Model and Algorithm for Multi-follower Tri-level Hierarchical Decision ...

Model and Algorithm for Multi-follower Tri-level Hierarchical Decision-Making Jialin Han1,2, Guangquan Zhang 2,*, Jie Lu2, Yaoguang Hu1, and Shuyuan Ma1 1

School of Mechanical Engineering, Beijing Institute of Technology, China Faculty of Engineering and Information Technology, University of Technology, Sydney, Australia {Hjl,hyg,bitmc}@bit.edu.cn, {Guangquan.Zhang,Jie.Lu}@uts.edu.au 2

Abstract. Tri-level decision-making addresses compromises among interacting decision entities that are distributed throughout a three-level hierarchy. Decision entities at the three hierarchical levels are respectively termed as the toplevel leader, the middle-level follower and the bottom-level follower. When multiple followers are involved at the middle and bottom levels, the leader’s decision will be affected not only by reactions of the followers but also by various relationships among them. To support such a multi-follower tri-level (MFTL) decision-making process, this study first proposes a general MFTL decision model for the situation involving both cooperative and uncooperative relationships among multiple followers. It then develops a MFTL Kth-Best algorithm to find an optimal solution to the model. Lastly, we use the proposed MFTL decision techniques to deal with a supply chain management problem in applications. Keywords: Hierarchical decision-making, multilevel programming, tri-level decision-making, Kth-Best algorithm, supply chain.

1

Introduction

Tri-level decision-making (also known as tri-level programming) technique is proposed to solve decentralized decision problems involving interacting decision entities distributed throughout a three-level hierarchy, which is a subfamily of multilevel programming [1] motivated by Stackelberg game theory [2]. Decision entities at the three hierarchical levels are respectively termed as the leader, the middle-level follower and the bottom-level follower, and make their decisions in sequence from the top level to the middle level and then to the bottom level seeking to optimize their individual objectives. The decision process means the higher level has the priority to make its decision and the lower-level decision entity reacts after and in view of a decision made by the higher level. However, the decision of each entity is affected by the actions of the others. The decision process is repeatedly executed until the Stackelberg equilibrium among them is achieved. This category of hierarchical *

This work is supported by the Australian Research Council (ARC) under discovery grant DP140101366, and the National High Technology Research and Development Program of China (NO. 2013AA040402).

C.K. Loo et al. (Eds.): ICONIP 2014, Part III, LNCS 8836, pp. 398–406, 2014. © Springer International Publishing Switzerland 2014

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decision-making process often appears in many decentralized management problems, such as supply chains management [3], resource allocation optimization [4,5] and hierarchical production operations [6]. Take the three-stage supply chain composing of a manufacturer, a distributor and a vendor as an example. The manufacturer, distributor and vendor are distributed throughout three hierarchical levels, respectively called the leader, the middle-level follower and the bottom-level follower. Within a marketing circle, each of them has to keep a certain amount of inventory to fulfill exceeded market requirements and meanwhile aims to minimize its own inventory cost. However, their total inventories must satisfy the exceeded market requirements, which means that one decision entity has to increase its holding inventory if the others reduce their inventories. Furthermore, the leader has the priority to determine its own inventory by considering market requirements and the implicit reactions of both followers. The middle-level follower then adjusts its holding inventory to respond to the leader likewise considering the implicit reactions of the bottom-level follower. In the light of the decisions of the top and middle levels, the bottom-level follower determines its inventory to optimize its own objective at last. The example describes a typical tri-level decision problem in which the execution of decisions is sequential, interactive and iterative among the three decision entities seeking to optimize their individual objectives until the Stackelberg equilibrium is achieved. In general, there are two fundamental issues in supporting the tri-level decisionmaking process in applications. One is how to model a real-world tri-level decision problem, which may manifest different characteristics at the three decision levels, and the other is how to find an optimal solution to the problem. Although tri-level decisionmaking has been attracting numerous investigations on models [7,8], solution algorithms [7,8] and applications [3,4],[6],[8], the existing research has been mainly limited to the situation that a single decision entity is involved at each level. Actually, two or more decision entities are often involved at the middle and bottom levels in real-world cases called multi-follower tri-level (MFTL) decision-making [9]. In the three-stage supply chain example, the manufacturer (the leader) may have multiple subordinate distributors (middle-level followers), and simultaneously, there may also be several vendors (bottom-level followers) attached to each distributor. Moreover, multiple followers at the same level may have a variety of relationships with one another, such as cooperative and uncooperative relationships. Such situations will make the MFTL decision complex and generate different decision processes, which need to be described and solved using different decision models and solution methods. This study considers a special situation that a cooperative and an uncooperative relationship appear at the middle and bottom levels respectively based on related definitions in our previous research [9]. The situation means that multiple followers at the middle level have the same decision variables but have individual optimization objectives and separate constraints, while multiple bottom-level followers attached to the same middle-level follower optimize their own objectives by controlling individual decision variables under their separate constraints. For example, the multiple distributers or vendors may collaborate with other counterparts by making joint decisions to enhance market competitiveness, called a cooperative relationship among them, or may consider their counterparts as competitors and make their decisions independently, known as an uncooperative relationship among them.

