# Mitigation of Bullwhip Effect in Closed-Loop Supply Chain Based on Fuzzy Robust Control Approach

Authors: Songtao Zhang; Min Zhang
Journal: Complexity
 (2020)

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## Abstract

Uncertainties and lead times make the closed-loop supply chain (CLSC) more complex, less stable, and then the bullwhip effect (BE) will become more intense. This paper will address a fuzzy robust control (FRC) approach to mitigate the BE in the uncertain CLSC with lead times. For the reverse channels for products in the CLSC, the customers’ used products are recycled by both the manufacturer and the third party recovery provider, and new products bought by customers within a certain period of time can be returned to the retailer. In the CLSC system, the state transformation equations of the inventories and the total operation cost are set up. A new FRC approach is proposed to mitigate the BE and realize the robust stability of the uncertain CLSC with lead times. A simulation example verifies the mitigation effect of the BE under the proposed FRC approach.


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## Body

## 1. Introduction

Whether governments construct green supply chains through governmental interventions [1] or enterprises undertake social responsibility through social work donation [2], the environment can be more friendly. Therefore, both governments and enterprises have been paying more and more attention to the closed-loop supply chain (CLSC) which can achieve sustainable development [3].The complexity of the business environment results in many uncertainties in the CLSC. Especially in the reverse supply chain, there are uncertainties in quality, quantity, and time of the recycled products, which lead to the uncertainty of remanufacturing profit. For example, ReCellular, one of the largest phone remanufacturers in the United States, divides used phones into six quality levels for remanufacturing. Uncertainty is one of the main reasons for the existence of bullwhip effect (BE), and the other reason is lead time. If BE cannot be effectively mitigated, the operating efficiency of the CLSC will be reduced, and then the operating cost will be increased. In serious cases, the CLSC will collapse. Therefore, for the mitigation issue of the BE in the CLSC with uncertainties and lead times, we will propose a fuzzy robust control (FRC) approach to reduce the BE and realize the robust stability of the CLSC.The remainder of this paper is formulated as follows: Section2 offers a review of the related literature. Section 3 puts forward a kind of Takagi–Sugeno fuzzy model for the CLSC with uncertainties and lead times. Section 4 addresses a FRC approach to mitigate the BE in the CLSC system. The simulation studies are carried out in Section 5. The conclusions and future research directions are given in Section 6.


## 2. Literature Review

In the CLSC, used products can be recycled by various subjects. For example, the used products can be recycled by the manufacturer [4] or by the retailer [5]. Wei and Zhao [6] studied the decision-making issue of the used products recycled by the manufacturer, the retailer, or the third party recovery provider (3PRP). Furthermore, recycling used products can be performed by multiple subjects at the same time to realize more convenient for customers and more efficient for enterprises. This recycling pattern is called hybrid recycling. Therefore, some scholars have been devoting themselves to the study of the CLSC with hybrid recycling channels; for example, based on hybrid recycling channels, Allah et al. [7] investigated the pricing strategies of the CLSCs with the single-channel forward supply chain and with the dual-channel forward supply chain; Shi and Ma [8] recycled the used medical equipment through hybrid recycling models; under hybrid recycling channels, Ma and Liu [9] analyzed the optimal profit of the CLSC.For the studies mentioned above, [7–9] did not consider the impacts of uncertainties and lead times on the CLSC system with hybrid recycling channels. However, uncertainties and lead times are two important factors leading to BE. Therefore, many scholars studied the BE caused by uncertainties or lead times; for example, by simulation-based approach, Do et al. [10] quantified the BE with different demands and stochastic lead times; for a two-echelon serial supply chain, Agrawal et al. [11] compared the effects of lead time reduction and information sharing on the mitigation of the BE; using the statistical method, Kim et al. [12] considered the stochastic lead time and provided expressions for quantifying the BE both with information sharing and without information sharing; Modak and Kelle [13] used a hybrid all-unit quantity discount along a franchise fee contract to mitigate the BE in the dual-channel supply chain with the delivery time and stochastic demand; Li and Liu [14] explored the mitigation of the BE in the supply chain with uncertainties and the vendor order placement lead time.In recent years, approaches based on control theory have been widely applied to mitigate the BE, such as Model Predictive Control (MPC) approach [15], Internal Model Control (IMC) approach [16], Proportional plus Integral Control (PIC) approach [17], and Common Robust Control (CRC) approach [18]. Among these mitigation approaches, the MPC approach, the IMC approach, and the PIC approach cannot implement the switching control, and the CRC approach can only perform the conventional switching control but cannot achieve the flexible switching control of the FRC approach.Up to now, research on the mitigation of the BE in the CLSC with uncertainties and lead times has not yet been found. But, a fuzzy control approach has been applied to mitigate the BE in uncertain CLSC with hybrid recycling channels by Zhang et al. [19]. Therefore, we will follow the research ideas in [19] to address a control approach for mitigating the BE of the CLSC with uncertainties and lead times. The critical contributions of this paper comparing to [19] are listed as follows.


### 2.1. Lead Times Are Included in the CLSC Models

There are the manufacturer’s production lead time and recycling lead time in the manufacturer’ inventory model; the retailer’s ordering lead time and the 3PRP’s recycling lead time are, respectively, considered in the retailer’s inventory model and in the 3PRP’s inventory model; all lead times are included in the total operation cost model of the CLSC.


### 2.2. An Additional Takagi–Sugeno Fuzzy Controller for Lead Times Is Designed

We will design an additional Takagi–Sugeno fuzzy negative feedback controller, which is the product of the inventory feedback gains with lead times and the inventory variables with lead times.


### 2.3. A New Fuzzy Control Approach Is Put Forward

We will put forward a new FRC approach which can effectively mitigate the BE caused by uncertainties and lead times and ensure the stability of the CLSC system. LMIs (linear matrix inequalities) to be solved in this paper are more complex than those in [19].


## 2.1. Lead Times Are Included in the CLSC Models

There are the manufacturer’s production lead time and recycling lead time in the manufacturer’ inventory model; the retailer’s ordering lead time and the 3PRP’s recycling lead time are, respectively, considered in the retailer’s inventory model and in the 3PRP’s inventory model; all lead times are included in the total operation cost model of the CLSC.


## 2.2. An Additional Takagi–Sugeno Fuzzy Controller for Lead Times Is Designed

We will design an additional Takagi–Sugeno fuzzy negative feedback controller, which is the product of the inventory feedback gains with lead times and the inventory variables with lead times.


## 2.3. A New Fuzzy Control Approach Is Put Forward

We will put forward a new FRC approach which can effectively mitigate the BE caused by uncertainties and lead times and ensure the stability of the CLSC system. LMIs (linear matrix inequalities) to be solved in this paper are more complex than those in [19].


