding methods. However, it is sometimes difficult to specify an appropriate membership function or accurate probability distribution in an uncertain environment. To overcome these difficulties, interval programming approaches [9] have appeared as a prominent tool for solving decision problems with interval parameter values. In interval programming the bounds of the uncertain coefficients are only required. The methodological aspects of interval programming studied in the past have been surveyed by Olivera et al [11]. The structure of this paper is organized as follows. Section 2 presents the TLP problem with interval coefficients in both of the objective functions and constraints. Section 3 presents treatment procedure of the uncertainty of the problem. Fuzzy models for TLP problem are provided in section 4. A numerical example is provided in section 5 to clarify the proposed approach. Section 6 contains the conclusion.

2. THREE-LEVEL PROGRAMMING PROBLEM WITH INTERVAL COEFFICIENTS The general form of TLP problem with interval coefficients in both of the objective functions and the constraints is as follows: max f 1 x  c11L ,c11R  x 1  c12L ,c12R  x 2  c13L ,c13R  x 3 ,

 

x1

(1) where

x 2 ,x 3

solve:

 

L R L R L  x 1  c 22  x 2  c 23 max f 2 x  c 21 , c 21 , c 22 , c 23R  x 3 , x2

(2) and

x3

solves:

 

max f 3 x  c 31L , c 31R  x 1  c 32L ,c 32R  x 2  c 33L ,c 33R  x 3 , x3 (3) subject to

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aiL1 , aiR1  x 1  aiL2 , aiR2  x 2  aiL3 , aiR3  x 3  b i , i  1, 2,..., m

314

Solving Multi-Level Programming Problem with Interval Coefficients by Using Fuzzy Programming (4)

x 1  0, x 2  0, x 3  0 where

x 1  x 11 , x 12 ,..., x 1n1  and n1

variables

controlled

by

controlled

by 2 3

n2 3

second

 and n

controlled by the third level.

level.

is the number of variables

the

x 3  x , x ,..., x 1 3

first

3

level.

is the number of variables



 is the decision

x  x 1,x 2 ,x 3

variables controlled by the decision maker on the three levels.

n  n1  n 2  n 3 L i

R i

a , a 

3

d 1 ,d 2 , d 3 , d 4 , d 5 and d 6

where

are

called

weighting

coefficients.

3.2 Treatment of the uncertain constraints The possibility degree of interval number represents certain degree that one interval number is larger or smaller than another.

 

Let g i x  aiL1 , aiR1  x 1  aiL2 , aiR2  x 2  aiL3 , aiR3  x 3       According to Jiang et al. [10], the possibility degree can be defined as follows:

and m is the number of constraints.

i  1,..., m

, for

6

d 5 , d 6  0, d 5  d 6  1

is the number of

the

x 2  x 21 , x 22 ,..., x 2n 2  and n 2

    d w  f  x 

 

f 3 x  d 5m f 3 x

(5)

and

c L , c R 

denote an

interval numbers and they represent a bounded set of real numbers between the bounds. The superscripts L and R denote lower and upper bounds of an interval number respectively.

3. TREATMENT PROCEDUOR In this section we will treat the uncertainty in both of the objective functions and the constraints.

3.1 Treatment of the uncertain objective function In interval mathematics, the uncertain objective function with interval numbers can be transformed into two deterministic objective functions as follows (Jiang et al., 2008):

    12  f  x   f  x  1 w  f  x   f  x   f  x  2 L

m f1 x

L

1

1

 

f 1L x

and

 

f 1R x

 

P b  E  i  

 

i  1, 2,..., m



b i  aiL1 x 1  aiL2 x 2  aiL3 x 3

a

R i1

x 1  a x 2  aiR3 x

 

3

  a

L i1



x 1  aiL2 x 2  aiL3 x 3



 i ,

0  i  1 3.3 The deterministic TLP problem

 

2

1

Now we have a deterministic form of TLP problem as follows

    d w f  x 

 

    d w  f  x 

max f 1 x  d 1m f 1 x x1

2

1

(6) ,

where

x 2 ,x 3

 

solve:

    d w f  x 

max f 2 x  d 3 m f 2 x x2

And by the same way the objective functions of the second and third levels can be written as follows: 4