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The main contribution of this paper is the provision of a general model and a solution method to describe and solve the proposed MFTL decision process. The paper first presents a linear MFTL decision model for the above situation. To find an optimal solution to the model, a MFTL Kth-Best algorithm is proposed. Lastly, a case study on three-stage supply chain decision illustrates the proposed MFTL decision techniques.

2

A MFTL Decision Model and Related Theoretical Properties

The organizational structure among decision entities in the three-level hierarchy that is studied in this paper is shown as Fig. 1. Let x ∈ X ⊂ Rk , y ∈ Yi ⊂ R k0 , z ij ∈ Z ij ⊂ R kij denote the decision variables of the leader, the middle-level follower i, and the bottom-level follower ij respectively where j = 1,2,, mi , i = 1,2,, n . We give detailed definitions of the cooperative and uncooperative relationships proposed in our previous research [9]. Definition 1. [9] If both the objective function and constraint conditions of the middle-level follower i only involve the shared decision variable y controlled by all of them in common apart from the decision variables x, z i1 , , zimi determined by the leader and the bottom-level followers, this means a cooperative relationship among multiple followers at the middle level. Definition 2. [9] If both the objective function and constraint conditions of the bottom-level follower ij only involve its own decision variable z ij apart from the decision variables x and y respectively determined by the leader and the middle-level follower i, this can be called a uncooperative relationship among multiple bottom-level followers attached to the same middle-level follower i. .

.

.

.

.

.

1m1

.

nmn

Fig. 1. The organizational structure of the three-level hierarchy

For

x ∈ X ⊂ Rk , y ∈ Yi ⊂ R k0

, Y = Y1    Yn , y ∈ Y ⊂ R k 0 , z ij ∈ Z ij ⊂ R kij ,

F : X × Y × Z 1 ×  × Z 1m1 ×  × Z n1  × Z nmn → R 1

,

f i ( 2 ) : X × Yi × Z i1 ×  × Z imi

→ R1

,

f ij(3) : X × Yi × Z ij → R1 and j = 1,2, , mi , i = 1,2,, n , a linear MFTL decision model in

which one leader, n cooperative middle-level followers and mi uncooperative bottom-level followers attached to the middle-level follower i are involved is defined as follows:

Model and Algorithm for Multi-follower Tri-level Hierarchical Decision-Making n

mi

min F ( x, y, z11 , , z1m , , z n1 , , z nm ) = cx + dy +  eij z ij x∈X

1

n

(Leader)

mi

n

(1a)

i =1 j =1

Ax + By +  Cij z ij ≤ b ,

s.t.

401

(1b)

i =1 j =1

where ( y, zi1 ,, zimi ) (i = 1,2,  , n) , for each given value x, solves (1c-1f): mi

min f i (2 ) ( x, y, zi1 , , z im ) = ci x + d i y +  g ij z ij y∈Yi

i

(Middle-level follower i)