## 3. CLSC Model

### 3.1. CLSC Model with Uncertainties and Lead Times

As shown in Figure1, this paper constructs a CLSC model that includes a manufacturer, a retailer, a 3PRP, and customers. Because products recycled by both the manufacturer and the 3PRP are the most effective pattern among hybrid recycling channels [20], for reverse supply chain in our model, the manufacturer and the 3PRP simultaneously recycle customers’ used products, and the retailer allows customers to return new products within a certain period of time.Figure 1
CLSC model.Based on Figure1, considering the uncertain system parameters and lead times, we set up the inventory equations of the CLSC as follows:(1)x1k+1=x1k+u1k+u1k−τ1+u3k+u3k−τ3+η+Δηx4k−u2k,x2k+1=x2k+u2k+u2k−τ2+μ+Δμx3k−w1k,x3k+1=x3k+w1k−u3k−u4k−μ+Δμx3k,x4k+1=x4k+u4k+u4k−τ4−η+Δηx4k−λ+Δλx4k.Accordingly, we set up the total cost equation of the CLSC as follows:(2)zk=ch1+Δch1x1k+ch2+Δch2x2k+ch3+Δch3x4k+cn+Δcnu1k+u1k−τ1+cr+Δcru3k+u3k−τ3+η+Δηx4k+cm+Δcmu3k+u3k−τ3+ct+Δctu4k+u4k−τ4+cmt+Δcmtη+Δηx4k+cq+Δcqμ+Δμx3k+cd+Δcdλ+Δλx4k+cs+Δcsu2k+u2k−τ2.Equations (1) and (2) are described by the deviation values (deviation value = actual value − nominal value).In addition, based on different inventory statuses, each node enterprise will design corresponding production patterns or ordering patterns, which leads to some different models in different periods. Then, theith model of the CLSC can be shown as follows:(3)xk+1=Ai+ΔAixk+Bi+ΔBiuk+∑e=14Bie+ΔBieuk−τe+Bwi+ΔBwiwk,zk=Ci+ΔCixk+Di+ΔDiuk+∑e=14Die+ΔDieuk−τe,where xTk=x1k,x2k,x3k,x4k, uTk=u1k,u2k,u3k,u4k, uTk−τe=u1k−τ1,u2k−τ2,u3k−τ3,u4k−τ4, and wTk=w1k,0,0,0; Ai denotes the inventory status coefficient matrix; Bi denotes the production, ordering, and recycling coefficient matrix; Bie denotes the production, ordering, and recycling coefficient matrix with lead times; Bwi denotes the coefficient matrix of the customers’ demand; Ci denotes the coefficient matrix of the inventory cost, the cost of the new products returned, the disposal cost, part of remanufacturing cost, and part of recycling cost; Di denotes the coefficient matrix of production cost, ordering cost, part of remanufacturing cost, and part of recycling cost; Die denotes the coefficient matrix of production cost, ordering cost, part of remanufacturing cost, and part of recycling cost with lead times; and ΔAi, ΔBi, ΔBie, ΔBwi, ΔCi, ΔDi, and ΔDie, respectively, denote the corresponding uncertain matrices.


### 3.2. Takagi–Sugeno Fuzzy Model of CLSC

In the process of wide variation of the inventory level, switching activities will take place among models to maintain a reasonable inventory level in each node enterprise. In order to effectively mitigate the BE of the CLSC, we will utilize Takagi–Sugeno fuzzy systems [21] to implement the flexible switching in the CLSC. Then, Model (3) is transformed into the following discrete Takagi–Sugeno fuzzy model:R
i: If x1k is M1i, …, xjk is Mji, …, and xnk is Mni, then(4)xk+1=Ai+ΔAixk+Bi+ΔBiuk+∑e=1gBie+ΔBieuk−τe+Bwi+ΔBwiwk,zk=Ci+ΔCixk+Di+ΔDiuk+∑e=1gDie+ΔDieuk−τe,xk=φk,k∈0,1,…,N,where wk∈l20,∞; φk denotes the initial condition of the CLSC; xTk=x1k,x2k,…,xnk; uTk=u1k,u2k,…,unk; uTk−τe=u1k−τ1,u2k−τ2,…,unk−τg; and wTk=w1k,w2k,…,wnk.By singleton fuzzifier, product inference, and centre-average defuzzifer, Model (4) can be expressed as follows:(5)xk+1=∑i=1rhixk⋅Ai+ΔAixk+Bi+ΔBiuk+∑e=1gBie+ΔBieuk−τe+Bwi+ΔBwiwk,zk=∑i=1rhixk⋅Ci+ΔCixk+Di+ΔDiuk+∑e=1gDie+ΔDieuk−τe,where hixk=μixk/∑i=1rμixk and μixk=∏j=1nMjixjk; since μixk≥0, we have hixk≥0 and ∑i=1rhixk=1. In the following expressions, xk will be elided from hixk for simplicity.Because of the existence of the membership degree function in Model (5), the flexible switching can be realized among the models of the CLSC.


## 3.1. CLSC Model with Uncertainties and Lead Times

As shown in Figure1, this paper constructs a CLSC model that includes a manufacturer, a retailer, a 3PRP, and customers. Because products recycled by both the manufacturer and the 3PRP are the most effective pattern among hybrid recycling channels [20], for reverse supply chain in our model, the manufacturer and the 3PRP simultaneously recycle customers’ used products, and the retailer allows customers to return new products within a certain period of time.Figure 1
CLSC model.Based on Figure1, considering the uncertain system parameters and lead times, we set up the inventory equations of the CLSC as follows:(1)x1k+1=x1k+u1k+u1k−τ1+u3k+u3k−τ3+η+Δηx4k−u2k,x2k+1=x2k+u2k+u2k−τ2+μ+Δμx3k−w1k,x3k+1=x3k+w1k−u3k−u4k−μ+Δμx3k,x4k+1=x4k+u4k+u4k−τ4−η+Δηx4k−λ+Δλx4k.Accordingly, we set up the total cost equation of the CLSC as follows:(2)zk=ch1+Δch1x1k+ch2+Δch2x2k+ch3+Δch3x4k+cn+Δcnu1k+u1k−τ1+cr+Δcru3k+u3k−τ3+η+Δηx4k+cm+Δcmu3k+u3k−τ3+ct+Δctu4k+u4k−τ4+cmt+Δcmtη+Δηx4k+cq+Δcqμ+Δμx3k+cd+Δcdλ+Δλx4k+cs+Δcsu2k+u2k−τ2.Equations (1) and (2) are described by the deviation values (deviation value = actual value − nominal value).In addition, based on different inventory statuses, each node enterprise will design corresponding production patterns or ordering patterns, which leads to some different models in different periods. Then, theith model of the CLSC can be shown as follows:(3)xk+1=Ai+ΔAixk+Bi+ΔBiuk+∑e=14Bie+ΔBieuk−τe+Bwi+ΔBwiwk,zk=Ci+ΔCixk+Di+ΔDiuk+∑e=14Die+ΔDieuk−τe,where xTk=x1k,x2k,x3k,x4k, uTk=u1k,u2k,u3k,u4k, uTk−τe=u1k−τ1,u2k−τ2,u3k−τ3,u4k−τ4, and wTk=w1k,0,0,0; Ai denotes the inventory status coefficient matrix; Bi denotes the production, ordering, and recycling coefficient matrix; Bie denotes the production, ordering, and recycling coefficient matrix with lead times; Bwi denotes the coefficient matrix of the customers’ demand; Ci denotes the coefficient matrix of the inventory cost, the cost of the new products returned, the disposal cost, part of remanufacturing cost, and part of recycling cost; Di denotes the coefficient matrix of production cost, ordering cost, part of remanufacturing cost, and part of recycling cost; Die denotes the coefficient matrix of production cost, ordering cost, part of remanufacturing cost, and part of recycling cost with lead times; and ΔAi, ΔBi, ΔBie, ΔBwi, ΔCi, ΔDi, and ΔDie, respectively, denote the corresponding uncertain matrices.


