The Kth-best approach for linear bilevel multifollower programming with partial shared variables among followers

The Kth-Best Approach for Linear Bilevel Multifollower Programming

with Partial Shared Variables among Followers

Chenggen Shi*, and Hong Zhou#, Jie Lu* and Guangquan Zhang* *Faculty of Information Technology

University of Technology, Sydney Sydney, NSW 2007, Australia Email:{cshi, jielu, zhangg}@it.uts.edu.au

#Electrical, Electronic & Computer Engineering The University of Southern Queensland Toowoomba, Queensland 4350, Australia

[email protected]

Abstract - In a real world bilevel decision-making, the lower level of a bilevel decision

usually involves multiple decision units. This paper proposes the Kth-best approach for linear bilevel multifollower programming problems with shared variables among followers. Finally a numeric example is given to show how the Kth-best approach works.

Keywords: Linear bilevel programming, mulitfollower, Kth-best approach, Von

Stackelberg game

1 Introduction

follower is involved. There have been nearly two dozen algorithms, such as, the Kth best approach [3, 4], Kuhn-Tucker approach [5, 6, 7], complementarity pivot approach [8], penalty function approach [9, 10], proposed for solving linear BLP problems since the field being caught the attention of researchers in the mid-1970s. Kuhn-Tucker approach has been proven to be a valuable analysis tool with a wide range of successful applications for linear BLP [2, 6, 7].

2 Model Overview

2.1 A model for linear BLMFP problems with partial shared variables among

followers

For x∈X ⊂Rn , mi i i Y R

y ∈ ⊂ , m

R Z

z∈ ⊂ 1

1

: X Y Y Z R

F × ×K× _K × → , and

1 1

: X Y Y Z R

fi × ×K× K × → , i=1,2,K,K , a linear BLMFP problem where K(≥2) followers are involved and there are shared partial decision variables, but separate objective functions and constraint functions among the followers is defined as follows [18]:

F x y y z cx d y dz

K

s s s K

X

x = +

∑

=

∈ ( , 1, , , ) ₁

min K

subject to Ax B y Bz b

K

s s

s + ≤

+

∑

f x y y z c x e y eiz

K

s s is i

K i

Z z Y

y_{i i} = +

∑

= ∈

∈ , ( , 1, , , ) ₁

min K

subject to _{i i} K

s

s is

ix C y C z b

A +

∑

=1

,

1

where c∈Rn , c_i ∈Rn , mi i R

d ∈ , d∈Rm , mi

is R

e ∈ , e_i∈Rm , b∈Rp , qi

i R

b ∈ ,

n p

R

A∈ × , p mi

i R

B ∈ × ,B∈Rp×m, _Ai∈_Rqi×n_, qi ms

is R

C ∈ × ,_Ci∈_Rqi×m_{, i,s 1,2, ,K}

K

= .

Definition 1 A topological space is compact if every open cover of the entire space has a

2.2 Model transformation for linear BLMFP with partial shared variables among

followers

The main idea to deal with linear BLMFP problems with partial shared variables among the followers is that an assumed third party controls the shared variable z . It means that the ith follower controls the variable y (_i i=1,2,K,K), and a third party

called a virtual follower: the (K +1)th follower controls the variable z . By using this splitting method, (1) can be rewritten as follows:

∑

+

= +

∈ = +

1 1 1

1, , , ) ,

( min

K

s s s K

K X

x F x y y y cx d y

K

subject to Ax B y b

K

s s s ≤

+

∑

1

1 2

∑

+

= +

∈ = +

1 1 1

1, , , ) ,

( min

K

s s is i

K K i

Y

y_{i i} f x y y y cx e y K

subject to _i K

s s is ix C y b

A +

∑

+

=

1 1

,

where i=1,K,K,K +1 , y z

K+1 = , dK+1 =d , BK+1 =B , el(K 1) =el(l=1,K,K)

+ ,

∑

= + =

K

s s

K c

c

1

1 , ( 1, , 1)

1 ) 1

( =

∑

=

+ e j K

e

K

s sj j

K K , ( 1, , )

) 1

( C l K

C_{l K}+ = _l = K ,

( )

1

0 1 = ₊

+ qK

K

A ,

( )

1

) 1

( l K

C

l K m q l

K = = K

×

+ + , bK+1 =

( )

variables among the followers. We can also find that all the variables of the followers parameterise into the objective functions and constraint functions of the followers.