(1c)

j =1

mi

s.t. Ai x + Bi y +  Dij z ij ≤ bi ,

(1d)

j =1

where z ij ( j = 1,2, , mi ) , for the given value ( x, y ) , solves (1e-1f): (Bottom-level follower ij) (1e) min f ( x, y, zij ) = cij x + dij y + hij zij (3) ij

zij∈Zij

s.t. Aij x + Bij y + Eij z ij ≤ bij ,

(1f)

where c, ci , cij ∈ R , d i , di , d ij ∈ R , eij , g ij , hij ∈ R k

k0

B ∈ R s×k0 , Bi ∈ R s ×k , Bij ∈ R i

0

sij ×k 0

, Cij ∈ R s×k , Dij ∈ R ij

kij

si ×kij

, A∈ R

s×k

, Ai ∈ R

, Eij ∈ R

sij ×kij

, b ∈ R s , bi ∈ R s , bij ∈ R

si ×k

, Aij ∈ R

sij ×k

i

,

sij

, j = 1,2,, mi , i = 1,2,, n . Concepts related to the solutions to the model (1) are defined as follows. Definition 3. 1 Constraint region of the MFTL decision model (1):

）

n

mi

n mi

S = {( x, y , z11 ,  , z1m1 ,  , z n1 ,  , z nmn ) ∈ X × Y × ∏ ∏ Z ij : Ax + By +   C ij z ij ≤ b, i =1 j =1

i =1 j =1

mi

Ai x + B i y +  D ij z ij ≤ bi , Aij x + B ij y + E ij z ij ≤ bij , j = 1, 2,  , m i , i = 1,2,  , n}.

）Feasible set of the ith middle-level follower and its bottom-level followers for j =1

2

each fixed x ∈ X :

mi

mi

Si ( x) = {( y, zi1 ,, zimi ) ∈ Yi × ∏ Z ij : Ai x + Bi y +  Dij zij ≤ bi , Aij x + Bij y + Eij zij ≤ bij , j = 1,2,, mi }, i = 1,2,, n.

）Feasible set of the ith middle-level follower’s jth bottom-level follower for j =1

3

j =1

each fixed ( x, y) ∈ X × Yi :

S ij ( x, y ) = {z ij ∈ Z ij : Aij x + Bij y + E ij z ij ≤ bij }, j = 1,2,  , m i , i = 1,2,  , n.

4

） Rational reaction set of the ith middle-level follower’s jth bottom-level follower:

Pij ( x, y ) = {z ij ∈ Z ij : z ij ∈ arg min[ f ij( 3) ( x, y , zˆ ij ), zˆ ij ∈ S ij ( x, y )]}, j = 1,2,  , mi , i = 1,2,  , n.

5

） Rational reaction set of the ith middle-level follower and its bottom-level followers:

mi

Pi ( x) = {( y, zi1 ,, zimi ) ∈ Yi × ∏ Zij : ( y, zi1 ,, zimi ) ∈ arg min fi ( 2) ( x, yˆ , zˆi1 ,, zˆimi ), ( yˆ , zˆi1 ,, zˆimi ) ∈ Si ( x), j =1

6

） Inducible region:

zˆ ij ∈ Pij ( x, yˆ ), j = 1,2, , mi }, i = 1,2,  , n.

IR = {( x , y , z 11 ,  , z 1m1 ,  , z nm1 ,  , z nm n ) : ( x , y , z 11 ,  , z 1m1 ,  , z nm1 ,  , z nm n ) ∈ S ,

( y, zi1 ,, zim ) ∈ Pi ( x), i = 1,2,, n}. i

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3

A MFTL Kth-Best Algorithm

The inducible region of the linear bi-level programming problem is composed of connected faces of S so that a vertex of the original polyhedron will provide the solution, which can be extended to the tri-level programming problem [7],[9]. Consider the following linear programming problem: (2) min{F ( x, y, z11 ,  , z1m1 ,  , z n1 ,  , z nmn ) : ( x, y, z11 ,  , z1m1 ,  , z n1 ,  , z nmn ) ∈ S} N and let ( x 1 , y 1 , z111 ,  , z11m1 ,  , z 1n1 ,  , z 1nm n ),  , ( x N , y N , z11N ,  , z1Nm1 ,  , z nN1 ,  , z nm ) denote the n

n mi

N-ranked basic feasible solution to (2), such that cx K + dy K +   eij z ijK ≤ i =1 j =1

cx

K +1

+ dy

K +1

n mi

  eij z i =1 j =1

K +1 ij

K = 1,2,  N − 1 .