## 3.2. Takagi–Sugeno Fuzzy Model of CLSC

In the process of wide variation of the inventory level, switching activities will take place among models to maintain a reasonable inventory level in each node enterprise. In order to effectively mitigate the BE of the CLSC, we will utilize Takagi–Sugeno fuzzy systems [21] to implement the flexible switching in the CLSC. Then, Model (3) is transformed into the following discrete Takagi–Sugeno fuzzy model:R
i: If x1k is M1i, …, xjk is Mji, …, and xnk is Mni, then(4)xk+1=Ai+ΔAixk+Bi+ΔBiuk+∑e=1gBie+ΔBieuk−τe+Bwi+ΔBwiwk,zk=Ci+ΔCixk+Di+ΔDiuk+∑e=1gDie+ΔDieuk−τe,xk=φk,k∈0,1,…,N,where wk∈l20,∞; φk denotes the initial condition of the CLSC; xTk=x1k,x2k,…,xnk; uTk=u1k,u2k,…,unk; uTk−τe=u1k−τ1,u2k−τ2,…,unk−τg; and wTk=w1k,w2k,…,wnk.By singleton fuzzifier, product inference, and centre-average defuzzifer, Model (4) can be expressed as follows:(5)xk+1=∑i=1rhixk⋅Ai+ΔAixk+Bi+ΔBiuk+∑e=1gBie+ΔBieuk−τe+Bwi+ΔBwiwk,zk=∑i=1rhixk⋅Ci+ΔCixk+Di+ΔDiuk+∑e=1gDie+ΔDieuk−τe,where hixk=μixk/∑i=1rμixk and μixk=∏j=1nMjixjk; since μixk≥0, we have hixk≥0 and ∑i=1rhixk=1. In the following expressions, xk will be elided from hixk for simplicity.Because of the existence of the membership degree function in Model (5), the flexible switching can be realized among the models of the CLSC.


## 4. Mitigation of BE

### 4.1. Measurement of BE

We use the following parameterγ to represent the mitigation degree of BE:(6)total cost of CLSC2customers'demand2≤γ.From the inequality above, we know that the mitigation degree of BE depends on the size of the total cost of the CLSC and the size of the customers’ demand. The lowerγ is, the smaller BE is.