2.3 Definition of solution

For simplification and convenience, we write model (2) as follows:

∑

=

∈ = +

K

s s s K

X

x F x y y cx d y 1 1, , )

, (

min K

(3 a)

subject to Ax B y b

K

s s s ≤

+

∑

=1 (3 b)

∑

=

∈ = +

K

s s is i

K i

Y

y_{i i} f x y y c x e y 1 1, , )

, (

min K

(3 c)

subject to _i K

s

s is ix C y b

A +

∑

=1

, _{(3 d)}

where c∈Rn , ci ∈Rn ,

i m i R

d ∈ , ms

is R

e ∈ , b∈Rp , qi

i R

b ∈ , A∈Rp×n , p mi

i R

B ∈ × , n

q i

i

R

A ∈ × , qi ms

is R

C ∈ × , i,s =1,2,K,K.

The formulation (3) is the same as (2) except the number of followers. They have the same solution algorithms. Corresponding to (3), [18] give following basic definition.

Definition 2

(a) Constraint region:

, ,

) , , , {(

1 1

1 y X Y Y Ax B y b

y x S

K

s s s k

K ∈ × × × + ≤

=

∑

=

K K

} , , 2 , 1 , 1

K i

b y C x

A _i

K

s s is

i +

∑

=

The constraint region refers to all possible combinations of choices that the leader and followers may make.

(b) Projection of S onto the leader’s decision space:

} , , 2 , 1 , , , : { ) ( 1 1 K i b y C x A b y B Ax Y y X x X S i K s s is i K s s s i

i∈ + ≤ + ≤ = K

∃ ∈ =

∑

∑

(c) Feasible set for each follower ∀x∈S( X): S_i(x)={y_i∈Y_i:(x,y₁,K,y_K)∈S}.

The feasible region for each follower is affected by the leader’s choice of x, and the allowable choices of each follower are the elements of S .

(d) Each follower’s rational reaction set forx∈S( X):

)]} ( ˆ : ) , , , 2 , 1 , , ˆ , ( min[ arg : { )

(x y Y y f x y y j K j i y S x

Pi = i∈ i i∈ i i j = K ≠ i∈ i ,

where i=1,2,K,K, argmin[f (x,yˆ ,y , j =1,2, ,K, j≠i): yˆ ∈S (x)]=

i i j i i K )} ( ˆ ), , , , 2 , 1 , , ˆ , ( ) , , , ( : ) (

{y_i∈S_i x f_i x y₁ K y_K ≤ f_i x y_i y_j j = K K j ≠i y_i∈S_i x .

The followers observe the leader’s action and simultaneously react by selecting

i

y from their feasible set to minimize their objective functions. (e) Inducible region:

} , , 2 , 1 ), ( , ) , , , ( : ) , , ,

{(x y1 y x y1 y S y P x i K

IR= K K K K ∈ i∈ i = K . Thus in terms of the above notations, (3) can be written as

min{F(x,y1,K,yK):(x,y1,K,yK)∈IR} (4) Shi proposed the following theorem to characterize the condition under which there

Theorem 1 If S is nonempty and compact, there exists an optimal solution for a linear

BLMFP problem.

3

An Extended Kth-best Algorithm for Linear Bilevel Multifollower

Programming with Partial Shared Variables among Followers

3.1 Properties of Linear Bilevel Multifollower Programming with Partial Shared

Variables among Followers

Theorem 2 The inducible region of the model (3) can be written equivalently as a

piecewise linear equality constraint comprised of supporting hyperplanes of constraint

regionS.