R i2

i  1, 2,..., m

are given as

d 1 , d 2  0, d 1  d 2  1

 

is a predetermined

then

    d w f  x 

f 2 x  d 3m f 2 x

i  1, 2,..., m

,

The linear combination method [8] is adopted to deal with the multi objective optimization. The linear combination method guarantees obtainment finite number of Pareto solutions. So it is appropriate for this type of problems, then we get:

 

,

 b i  ai 1 ,ai 1  x 1  ai 2 ,ai 2  x 2  ai 3 ,ai 3  x 3         

f 1L x  min f 1 x , f 1R x  max f 1 x

f 1 x  d 1m f 1 x

is the interval of the ith

is the possibility degree of the ith constraint

0  i  1

where

follows:

 

 

E   g iL x , g iR x   

constraint function.

where m is called the midpoint value and w is called the radius of the interval number. The two functions

b ER

,

1

R

1

where

E L b  E R

possibility degree level. By applying the possibility degree definition, we can transform the uncertain form of constraints (4) into certain form as follows: P  L R   L R   L R    i ,

R

1

b EL

0   b E L P b E   R   E  E L 1 

4

2

(7) and

x3

solves:

2

d 3 , d 4  0, d 3  d 4  1

 

     d w  f  x 

max f 3 x  d 5 m f 3 x

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x3

6

3

315

International Journal of Innovative Research in Engineering & Management (IJIREM) ISSN: 2350-0557, Volume-3, Issue-4, July-2016 (8)

Equations (13), (14) and (15) will be used to build the membership functions of fuzzy set theory [13].

subject to



bi  aiL1 x 1  aiL2 x 2  aiL3 x 3





 

aiR1 x 1  aiR2 x 2  aiR3 x 3  aiL1 x 1  aiL2 x 2  aiL3 x 3



 i ,

i  1, 2,..., m

4.1. FLDM problem By using (13) the following the membership functions of fuzzy set theory can be formulated:

(9)

0  i  1

(10)

d 1 , d 2 , d 3 , d 4 , d 5 , d 6  0, d 1  d 2  1

, d 3  d 4  1, d 5  d 6  1

(11)

x 1  0, x 2  0, x 3  0 For some values of

f 1  f 1 

i , d 1 , d 2 , d 3 , d 4 , d 5 and d 6 we

can

4. FUZZY MODELS FOR TLP PROBLEM To solve the TLP problem one first gets the satisfactory solution that is acceptable to FLDM, and then gives decision variables and goals of the FLDM with some leeway to the SLDM for him/her to seek the satisfactory solution. Then give the decision variables and goals of the SLDM with some leeway to the TLDM for him/her to seek the satisfactory solution, and to arrive at the solution which is closest to the satisfactory solution of the FLDM. This due to, the TLDM who should not only optimize his/her objective functions but also try to satisfy the goals and preferences of SLDM as much as possible, SLDM also do the same action to satisfy the goals and preferences of FLDM as much as possible. The TLP problem will be solved as a single objective problem considering one objective at a time and ignoring the rest. By using the obtained solutions, we find the values of all objective functions at each solution and construct a payoff matrix as follows [15]:

 

f1 x

o

f2 x1

f3 x 1

o

o

2

3

2

o

2

2

o

3

3

3

      

The minimum and maximum of each column gives us lower bound and upper bound of each objective function then we have:

 

l 1

 

 

 



bi  aiL1 x 1  aiL2 x 2  aiL3 x

a

R i1

3

(18)



 

R i2

x 1  a x 2  aiR3 x 3  aiL1 x 1  aiL2 x 2  aiL3 x 3

 i ,

i  1, 2,..., m

(20) (21)

x 1  0, x 2  0, x 3  0 Whose F

solutions F

(22) assumed

are

to

be

F

 x , x , x , f F ,  F  satisfactory level   .  1 2 3 1 

4.2. SLDM problem By using (14) the following the membership functions can be formulated:

 f 2 f 2 

 

0   f x f l x 2  2 x  u l  f 2 x  f 2 x  1

   

 

 

f 2 x  f 2l x ,

   

 

 

 

f 2l x  f 2 x  f 2u x ,

 

 

f 2 x  f 2u x .

