A fuzzy tri-level decision making algorithm and its application in supply ...

8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2013)

A fuzzy tri-level decision making algorithm and its application in supply chain Zhen Zhang1,2 , Guangquan Zhang2 , Jie Lu2 , Chonghui Guo1 1

2

Institute of Systems Engineering, Dalian University of Technology, Dalian 116024, P. R. China Lab of Decision Systems and e-Service Intelligence, Centre for Quantum Computation and Intelligent Systems, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia

Abstract

faculty and the department need to be considered, but the three decision entities are in a hierarchical organization. Each decision entity independently optimizes his/her objective, but is affected by the decision of the other two decision entity through external influences [14]. For decision making problems within a tri-level hierarchical organization, Bard and Falk [15] first proposed the necessary conditions for tri-level decision making (TLDM) problems based on Stackelberg game theory and then developed rational reaction sets for each of the decision entities and a cutting-plane algorithm to solve the TLDM problems. Followed by their research, some other studies, such as the penalty-function approach [16], the satisfactory solution-based approach [9] and interactive balance space approach [17], are developed to deal with TLDM problems. In a recent work, Zhang et al. [14] presented a general tri-level decision making model, defined the solution concept of the model and developed a Kth-best algorithm to solve the TLDM model. Considering that multiple decision entities may be involved in the middle level and the bottom level for a tri-level decision making problem, Lu et al. [18] proposed a multi-follower tri-level (MFTL) decision making framework, in which 64 standard MFTL decision situations and their possible combinations are identified. The previous studies have significantly advanced the field of tri-level decision making. However, in most situations the coefficients of the optimization problems are often imprecisely known to the decision makers and can’t be provided by precise values [19]. For such situations, it may be more appropriate to denote the coefficients of the optimization models with the use of fuzzy sets [20] and optimization problems with fuzzy coefficients have received much attentions in the last few decades [21–24]. The fuzzy coefficients may also appear in a TLDM problem, therefore, there is a need to develop efficient algorithms to deal with the fuzzy tri-level decision making (FTLDM) problems. In order to do so, the rest of this paper is organized as follows. In section 2, some preliminaries related to fuzzy sets are reviewed. Afterwards, section 3 develops a fuzzy tri-level decision making model and proposes a Kth-best algorithm to solve the model based on the λ-cut of fuzzy sets. In sec-

In this paper, we develop a fuzzy tri-level decision making (FTLDM) model to deal with decentralized decision making problems with three levels of decision makers. Based on the λ-cut of fuzzy set, we transform an FTLDM problem into a multiobjective tri-level decision making problem. Based on the linear tri-level Kth-best algorithm, the global optimal solution can be obtained. A case study for third-party logistics decision making in supply chain is utilized to illustrate the effectiveness of the proposed algorithm. Keywords: Fuzzy tri-level decision making, Kthbest algorithm, Supply chain 1. Introduction Multi-level decision making models, initiated by Von Stackelberg [1], are used to character decentralized decision making problems where decision makers are in a hierarchical organization. Many real-world decision making problem, such as decentralized resource planning [2], highway pricing [3], electronic power market [4] and logistics planning [5] can be formulated as multi-level decision making models. In the last decades, multi-level decision making problems have received more and more attentions [6–8]. Most of the research on multi-level decision making has focused on the bi-level version. In a bi-level decision making model, decision makers in the top level are called the leader and decision makers in the bottom level are called the follower. Decision makers in each level try to optimize their objective functions with partially or without considering the objective of the other level, but the final decision of each level will affect the objective of the other level [9]. In order to get a global optimal solution, a lot of algorithms have been developed, such as the Kth-best algorithm [10], the branch and bound algorithm [11], the Kuhn-Tucker condition-based algorithm [12] and heuristics algorithms [13]. However, many decision making problems sometimes involve decision entities at three levels instead of two levels, i.e. the top level, the middle level and the bottom level. For instance, when making a decision in a university, the objectives of the university, the © 2013. The authors - Published by Atlantis Press

154

tion 4, a case study of third-party logistics decision making is utilized to illustrate the proposed model and algorithm. Finally, some conclusions are provided in section 5.