Proof: Let us begin by writing the inducible region of Definition 2(e) explicitly as

follower:

, ~

: ~ min[ ,

) , , , ( : ) , , , {(

, 1 1

1

∑

≠ =

− − ≤ =

∈

= K

i s s

s s i

i i ii i

ii K

K x y y S e y e y B y b Ax B y

y y x

IR K K

~ , 1,2, , ,~ 0], 1,2, , } ,

1

K i

y K j

y C x

A b y

C i

K

i s s

s is i

i i

ji ≤ − −

∑

≠ =

.

Let us define

T K

b b b

b ( , 1, , )

'

K

= , A (A,A1, ,AK)T '

K

= , Bi (Bi,C1i, ,CKi)T '

K

= ,

− ≤ = ∈ = ' ' 1

1, , ):( , , , ) , min[ ~ : ~ ,

{(x y y_K x y y_K S e_iiy_i e_iiy_i B_iy_i b_i

IR K K

} , , 2 , 1 ], 0 ~ , , 1 ' ' K i y y B x A i K i s s s

s ≥ = K

−

∑

.

Let us define

− ≤ = ≠ = '~ ' : ~ min[ ) , , 2 , 1 , ,

( j ii i i i i i x y j K j i e y B y b

Q K ] 0 ~ , , 1 ' ' −

∑

B x

A ,

(5)

where i =1,2,K,K. For each value ofx∈S( X), the resulting feasible region to problem

(3) is nonempty and compact. Thus, forQ , which is a linear program parameterized in _i

j

y

x, , j=1,2,K,K and j ≠i, always has a solution. From duality theory we get

} 0 , : ) (

max{ ' ' , 1 ' ' +

∑

u _{i i ii}

K

i s s

s

s , (6)

which has the same optimal value as (5) at the solution u*. Let u1,K,usbe a listing of all

the vertices of the constraint region of (6) given by U ={u:uBi' ≥−eii,u≥0}. Because we know that a solution to (6) occurs at a vertex of U, we get the equivalent problem

}} , , { : ) (

max{ ' 1

, 1

'

' l s

i K i s s s s l u u u b y B x A

u +

∑

≠ =

,

which demonstrates that Qi(x,yj, j=1,2,K,K, j≠i), is a piecewise linear function. Rewriting IR as

0 ) , , , 2 , 1 , , ( : ) , , ,

{( ₁ ∈ = ≠ − =

= x y yk S Qi x yj j K j i eiiyi

IR K K ,i=1,2,K,K}

Corollary 1 The problem (3) is equivalent to minimizing F over a feasible region

comprised of a piecewise linear equality constraint.

Proof: By (4) and 2, we have the desired result.

The each function Q defined by (5) is convex and continuous. In general, because i

we are minimizing a linear function

∑

= +

= K

s s sy

d cx F

1

over IR , and because F is

bounded below S by, say, cx d y x y y_K S

K

s s

s ∈

+

∑

) , , , ( :

min{ ₁

1

K }, the following can be concluded.

Corollary 2 A solution for the linear BLMFP problem occurs at a vertex of IR .

Proof: A linear BLMFP problem can be written as in (4). Since

∑

= +

= K

s s sy

d cx F

1

is

linear, if a solution exists, one must occur at a vertex of IR . The proof is completed.

Theorem 3 The solution (x*,y₁*,K,y*_K)of the linear BLMFP problem occurs at a vertex of S .

Proof: Let ( , , , ), ,( , 1, , ) 1

1 1

1 r

K r

r

K x y y

y y

x K K K be the distinct vertices of S . Since any point in S can be written a convex combination of these vertices, let

∑

j

j K j

j j

K x y y

y y

x ₁ 1

* * 1 *

) , , , ( )

, , ,

( K

α

j j

r

j 1 =1, ≥0, =1,2,K,

∑

α

α

r

r ≤ . It must be shown that r =1. To see this let us write the constraints to (3) at )

, , ,

( 1* * *

K

y y

x K in their piecewise linear form (7).

