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

Uncertainties and lead times make the closed-loop supply chain (CLSC) more complex, less stable, and then the bullwhip eect (BE) will become more intense. is 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 veries the mitigation eect of the BE under the proposed FRC approach.


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. erefore, both governments and enterprises have been paying more and more attention to the closed-loop supply chain (CLSC) which can achieve sustainable development [3]. e 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 pro t. 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 e ect (BE), and the other reason is lead time. If BE cannot be effectively mitigated, the operating e ciency of the CLSC will be reduced, and then the operating cost will be increased. In serious cases, the CLSC will collapse. erefore, 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. e remainder of this paper is formulated as follows: Section 2 o ers 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. e simulation studies are carried out in Section 5. e conclusions and future research directions are given in Section 6.

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 e cient for enterprises.
is recycling pattern is called hybrid recycling. erefore, 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][8][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. erefore, many scholars studied the BE caused by uncertainties or lead times; for example, by simulationbased 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]. erefore, 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. e critical contributions of this paper comparing to [19] are listed as follows.

Lead Times Are Included in the CLSC Models.
ere 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.

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.

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].

CLSC Model with Uncertainties and Lead Times.
As shown in Figure 1, 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.
Based on Figure 1, considering the uncertain system parameters and lead times, we set up the inventory equations of the CLSC as follows: 2 Complexity Accordingly, we set up the total cost equation of the CLSC as follows: 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. en, the ith model of the CLSC can be shown as follows: , and w T (k) � w 1 (k), 0, 0, 0 ; A i denotes the inventory status coefficient matrix; B i denotes the production, ordering, and recycling coefficient matrix; B ie denotes the production, ordering, and recycling coefficient matrix with lead times; B wi denotes the coefficient matrix of the customers' demand; C i 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; D i denotes the coefficient matrix of production cost, ordering cost, part of remanufacturing cost, and part of recycling cost; D ie denotes the coefficient matrix of production cost, ordering cost, part of remanufacturing cost, and part of recycling cost with lead times; and ΔA i , ΔB i , ΔB ie , ΔB wi , ΔC i , ΔD i , and ΔD ie , respectively, denote the corresponding uncertain matrices.

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. en, Model (3) is transformed into the following discrete Takagi-Sugeno fuzzy model: and By singleton fuzzifier, product inference, and centreaverage defuzzifer, Model (4) can be expressed as follows: In the following expressions, x(k) will be elided from h i (x(k)) 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.

Measurement of BE.
We use the following parameter c to represent the mitigation degree of BE: 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. e lower c is, the smaller BE is.