### 4.2. FRC Approach

We introduce known constant matricesH1i, H2i, E11i, E12i, E13i, E21i, E22i, Lei, and Oei and time-varying uncertain matrices F1ik and F2ik to describe the uncertain parameters in the CLSC; F1ik, and F2ik are Lebesgue-measurable and satisfy F1iTkF1ik≤I and F2iTkF2ik≤I. Then, we represent the uncertain parameter matrices in Model (5) as follows:(7)ΔAi,ΔBi,ΔBwi,ΔBi1,…,ΔBie,…,ΔBig=H1iF1ikE11i,E12i,E13i,L1i,…,Lei,…,Lgi;ΔCi,ΔDi,ΔDi1,…,ΔDie,…,ΔDig=H2iF2ikE21i,E22i,O1i,…,Oei,…,Ogi.For the fuzzy CLSC system in Model (5), we design the Takagi–Sugeno fuzzy controller as follows.Controller ruleKi:Ifx1k is M1i, …, xjk is Mji, …, and xnk is Mni, then(8)uk=−Kixk,i=1,2,…,r,uk−τe=−Kiexk−τe,e=1,2,…,g,where Ki is the inventory feedback gain matrix and Kie is the inventory feedback gain matrix with lead times. Furthermore, we can obtain the following overall model of Model (8):(9)uk=−∑i=1rhiKixk,uk−τe=−∑i=1rhiKiexk−τe.Therefore, introducing Controller (9) into Model (5), we have (10)xk+1=∑i=1r∑j=1rhihjA¯i−B¯iKjxk−∑e=1gB¯ieKjexk−τe+B¯wiwk,zk=∑i=1r∑j=1rhihjC¯i−D¯iKjxk−∑e=1gD¯ieKjexk−τe.where A¯i=Ai+ΔAi, B¯i=Bi+ΔBi, B¯ie=Bie+ΔBie, B¯wi=Bwi+ΔBwi, C¯i=Ci+ΔCi, D¯i=Di+ΔDi, and D¯ie=Die+ΔDie.For the further analysis, we introduce the following Definitions, Property, and Lemma.Definition 1 (see [22]).
A cluster of fuzzy setsFjm,m=1,2,…,qj are said to be a standard fuzzy partition in the universe X if each Fjm is a normal fuzzy set and Fjmm=1,2,…,qj are full-overlapped in the universe X. qj is said to be the number of fuzzy partitions of the jth input variable on X.Definition 2 (see [22]).
For a given fuzzy system, an overlapped-rule group (ORG) with the largest amount of rules is said to be a maximal overlapped-rule group (MORG).Definition 3 (see [23]).
Given a scalarγ>0, discrete switched system (10) is said to be robustly stable with the disturbance attenuation level γ constraint under the H∞ norm if the following conditions are satisfied:(1)
Whenwk≡0, System (10) is asymptotically stable(2)
Whenwk≠0, under the condition of the initial value of zero, any uncertain customers’ demand meets zk22≤γwk22Property 1 (see [22]).
If the input variables of a fuzzy system adopt standard fuzzy partitions, then all the rules in an ORG must be included in a MORG.Lemma 1 (see [24]).
For any real matricesXij 1≤i,j≤r and P>0 with appropriate dimensions, the following inequality holds:(11)∑i=1r∑j=1r∑p=1r∑q=1rhihjhphqXijTPXpq≤∑i=1r∑j=1rhihjXijTPXij.
The FRC approach for the CLSC system will be presented in the following Theorem1.Theorem 1.
System (10) with a certain γ and fuzzy sets of inventories satisfying standard fuzzy partitions is robustly asymptotically stable if local common positive definite matrices Pc and Qec can be found in the following inequalities:(12)−P¯¯∗∗M¯ii−Pc−1∗N¯ii0−I<0,i∈Ic,(13)−4P¯¯∗∗2M¯¯ij−Pc−1∗2N¯¯ij0−I<0,i<j,i,j∈Ic,where(14)P¯¯=Pc−∑e=1gQec∗∗0Q^∗00γ2I,M¯¯ij=M¯ij+M¯ji2,Q^=diagQ1c⋯Qec⋯Qgc,N¯¯ij=N¯ij+N¯ji2.Proof.
Suppose there aref ORGs in System (10), vdd=1,2,…,f denotes the operating region of the dth ORG and Ld=the rule numbers involved in thedth ORG.(1)
The first part of the proof:Forxk and xk+1 in the same ORG, we express the local model of the dth ORG as follows:(15)xk+1=∑i∈Ld∑j∈LdhihjMijxk−∑e=1gB¯ieKjecxk−τe+B¯wiwk,zk=∑i∈Ld∑j∈LdhihjNijxk−∑e=1gD¯ieKjecxk−τe,where Mij=A¯i−B¯iKjc, Nij=C¯i−D¯iKjc, and Kjec denotes the state feedback gain matrix with lead times in the cth MORG.Furthermore, Model (15) is described further as follows:(16)xk+1=∑i∈Ld∑j∈LdhihjM¯ijx¯k,zk=∑i∈Ld∑j∈LdhihjN¯ijx¯k,where(17)M¯ij=Mij,−B¯i1Kj1c,…,−B¯ieKjec,…,−B¯igKjgc,B¯wi,N¯ij=Nij,−D¯i1Kj1c,…,−D¯ieKjec,…,−D¯igKjgc,0,x¯k=xk,xk−τ1,…,xk−τe,…,xk−τg,wkT.For System (16), a Lyapunov function is defined as follows:(18)Vdxk=xTkPcxk+∑e=1g∑ξ=k−τek−1xTξQecxξ.Then, based on Lemma1, we obtain the following