Reverse Logistics Network in Uncertain Environment

(1)

Reverse Logistics Network in Uncertain Environment

Jianjun Liu, Yufu Ning

^∗

, Xuedou Yu

Department of Computer Science, Dezhou University, Dezhou 253023, China

∗

Corresponding author: [email protected]

Abstract

This paper mainly studies reverse logistics network in uncertain environment, and proposes an expected value model for it. Then the proposed model is converted into a crisp equivalent model. A numerical example is presented to illustrate the given model.

Keywords: Reverse logistics, Uncertainty theory, Uncertain programming

1 Introduction

Reverse logistics industry aims at designing a network for returning products with minimal cost. Many researchers have proposed some models in this area. In 2000, Fleischmann et. al. [3] considered the general characteristics of reverse logistics network, and proposed a network for product recovery. Then Beamon and Fernandes [1] designed a closed-loop network for product recovery and formulated a multi-period integer programming model in 2004.

Indeterminacy factors exist in the system of reverse logistics network. In order to deal with this problem, some researchers proposed stochastic models for reverse logistics. Salema et. al. [16] studied a reverse logistics network with stochastic demands and returns. Chouinard et. al. [2] proposed a stochastic programming model for designing supply loops.

As we know, probability theory is applicable only when the obtained distribution is close enough to the real frequency. However, we are often lack of observed data to estimate the probability distribution via statistics.

Then we have to invite some experts to estimate the real frequency, and evaluate their belief degree which usually has a much larger range than the real frequency. In order to deal with the belief degree, an uncertainty theory was founded by Liu [6] in 2007 and refined by Liu [12] in 2011.

In the framework of uncertainty theory, uncertain programming was proposed by Liu [8] as a spectrum of mathematical programming involving uncertain variables. Gao [4] studied shortest path problems in uncertain environment, and Gao [5] studied facility location problems with uncertain parameters. Sheng and Yao [17]

proposed an uncertain transportation model.

This paper will propose an uncertain reverse logistic model. This rest of this paper is organized as follows.

In Section 2, we will review some basic concepts in uncertainty theory which will be used throughout the paper. Then we introduce reverse logistic problem in Section 3. Then the uncertain reverse logistics model is formulated and converted to a crisp model in Section 4. Then an example was given to illustrate the model in Section 5. At last, some remarks are given in Section 6.

2 Preliminary

Uncertainty theory, founded by Liu [6] in 2007 and refined by Liu [12] in 2011, is a branch of axiomatic mathematics to deal with human uncertainty. It has brought about many branches such as uncertain process (Liu [7]), uncertain calculus (Liu [9], Yao [18]), uncertain differential equation (Liu [7]), uncertain set (Liu [10, 15]), uncertain inference (Liu [10]), uncertain logic (Liu [13]) and uncertain risk analysis (Liu [11]). In this section, we will introduce some basic concepts about uncertainty theory which will be used throughout the paper.

Definition 1 (Liu [6]) Let L be a σ-algebra on a nonempty set Γ. A set function M : L → [0, 1] is called an uncertain measure if it satisfies the following axioms:

Axiom 1: (Normality Axiom) M{Γ} = 1 for the universal set Γ.

(2)

Axiom 2: (Duality Axiom) M{Λ} + M{Λ

^c

} = 1 for any event Λ.

Axiom 3: (Subadditivity Axiom) For every countable sequence of events Λ

₁

, Λ

₂

, · · · , we have M

(

_∞

[

i=1

Λ

i

)

≤

∞

X

i=1

M {Λ

i

} .

The triplet (Γ, L, M) is called an uncertainty space. Besides, Liu [9] defined the product uncertain measure on the product σ-algebra L as follows,

Axiom 4: (Product Axiom) Let (Γ

k

, L

k

, M

k

) be uncertainty spaces for k = 1, 2, · · · Then the product uncertain measure M is an uncertain measure satisfying

M (

_∞

Y

i=1

Λ

k

)

=

∞

^

k=1

M

k

{Λ

k

}

where Λ

k

are arbitrarily chosen events from L

k

for k = 1, 2, · · · , respectively.

Definition 2 (Liu [6]) An uncertain variable ξ is a measurable function from an uncertainty space (Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set

{ξ ∈ B} = {γ ∈ Γ|ξ(γ) ∈ B}

is an event.

Definition 3 (Liu [6]) The uncertainty distribution Φ of an uncertain variable ξ is defined by Φ(x) = M{ξ ≤ x}

for any real number x.