* *

) , , , 2 , 1 , , (

) ( )) , , , 2 , 1 , , ( (

0=

∑

∑

j j i j ii j l j j j

i x y l K l i e y

Q α K α

j i ii j j j l j i j

iQ x y l K l i

∑

∑

≤ α ( , , 1,2,K, , ) α ,

where i=1,2,K,K.

By convexity of Q_i(x,y_l,l =1,2,K,K,l ≠i), we have 0 ( _i( j, _lj, 1,2, , , ) _{ii i}j)

j

j Q x y l= K l ≠i −e y

≤

∑

where i=1,2,K,K. But by a definition,

j i ii i ii x S y j l j

i x y l K l i e y e y

Q j i ≤ = ≠ = ∈ ( ) min ) , , , 2 , 1 , ,

( K , i=1,2,K,K.

Therefore, Q_i(xj,y_lj,l =1,2,K,K,l ≠i)−e_iiy_ij ≤0, j =1,2,K,r , i=1,2,K,K . Noting that αj ≥0, j =1,2,K,r, the equality in the preceding expression must hold or

else a contradiction would result in the sequence above. Consequently, 0 ) , , , 2 , 1 , ,

( j _lj = ≠ − _{ii i}j =

i x y l K l i e y

Q K , j =1,2,K,r , i=1,2,K,K . This implies that

IR y

y

xj, j, , _Kj)∈

( ₁ K , j =1,2,K,r and ( , , , ) * * 1 * K y y

x K can be written as a convex combination of points in IR . Because ( , 1*, , * )

*

K

y y

x K is a vertex of IR , a contradiction results unless r =1. The proof is completed.

Corollary 1 If x is an extreme point of IR , it is an extreme point of S . Proof: Let ( , , , ), ,( , 1, , )

1 1 1 1 r K r r

K x y y

y y

x K K K be the distinct vertices of S . Since any point in S can be written a convex combination of these vertices, let

∑

K x y y

y y x 1 1 * * 1 * ) , , , ( ) , , ,

( K

α

j j

r

j 1 =1, ≥0, =1,2,K,

r

r ≤ . It must be shown that r =1. To see this let us write the constraints to (3) at ) , , , ( * * 1 * K y y

x K in their piecewise linear form (7).

* * ) , , , 2 , 1 , , (

0=Q_i x y_l l = K K l≠i −e_iiy_i, i=1,2,K,K . Rewrite the above formulation as follows

) ( )) , , , 2 , 1 , , ( (

0=

∑

∑

j j i j ii j l j j j

i x y l K l i e y

Q α K α

j i ii j j j l j i j

iQ x y l K l i

∑

∑

≤ α ( , , 1,2,K, , ) α ,

where i=1,2,K,K.

By convexity of Qi(x,yl,l =1,2,K,K,l ≠i), we have 0 ( _i( j, _lj, 1,2, , , ) _{ii i}j)

j

j Q x y l= K l ≠i −e y

≤

∑

where i=1,2,K,K. But by a definition,

j i ii i ii x S y j l j

i x y l K l i e y e y

Q j i ≤ = ≠ = ∈ ( ) min ) , , , 2 , 1 , ,

( K , i=1,2,K,K.

Therefore, Q x y l K l i eiiyij j r

j l j

i( , , =1,2,K, , ≠ )− ≤0, =1,2,K, , i=1,2,K,K . Noting that α_j ≥0, j =1,2,K,r, the equality in the preceding expression must hold or

else a contradiction would result in the sequence above. Consequently, 0 ) , , , 2 , 1 , ,

( = ≠ − ii ij =

j l j

i x y l K l i e y

Q K , j =1,2,K,r , i=1,2,K,K . This implies that

IR y

y

x Kj

j j ∈ ) , , ,

( 1 K , j =1,2,K,r and ( , , , ) * * 1 * K y y

4. An Extended Kth-best Algorithm for Linear Bilevel Multifollower

Programming with Partial Shared Variables among Followers

Theorem and Corollary have provided theoretical foundation for our new algorithm. It means that by searching extreme points on the constraint region S , we can efficiently find an optimal solution for a linear BLMFP problem. The basic idea of our algorithm is that according to the objective function of the upper level, we arrange all the extreme points in S in descending order, and select the first extreme point to check if it is on the inducible region IR . If yes, the current extreme point is the optimal solution. Otherwise, the next one will be selected and checked.