FRC Approach.
We introduce known constant matrices H 1i , H 2i , E 11i , E 12i , E 13i , E 21i , E 22i , L ei , and O ei and timevarying uncertain matrices F 1i (k) and F 2i (k) to describe the uncertain parameters in the CLSC; F 1i (k), and F 2i (k) are Lebesgue-measurable and satisfy F T 1i (k)F 1i (k) ≤ I and F T 2i (k)F 2i (k) ≤ I. en, we represent the uncertain parameter matrices in Model (5) as follows: For the fuzzy CLSC system in Model (5), we design the Takagi-Sugeno fuzzy controller as follows.
Controller rule K i : where K i is the inventory feedback gain matrix and K ie is the inventory feedback gain matrix with lead times. Furthermore, we can obtain the following overall model of Model (8): 4 Complexity erefore, introducing Controller (9) into Model (5), we have For the further analysis, we introduce the following Definitions, Property, and Lemma.
Definition 1 (see [22]). A cluster of fuzzy sets F m j , m � 1, 2, . . . , q j are said to be a standard fuzzy partition in the universe X if each F m j is a normal fuzzy set and F m j (m � 1, 2, . . . , q j ) are full-overlapped in the universe X. q j 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 scalar c > 0, discrete switched system (10) is said to be robustly stable with the disturbance attenuation level c constraint under the H ∞ norm if the following conditions are satisfied: (1) When w(k) ≡ 0, System (10) is asymptotically stable (2) When w(k) ≠ 0, under the condition of the initial value of zero, any uncertain customers' demand meets ‖z(k)‖ 2 2 ≤ c‖w(k)‖ 2 2 Property 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 matrices X ij (1 ≤ i, j ≤ r) and P > 0 with appropriate dimensions, the following inequality holds: e FRC approach for the CLSC system will be presented in the following eorem 1. (10) with a certain c and fuzzy sets of inventories satisfying standard fuzzy partitions is robustly asymptotically stable if local common positive definite matrices P c and Q ec can be found in the following inequalities: where Q� diag Q 1c · · · Q ec · · · Q gc , Proof. Suppose there are f ORGs in System (10), v d (d � 1, 2, . . . , f) denotes the operating region of the dth ORG and L d � the rule numbers involved in the dth ORG { }.
(1) e first part of the proof: For x(k) and x(k + 1) in the same ORG, we express the local model of the dth ORG as follows:  (15) is described further as follows: For System (16), a Lyapunov function is defined as follows: en, based on Lemma 1, we obtain the following ΔV d (x(k)): en, ΔV d (x(k)) is described further as For w(k) ≠ 0, the H ∞ performance index function is expressed as follows: J 1 can be expressed further as After Inequality (21) is introduced into (23), we have where P � By the Schur complement, we have M ij < 0 which are equivalent to Inequalities (12) and (13), then J 1 < 0 holds, i.e., z T (k)z(k) < c 2 w T (k)w(k); furthermore, if let N ⟶ +∞, ‖z(k)‖ 2 2 < c 2 ‖w(k)‖ 2 2 holds. erefore, CLSC system (16) is asymptotically stable under w(k) ≠ 0. If w(k) ≡ 0, Inequality (21) can be expressed as Based on Inequalities (12) and (13), we have M T ii P c M ii − P < 0 and M T ij P c M ij − P < 0, and then ΔV d (x(k)) < 0 can be obtained. erefore, CLSC system (16) is robustly asymptotically stable under w(k) ≡ 0.
(2) e second part of the proof: For x(k) and x(k + 1) in the different ORGs, we first construct the following characteristic function in any ORG: en, the overall system of local system (16) can be expressed as After P m � f d�1 λ d P c and Q em � f d�1 λ d Q ec are defined, we present the following Lyapunov function: For w(k) ≠ 0 in System (28), considering In the same Complexity 7 way, we have J 2 < 0, i.e., z T (k)z(k) < c 2 w T (k)w(k); if let N ⟶ +∞, ‖z(k)‖ 2 2 < c 2 ‖w(k)‖ 2 2 can be obtained. erefore, System (28) is asymptotically stable under w(k) ≠ 0. For w(k) ≡ 0 in System (28), we have (30) erefore, System (28) under w(k) ≡ 0 is asymptotically stable in any ORG. From Proposition 1, CLSC system (10) is robustly asymptotically stable if P c and Q ac 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 eorem 1 into the following eorem 2. □ Theorem 2. System (10) with a certain c and fuzzy sets of inventories satisfying standard fuzzy partitions is robustly asymptotically stable if local common positive definite matrices P c and Q ec , matrices K ic , K jc , K iec , and K jec , and constants ε ijc > 0 and ε jic > 0 can be found in the following Inequalities (31) and (32): Proof. eorem 2 can be proved in the same idea as eorem 1. erefore, the proof process of eorem 2 is not shown. Q.E.D.

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 Section 4. e television manufacturer's fuzzy membership functions and the television retailer's fuzzy membership functions are shown in Figure 2. In Figure 2, both F t 1 (x 1 (k)) and F s 2 (x 2 (k)) satisfy the conditions of standard fuzzy partition. Figure 2, 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:    erefore, under the different rules, the uncertain CLSC model with hybrid recycling channels and lead times can be expressed as follows: Complexity 9 R 1 : + c mt + Δc mt (η + Δη)x 4 (k) + c q + Δc q (μ + Δμ)x 3 (k) + c d + Δc d (λ + Δλ)x 4 (k) + c s + Δc s u 2 (k) + u 2 k − τ 2 , R 2 : R 3 : R 4 : Furthermore, the fuzzy CLSC model can be obtained as follows: According to the Takagi-Sugeno fuzzy models mentioned above, we design the following fuzzy feedback controller.
12 Complexity e FRC approach proposed in Section 4 can guarantee the television CLSC system to be robustly stable after the following results, which meet Conditions (31) and (32) in eorem 2, are obtained.