Definition 4 (Liu [6]) Let ξ be an uncertain variable. Then the expected value of ξ is defined by

E[ξ] = Z

+∞

0

M{ξ ≥ r}dr − Z

0

−∞

M{ξ ≤ r}dr

provided that at least one of the two integrals is finite.

Theorem 1 (Liu [12]) Let ξ and η be independent uncertain variables with finite expected values. Then for any real numbers a and b, we have

E[aξ + bη] = aE[ξ] + bE[η].

3 Problem statement

The key step of a reverse logistics network is to establish some recovery facilities as the links between consumers and the manufactory. The decision-making targets are as follows: (i) choosing the sites of recovery facilities from the potential sets, (ii) determining the amount of products returned by each recovery facility from consumers. Figure 1 is a visual representation of this network.

The fixed cost (for short, FC) for reprocessing capital investment rises with the capacity level, and the variable cost (for short, VC) for processing declines due to economy scale. In addition, an inventory cost (for short, IC) associated with the inventory levels may become significant. Furthermore, the transportation cost (for short, TC) is assumed to be a linear function of volume. The cost of uncollected fraction is named as penalty cost (for short, PC). To conclude, the total cost of recovery network can be presented by C = F C + V C + IC + T C + P C. The parameters of the reverse logistics network are listed as follows.

Index sets

i = 1, 2, . . . , I, set of consumer zones.

j = 1, 2, . . . , J , set of recovery facilities.

(3)

1

I 2

1

J

2 M

Consumers Recovery facilities

Manufactory

Figure 1: Reverse logistic network graph

Decision variables

x

ij

: volume of returned products collected by recovery facilities j from consumer i.

y

j

=

1, if a recovery facility j is opened, 0, otherwise.

Model parameters

ξ

i

: volume of products returned by consumer i, 1 ≤ i ≤ I;

F IX

_j

: fixed cost to open facility j, 1 ≤ j ≤ J ; ζ

_j

: unit reprocessing cost at facility j, 1 ≤ j ≤ J ; h

j

: unit holding cost at facility j, 1 ≤ j ≤ I;

c

ij

: unit transportation cost between consumer i and facility location j, 1 ≤ i ≤ I, 1 ≤ j ≤ J ; q

_j

: unit transportation cost from facility location j to manufactory, 1 ≤ j ≤ J ;

c

^p_i

: unit penalty cost of non-returned products from consumer i, 1 ≤ i ≤ I;

V

j

: maximum capacity of facility j, 1 ≤ j ≤ J ;

M : maximum number of established reprocessing facilities;

λ: service level given by the manufactory.

Therefore the cost structures are

F C =

J

X

j=1

F IX

j

y

j

; V C =

J

X

j=1

ζ

j

X

^I

i=1

x

ij

; IC =

J

X

j=1

h

j

X

^I

i=1

x

ij

;

T C =

I

X

i=1 J

X

j=1

c

ij

x

ij

+

J

X

j=1

q

j

I

X

i=1

x

ij

; P C =

I

X

i=1

c

^p_i

ξ

i

−

J

X

j=1

x

ij

.

The total cost of the reverse logistics network is C = P

J

j=1

F IX

j

y

j

+ P

J j=1

ζ

j

P

I i=1

x

ij

+ P

I

i=1

P

J

j=1

c

ij

x

ij

+ P

J j=1

q

j

P

I

i=1

x

ij

+ P

J

j=1

h

j

P

I i=1

x

ij

+ P

I

i=1

c

^p_i

ξ

i

− P

J j=1

x

ij

.

(1)

In this paper, the parameters ξ

i

, F IX

j

, ζ

j

and h

j

are all assumed to be uncertain variables. Therefore, the total cost C is also an uncertain variable.

4 The reverse logistics network model

Expected value model is the most commonly used approach to solve indeterminacy problems. In this section,

it is employed to design a reverse logistics network with uncertain parameters.

(4)

The expected value model for reserve logistics network is formulated as follows,



 

 

 

 

min E[C]

s.t.

E[λξ

_i

] ≤ P

J

j=1

x

_ij

(A1)

P

I

i=1

x

ij

≤ y

j

V

j

(A2)

1 ≤ P

J

j=1

y

j

≤ M (A3)

x

ij

≥ 0 (A4)

y

_j

∈ {0, 1} (A5)

0 ≤ i ≤ I, 1 ≤ j ≤ J. (A6)

(2)

The objective function is the expected value of the total cost of the product recovery network. Constraint (A1) means that the total volume of products collected from all consumers must satisfy the demand of each consumer at the service level λ in expected value sense. Constraint (A2) ensures the returned products by each recovery facility does not exceed its maximum capacity. Constraint (A3) maintains at least one recovery facility and at most M recovery facilities. Constraint (A4) preserves the nonnegativity of decision variables x

ij

. Constraint (A5) assures the binary integrality of decision variables y

j

.