ΔVdxk:(19)ΔVdxk=Vdxk+1−Vdxk=xTk+1Pcxk+1−xTkPcxk+∑e=1gxTkQecxk−xTk−τeQecxk−τe=∑i∈Ld∑j∈Ldhihj∑p∈Ld∑q∈Ldhphqx¯TkM¯ijTPcM¯pqx¯k−xTkPcxk+∑e=1gxTkQecxk−xTk−τeQecxk−τe=∑i∈Ld∑j∈Ldhihj∑p∈Ld∑q∈Ldhphqx¯TkM¯ijTPcM¯pq−P¯x¯k=∑i∈Ld∑j∈Ldhihj∑p∈Ld∑q∈Ldhphqx¯TkM¯ij+M¯ji2TPcM¯pq+M¯qp2−P¯x¯k=∑i∈Ld∑j∈Ldhihj∑p∈Ld∑q∈Ldhphqx¯TkM¯¯ijTPcM¯¯pq−P¯x¯k≤∑i∈Ld∑j∈Ldhihjx¯TkM¯¯ijTPcM¯¯ij−P¯x¯k,where(20)P¯=Pc−∑e=1gQec∗∗0Q^∗000,M¯¯pq=M¯pq+M¯qp2,Q^=diagQ1c,Q2c,…,Qec,…,Qgc,M¯¯ij=M¯ij+M¯ji2.Then,ΔVdxk is described further as(21)ΔVdxk≤∑i=j,i∈Ldhi2x¯TkM¯iiTPcM¯ii−P¯x¯k+2∑i<ji∈Ld,j∈Ldhihjx¯TkM¯¯ijTPcM¯¯ij−P¯x¯k.Forwk≠0, the H∞ performance index function is expressed as follows:(22)J1=∑k=0N−1zTkzk−γ2wTkwk.J1 can be expressed further as(23)J1=∑k=0N−1zTkzk−γ2wTkwk+ΔVdxk−VdxN≤∑k=0N−1zTkzk−γ2wTkwk+ΔVdxk.After Inequality (21) is introduced into (23), we have(24)J1≤∑k=0N−1∑i=j,i∈Ldhi2x¯TkM¯iiTPcM¯ii−P¯¯+N¯iiTN¯iix¯k+2∑k=0N−1∑i<j,i∈Ldj∈Ldhihjx¯TkM¯¯ijTPcM¯¯ij−P¯¯+N¯¯ijTN¯¯ijx¯k,where(25)P¯¯=Pc−∑e=1gQec∗∗0Q^∗00γ2I,N¯¯ij=N¯ij+N¯ji2.By the Schur complement, we haveM¯iiTPcM¯ii−P¯¯+N¯iiTN¯ii<0 and M¯¯ijTPcM¯¯ij−P¯¯+N¯¯ijTN¯¯ij<0 which are equivalent to Inequalities (12) and (13), then J1<0 holds, i.e., zTkzk<γ2wTkwk; furthermore, if let N⟶+∞, zk22<γ2wk22 holds. Therefore, CLSC system (16) is asymptotically stable under wk≠0.Ifwk≡0, Inequality (21) can be expressed as(26)ΔVdxk≤∑i=j,i∈Ldhi2x¯TkM¯iiTPcM¯ii−P¯¯x¯k+2∑i<ji∈Ld,j∈Ldhihjx¯TkM¯¯ijTPcM¯¯ij−P¯¯x¯k.Based on Inequalities (12) and (13), we have M¯iiTPcM¯ii−P¯¯<0 and M¯¯ijTPcM¯¯ij−P¯¯<0, and then ΔVdxk<0 can be obtained. Therefore, CLSC system (16) is robustly asymptotically stable under wk≡0.(2)
The second part of the proof:Forxk and xk+1 in the different ORGs, we first construct the following characteristic function in any ORG:(27)λd=1,xk∈vd,0,xk∉vd,∑d=1fλd=1.Then, the overall system of local system (16) can be expressed as(28)xk+1=∑d=1fλd∑i∈Ld∑j∈LdhihjM¯ijx¯k,zk=∑d=1fλd∑i∈Ld∑j∈LdhihjN¯ijx¯k.After Pm=∑d=1fλdPc and Qem=∑d=1fλdQec are defined, we present the following Lyapunov function:(29)Vxk=xTkPmxk+∑e=1g∑ξ=k−τek−1xTξQemxξ=xTk∑d=1fλdPcxk+∑e=1g∑ξ=k−τek−1xTξ∑d=1fλdQecxξ=∑d=1fλdxTkPcxk+∑e=1g∑ξ=k−τek−1xTξQecxξ=∑d=1fλdVdxk.For wk≠0 in System (28), considering J1=∑k=0N−1zTkzk−γ2wTkwk, we know J2=∑k=0N−1∑d=1fλdzTkzk−γ2wTkwk. In the same way, we have J2<0, i.e., zTkzk<γ2wTkwk; if let N⟶+∞, zk22<γ2wk22 can be obtained. Therefore, System (28) is asymptotically stable under wk≠0.Forwk≡0 in System (28), we have(30)ΔVxk=Vxk+1−Vxk=∑d=1fλdVdxk+1−∑d=1fλdVdxk=∑d=1fλdVdxk+1−Vdxk=∑d=1fλdΔVdxk<0.Therefore, System (28) under wk≡0 is asymptotically stable in any ORG.From Proposition 1, CLSC system (10) is robustly asymptotically stable if Pc and Qac can be solved in Inequalities (12) and (13) Q.E.D.In order for Inequalities (12) and (13) to be easily solvable LMIs, we transform Theorem 1 into the following Theorem 2.Theorem 2.
System (10) with a certain γ and fuzzy sets of inventories satisfying standard fuzzy partitions is robustly asymptotically stable if local common positive definite matrices Pc and Qec, matrices Kic, Kjc, Kiec, and Kjec, and constants εijc>0 and εjic>0 can be found in the following Inequalities (31) and (32):(31)−Pc+∑e=1gQec∗∗∗∗∗∗0−Q^∗∗∗∗∗00−γ2I∗∗∗∗Ω1−Ω3Bwi−I+εiicH1iH1iT∗∗∗Ω2−Ω400−I+εiicH2iH2iT∗∗Δ1−Δ3E13i00−εiicI∗Δ2−Δ40000−εiicI<0,i∈Ic,(32)−4Pc+4∑e=1gQec∗∗∗∗∗∗∗∗0−4Q^∗∗∗∗∗∗∗00−4γ2I∗∗∗∗∗∗Ω5−Ω7Bwi+BwjΔ11∗∗∗∗∗Ω6−Ω800Δ12∗∗∗∗Δ5−Δ3E13i00−εijcI∗∗∗Δ6−Δ40000−εijcI∗∗Δ7−Δ9E13j0000−εjicI∗Δ8−Δ10000000−εjicI<0,i<j,i,j∈Ic,where(33)Q^=diagQ1c,…,Qec,…,Qgc,Ω1=Ai−BiKic,Ω2=Ci−DiKic,Ω3=Bi1Ki1c,…,BieKiec,…,BigKigc,Ω4=Di1Ki1c,…,DieKiec,…,DigKigc,Ω5=Ai−BiKjc+Aj−BjKic,Ω6=Ci−DiKjc+Cj−DjKic,Ω7=Bi1Kj1c+Bj1Ki1c,…,BieKjec+BjeKiec,…,BigKjgc+BjgKigc,Ω8=Di1Kj1c+Dj1Ki1c,…,DieKjec+DjeKiec,…,DigKjgc+DjgKigc,Δ1=E11i−E12iKic,Δ2=E21i−E22iKic,Δ3=L1iKi1c,…,LeiKiec,…,LgiKigc,Δ4=O1iKi1c,…,OeiKiec,…,OgiKigc,Δ5=E11i−E12iKjc,Δ6=E21i−E22iKjc,Δ7=E11j−E12jKic,Δ8=E21j−E22jKic,Δ9=L1jKjic,…,LejKjec,…,LgjKjgc,Δ10=O1jKj1c,…,OejKjec,…,OgjKjgc,Δ11=−Ic+εijcH1iH1iT+εjicH1jH1jT,Δ12=−I+εijcH2iH2iT+εjicH2jH2jT.Proof.
Theorem2 can be proved in the same idea as Theorem 1. Therefore, the proof process of Theorem 2 is not shown. Q.E.D.