3. The proposed algorithm Let’s consider the following fuzzy tri-level decision making (FTLDM) problem: For x ∈ X ⊂ Rn , y ∈ Y ⊂ Rm , z ∈ Z ⊂ Rp , fi : X × Y × Z → F ∗ (R), i = 1, 2, 3,

2. Preliminaries

min f1 (x, y, z) = c˜1 x + d˜1 y + e˜1 z

Let R denote the set of all real numbers and Rn be n−dimensional Euclidean space, and the following definitions are given.

x∈X

˜1 y + C˜1 z ≤ ˜b1 s.t. A˜1 x + B min f2 (x, y, z) = c˜2 x + d˜2 y + e˜2 z

Definition 1. [25] A fuzzy number a ˜ is defined as a fuzzy set on R, whose membership function µa˜ satisfies: (1) µa˜ is a mapping from R to the closed interval [0, 1]; (2) It is normal if there exists x ∈ R such that µa˜ (x) = 1; (3) For any λ ∈ [0, 1], the λ-cut of a ˜ is defined as aλ = {x : µa˜ (x) ≥ λ}, which is a closed interval, R denoted by [aL λ , aλ ].

y∈Y

˜2 y + C˜2 z ≤ ˜b2 s.t. A˜2 x + B min f3 (x, y, z) = c˜3 x + d˜3 y + e˜3 z z∈Z

˜3 y + C˜3 z ≤ ˜b3 , s.t. A˜3 x + B where c˜i ∈ F ∗ (Rn ), d˜i ∈ F ∗ (Rm ), e˜i ∈ F ∗ (Rp ), ˜bi ∈ F ∗ (Rqi ), A˜i = (˜ ajk,i )qi ×n , a ˜jk,i ∈ F ∗ (R), ∗ ˜ ˜ ˜ ˜ Bi = (bjk,i )qi ×m , bjk,i ∈ F (R), Ci = (˜ cjk,i )qi ×p , c˜jk,i ∈ F ∗ (R), i = 1, 2, 3. In the FTLDM problem (2), x, y, z denote the decision variables of the top-level, middle-level and bottom-level, respectively, and f1 (x, y, z), f2 (x, y, z) and f3 (x, y, z) denote the objective functions of the top-level, middle-level and bottom-level, respectively. Associated with the FTLDM problem, we consider the following multi-objective tri-level decision making (MOTLDM) problem: For x ∈ X ⊂ Rn , y ∈ Y ⊂ Rm , z ∈ Z ⊂ Rp , fi : X × Y × Z → F ∗ (R), i = 1, 2, 3,

Let F (R) be the set of all fuzzy numbers. By the decomposition theorem of fuzzy set, we have ∪ R a ˜ = λ[aL , a ], for every a ˜ ∈ F (R). A fuzzy λ λ λ∈[0,1]

number a ˜ is called finite if its λ-cut is a closed interval when λ = 0. Let F ∗ (R) denote the set of all finite fuzzy numbers on R. For any a ˜, ˜b ∈∪F (R) and α ≥ 0, we have [25] L R R ˜ (1) a ˜+b= λ[aL λ + bλ , aλ + bλ ], λ∈[0,1] ∪ R R L (2) a ˜ − ˜b = λ[aL λ − bλ , aλ − bλ ], λ∈[0,1] ∪ R (3) α˜ a= λ[αaL λ , αaλ ].

L L L min (f1 (x, y, z))L λ = c1 λ x + d1 λ y + e1 λ z, λ ∈ [0, 1]

x∈X

R R R min (f1 (x, y, z))R λ = c1 λ x + d1 λ y + e1 λ z, λ ∈ [0, 1]

λ∈[0,1]

x∈X

Definition 2. [25] Let a ˜i ∈ F (R), i = 1, 2 . . . , n. We define a ˜ = (˜ a1 , a ˜2 , . . . , a ˜n ) µa˜ : Rn → [0, 1] n ∧ x 7→ µa˜i (xi ),

L L L s.t. A1 L λ x + B1 λ y + C1 λ z ≤ b1 λ , λ ∈ [0, 1] R R R A1 R λ x + B1 λ y + C1 λ z ≤ b1 λ , λ ∈ [0, 1] L L L min (f2 (x, y, z))L λ = c2 λ x + d2 λ y + e2 λ z, λ ∈ [0, 1]

x∈X

(1)