More specifically, let ( , , , ), ,( , 1 , , ), 1

1 1

1 N

K N

N

K x y y

y y

x K K K denote the N ordered

extreme points to the linear BLMFP problem

} ) , , , ( :

min{ 1

1

S y y x y d

cx K

K

s s

s ∈

+

∑

K ,

(8)

such that , 1,2, , 1.

1 1 1

1

− =

+ ≤

+

∑

∑

= + +

=

N j

y d cx

y d cx

K

s j s s j

K

s j s s j

K Let )

~ , , ~ , ~

(y₁ y₂ K y_K denote

the optimal solution to the following problem.

) , , 2 , 1 ), ( :

) , , , (

min(f x y1 y y S x i K

j i i K j

i K ∈ = K . (9)

We only need to find the smallest j , j =1,2,K,N under which

i j i y

y = ~ ,

K

i=1,2,K, . Let us write (9) as follows )

, , , (

min fi x y1 K yK

where i=1,2,K,K . We only need to find the smallest j under which

i j i y

y = ~ ,

K

i=1,2,K, . From Definition 2(b), we have

∑

= +

= K

s s is i

K

i x y y cx e y

f

1 1, , )

, (

min K

(10a)

subject to Ax B y b

K

s s s ≤

+

∑

(10b)

l K

s

s ls lx C y b

A +

∑

=1

, l=1,2,K,K

(10c)

j

x

x= _(10d)

0 ,

, 0 , 0 2

1 ≥ y ≥ yK ≥

y K ,

(10f)

where i=1,2,K,K.

The solving is equivalent to select one ordered extreme point (xj,y₁j,K,y_Kj), then solve (10) to obtain the optimal solution y~ . If for all i , _i y_ij =~y_i, then (xj,y₁j,K,y_Kj) is the global optimum to (3). Otherwise, check the next extreme point. It can be accomplished with the following procedure.

Step 1: Put j←1. Solve (8) with the simplex method to obtain the optimal solution )

, , ,

( 11 1

1

K

y y

x K . Let ( , , , )

1 1 1 1

K

y y x

W = K and T =φ . Go to Step 2.

Step 2: Solve (10) with the bounded simplex method. Let y~ denote the optimal _i

solution to (10). If y_ij = ~y_i for all i ,i=1,K,K , stop; ( , , , )

1

j K j

j

y y

x K is

Step 3: Let W_{[ j]} denote the set of adjacent extreme points of (xj,y₁j,K,y_Kj) such

that (x,y₁,K,y_K)∈W_[j] implies

∑

∑

= =

+ ≤

+ K

s j s s j

K

s s

sy cx d y

d cx

1 1

. Let

)} , , ,

{( 1

j K j

j

y y x T

T = ∪ K and W W W T

j )\ ( ∪ [ ]

= . Go to Step 4.

Step 4: Set j← j+1 and choose (xj,y₁j,K,y_Kj) so that

} ) , , ( :

min{ 1

1 1

W y y x y d cx y

d

cx K

K

s s s K

s j s s

j +

∑

∑

= =

K .

Go to Step 2.

4. A Numeric Example

Let us give a following example to show how the Kth-best approach works.