Theorem 2 Let ξ

_i

, F IX

_j

, ζ

_j

and h

_j

be independent uncertain variables for all i and j. Then model (2) is equivalent to the following deterministic programming model



 

 

 

 

min P

J

j=1

E[F IX

_j

]y

_j

+ P

J

j=1

E[ζ

_j

] P

I

i=1

x

_ij

+ P

I

i=1

P

J

j=1

c

_ij

x

_ij

+ P

J

j=1

q

_j

P

I

i=1

x

_ij

+ P

J

j=1

E[h

_j

] P

I

i=1

x

_ij

+ P

I

i=1

c

^p_i

E[ξ

_i

] − P

J j=1

x

_ij

s.t.

(A1) − (A6).

(3)

Proof: Note that C is a linear function of independent uncertain variables ξ

_i

, F IX

_j

, ζ

_j

, h

_j

. We obtain that E[C] = P

J

j=1

E[F IX

j

]y

j

+ P

J

j=1

E[ζ

j

]( P

I

i=1

x

ij

) + P

I i=1

P

J

j=1

c

ij

x

ij

+ P

J

j=1

(q

_j

P

I

i=1

x

_ij

) + P

J

j=1

E[h

_j

]( P

I

i=1

x

_ij

) + P

I

i=1

c

^p_i

(E[ξ

_i

] − P

J j=1

x

_ij

) which is the objective function. The proof is completed.

When the parameters ξ

i

, F IX

j

, ζ

j

and h

j

are normal uncertain variables N(e

ξ_i

, σ

ξ_i

), N(e

F IX_j

, σ

F IX_j

), N(e

ζ_j

, σ

ζ_j

) and N(e

h_j

, σ

h_j

), the expected value model (3) is converted into the following model,



 

 

 

 

min C

_E

s.t.

λe

_ξ_i

≤ P

J

j=1

x

_ij

(A1

⁰

) (A2) − (A6)

(4)

where

C

E

= P

J

j=1

e

F IX_j

y

j

+ P

J j=1

e

ζ_j

P

I i=1

x

ij

+ P

I

i=1

P

J

j=1

c

ij

x

ij

+ P

J j=1

q

j

P

I

i=1

x

ij

+ P

J

j=1

e

h_j

P

I i=1

x

ij

+ P

I

i=1

c

^p_i

e

ξ_i

− P

J j=1

x

ij

.

5 Numerical example

In this section, we consider an example to illustrate the expected value model of reverse logistics network.

Suppose that the manufacturer only chooses 5 recovery facilities to serve 3 consumer zones. Table 1 summarizes

the parameters ξ

i

and c

^p_i

related to consumer zones, in which c

^p_i

are crisp quantities, but ξ

i

are assumed to

(5)

Table 1: Parameters related to consumer zones

i ξ

_i

c

^p_i

1 N(75, 5) 1.78 2 N(90, 10) 1.00 3 N(100, 5) 1.77

be normal uncertain variables. In Table 2, parameters V

_j

and q

_j

are assumed to be crisp numbers, F IX

_j

, h

_j

and ζ

_j

are considered as normal uncertain variables. The unit transportation costs c

_ij

are given in Table 3 as crisp quantities. The parameter λ is chosen as 0.8.

Table 2: Parameters related to recovery facilities

j F IX

j

ζ

j

h

j

q

j

V

j

1 N(180, 4) N(0.54, 0.02) N(12, 2) 1.44 490 2 N(202, 2) N(0.36, 0.03) N(16, 3) 1.23 502 3 N(188, 4) N(0.69, 0.03) N(14, 1) 1.58 487 4 N(208, 6) N(0.54, 0.04) N(16, 2) 1.53 506 5 N(200, 3) N(0.45, 0.03) N(10, 1) 1.63 503

Table 3: Unit transportation cost c

ij

i/j 1 2 3 4 5

1 0.63 0.56 0.39 0.81 0.59 2 0.27 0.89 0.71 0.51 0.30 3 0.97 0.45 0.22 0.40 0.51

The objective function of the expected value model (4) is converted into the following form based on the data in Tables 1-3,

180y

₁

+ 202y

₂

+ 188y

₃

+ 208y

₄

+ 200y

₅

+ 12.83x

₁₁

+ 16.37x

₁₂

+ 14.88x

₁₃

+17.1x

14

+ 10.89x

15

+ 13.25x

21

+ 17.48x

22

+ 15.98x

23

+ 17.58x

24

+11.38x

₂₅

+ 13.18x

₃₁

+ 16.27x

₃₂

+ 14.72x

₃₃

+ 16.7x

₃₄

+ 10.82x

₃₅

.