R R R min (f2 (x, y, z))R λ = c2 λ x + d2 λ y + e2 λ z, λ ∈ [0, 1]

x∈X

i=1

L L L s.t. A2 L λ x + B2 λ y + C2 λ z ≤ b2 λ , λ ∈ [0, 1]

where x = (x1 , x2 . . . , xn )T ∈ Rn and a ˜ is called an n-dimensional fuzzy number on Rn . If a ˜i ∈ F ∗ (R), i = 1, 2, . . . , n, a ˜ is called an n-dimensional finite fuzzy number on Rn . n

∗

(2)

R R R A2 R λ x + B2 λ y + C2 λ z ≤ b2 λ , λ ∈ [0, 1] L L L min (f3 (x, y, z))L λ = c3 λ x + d3 λ y + e3 λ z, λ ∈ [0, 1]

x∈X

R R R min (f3 (x, y, z))R λ = c3 λ x + d3 λ y + e3 λ z, λ ∈ [0, 1]

x∈X

n

Let F (R ) and F (R ) be the set of all ndimensional fuzzy numbers and the set of all ndimensional finite fuzzy numbers on Rn , respectively.

L L L s.t. A3 L λ x + B3 λ y + C3 λ z ≤ b3 λ , λ ∈ [0, 1] R R R A3 R λ x + B3 λ y + C3 λ z ≤ b3 λ , λ ∈ [0, 1],

(3) R L m L R p R n where ci L λ , ci λ ∈ R , di λ , di λ ∈ R , ei λ , ei λ ∈ R , R L R L R L qi ×n qi , B i λ , Bi λ ∈ bi λ , bi λ ∈ R , Ai λ , Ai λ ∈ R R qi ×p , i = 1, 2, 3. Rqi ×m , Ci L λ , Ci λ ∈ R

Definition 3. [25] For any n-dimensional fuzzy numbers a ˜, ˜b ∈ F (Rn ), we define L R R (1) a ˜ ≻ ˜b iff aL iλ = biλ and aiλ = biλ , i = 1, 2 . . . , n, λ ∈ [0, 1]; L R R (2) a ˜ ≽ ˜b iff aL iλ ≥ biλ and aiλ ≥ biλ , i = 1, 2 . . . , n, λ ∈ [0, 1]; L R R (3) a ˜ ≻ ˜b iff aL iλ > biλ and aiλ > biλ , i = 1, 2 . . . , n, λ ∈ [0, 1].

Theorem 1. Let (x∗ , y ∗ , z ∗ ) be the solution of the problem (3), then it is also a solution of the FTLDM problem (2).

We call ≻, ≽ and ≻ a fuzzy max order, a strict fuzzy max order and a strong fuzzy max order, respectively.

Lemma 1. If there is a (x∗ , y ∗ , z ∗ ) such that cL αx + L L ∗ L ∗ L ∗ R R R dL α y + eα z ≥ cα x + dα y + eα z , cα x + dα y + eα z ≥ ∗ R ∗ R ∗ L L L L ∗ L ∗ cR α x +dα y +eα z , cβ x+dβ y+eβ z ≥ cβ x +dβ y +

Proof. The proof is obvious from Definition 3.

155

∗ R R R R ∗ R ∗ R ∗ eL β z , and cβ x+dβ y+eβ z ≥ cβ x +dβ y +eβ z for any (x, y, z) (0 ≤ β ≤ α ≤ 1), and the coefficients c˜, d˜ and e˜ have trapezoidal fuzzy membership function as  0 t < aL β    α−β  L L  (t − aβ ) + β aβ ≤ t < aL  α L  aL   α − aβ R α aL µa˜ (t) = , (4) α ≤ t < aα   α−β  R R R  (t − aβ ) + β aα ≤ t < aβ  R  aR  α − aβ   R

L bi L α −bi β α−β (λ

− β) + bi L β , i = 1, 2, . . . , q. Then we have L aij L α − aij β L ∗ ∗ L ∗ aij L x + b y + a z = ( (λ − β)+ ij λ ij λ λ α−β aij L β )x + (

α−β eL α

eL λ =

−

eL β

α−β

L (λ − β) + cL β , dλ =

L dL α − dβ

α−β

(λ − β) + dL β,

L L L min (f1 (x, y, z))L α = c1 α x + d1 α y + e1 α z

x∈X

(λ − β) + eL β.