Example 1

Consider a following linear BLMFP problem with x∈R1 , y₁,y₂∈R1 , z∈R1 and }

0 { ≥

= x

X , Y ={y₁ ≥0,y₂ ≥0}, Z ={z≥0}

z y y x z

y y x F

X

x∈ ( , 1, 2, )=−8 + 1 +2 2 − min

subject to x≤1

f x y y z x y y z

Z z Y

y1min∈ ,∈ 1( , 1, 2, )= −2 1+ 2 + subject to y₁ ≤1

f x y y z x y y z

Z z Y

y₂min∈ ,∈ 2( , 1, 2, )= + 1−2 2+ subject to y2 ≤1

The followers share the variable z. According to the way of model transformation,

(1), (2) and (3), we have as follows:

F x y y z x y y z

X

x∈ ( , 1, 2, )=−8 + 1+2 2 − min

subject to x≤1

f x y y z x y y z

Y

y∈ 1( , 1, 2, )= −2 1+ 2 + min

1

f x y y z x y y z

Y

y∈ 2( , 1, 2, )= + 1−2 2 + min

2

f x y y z x y y z

Z

z ( , , , ) 2 2

min _{3 1 2} = − ₁− ₂ +

∈

subject to y1 ≤1

y2 ≤1

z ≤1.

According to the extended Kth-best approach, the transferred form of Example can

be rewritten as follow in the format of (8),

z y y x z

y y x

F( , ₁, ₂, )=−8 + ₁ +2 ₂ − min

subject to x≤1 y₁ ≤1

y2 ≤1

z ≤1

x≥0,y₁ ≥0,y₂ ≥0,z ≥0.

Loop 1:

Setting i←1 and by (10), we have

z y y x z y y x

f1( , 1, 2, )= −2 1+ 2 + min

subject to x≤1 y₁ ≤1

y₂ ≤1

z ≤1 x=1 y1 ≥0

y2 ≥0

z≥0.

Using the bounded simplex method, we have ~y_1j =1. Because of ~y_1j ≠ y_1[j], we go to Step 3. We have: W_[j] ={(1,0,1,1),(1,1,0,1),(1,0,0,0),(0,0,0,1)} , T ={(1,0,0,1)} and

)} 1 , 0 , 0 , 0 ( ), 0 , 0 , 0 , 1 ( ), 1 , 0 , 1 , 1 ( ), 1 , 1 , 0 , 1 {(

=

W , then go to Step 4. Update j=2, and choose )

1 , 0 , 1 , 1 ( ) , , ,

(x[j] y1[j] y2[j] z[j] = , then go to Step 2. Loop 2:

Setting i←1 and by (10), we have

z y y x z y y x

f₁( , ₁, ₂, )= −2 ₁+ ₂ + min

z ≤1 x=1 y₁ ≥0

y2 ≥0

z≥0.

Using the bounded simplex method, we have ~y_1j =1 and ~y_1j = y_1[j]. Setting 1

+ ←i

i and by (10), we have

z y y x z y y x f

Y

y∈ 2( , 1, 2, )= + 1−2 2 + min

2

subject to x≤1 y₁ ≤1

y₂ ≤1

z ≤1 x=1 y1 ≥0

y₂ ≥0

z≥0.

Using the bounded simplex method, we have ~y_2j =1. Because of ~y_2j ≠ ~y_2[j], we go

to Step 3. We have W[j] ={(0,1,0,1),(1,1,1,1),(1,0,0,0),(1,1,0,0)}, T ={(1,0,0,1),(1,1,0,1)} and

)} 0 , 0 , 1 , 1 ( ), 1 , 1 , 1 , 1 ( ), 1 , 0 , 1 , 0 ( ), 1 , 1 , 0 , 1 ( ), 0 , 0 , 0 , 1 ( ), 1 , 0 , 0 , 0 {(

=

W , then go to Step 4. Update

1

+ = j

Setting i←1 and by (10), we have

z y y x z y y x

f₁( , ₁, ₂, )= −2 ₁+ ₂ + min

subject to x≤1 y1 ≤1

y₂ ≤1

z ≤1 x=1 y₁ ≥0

y₂ ≥0

z≥0.