## 4.1. Measurement of BE

We use the following parameterγ to represent the mitigation degree of BE:(6)total cost of CLSC2customers'demand2≤γ.From the inequality above, we know that the mitigation degree of BE depends on the size of the total cost of the CLSC and the size of the customers’ demand. The lowerγ is, the smaller BE is.


## 4.2. FRC Approach

We introduce known constant matricesH1i, H2i, E11i, E12i, E13i, E21i, E22i, Lei, and Oei and time-varying uncertain matrices F1ik and F2ik to describe the uncertain parameters in the CLSC; F1ik, and F2ik are Lebesgue-measurable and satisfy F1iTkF1ik≤I and F2iTkF2ik≤I. Then, we represent the uncertain parameter matrices in Model (5) as follows:(7)ΔAi,ΔBi,ΔBwi,ΔBi1,…,ΔBie,…,ΔBig=H1iF1ikE11i,E12i,E13i,L1i,…,Lei,…,Lgi;ΔCi,ΔDi,ΔDi1,…,ΔDie,…,ΔDig=H2iF2ikE21i,E22i,O1i,…,Oei,…,Ogi.For the fuzzy CLSC system in Model (5), we design the Takagi–Sugeno fuzzy controller as follows.Controller ruleKi:Ifx1k is M1i, …, xjk is Mji, …, and xnk is Mni, then(8)uk=−Kixk,i=1,2,…,r,uk−τe=−Kiexk−τe,e=1,2,…,g,where Ki is the inventory feedback gain matrix and Kie is the inventory feedback gain matrix with lead times. Furthermore, we can obtain the following overall model of Model (8):(9)uk=−∑i=1rhiKixk,uk−τe=−∑i=1rhiKiexk−τe.Therefore, introducing Controller (9) into Model (5), we have (10)xk+1=∑i=1r∑j=1rhihjA¯i−B¯iKjxk−∑e=1gB¯ieKjexk−τe+B¯wiwk,zk=∑i=1r∑j=1rhihjC¯i−D¯iKjxk−∑e=1gD¯ieKjexk−τe.where A¯i=Ai+ΔAi, B¯i=Bi+ΔBi, B¯ie=Bie+ΔBie, B¯wi=Bwi+ΔBwi, C¯i=Ci+ΔCi, D¯i=Di+ΔDi, and D¯ie=Die+ΔDie.For the further analysis, we introduce the following Definitions, Property, and Lemma.Definition 1 (see [22]).
A cluster of fuzzy setsFjm,m=1,2,…,qj are said to be a standard fuzzy partition in the universe X if each Fjm is a normal fuzzy set and Fjmm=1,2,…,qj are full-overlapped in the universe X. qj is said to be the number of fuzzy partitions of the jth input variable on X.Definition 2 (see [22]).
For a given fuzzy system, an overlapped-rule group (ORG) with the largest amount of rules is said to be a maximal overlapped-rule group (MORG).Definition 3 (see [23]).
Given a scalarγ>0, discrete switched system (10) is said to be robustly stable with the disturbance attenuation level γ constraint under the H∞ norm if the following conditions are satisfied:(1)
Whenwk≡0, System (10) is asymptotically stable(2)
Whenwk≠0, under the condition of the initial value of zero, any uncertain customers’ demand meets zk22≤γwk22Property 1 (see [22]).
If the input variables of a fuzzy system adopt standard fuzzy partitions, then all the rules in an ORG must be included in a MORG.Lemma 1 (see [24]).
For any real matricesXij 1≤i,j≤r and P>0 with appropriate dimensions, the following inequality holds:(11)∑i=1r∑j=1r∑p=1r∑q=1rhihjhphqXijTPXpq≤∑i=1r∑j=1rhihjXijTPXij.
The FRC approach for the CLSC system will be presented in the following Theorem1.Theorem 1.
System (10) with a certain γ and fuzzy sets of inventories satisfying standard fuzzy partitions is robustly asymptotically stable if local common positive definite matrices Pc and Qec can be found in the following inequalities:(12)−P¯¯∗∗M¯ii−Pc−1∗N¯ii0−I<0,i∈Ic,(13)−4P¯¯∗∗2M¯¯ij−Pc−1∗2N¯¯ij0−I<0,i<j,i,j∈Ic,where(14)P¯¯=Pc−∑e=1gQec∗∗0Q^∗00γ2I,M¯¯ij=M¯ij+M¯ji2,Q^=diagQ1c⋯Qec⋯Qgc,N¯¯ij=N¯ij+N¯ji2.Proof.
Suppose there aref ORGs in System (10), vdd=1,2,…,f denotes the operating region of the dth ORG and Ld=the rule numbers involved in thedth ORG.(1)
The first part of the proof:Forxk and xk+1 in the same ORG, we express the local model of the dth ORG as follows:(15)xk+1=∑i∈Ld∑j∈LdhihjMijxk−∑e=1gB¯ieKjecxk−τe+B¯wiwk,zk=∑i∈Ld∑j∈LdhihjNijxk−∑e=1gD¯ieKjecxk−τe,where Mij=A¯i−B¯iKjc, Nij=C¯i−D¯iKjc, and Kjec denotes the state feedback gain matrix with lead times in the cth MORG.Furthermore, Model (15) is described further as follows:(16)xk+1=∑i∈Ld∑j∈LdhihjM¯ijx¯k,zk=∑i∈Ld∑j∈LdhihjN¯ijx¯k,where(17)M¯ij=Mij,−B¯i1Kj1c,…,−B¯ieKjec,…,−B¯igKjgc,B¯wi,N¯ij=Nij,−D¯i1Kj1c,…,−D¯ieKjec,…,−D¯igKjgc,0,x¯k=xk,xk−τ1,…,xk−τe,…,xk−τg,wkT.For System (16), a Lyapunov function is defined as follows:(18)Vdxk=xTkPcxk+∑e=1g∑ξ=k−τek−1xTξQecxξ.Then, based on Lemma1, we obtain the