R R R min (f1 (x, y, z))R α = c1 α x + d1 α y + e1 α z

x∈X

L L L min (f1 (x, y, z))L β = c1 β x + d1 β y + e1 β z

Thus we have

x∈X

L L cL λ x + dλ y + eλ z = (

(λ − β) +

dL β )y

+(

L cL α − cβ

α−β

L eL α − eβ

α−β

(λ − β) + cL β )x + (

(λ − β) +

R R R min (f1 (x, y, z))R β = c1 β x + d1 β y + e1 β z

L dL α − dβ

x∈X

α−β

L L L s.t. A1 L α x + B1 α y + C1 α z ≤ b1 α

eL β )z

R R R A1 R α x + B1 α y + C1 α z ≤ b1 α

λ−β L λ−β L L (c x + dL )(cL x + dL α y + eα z) + (1 − β y + eβ z) α−β α α−β β λ−β L ∗ λ−β ∗ L ∗ ∗ ≥ (c x + dL )(cL x∗ + dL α y + eα z ) + (1 − βy α−β α α−β β

L L L A1 L β x + B1 β y + C1 β z ≤ b1 β

=

R R R A1 R β x + B1 β y + C1 β z ≤ b1 β L L L min (f2 (x, y, z))L α = c2 α x + d2 α y + e2 α z

∗ + eL βz )

=( +(

x∈X

L cL α − cβ

α−β L eL α − eβ

α−β

∗ (λ − β) + cL β )x + (

L dL α − dβ

α−β

R R R min (f2 (x, y, z))R α = c2 α x + d2 α y + e2 α z

∗ (λ − β) + dL β )y

x∈X

L L L min (f2 (x, y, z))L β = c2 β x + d2 β y + e2 β z

x∈X

∗ (λ − β) + eL β )z

R R R min (f2 (x, y, z))R β = c2 β x + d2 β y + e2 β z

x∈X

∗ L ∗ L ∗ = cL λ x + dλ y + eλ z , ∀λ ∈ [β, α].

R R cR λ x+dλ y +eλ z

Similarly, we can prove ∗ R ∗ dR λ y + eλ z , ∀λ ∈ [β, α].

∗

∗

≥

L L L s.t. A2 L α x + B2 α y + C2 α z ≤ b2 α

∗ cR λx +

R R R A2 R α x + B2 α y + C2 α z ≤ b2 α L L L A2 L β x + B2 β y + C2 β z ≤ b2 β

∗

Lemma 2. If there is a (x , y , z ) such that ∗ L ∗ L ∗ L R ∗ R ∗ R ∗ AL α x +Bα y +Cα z ≤ bα , Aα x +Bα y +Cα z ≤ R L ∗ L ∗ L ∗ L R ∗ R ∗ bα , Aβ x + Bβ y + Cβ z ≤ bβ , Aβ x + Bβ y + CβR z ∗ ≤ bR β for trapezoidal fuzzy number matrices ˜ ˜ ˜ then AL x∗ + B L y ∗ + C L z ∗ ≤ bL and A, B and C, λ λ λ λ R ∗ R ∗ Aλ x + Bλ y + CλR z ∗ ≤ bR λ , ∀λ ∈ [β, α].