(23) Now, we can get the solution of the SLDM problem by solving the following Tchebycheff problem: max  (24) subject to:

  and f  x   min f  x  ,

 



0  i  1 d 1 , d 2  0, d 1  d 2  1

 

f 2  f 2 x      

L i1

L i2

L i3

bi  a x 1  a x 2  a x 3

a

R i1

 

(25)



x 1  aiR2 x 2  aiR3 x 3  aiL1 x 1  aiL2 x 2  aiL3 x 3



 i ,

i  1, 2,..., m

(26)

0  i  1

(27)

d 3 , d 4  0, d 3  d 4  1

(28)

x 1  0, x 2  0, x 3  0

(29)

2

(14)

 

 

(19)

1

l 2

 

f 1 x  f 1u x .

(13)

f 2u x  max f 2 x

 

f 1l x  f 1 x  f 1u x ,

 

  and f  x   min f  x  ,

f 1u x  max f 1 x

   

 f 1 f 1 x     

 

o

 

f 1 x  f 1l x ,

(16) Now, we can get the solution of the FLDM problem by solving the following Tchebycheff problem [1, 7, 14]: max  (17) subject to:

f3 x

        f x  f x    f x  f x 

f x o x1  1 1 o  o x 2 f 1 x 2 o  o x3 f x 3 1  o

 

f2 x

   

 

(12)

treat problem (6) – (12) by using fuzzy programming.

 

0   f x f l x 1  1 x  u l  f 1 x  f 1 x  1

 

 

f 3u x  max f 3 x and f 3l x  min f 3 x

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Whose solutions are assumed to be: 316

Solving Multi-Level Programming Problem with Interval Coefficients by Using Fuzzy Programming

 x S , x S , x S , f S ,  S  satisfactory level   .  1 2 3 2 

F

where

4.3. TLDM problem By using (15) the following the membership functions can be formulated:

 f 3 f 3 

 

0   f x f l x 3  3 x  u l  f 3 x  f 3 x  1

   

 

   

f

l 3





 



 

x 2 x 2

 

L i3



 

x 1  aiR2 x 2  aiR3 x 3  aiL1 x 1  aiL2 x 2  aiL3 x

3



 i ,

x  x t 2 2  2  t  2  S  x 2 t2  x 2   t2



where S 2

where

(35)

x2

T

T

 x , x , x , f T ,  T  satisfactory level   .  1 2 3 3 

 t2





F

f1  f1

t1 and t 2

x 1 and x

2

S

x 1 and x 2

S

S

S 3



\

x

S 2

t2



and

f 1  f 1F

is

absolutely unacceptable

and that the preference with

 

with maximum tolerances

respectively. The following membership function

 

f 1 x  f 1\ x ,

0   f x f \ x 1  1 x  F \  f x  f  1 1 x  1 

       

 

 

 

f 1 \ x  f 1 x  f 1F x ,

 

 

f 1 x  f 1F x .

(39)

S

SLDM knows that using the optimal decisions x 1 , x 2 as a control variables for the TLDM are not practical. It is more reasonable to have some tolerance that gives the TLDM an extent feasible region to search for his/her optimal solution, and also reduce searching time or interactions.

should be around

S

is linearly increasing.

 

Now the solution of the FLDM, SLDM, and TLDM are disclosed. However, three solutions are usually different because of nature between three levels objective functions. The FLDM and

F

S

x 2  x 2  x 2 t2,

is the worst acceptable decision.

f 1\  f 1 x 1 , x 2 , x

 f\1 f 1 

4.4 TLP problem

In this way, the range of decision variables

S

Then the following membership function of the FLDM can be stated as:

Whose solutions are assumed to be: T

:

S

is the most preferred solution, the

 f 1\ , f 1F 

(36)

and

x 2 t2  x 2  x 2 ,



absolutely acceptable and

x 1  0, x 2  0, x 3  0

x2

First, the FLDM goals may reasonably consider

(34)



(38)

i  1, 2,..., m

(33)

0  i  1 d 5 , d 6  0, d 5  d 6  1



S

S

x

(32)

 t1

1

is the worst acceptable decision.

x   f x   f x , 3

F

S

f 3 x  f 3u x .

bi  a x 1  a x 2  a x 3 R i1

 t1

u 3

 

a

1

x

Also the following membership function specify

 

 f 3 f 3 x      L i2

F

is the most preferred solution, the

f 3 x  f 3l x ,

(30) Now, we can get the solution of the TLDM problem by solving the following Tchebycheff problem: (31) max  subject to:

L i1

x

x1

Second, the SLDM goals may reasonably consider is absolutely acceptable and where



T

T

f 2  f 2\ T

f 2\  f 2 x 1 , x 2 , x 3

 f 2\ , f 2S 



f 2  f 2S

absolutely unacceptable

and that the preference with

is linearly increasing.