(5)

The constraints are given as follows, P

5

j=1

x

_1j

≥ 60; P

5

j=1

x

_2j

≥ 72; P

5

j=1

x

_3j

≥ 80; (D1) 1 ≤ P

5

j=1

y

_j

≤ 5; (D2)

P

3

i=1

x

i1

≤ 490y

1

; P

3

i=1

x

i2

≤ 502y

2

; P

3

i=1

x

i3

≤ 487y

3

; (D3) P

3

i=1

x

i4

≤ 506y

4

; P

3

i=1

x

i5

≤ 503y

5

; (D4)

x

ij

≥ 0; (D5)

y

j

∈ {0, 1}; (D6)

i = 1, 2, 3; j = 1, 2, 3, 4, 5. (D7)

(6)

Then the model (4) is converted into a crisp programming model with the objective function (5) and constraints

(D1) − (D7). The optimal objective value of the model (4) is 2938.86, and the optimal solution is shown in

Table 4.

(6)

Table 4: Optimal production plan

variable x

₁₁

x

₁₂

x

₁₃

x

₁₄

x

₁₅

x

₂₁

x

₂₂

x

₂₃

x

₂₄

x

₂₅

x

₃₁

x

₃₂

x

₃₃

x

₃₄

x

₃₅

volume 0 0 0 0 60 0 0 0 0 72 0 0 0 0 80

unit cost 1.94 5.48 3.99 6.21 1.2 1.87 6.10 4.60 6.20 0.9 2.36 5.45 3.9 5.88 1.4

6 Conclusions

The reverse logistics network with uncertain costs was first considered in this paper. An expected value model was proposed to describe the uncertain reverse logistics problem and was converted to a crisp programming model. Besides, an example was given to illustrate the proposed model.

Acknowledgement

This paper is supported by Shandong Provincial Scientific and Technological Research Plan Project (No.

2009GG20001029 and No. 2011YD01069).

References

[1] Beamon B, Fernandes C, Supply-chain network configuration for product recovery, Production Planning and Control, Vol.15, No.3, 270-281, 2004.

[2] Chouinard M, Dmoursa S, At-Kadi D, A stochastic programming approach for designing supply loops, Interna- tional Journal Production Economics, Vol.113, No.2, 657-677, 2008.

[3] Fleischmann M, Krikke HR, Dekker R, Flapper S D, A characterisation of logistics networks for product recovery, Omega, Vol.28, No.6, 653-666, 2000.

[4] Gao Y., Shortest path problem with uncertain arc lengths, Computers and Mathematics with Applications, Vol.62, No.6, 2591-2600, 2011.

[5] Gao Y., Uncertain models for single facility location problems on networks, Applied Mathematical Modelling, Vol.36, No.6, 2592-2599, 2012.

[6] Liu B., Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007.

[7] Liu B., Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, Vol.2, No.1, 3-16, 2008.

[8] Liu B., Theory and Practice of Uncertain Programming, 2nd ed., Springer-Verlag, Berlin, 2009.

[9] Liu B., Some research problems in uncertainty theory, Journal of Uncertain Systems, Vol.3, No.1, 3-10, 2009.

[10] Liu B., Uncertain set theory and uncertain inference rule with application to uncertain control, Journal of Un- certain Systems, Vol.4, No.2, 83-98, 2010.

[11] Liu B., Uncertain risk analysis and uncertain reliability analysis, Journal of Uncertain Systems, Vol.4, No.3, 163-170, 2010.

[12] Liu B., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2011.

[13] Liu B., Uncertain logic for modeling human language, Journal of Uncertain Systems, Vol.5, No.1, 3-20, 2011.

[14] Liu B., Why is there a need for uncertainty theory? Journal of Uncertain Systems, Vol.6, No.1, 3-10, 2012.

[15] Liu B., Membership functions and operational law of uncertain sets, Fuzzy Optimization and Decision Making, to be published.

[16] Salema M, Barbosa-Povoa A, Novais Q, An optimization model for the design of a capacitated multi-product reverse logistics network with uncertainty, European Journal of Operational Research, Vol.179, No.3, 1063-1077, 2007.

[17] Sheng Y.H., and Yao K., A transportation model with uncertain costs and demands, Information: An Interna- tional Interdisciplinary Journal, Vol.15, No.8, 3179-3186, 2012.

Reverse Logistics Network in Uncertain Environment