Using the bounded simplex method, we have ~y1j =1. Because of ~y1j ≠ y1[j], we go to Step 3. We have: W_[j] ={(1,0,1,0),(0,0,0,0)} , T ={(1,0,0,1),(1,1,0,1),(1,0,0,0)} and

)} 0 , 0 , 0 , 0 ( ), 0 , 1 , 0 , 1 ( ), 0 , 0 , 1 , 1 ( ), 1 , 1 , 1 , 1 ( ), 1 , 0 , 1 , 0 ( ), 1 , 1 , 0 , 1 ( ), 1 , 0 , 0 , 0 {(

=

W , then go to Step 4.

Update j= j+1, and choose (x_[j],y_1[j],y_2[j],z_[j])=(1,0,1,1), then go to Step 2. Loop 4:

Setting i←1 and by (10), we have

z y y x z y y x

f1( , 1, 2, )= −2 1+ 2 + min

subject to x≤1 y1 ≤1

y1 ≥0

y2 ≥0

z≥0.

Using the bounded simplex method, we have ~y_1j =1. Because of ~y_1j ≠ y_1[j], we go to Step 3. We have: W_[j] ={(0,0,1,1)} , T ={(1,0,0,1),(1,1,0,1),(1,0,0,0),(1,0,1,1)} and

} 1 , 1 , 0 , 0 ( ), 0 , 0 , 0 , 0 ( ), 0 , 1 , 0 , 1 ( ), 0 , 0 , 1 , 1 ( ), 1 , 1 , 1 , 1 ( ), 1 , 0 , 1 , 0 ( ), 1 , 0 , 0 , 0 {(

=

W , then go to Step 4.

Update j= j+1, and choose (x_[j],y_1[j],y_2[j],z_[j])=(1,1,0,0), then go to Step 2. Loop 5:

Setting i←1 and by (10), we have

z y y x z y y x

f₁( , ₁, ₂, )= −2 ₁+ ₂ + min

subject to x≤1 y₁ ≤1

y₂ ≤1

z ≤1 x=1 y1 ≥0

y2 ≥0

z≥0.

Using the bounded simplex method, we have ~y_1j =1 and ~y_1j = y_1[j]. Setting 1

+ ←i

i and by (10), we have

z y y x z y y x f

Y

y∈ 2( , 1, 2, )= + 1−2 2 + min

subject to x≤1 y₁ ≤1

y2 ≤1

z ≤1 x=1 y₁ ≥0

y₂ ≥0

z≥0.

Using the bounded simplex method, we have ~y2j =1. Because of ~y2j ≠ ~y2[j], we go to Step 3. We have

)} 0 , 1 , 1 , 1 ( ), 0 , 0 , 1 , 0 {( ] [j =

W , T ={(1,0,0,1),(1,1,0,1),(1,0,0,0),(1,0,1,1),(1,1,0,0)} and )}

0 , 1 , 1 , 1 ( ), 0 , 0 , 1 , 0 ( }, 1 , 1 , 0 , 0 ( ), 0 , 0 , 0 , 0 ( ), 0 , 1 , 0 , 1 ( ), 1 , 1 , 1 , 1 ( ), 1 , 0 , 1 , 0 ( ), 1 , 0 , 0 , 0 {(

=

W , then go to

Step 4. Update j= j+1, and choose (x_[j],y_1[j],y_2[j],z_[j])=(1,1,1,1), then go to Step 2. Loop 6:

Setting i←1 and by (10), we have

z y y x z y y x

f1( , 1, 2, )= −2 1+ 2 + min

subject to x≤1 y₁ ≤1

y₂ ≤1

y2 ≥0

z≥0.

Using the bounded simplex method, we have ~y_1j =1 and ~y_1j = y_1[j]. Setting 1

+ ←i

i and by (10), we have

z y y x z y y x f

Y

y∈ 2( , 1, 2, )= + 1−2 2 + min

2

subject to x≤1 y₁ ≤1

y₂ ≤1

z ≤1 x=1 y₁ ≥0

y2 ≥0

z≥0.