Then the following membership function of the SLDM can be stated as:

F

specify

x1

:





x

1

x  1

 x  x F t 1 1  1  t1  F  x 1  t1  x 1   t1





F

f\2 f 2 

F

x 1  t1  x 1  x 1 , F

F

x 1  x 1  x 1  t1,

(37)

0   f x f \ x 2  2 x  S \  f 2 x  f 2 x  1

 

   

   

 

 

f 2 x  f 2\ x ,

 

 

 

f 2\ x  f 2 x  f 2S x ,

 

 

f 2 x  f 2S x .

(40) Third, the TLDM may be willing to build a membership function for his/her objective function as follows:

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317

International Journal of Innovative Research in Engineering & Management (IJIREM) ISSN: 2350-0557, Volume-3, Issue-4, July-2016

 f\3 f 3 

 

   

 

 

f 3 x  f 3\ x ,

0   f x f \ x 3  3 x  T \  f 3 x  f 3 x  1

   

 

 

 

f 3\ x  f 3 x  f 3T x ,

 

 



S

S

f 3  f 3 x 1 ,x 2 ,x

where

S 3



F

x 1  x 1  t1 t1



where

x 1  t1  x 1 t1



S

x 2  x 2 t2 t2

x

S 2

 I



t2  x 2 t2



bi  aiL1 x 1  aiL2 x 2  aiL3 x 3

a

R i3

 

L i1

max f 2  x 1 , x 2 , x 3   9,14  x 1   4,8 x 2   2, 7  x 3 , x2

x3

and

solves:

max f 3  x 1 , x 2 , x 3   1,5 x 1  11,14 x 2

(43)

I

(44)

 I

(45)

I

(46)

x3

L i2

3, 6 x 1  5, 7 x 2   2,5 x 3  9 1, 2 x 1   2, 7 x 2  3,5 x 3  12 x 1, x 2 , x 3  0 d 1  0.5, d 2  0.5, d 3  0.4, d 4  0.6,

For

     

R i2

solve:

subject to

f\1  f 1 x      f\2  f 2 x      \  f 3 f 3 x      R i1

x 2, x 3

 6,10 x 3 ,



F

max f 1  x 1 , x 2 , x 3    2, 4 x 1  3,6 x 2  5,9 x 3 , x1

.

Finally, we can generate the satisfactory solution by solving the following Tchebycheff problem. max  (42) subject to:



To clarify the solution method for TLP problem with interval coefficients in both of the objective function and constraints. Let us consider the following example:

f 3 x  f 3T x .

(41) \

5. NUMERICAL EXAMPLE

(47)

x 1 a x 2  a x 3  a x 1 a x 2 a x 3

max f 1  x 1 , x 2 , x 3   2x 1  3x 2  4.5x 3 , x1

where

x 2, x 3

solve:

max f 2  x 1 , x 2 , x 3   6.1x 1  3.6x 2  3.3x 3 ,

 L i3

in section 3 the above problem can be formulated as the following deterministic TLP problem:

(48) (49)

and

d 5  0.3, d 6  0.7, 1  2  0.5 and by using mathematical treatment described



 i ,

x2

i  1, 2,..., m

x3

and

(50)

0  i  1 (51) d 1 , d 2 , d 3 , d 4 , d 5 , d 6  0, d 1  d 2  1 , d 3  d 4  1, d 5  d 6  1 (52) x 1  0, x 2  0, x 3  0

solves:

max f 3  x 1 , x 2 , x 3   2.3x 1  4.8x 2  3.8x 3 , x3

subject to

4.5x 1  6x 2  3.5x 3  9 1.5x 1  4.5x 2  4x 3  12

(53)

where  is the overall satisfaction, and I is the column vector whose elements are equal to 1. By solving problem (42)-(53) if the FLDM is satisfied with the obtained solution then satisfactory solution is reached. Otherwise, he/she should provide new membership function for the control variable and objective to the SLDM, who also should provide new membership functions for the control variable and objective to the TLDM until a satisfactory solution is reached.