following ΔVdxk:(19)ΔVdxk=Vdxk+1−Vdxk=xTk+1Pcxk+1−xTkPcxk+∑e=1gxTkQecxk−xTk−τeQecxk−τe=∑i∈Ld∑j∈Ldhihj∑p∈Ld∑q∈Ldhphqx¯TkM¯ijTPcM¯pqx¯k−xTkPcxk+∑e=1gxTkQecxk−xTk−τeQecxk−τe=∑i∈Ld∑j∈Ldhihj∑p∈Ld∑q∈Ldhphqx¯TkM¯ijTPcM¯pq−P¯x¯k=∑i∈Ld∑j∈Ldhihj∑p∈Ld∑q∈Ldhphqx¯TkM¯ij+M¯ji2TPcM¯pq+M¯qp2−P¯x¯k=∑i∈Ld∑j∈Ldhihj∑p∈Ld∑q∈Ldhphqx¯TkM¯¯ijTPcM¯¯pq−P¯x¯k≤∑i∈Ld∑j∈Ldhihjx¯TkM¯¯ijTPcM¯¯ij−P¯x¯k,where(20)P¯=Pc−∑e=1gQec∗∗0Q^∗000,M¯¯pq=M¯pq+M¯qp2,Q^=diagQ1c,Q2c,…,Qec,…,Qgc,M¯¯ij=M¯ij+M¯ji2.Then,ΔVdxk is described further as(21)ΔVdxk≤∑i=j,i∈Ldhi2x¯TkM¯iiTPcM¯ii−P¯x¯k+2∑i<ji∈Ld,j∈Ldhihjx¯TkM¯¯ijTPcM¯¯ij−P¯x¯k.Forwk≠0, the H∞ performance index function is expressed as follows:(22)J1=∑k=0N−1zTkzk−γ2wTkwk.J1 can be expressed further as(23)J1=∑k=0N−1zTkzk−γ2wTkwk+ΔVdxk−VdxN≤∑k=0N−1zTkzk−γ2wTkwk+ΔVdxk.After Inequality (21) is introduced into (23), we have(24)J1≤∑k=0N−1∑i=j,i∈Ldhi2x¯TkM¯iiTPcM¯ii−P¯¯+N¯iiTN¯iix¯k+2∑k=0N−1∑i<j,i∈Ldj∈Ldhihjx¯TkM¯¯ijTPcM¯¯ij−P¯¯+N¯¯ijTN¯¯ijx¯k,where(25)P¯¯=Pc−∑e=1gQec∗∗0Q^∗00γ2I,N¯¯ij=N¯ij+N¯ji2.By the Schur complement, we haveM¯iiTPcM¯ii−P¯¯+N¯iiTN¯ii<0 and M¯¯ijTPcM¯¯ij−P¯¯+N¯¯ijTN¯¯ij<0 which are equivalent to Inequalities (12) and (13), then J1<0 holds, i.e., zTkzk<γ2wTkwk; furthermore, if let N⟶+∞, zk22<γ2wk22 holds. Therefore, CLSC system (16) is asymptotically stable under wk≠0.Ifwk≡0, Inequality (21) can be expressed as(26)ΔVdxk≤∑i=j,i∈Ldhi2x¯TkM¯iiTPcM¯ii−P¯¯x¯k+2∑i<ji∈Ld,j∈Ldhihjx¯TkM¯¯ijTPcM¯¯ij−P¯¯x¯k.Based on Inequalities (12) and (13), we have M¯iiTPcM¯ii−P¯¯<0 and M¯¯ijTPcM¯¯ij−P¯¯<0, and then ΔVdxk<0 can be obtained. Therefore, CLSC system (16) is robustly asymptotically stable under wk≡0.(2)
The second part of the proof:Forxk and xk+1 in the different ORGs, we first construct the following characteristic function in any ORG:(27)λd=1,xk∈vd,0,xk∉vd,∑d=1fλd=1.Then, the overall system of local system (16) can be expressed as(28)xk+1=∑d=1fλd∑i∈Ld∑j∈LdhihjM¯ijx¯k,zk=∑d=1fλd∑i∈Ld∑j∈LdhihjN¯ijx¯k.After Pm=∑d=1fλdPc and Qem=∑d=1fλdQec are defined, we present the following Lyapunov function:(29)Vxk=xTkPmxk+∑e=1g∑ξ=k−τek−1xTξQemxξ=xTk∑d=1fλdPcxk+∑e=1g∑ξ=k−τek−1xTξ∑d=1fλdQecxξ=∑d=1fλdxTkPcxk+∑e=1g∑ξ=k−τek−1xTξQecxξ=∑d=1fλdVdxk.For wk≠0 in System (28), considering J1=∑k=0N−1zTkzk−γ2wTkwk, we know J2=∑k=0N−1∑d=1fλdzTkzk−γ2wTkwk. In the same way, we have J2<0, i.e., zTkzk<γ2wTkwk; if let N⟶+∞, zk22<γ2wk22 can be obtained. Therefore, System (28) is asymptotically stable under wk≠0.Forwk≡0 in System (28), we have(30)ΔVxk=Vxk+1−Vxk=∑d=1fλdVdxk+1−∑d=1fλdVdxk=∑d=1fλdVdxk+1−Vdxk=∑d=1fλdΔVdxk<0.Therefore, System (28) under wk≡0 is asymptotically stable in any ORG.From Proposition 1, CLSC system (10) is robustly asymptotically stable if Pc and Qac can be solved in Inequalities (12) and (13) Q.E.D.In order for Inequalities (12) and (13) to be easily solvable LMIs, we transform Theorem 1 into the following Theorem 2.Theorem 2.
System (10) with a certain γ and fuzzy sets of inventories satisfying standard fuzzy partitions is robustly asymptotically stable if local common positive definite matrices Pc and Qec, matrices Kic, Kjc, Kiec, and Kjec, and constants εijc>0 and εjic>0 can be found in the following Inequalities (31) and (32):(31)−Pc+∑e=1gQec∗∗∗∗∗∗0−Q^∗∗∗∗∗00−γ2I∗∗∗∗Ω1−Ω3Bwi−I+εiicH1iH1iT∗∗∗Ω2−Ω400−I+εiicH2iH2iT∗∗Δ1−Δ3E13i00−εiicI∗Δ2−Δ40000−εiicI<0,i∈Ic,(32)−4Pc+4∑e=1gQec∗∗∗∗∗∗∗∗0−4Q^∗∗∗∗∗∗∗00−4γ2I∗∗∗∗∗∗Ω5−Ω7Bwi+BwjΔ11∗∗∗∗∗Ω6−Ω800Δ12∗∗∗∗Δ5−Δ3E13i00−εijcI∗∗∗Δ6−Δ40000−εijcI∗∗Δ7−Δ9E13j0000−εjicI∗Δ8−Δ10000000−εjicI<0,i<j,i,j∈Ic,where(33)Q^=diagQ1c,…,Qec,…,Qgc,Ω1=Ai−BiKic,Ω2=Ci−DiKic,Ω3=Bi1Ki1c,…,BieKiec,…,BigKigc,Ω4=Di1Ki1c,…,DieKiec,…,DigKigc,Ω5=Ai−BiKjc+Aj−BjKic,Ω6=Ci−DiKjc+Cj−DjKic,Ω7=Bi1Kj1c+Bj1Ki1c,…,BieKjec+BjeKiec,…,BigKjgc+BjgKigc,Ω8=Di1Kj1c+Dj1Ki1c,…,DieKjec+DjeKiec,…,DigKjgc+DjgKigc,Δ1=E11i−E12iKic,Δ2=E21i−E22iKic,Δ3=L1iKi1c,…,LeiKiec,…,LgiKigc,Δ4=O1iKi1c,…,OeiKiec,…,OgiKigc,Δ5=E11i−E12iKjc,Δ6=E21i−E22iKjc,Δ7=E11j−E12jKic,Δ8=E21j−E22jKic,Δ9=L1jKjic,…,LejKjec,…,LgjKjgc,Δ10=O1jKj1c,…,OejKjec,…,OgjKjgc,Δ11=−Ic+εijcH1iH1iT+εjicH1jH1jT,Δ12=−I+εijcH2iH2iT+εjicH2jH2jT.Proof.
Theorem2 can be proved in the same idea as Theorem 1. Therefore, the proof process of Theorem 2 is not shown. Q.E.D.