Using the bounded simplex method, we have ~y_2j =1and ~y_2j = ~y_2[j] . Setting 1

+ ←i

i and by (10), we have

z y y x z y y x f

Z

z ( , , , ) 2 2

min _{3 1 2} = − ₁− ₂ +

∈

subject to x≤1 y₁ ≤1

y2 ≤1

y1 ≥0

y2 ≥0

z≥0.

Using the bounded simplex method, we have ~zj =0. Because of ~zj ≠z[j], we go to Step 3. We have

)} 1 , 1 , 1 , 0 {( ] [j =

W , T ={(1,0,0,1),(1,1,0,1),(1,0,0,0),(1,0,1,1),(1,1,0,0),(1,1,1,1)} and )}

1 , 1 , 1 , 0 ( ), 0 , 1 , 1 , 1 ( ), 0 , 0 , 1 , 0 ( }, 1 , 1 , 0 , 0 ( ), 0 , 0 , 0 , 0 ( ), 0 , 1 , 0 , 1 ( ), 1 , 0 , 1 , 0 ( ), 1 , 0 , 0 , 0 {(

=

W , then go to

Step 4. Update j= j+1, and choose (x_[j],y_1[j],y_2[j],z_[j])=(1,0,1,0), then go to Step 2. Loop 7:

Setting i←1 and by (10), we have

z y y x z y y x

f1( , 1, 2, )= −2 1+ 2 + min

subject to x≤1 y₁ ≤1

y₂ ≤1

z ≤1 x=1 y₁ ≥0

y₂ ≥0

)} 0 , 1 , 0 , 0 {( ] [j =

W , T ={(1,0,0,1),(1,1,0,1),(1,0,0,0),(1,0,1,1),(1,1,0,0),(1,1,1,1),(1,0,1,0)} and )}

0 , 1 , 0 , 0 ( ), 1 , 1 , 1 , 0 ( ), 0 , 1 , 1 , 1 ( ), 0 , 0 , 1 , 0 ( }, 1 , 1 , 0 , 0 ( ), 0 , 0 , 0 , 0 ( ), 1 , 0 , 1 , 0 ( ), 1 , 0 , 0 , 0 {(

=

W , then go to

Step 4. Update j= j+1, and choose (x[j],y1[j],y2[j],z[j])=(1,1,1,0), then go to Step 2. Loop 8:

Setting i←1 and by (10), we have

z y y x z y y x

f₁( , ₁, ₂, )= −2 ₁+ ₂ + min

subject to x≤1 y1 ≤1

y2 ≤1

z ≤1 x=1 y₁ ≥0

y₂ ≥0

z≥0.

Using the bounded simplex method, we have ~y1j =1 and ~y1j = y1[j]. Setting 1

+ ←i

i and by (10), we have

z y y x z y y x f

Y

y∈ 2( , 1, 2, )= + 1−2 2 + min

2

subject to x≤1 y1 ≤1

y₂ ≤1

x=1 y₁ ≥0

y₂ ≥0

z≥0.

Using the bounded simplex method, we have ~y2j =1and ~y2j = ~y2[j] . Setting 1

+ ←i

i and by (10), we have

z y y x z y y x f

Z

z ( , , , ) 2 2

min 3 1 2 = − 1− 2 +

∈

subject to x≤1 y₁ ≤1

y₂ ≤1

z ≤1 x=1 y₁ ≥0

y2 ≥0

z≥0.

Using the bounded simplex method, we have ~z_j =1 and ~z_j = z_[j] . Solution )

0 , 1 , 1 , 1 ( ) , , ,

(x_[j] y_1[j] y_2[j] z_[j] = is the global solution to Example .

By examining above procedure, we found that the solution occurs at the point

) 0 , 1 , 1 , 1 ( ) , , ,

5. Conclusion and further study

This paper proposes the Kth-best approach for linear bilevel multifollower programming

problems with shared variables among followers. A numeric example is given to show

how the Kth-best approach works. The further study of the research is to integrate this

method into decision support system (DSS) technology.

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