,

Then solving the problem (1) is equivalent to

K searching the index K ∗ = min{K ∈{1,2,, N} : (xK , yK , z11K ,, z1Km ,, znK1,, znm ) ∈ IR} , which ensures 1

n

K∗ that ( x K ∗ , y K ∗ , z11K ∗ ,  , z1Km∗1 ,  , z nK1∗ ,  , z nm ) is an optimal solution to the model (1). As n

this requires finding the K ∗ th best vertex solution to problem (2) to obtain an optimal solution to model (1), the algorithm therefore is named Kth-Best algorithm. Procedures of the MFTL Kth-Best algorithm are developed as follows. The MFTL Kth-Best algorithm [Begin] Step 1: Set k=1, adopt the simplex method to obtain an optimal solution 1 ( x1 , y1 , z11 ,, z11m ,, z1n1 ,, z1nm ) to the linear programming problem (2). Let T be a set of n

1

feasible vertices of problem (2) that has been searched and W be a set of feasible vertices to be searched. Let T = ∅ and W = {(x1 , y1 , z111 ,, z11m ,, z 1n1 ,, z 1nm )} . Set i = 1 and 1

n

go to Step 2. Step 2: Put x = x k and solve the problem (1c-1f) and obtain an optimal solution ( yˆ, zˆi1,, zˆim ) using the bi-level Kth-Best algorithm [10]. Then go to Step 3. i

Step 3: If ( yˆ , zˆi1 ,, zˆim ) ≠ ( y k , zik1 ,, zimk ) , go to Step 4. If ( yˆ , zˆi1 ,, zˆim ) = ( y k , zik1 ,, zimk ) i

i

i

i

and i ≠ n , set i=i+1 and go to Step 2. If ( yˆ , zˆi1 ,, zˆim ) = ( y k , zik1 ,, z imk ) and i = n , i

i

k stop and ( x k , y k , z11k ,, z1km ,, z nk1 ,, z nm ) is an optimal solution to the MFTL decision 1

n

∗

model (1) and K = k . Step 4: Let W k denote ( x , y , z ,, z , , z ,, z k

k

k 11

n

k 1m1

k n1

k nmn

mi

) n

the

of

adjacent

extreme

points

such that ( x, y, z11 ,  , z1m ,  , z n1 ,  , z nm ) ∈ Wk 1

mi

cx + dy +  eij z ij ≥cx k + dy k +  eij z ijk i =1 j =1

set

n

of

implies

k . Let T = T  {( x k , y k , z11k , , z1km , , z nk1 , , z nm )} 1

n

i =1 j =1

and W = (W  W k ) \ T . Go to Step 5. k k k Step 5: Set k=k+1 and choose ( x k , y k , z11k ,, z1km ,, znk1 ,, znm ) such that cx + dy + 1

n mi

n mi

i =1 j =1

i =1 j =1

n

  eij z ijk = min{cx + dy +   eij z ij : ( x, y, z11 ,  , z1m1 ,  , z n1 ,  , z nmn ) ∈W } . Set i=1 and go to Step 2. [End]

Model and Algorithm for Multi-follower Tri-level Hierarchical Decision-Making

4

403

A Case Study

Following the three-stage supply chain decision example proposed in the introduction of the paper, we give more detailed decision information as follows. In the MFTL decision case, the manufacturer (leader) and the vendors (bottom-level followers) keep their individual inventories using their respective warehouses and determine their inventories independently to optimize their individual objective under separate constraints. Clearly, there exists the uncooperative relationship among multiple bottom-level followers. However, to reduce the inventory cost, all the distributers at the middle level share a common warehouse and determine the inventory in common, which means they have the same decision variable even though they may have individual objectives and separate constraints called a cooperative relationship among them. Therefore, the leader seeks to minimize its inventory cost F ( x, y , z11 , , z1m , , z n1 , , z nm ) by controlling its own decision variable x (its invento1

n

ry). The middle-level followers want to optimize their individual objectives f i ( 2) ( x, y, zi1 ,, zim ) ( i = 1,2,, n ) by determining the shared decision variable y (the i

common inventory) for the given x determined by the leader. The bottom-level follower ij attached to the middle-level follower i( i = 1,2, , n ) optimize its objective function f ij(3) ( x, y, z ij ) ( j = 1,2,, mi ) by choosing its individual decision variable z ij in view of the given x and y. In this paper, we simplify the three-stage supply chain decision problem as the following numerical model in the format of model (1). For x ∈ X ⊂ R1 , y ∈ Yi ⊂ R 1 , z ij ∈ Z ij ⊂ R1 and X = { x : x ≥ 0}, Y i = { y : y ≥ 0} , Z ij = { z ij : z ij ≥ 0}, j = 1,2,  , mi , i = 1,2 , mi = 2 .