R R R A2 R β x + B2 β y + C2 β z ≤ b2 β L L L min (f3 (x, y, z))L α = c3 α x + d3 α y + e3 α z

x∈X

R R R min (f3 (x, y, z))R α = c3 α x + d3 α y + e3 α z

x∈X

L L L min (f3 (x, y, z))L β = c3 β x + d3 β y + e3 β z

x∈X

˜ = (˜bij )q×m , Proof. Let A˜ = (˜ aij )q×n , B ˜ ˜ ˜ C = (˜ cij )q×p and b = (bi )q×1 . By the definition of λ-cut, we have aij L λ = i

=

1, 2, . . . , q,

L bij L α −bij β (λ α−β

j

=

L aij L α −aij β (λ α−β

=

1, 2, . . . , q,

j

bij L λ

L L L s.t. A3 L α x + B3 α y + C3 α z ≤ b3 α

1, 2, . . . , q,

L cij L α −cij β (λ α−β

− β) + cij L β,

1, 2, . . . , p;

bi L λ

R R R A3 R α x + B3 α y + C3 α z ≤ b3 α

=

=

=

i

R R R min (f3 (x, y, z))R β = c3 β x + d3 β y + e3 β z

x∈X

− β) + aij L β,

1, 2, . . . , n;

− β) + bij L β,

j = 1, 2, . . . , m; cij L λ = i

α−β

Theorem 2. Let (x∗ , y ∗ , z ∗ ) be the solution of the following MOTLDM decision making problem (5):

Proof. By the definition of λ-cut, we have L cL α − cβ

L cij L α − cij β

·

λ−β L ∗ L ∗ ∗ (aij L α x + bij α y + cij α z ) α−β λ−β L λ−β L ∗ L ∗ ∗ + (1 − )(aij L bi α x + bij α y + cij α z ) ≤ α−β α−β α λ−β L + (1 − )bi = bi L λ. α−β β L ∗ L ∗ Thus Aλ x + Bλ y + CλL z ∗ ≤ bL λ , ∀λ ∈ [β, α]. ∗ Similarly, we can prove AR x + BλR y ∗ + CλR z ∗ ≤ λ R bλ , ∀λ ∈ [β, α].

where a ˜ denotes c˜, d˜ and e˜, respectively, then cL λx+ L L ∗ L ∗ L ∗ R R dλ y + eL z ≥ c x + d y + e z and c x + d λ λ λ λ λ λy + R ∗ R ∗ R ∗ eR z ≥ c x + d y + e z , ∀λ ∈ [β, α]. λ λ λ λ

cL λ =

α−β

(λ − β) + bij L β )y + (

(λ − β) + cij L β )z =

aβ ≤ t

0

L bij L α − bij β

L L L A3 L β x + B3 β y + C3 β z ≤ b3 β R R R A3 R β x + B3 β y + C3 β z ≤ b3 β .

(5) If the coefficients of the FTLDM problem (2) have trapezoidal membership functions as Eq. (4), the

=

156

solution is also the solution of the FTLDM problem (2).

x∈X

Proof. By Lemma 1, if (x∗ , y ∗ , z ∗ ) satisfies

R L R L R + d1 L β + d1 β )y + e1 α + e1 α + e1 β + e1 β )z

min

x∈X

(f1 (x, y, z))L α

=

∗ c1 L αx

+

∗ d1 L αy

+

R L R L R min g1 (x, y, z) = (c1 L α + c1 α + c1 β + c1 β )x + (d1 α + d1 α

L L L s.t. A1 L α x + B1 α y + C1 α z ≤ b1 α

∗ e1 L αz ,

R R R A1 R α x + B1 α y + C1 α z ≤ b1 α L L L A1 L β x + B1 β y + C1 β z ≤ b1 β

R ∗ R ∗ R ∗ min (f1 (x, y, z))R α = c 1 α x + d 1 α y + e1 α z ,

x∈X

R R R A1 R β x + B1 β y + C1 β z ≤ b1 β

L ∗ L ∗ L ∗ min (f1 (x, y, z))L β = c1 β x + d1 β y + e1 β z ,

R L R L min g2 (x, y, z) = (c2 L α + c2 α + c2 β + c2 β )x + (d2 α +

x∈X

y∈Y

R ∗ ∗ R ∗ min (f1 (x, y, z))βR = c1 R β x + d 1 β y + e1 β z ,

L R L R L R d2 R α + d2 β + d2 β )y + (e2 α + e2 α + e2 β + e2 β )z

x∈X

L L L s.t. A2 L α x + B2 α y + C2 α z ≤ b2 α

then we have

R R R A2 R α x + B2 α y + C2 α z ≤ b2 α

L ∗ L ∗ L ∗ min (f1 (x, y, z))L λ = c1λ x + d1λ y + e1λ z ,

L L L A2 L β x + B2 β y + C2 β z ≤ b2 β

x∈X

R ∗ R ∗ R ∗ min (f1 (x, y, z))R λ = c1λ x + d1λ y + e1λ z , λ ∈ [β, α].