## 5. Simulation Analysis

We choose a type of television CLSC composed of a television manufacturer, a television retailer, and a 3PRP as the simulation object to evaluate the mitigation effect of the BE under the FRC approach proposed in Section4.The television manufacturer’s fuzzy membership functions and the television retailer’s fuzzy membership functions are shown in Figure2. In Figure 2, both F1tx1k and F2sx2k satisfy the conditions of standard fuzzy partition. We set M11=M12=F11, M13=M14=F12, M21=M24=F21, and M22=M23=F22.Figure 2
Fuzzy membership functions.From Figure2, we know there are 4 fuzzy rules included in one MORG called S. Referring to the product-design strategies in [25], we apply 4 fuzzy rules to describe the manufacturer’s production patterns and the retailer’s ordering patterns for the different inventory levels as follows:R1: the television manufacturer produces new televisions and remanufactures recycled televisions simultaneously. The television retailer allows return of new televisions and orders new televisions simultaneously.R2: the television manufacturer produces new televisions and remanufactures recycled televisions simultaneously. The television retailer only allows return of new televisions.R3: the television manufacturer only remanufactures recycled televisions. The television retailer only allows return of new televisions.R4: the television manufacturer only remanufactures recycled televisions. The television retailer allows return of new televisions and orders new televisions simultaneously.Therefore, under the different rules, the uncertain CLSC model with hybrid recycling channels and lead times can be expressed as follows:(34)R1:x1k+1=x1k+u1k+u1k−τ1+u3k+u3k−τ3+η+Δηx4k−u2k,x2k+1=x2k+u2k+u2k−τ2+μ+Δμx3k−w1k,x3k+1=x3k+w1k−u3k−u4k−μ+Δμx3k,x4k+1=x4k+u4k+u4k−τ4−η+Δηx4k−λ+Δλx4k,zk=ch1+Δch1x1k+ch2+Δch2x2k+ch3+Δch3x4k+cn+Δcnu1k+u1k−τ1+cr+Δcru3k+u3k−τ3+η+Δηx4k+cm+Δcmu3k+u3k−τ3+ct+Δctu4k+u4k−τ4+cmt+Δcmtη+Δηx4k+cq+Δcqμ+Δμx3k+cd+Δcdλ+Δλx4k+cs+Δcsu2k+u2k−τ2,R2:x1k+1=x1k+u1k+u1k−τ1+u3k+u3k−τ3+η+Δηx4k,x2k+1=x2k+μ+Δμx3k−w1k,x3k+1=x3k+w1k−u3k−u4k−μ+Δμx3k,x4k+1=x4k+u4k+u4k−τ4−η+Δηx4k−λ+Δλx4k,zk=ch1+Δch1x1k+ch2+Δch2x2k+ch3+Δch3x4k+cn+Δcnu1k+u1k−τ1+cr+Δcru3k+u3k−τ3+η+Δηx4k+cm+Δcmu3k+u3k−τ3+ct+Δctu4k+u4k−τ4+cmt+Δcmtη+Δηx4k+cq+Δcqμ+Δμx3k+cd+Δcdλ+Δλx4k,R3:x1k+1=x1k+u3k+u3k−τ3+η+Δηx4k,x2k+1=x2k+μ+Δμx3k−w1k,x3k+1=x3k+w1k−u3k−u4k−μ+Δμx3k,x4k+1=x4k+u4k+u4k−τ4−η+Δηx4k−λ+Δλx4k,zk=ch1+Δch1x1k+ch2+Δch2x2k+ch3+Δch3x4k+cr+Δcru3k+u3k−τ3+η+Δηx4k+cm+Δcmu3k+u3k−τ3+ct+Δctu4k+u4k−τ4+cmt+Δcmtη+Δηx4k+cq+Δcqμ+Δμx3k+cd+Δcdλ+Δλx4k,R4:x1k+1=x1k+u3k+u3k−τ3+η+Δηx4k−u2k,x2k+1=x2k+u2k+u2k−τ2+μ+Δμx3k−w1k,x3k+1=x3k+w1k−u3k−u4k−μ+Δμx3k,x4k+1=x4k+u4k+u4k−τ4−η+Δηx4k−λ+Δλx4k.zk=ch1+Δch1x1k+ch2+Δch2x2k+ch3+Δch3x4k+cr+Δcru3k+u3k−τ3+η+Δηx4k+cm+Δcmu3k+u3k−τ3+ct+Δctu4k+u4k−τ4+cmt+Δcmtη+Δηx4k+cq+Δcqμ+Δμx3k+cd+Δcdλ+Δλx4k+cs+Δcsu2k+u2k−τ2Furthermore, the fuzzy CLSC model can be obtained as follows:R1: If x1 is M11 and x2 is M21, then(35)xk+1=h1A1+ΔA1xk+B1+ΔB1uk+∑e=14B1e+ΔB1euk−τe+Bw1+ΔBw1wk,zk=h1C1+ΔC1xk+D1+ΔD1uk+∑e=14D1e+ΔD1euk−τe.R2: If x1 is M12 and x2 is M22, then(36)xk+1=h2A2+ΔA2xk+B2+ΔB2uk+∑e=14B2e+ΔB2euk−τe+Bw2+ΔBw2wk,zk=h2C2+ΔC2xk+D2+ΔD2uk+∑e=14D2e+ΔD2euk−τe.R3: If x1 is M13 and x2 is M23, then(37)xk+1=h3A3+ΔA3xk+B3+ΔB3uk+∑e=14B3e+ΔB3euk−τe+Bw3+ΔBw3wk,zk=h3C3+ΔC3xk+D3+ΔD3uk+∑e=14D3e+ΔD3euk−τe.R4: If x1 is M14 and x2 is M24, then(38)xk+1=h4A4+ΔA4xk+B4+ΔB4uk+∑e=14B4e+ΔB4euk−τe+Bw4+ΔBw4wk,zk=h4C4+ΔC4xk+D4+ΔD4uk+∑e=14D4e+ΔD4euk−τe.According to the Takagi–Sugeno fuzzy models mentioned above, we design the following fuzzy feedback controller.K
i: If x1k is M1i and x2k is M2i, then(39)uk=−∑i=14hiKi1xk,uk−τ1=−∑i=14hiKi11xk−τ1,uk−τ2=−∑i=14hiKi21xk−τ2,uk−τ3=−∑i=14hiKi31xk−τ3,uk−τ4=−∑i=14hiKi41xk−τ4.Considering the actual operation situation of the television CLSC, we setD0m=120, D1m=160, D0r=110, and D1r=155 (×103 sets); ch1 = 0.015, ch2 = 0.020, ch3 = 0.015, cn = 0.150, cr = 0.005, cm = 0.100, cmt = 0.100, ct = 0.095, cq = 0.020, cd = 0.007, cs = 0.180 (×103 Yuan); η=0.98, μ=0.01, and λ=0.02.(40)A1=A2=A3=A4=1000.98010.010000.9900000,B1=1−110010000−1−10001,B2=1010000000−1−10001,B3=0010000000−1−10001,B4=0−110010000−1−10001,B11=B21=1000000000000000,B31=B41=0000000000000000,B12=B42=0000010000000000,B22=B32=0000000000000000,B13=B23=B33=B43=0010000000000000,B14=B24=B34=B44=0000000000000001,Bw1=Bw2=Bw3=Bw4=0000−100010000000,(41)C1=C2=C3=C4=ch1ch2μcqch3+ηcr+ηcmt+λcd,D1=cncscr+cmct,D2=cn0cr+cmct,D3=00cr+cmct,D4=0cscr+cmct,D11=D21=cn000,D31=D41=0000,D12=D42=0cs00,D22=D32=0000,D13=D23=D33=D43=00cr+cm0,D14=D24=D34=D44=000ct,E11i=0000.02000.02000−0.020000−0.03,E12i=E13i=0,L1i=L2i=L3i=L4i=0,E21i=0.0040.0040.0020.007,E221=0.0030.0020.0150.015,E222=0.00300.0150.015,E223=000.0150.015,E224=00.0020.0150.015,O11=O21=0.001000,O31=O41=0000,O12=O42=00.00200,O22=O32=0000,O13=O23=O33=O43=000.0020,O14=O24=O34=O44=0000.001,H1i=0.1,H2i=0.2,F1i=F2i=sink,i=1,2,3,4,γ=0.5.The FRC approach proposed in Section4 can guarantee the television CLSC system to be robustly stable after the following results, which meet Conditions (31) and (32) in Theorem 2, are obtained.(42)P1=121.88980.00280.15110.14850.0028122.79890.0013−0.01130.15110.0013121.88290.15090.1485−0.01130.1509121.8811,Q11=Q21=Q31=Q41=24.3455−0.0000−0.0006−0.0006−0.000024.3419−0.00000.0000−0.0006−0.000024.3455−0.0006−0.00060.0000−0.000624.3455,Then, we will perform the simulation tests to compare the mitigation effect of the BE between the proposed FRC approach and the CRC approach. Set the initial values as follows:x10=14, x20=12, x30=8, and x40=6 (unit: ×103 sets). Set the nominal values as follows: x⟶1=155, x⟶2=140, x⟶3=120, and x⟶4=80 (unit: ×103 sets); u⟶1=130, u⟶2=120, u⟶3=50, and u⟶4=70 (unit: ×103 sets); we let τ1=τ2=τ3=τ4=4 (unit: day), and the customers’ demand meets the normal distribution N6,0.32. Then, the simulation results are shown in Figures 3–8, in which Figures 3–5 are the simulation diagrams under CRC, and Figures 6–8 are the simulation diagrams under FRC.Figure 3
Simulation diagram of the inventories under CRC (unit:×103 sets).Figure 4
Simulation diagram of the control variables under CRC (unit:×103 sets).Figure 5
Simulation diagram of the total cost under CRC (unit:×105 Yuan).Figure 6
Simulation diagram of the inventories under FRC (unit:×103 sets).Figure 7
Simulation diagram of the control variables under FRC (unit:×103 sets).Figure 8
Simulation diagram of the total cost under FRC (unit:×105 Yuan).As shown in Figures3–5, by utilizing the CRC approach to mitigate the BE of the uncertain CLSC with hybrid recycling channels and lead times, there are larger fluctuations for the variables of the CLSC system, which result in the worse mitigation level of the BE and the higher total cost of the CLSC. On the contrary, it can be seen from Figures 6–8 that the FRC approach proposed in this paper can mitigate the BE more effectively and make the CLSC system to be robustly stable. Also, the fluctuation ranges of all variables under the FRC approach are less that those under the CRC approach. Especially, compared with the CRC approach, the FRC approach realizes the total cost of the CLSC system in the stable and lower levels. Therefore, for the mitigation of the BE caused by uncertainties and lead times, the FRC approach proposed in this paper can more effectively guarantee the interests of enterprises in the supply chain and ensure the long-term and stable development of the CLSC.


## 6. Conclusions

BE is a universal phenomenon in the operation process of the supply chain system. Therefore, the mitigation of the BE has become one of the most focused and urgent issues for enterprises in the supply chain. In this paper, we have set up an uncertain model of the CLSC with hybrid recycling channels and lead times. To implement the flexible switching in the CLSC system, the aforementioned model is transformed into a CLSC fuzzy model. Then, the manufacturer’s production patterns and the retailer’s ordering patterns are put forward according to their respective inventory levels. A new FRC approach is proposed to mitigate the BE of the uncertain CLSC with lead times. Simulation results show that this approach can effectively mitigate the impacts of uncertainties and lead times on the BE of the CLSC and guarantee the CLSC system in a robust stable state, which will reduce the loss of the CLSC caused by the BE. It is known from this paper and [19] that the FRC approach is an effective approach to mitigate the BE and can improve the mitigation effect of the CRC approach. For the future research directions, one is how to coordinate the cost sharing among the manufacturer, the retailer, and the 3PRP; another is that the FRC approach is combined with other approaches to more effectively mitigate the BE.



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