min F ( x, y, z11 , z12 , z 21 , z 22 ) = −1.5x − y + 2 z11 + z12 − z 21 − 0.5z 22 x∈ X

s.t.

x + y + z11 + z12 + z 21 + z 22 ≥ 10, x ≤ 1.5, min f1( 2 ) ( x , y , z11 , z12 ) = x + y + z11 + z12 y∈Y1

s.t.

min f 2( 2 ) ( x , y , z 21 , z 22 ) = x − y + 2 z 21 + 3 z 22 y∈Y2

x + y + z11 + z12 ≥ 6.5, x + y ≤ 2, min f

z11∈Z 11

( 3) 11

( x , y , z11 ) = x + y + 3 z11

s.t. x + y + z11 ≥ 3.5,

s.t. x + y + z 21 + z 22 ≥ 5.5, x + y ≤ 2, min f 21( 3) ( x, y , z 21 ) = x + y + 2 z 21

z 21∈Z 21

s.t. x + y + z 21 ≥ 3, z 21 ≤ 2,

z11 ≤ 2, min f 12( 3 ) ( x , y1 , y 0 , z12 ) = x + y + 2 z12

z12 ∈Z12

s.t. x + y + z12 ≥ 5, z12 ≤ 4,

min f 22( 3) ( x, y, z 21 ) = x + y + z 22

z 22 ∈Z 22

s.t. x + y + z 22 ≥ 4.5, z 22 ≤ 3.

We can adopt the MFTL Kth-Best algorithm to solve the MFTL decision problem. First, we have to solve a linear programming problem in the format (2). Step 1: Set k=1 and adopt the simplex method to obtain an optimal solution to the problem (2). The optimal solution to (2) is ( x1 , y 1 , z111 , z121 , z 121 , z 122 ) = (1.5,0.5,1.5,3,2,3) and

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now W = {(1.5,0.5,1.5,3,2,3)} , T = ∅ . Set i=1 and go to Step 2 and the iteration 1 will begin. Step 2: Put x = x1 = 1.5 , and solve the problem in the form of (1c-1f). We can get the optimal solution ( yˆ , zˆ11 , zˆ12 ) = (0.5,1.5,3) to (1c-1f) by bi-level Kth-Best algorithm and go to Step 3. Step 3: Obviously, ( yˆ , zˆ11 , zˆ12 ) = ( y 1 , z111 , z121 ) = (0.5,1.5,3) , i ≠ n , set i=2 and go to Step 2. Step 2: Put x = x1 = 1.5 and i=2, and solve the problem in the form of (1c-1f). We can get the optimal solution ( yˆ , zˆ 21 , zˆ 22 ) = (0.5,1,2.5) to (1c-1f) by bi-level Kth-Best algorithm and go to Step 3. Step 3: Now, ( yˆ , zˆ 21 , zˆ 22 ) ≠ ( y1 , z 121 , z 122 ) and go to Step 4. Step 4: Find the adjacent extreme points of s 1 = ( x 1 , y 1 , z111 , z121 , z 121 , z 122 ) =(1.5,0.5,1.5,3,2,3) and now the set of adjacent extreme points W1 = {(0,2,1.5,3,2,3), (1.5,0.5,1.5,3,1,3), (1.5,0.5, 1.5,3,2,2.5)} , T = {s 1 } = {(1.5,0.5,1.5,3,2,3)} , W = W1 . Go to Step 5. Step 5: Set k=k+1=2 and choose ( x 2 , y 2 , z112 , z122 , z 212 , z 222 ) = (1.5,0.5,1.5,3,2,2.5) from the ver2 2 that F ( x 2 , y 2 , z112 , z122 , z 21 , z 22 ) = min{F ( x, y, z11 , z12 , z 21 , z 22 ) : ( x , y , z 11 , z12 , z 21 , z 22 ) ∈ W } , set i=1 and go to Step 2. This step means the iteration 1 has stopped and we cannot get an optimal solution through the iteration so the second iteration will then be executed.