x∈X

R R R A2 R β x + B2 β y + C2 β z ≤ b2 β R L R L min g3 (x, y, z) = (c3 L α + c3 α + c3 β + c3 β )x + (d3 α

z∈Z

Similarly, we have

L R L R L R + d3 R α + d3 β + d3 β )y + (e3 α + e3 α + e3 β + e3 β )z L L L s.t. A3 L α x + B3 α y + C3 α z ≤ b3 α

L ∗ L ∗ L ∗ min (f2 (x, y, z))L λ = c2λ x + d2λ y + e2λ z , y∈Y

R ∗ R ∗ R ∗ min (f2 (x, y, z))R λ = c2λ x + d2λ y + e2λ z , λ ∈ [β, α]; y∈Y

R R R A3 R α x + B3 α y + C3 α z ≤ b3 α L L L A3 L β x + B3 β y + C3 β z ≤ b3 β R R R A3 R β x + B3 β y + C3 β z ≤ b3 β .

L ∗ L ∗ L ∗ min (f3 (x, y, z))L λ = c3λ x + d3λ y + e3λ z ,

(6)

z∈Z

min z∈Z

(f3 (x, y, z))R λ

=

∗ cR 3λ x

+

∗ dR 3λ y

+

∗ eR 3λ z ,

λ ∈ [β, α].

By Lemma 2, if (x∗ , y ∗ , z ∗ ) satisfies L L L Ai L α x + Bi α y + Ci α z ≤ bi α , R R R Ai R α x + Bi α y + Ci α z ≤ bi α , L L L Ai L β x + Bi β y + Ci β z ≤ bi β ,

L R L R R B3 L α y + C3 α z ≤ b3 α , A3 α x + B3 α y + C3 α z ≤ R L L L L b3 α , A3 β x + B3 β y + C3 β z ≤ b3 β , A3 R βx + R R R B3 β y + C3 β z ≤ b3 β } and P (x) = {(y, z) : (y, z) ∈ arg min{g2 (x, yˆ, zˆ) : (ˆ y , zˆ) ∈ S(x), zˆ ∈ arg min{g3 (x, yˆ, z˜) : z˜ ∈ S(x, yˆ)}}}, then the algorithm can be described as follows:

[Algorithm 1]

R R R Ai R β x + Bi β y + Ci β z ≤ bi β , i = 1, 2, 3,

Step 1: Transform the FTLDM problem (2) into problem (6).

then ∗ L ∗ L ∗ L R ∗ R ∗ AL iλ x +Biλ y +Ciλ z ≤ biλ , and Aiλ x +Biλ y + R ∗ R Ciλ z ≤ biλ will hold, ∀λ ∈ [β, α], i = 1, 2, 3. To summarize, the proof of Theorem 2 is completed.

Step 2: Set i ← 1. Solve the linear programming problem min{g1 (x, y, z) : (x, y, z) ∈ S} by the simplex method to obtain the optimal solution (x[1] , y[1] , z[1] ), and then let W = {(x[1] , y[1] , z[1] )} and T = ϕ.

By Theorem 2, solving the FTLDM problem (2) is equivalent to solving the MOTLDM problem (5). In order to solve the problem (5), the linear weighted method can be used. Therefore, the problem (5) can be transformed into the problem (6). The problem (6) is a TLDM problem. According to the theorem provided by [14], the solution to a TLDM problem occurs at a vertex of the constraint region, thus the linear tri-level Kth-best algorithm [14] can be used to solve the problem. Let S = {(x, y, z) : x ∈ X, y ∈ Y, z ∈ Z, Ai L αx + L R R R L y + C x + B , A z ≤ b y + C Bi L i i i i i αz ≤ α α α α α L R R L L L x + B , A z ≤ b y + C x + B , A bi R iβ y + i i i i i β β β β β α R R Ci β z ≤ bi β , i = 1, 2, 3}, S(x) = {(y, z) ∈ Y × Z : L R R R L L Ai L α x + Bi α y + Ci α z ≤ bi α , Ai α x + Bi α y + Ci α z ≤ L R R R L L L bi α , Ai β x + Bi β y + Ci β z ≤ bi β , Ai β x + Bi β y + R L Ci R β z ≤ bi β , i = 2, 3}, S(x, y) = {z ∈ Z : A3 α x+