x 1, x 2 , x 3  0 By using the fuzzy method proposed in section 4, the solution

x

of the first level is

f 1F  11.569

and

x

S 1

1

of



F

. The solution of the second level is



S

F

, x 2 , x 3 ,  F   0, 0, 2.571,1

, x 2 , x 3 ,  S   2, 0, 0,1

solution

Copyright © 2016. Innovative Research Publications. All Rights Reserved

S

F

the

and third

f 2S  12.2 level

. The is

318

Solving Multi-Level Programming Problem with Interval Coefficients by Using Fuzzy Programming

x

T 1

T

and

[5] Cassidy, R., Kirby, M. and Raike, W., 1971. Efficient distribution of resources through three levels of government. Management Science, 17, 462–473.

is around 0 with tolerance

[6] Emam, O., El-Araby, E. M. and Belal, M. A., 2015. On Rough Multi-Level Linear Programming Problem. Information Sciences Letters, 4 (1), 41-49.



T

, x 2 , x 3 ,  T   0,0.0008, 2.57,1

f 3T  9.769 . F

Let the FLDM control decision

t 1  1 and tolerance

x1

S

the SLDM control decision

t2  1.

x2

is around 0 with

Then the Tchebycheff problem (42)-(53) take

the following form:

[8] Hu, Y.D., 1990. Applied Multiobjective Optimization. Shanghai Science and Technology Press. Shanghai. China.

max  subject to:

x 1 1   I 1 x 1   I x 2 1   I 1 x 2   I 2x 1  3x 2  4.5x 3  7.567  4 6.1x 1  3.6x 2  3.3x 3  3.72  8.48 2.3x 1  4.8x 2  3.8x 3  5.169  4.6 4.5x 1  6x 2  3.5x 3  9 1.5x 1  4.5x 2  4x 3  12 x 1, x 2 , x 3  0

   0,1

[9] Inuiguchi, M. and Kume, Y., 1991. Goal Programming Problems with interval coefficients and target intervals. European Journal of Operational Research, 52, 345-360. [10] Jiang, C., Han, X., Liu, G.R. and Liu, G.P., 2008. A nonlinear interval number programming method for uncertain optimization problems. European Journal of Operational Research, 188, 1–13. [11] Olivera, C. and Antunes, C.H., 2007. Multiple objective Linear Programming Models with Interval Coefficients – an illustrated overview. European Journal Of Operational Research, 181, 1434-1463. [12] Osman, M.S., Abo-Sinna, M.A., Amer, A.H. and Emam, O.E., 2004. A multi-level non-linear multi-objective decisionmaking under fuzziness. Applied Mathematics and Computation, 153, 239–252. [13] Sakawa, M., 1993. Fuzzy Sets and Interactive MultiObjective Optimization. New York : Plenum Press.

Whose solution is:

 x 1 , x 2 , x 3    0.666, 0,1.7152  ,

[7] Hsu, S.S., Lai, Y.J. and Stanley, L.E., 2001. Fuzzy and Multi Level Decision-Making. Springer-Verlag Ltd, London.

  0.334,

.

f 1  9.0504, f 2  9.7228 and f 3  8.0496

6. CONCLUSION A solution method for TLP problem with interval coefficients has been introduced. These coefficients are located in both of objective functions and constraints. A fuzzy approach is used in the introduced solution method. The concept of tolerance membership function is used. The investigation of other solution methods for multi-level programming with interval coefficients would be future topics.

[14] Shi, X. and Xia, H., 1997. Interactive bi-level multi-objective decision-making. Journal of Operational Research Society, 48, 943–949. [15] Sinha, S., 2003. Fuzzy mathematical Programming applied to multi-level programming problems. Computers and Operations Research, 30, 1259-1268. [16] Wen, U.P. and Bialas, W.F., 1986. The hybrid algorithm for solving the three-level linear programming problem. Computers and Operations Research, 13, 367–377.

REFERENCES [1] Abo-Sinna, M.A., 2001. A bi-level non-linear multi-objective decision-making under Fuzziness. Journal of Operational Research Society (OPSEARCH), 38 (5), 484–495. [2] Anandolingam, G., 1988. A mathematical programming model of decentralized multi-level system. Journal of Operational Research Society, 39, 1021–1033. [3] Anandalingam, G. and Apprey, V., 1991. Multi-level programming and conflict resolution. European Journal of Operational Research, 51, 233–247. [4] Bialas, W.F. and Karwan, M.H., 1984. Two-level linear programming, Management Science, 30, 1004–1020.

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