tices

set

W

such

Table 1. The detailed computing process by the MFTL Kth-Best algorithm Iteration k

s k = ( x k , y k , z11k ,

Wk

k k z12k , z 21 , z 22 )

T

2

(1.5,0.5,1.5,3,2,2.5)

{(1.5,0.5,1.5,3,1,2.5)}

{ s1 , s 2 }

3

(0,2,1.5,3,2,3)

{(0,2,1.5,3,1,3), (0,2,1.5,3,2,2.5)}

{ s1 , s 2 , s 3 }

4

(1.5,0.5,1.5,3,1,3)

5

(0,2,1.5,3,2,2.5)

6

(0,2,1.5,3,1,3)

7

(1.5,0.5,1.5,3,1,2.5)

∅

{(0,2,1.5,3,1,2.5)} ∅

---

{ s1 , s 2 , s 3 , s 4 } { s1 , s 2 , s 3 , s 4 , s 5 } { s1 , s 2 , s 3 , s 4 , s 5 , s 6 } ---

W

{(0,2,1.5,3,2,3), (1.5,0.5,1.5,3,1,3), (1.5,0.5,1.5,3,1,2.5)} {(1.5,0.5,1.5,3,1,3), (1.5,0.5,1.5,3,1,2.5), (0,2,1.5,3,1,3), (0,2,1.5,3,2,2.5)} {(1.5,0.5,1.5,3,1,2.5), (0,2,1.5,3,1,3), (0,2,1.5,3,2,2.5)} {(1.5,0.5,1.5,3,1,2.5), (0,2,1.5,3,1,3), (0,2,1.5,3,1,2.5)} {(1.5,0.5,1.5,3,1,2.5), (0,2,1.5,3,1,2.5)} ---

In this way, we finally get an optimal solution through seven iterations. The searched extreme points and the detailed computing process of iterations 2-7 are shown as Table 1. In iteration 7, ( x 7 , y 7 , z117 , z127 , z 217 , z 227 ) = (1.5,0.5,1.5,3,1,2.5) is an optimal solution to the MFTL decision problem, which implies that all decision entities achieve the equilibrium at the vertex solution. The objective function values of all

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decision entities are F ( x, y , z11 , z12 , z 21 , z 22 ) = 1.0 , f1( 2) ( x, y, z11, z12 ) = 6.5 , f 2( 2) ( x, y, z21 , z22 ) = 10.5 , (3 ) f 11( 3 ) ( x , y , z 11 ) = 6 . 5 , f 12( 3) ( x , y , z 12 ) = 8.0 , f 21( 3) ( x , y , z 21 ) = 4.0 , f 22 ( x , y , z 22 ) = 4 .5 . It is noticeable that W 4 = ∅ and W 6 = ∅ in Table 1 do not mean there does not exist adjacent extreme points of ( x 4 , y 4 , z114 , z124 , z 214 , z 224 ) and ( x 6 , y 6 , z116 , z126 , z 216 , z 226 ) but may imply their adjacent extreme points have been found in previous iterations and have been involved in W. However, when plenty of followers are involved or a large number of decision variables and constraints exist, the execution efficiency of the algorithm may experience a steep decline as superabundant vertices are needed to complete the search.

5

Conclusions and Further Study

In a tri-level decision problem, multiple followers are often involved at the middle and bottom levels. Various relationships among multiple followers at the same level can result in different decision processes in a three-level hierarchical system. To support MFTL decision problems involving both cooperative and uncooperative relationships among multiple followers, this paper proposes a decision model and then develops a Kth-Best algorithm to find an optimal solution to the model. Lastly, a case study on three-stage supply chain decision illustrates the effectiveness of the proposed MFTL decision techniques. The results show that the MFTL decision model and KthBest algorithm provide an available way in describing and solving the proposed MFTL decision process. Our future research will develop a decision support system driven by the proposed decision techniques to explore the performance of the MFTL Kth-Best algorithm through sufficient numerical experiments and to handle largescale or more complex MFTL decision problems in applications.

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