Step 3: Let x = x[i] and utilize the bi-level Kthbest algorithm [10] to solve the bi-level decision making problem min{g2 (x, y, z) : (y, z) ∈ P (x[i] )}

(7)

Let (ˆ y , zˆ) be the optimal solution to (7). If yˆ = y[i] and zˆ = z[i] , (x[i] , y[i] , z[i] ) is the global optimum of the FTLDM problem (2) and K ∗ = i, go to Step 6; otherwise, go to Step 4. Step 4: Let W[i] denote the set of adjacent extreme points of (x[i] , y[i] , z[i] ) such that g1 (x, y, z) ≥ g1 (x[i] , y[i] , z[i] ). Let T = T ∪{(x[i] , y[i] , z[i] )} and W = (W ∪W[i] )\T . Go to Step 5.

157

Step 5: Set i ← i + 1 and choose {(x[i] , y[i] , z[i] )} from W such that g1 (x[i] , y[i] , z[i] ) = min{g1 (x, y, z) : (x, y, z) ∈ W }. Go to Step 3.

chain as follows: min f1 (x, y, z) = ˜1x − ˜4y + ˜2z + ˜2

x∈X

s.t. − ˜1x + ˜3y + ˜1z ≤ ˜4 min f2 (x, y, z) = ˜1x + ˜1y − ˜1z + ˜2

Step 6: Output the solution of the FTLDM problem (2). [End]

y∈Y

s.t. 1˜x − ˜1y − ˜2z ≤ ˜0 max f3 (x, y, z) = −˜1x + ˜2y + ˜1z + ˜ 3

4. A case study on third-party logistics decision making

(8)

z∈Z

s.t. − ˜1x − ˜1y + ˜3z ≤ ˜0, where

For the purpose of this research, a case study on third-party logistics decision making is developed based on the FTLDM model. Nowadays companies are focusing on enhancing their competitive advantage in order to meet the economy globalization. One effective way is developing their core businesses in the supply chains, while outsourcing non-core businesses, among which logistics is the main operation to be outsourced [26]. Especially with the rapid development of the Internet of Things techniques, companies can grasp the information of the products in real time online and more and more companies have begun to outsource their main logistics operation to third-party logistics (3PL) companies in recent years. In this case, the vendor, the logistics company and the distributor form a hierarchical organization. The three decision entities have their own objectives, for instance, maximize the profits or minimize the production costs. In addition, each decision entity has its own constraints and variables. When making a decision in the supply chain, the logistics company fully considers the decision of the vendor and the distributor also considers the decision of the logistics company. At the same time, the vendor also takes into account the reaction of the logistics company and the logistics company likewise considers the reaction of the distributor. Due to the uncertainty of judgments, the coefficients of the three entities cannot be expressed with exact numerical numbers. For instance, the unit profit of a product may be “about $10". Therefore the decision making problem faced to the decision entities is usually with fuzzy coefficients. In order to make an optimal decision, the FTLDM models can be utilized. In a supply chain, a high-tech product vendor wants to minimize its production cost (represented by min f1 (x, y, z)) and x (the number of orders on the planning horizon) is the decision variable. The logistics company needs to determine the number of trucks to be rent for the distribution (y), with the objective of minimizing the overall transportation costs (min f2 (x, y, z)). The distributor has the objective “maximizing the overall profit” (max f3 (x, y, z)) and z (the order quantity per lot-size) is its decision variable. The objectives at the three levels are subject to their particular constraints, for instance, the cost of inventory and the supply of materials. We simplify the fuzzy tri-level decision making problem of the supply

 0 t < −1     t + 1 −1 ≤ t < 0 µ˜0 (t) = ,  1−t 0≤t