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Synchronization and control of chaos in supply chain management
Accepted Manuscript Synchronization and Control of Chaos in Supply Chain Management Alper Göksu, Uğur Erkin Kocamaz, Yılmaz Uyaroğlu PII: DOI: Reference:
S0360-8352(14)00292-7 http://dx.doi.org/10.1016/j.cie.2014.09.025 CAIE 3817
To appear in:
Computers & Industrial Engineering
Please cite this article as: Göksu, A., Kocamaz, U.E., Uyaroğlu, Y., Synchronization and Control of Chaos in Supply Chain Management, Computers & Industrial Engineering (2014), doi: http://dx.doi.org/10.1016/j.cie.2014.09.025
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Synchronization and Control of Chaos in Supply Chain Management Alper Göksu 1*, Uğur Erkin Kocamaz 2, and Yılmaz Uyaroğlu 3 1
Department of Industrial Engineering, Engineering Faculty, Sakarya University, Esentepe Campus, 54187 Serdivan, Sakarya, Turkey [email protected] 2 Department of Computer Technologies, Vocational School of Karacabey, Uludağ University, 16700 Karacabey, Bursa, Turkey [email protected] 3 Department of Electrical & Electronics Engineering, Engineering Faculty, Sakarya University, Esentepe Campus, 54187 Serdivan, Sakarya, Turkey [email protected] *
Corresponding Author: Alper Göksu E-mail: [email protected] Tel: +90 264 295 5492 Fax: +90 264 295 5664 Abstract This paper presents the synchronization and control of a chaotic supply chain management system based on its mathematical model. For this purpose, active controllers are applied for the synchronization of two identical chaotic supply chain management systems. Also, linear feedback controllers are designed and added to the nonlinear supply chain management system to achieve the control of the system. In these methods, synchronization and control are established by using Lyapunov stability theory. As a result, the synchronization and control of chaotic supply chain management system are realized numerically. Computer simulations are performed to verify the robustness of proposed synchronization and control methods. Keywords: Supply chain management, Chaos, Synchronization, Control, Active control, Linear feedback control.
Synchronization and Control of Chaos in Supply Chain Management Abstract This paper presents the synchronization and control of a chaotic supply chain management system based on its mathematical model. For this purpose, active controllers are applied for the synchronization of two identical chaotic supply chain management systems. Also, linear feedback controllers are designed and added to the nonlinear supply chain management system to achieve the control of the system. In these methods, synchronization and control are established by using Lyapunov stability theory. As a result, the synchronization and control of chaotic supply chain management system are realized numerically. Computer simulations are performed to verify the robustness of proposed synchronization and control methods. Keywords: Supply chain management, Chaos, Synchronization, Control, Active control, Linear feedback control.
1. Introduction A chaotic system is a nonlinear dynamical system which sensitively depends on initial conditions. Chaos has been intensively found and investigated in a variety of fields since Lorenz discovered the first chaotic attractor (Lorenz, 1963). Because of the undesired complex behavior of chaos, the synchronization and control of chaotic systems have been one of the major issues in engineering. Hubler was the first to introduce an adaptive control for chaotic systems (Hubler, 1989) and then, Ott, Grebogi and Yorke developed a method called OGY to control chaotic systems (Ott et al., 1990). Along with these, Pecora and Carroll introduced the idea of synchronizing two identical chaotic systems (Pecora & Carroll, 1990). Since these pioneering studies, various types of chaos control and synchronization methods such as active control, linear feedback control, sliding mode control, impulsive control, passive control and backstepping design have been proposed and applied to the chaotic systems. Among them, the most preferred and effective one for the synchronization is the active control method. It has been used for the synchronization of many chaotic systems such as Lorenz (Bai & Lonngren, 1997; Bai & Lonngren, 2000), Rossler (Agiza & Yassen, 2001), Chen (Agiza & Yassen, 2001), unified (Ucar et al., 2006), Bonhöffer–Van der Pol (Njah & Vincent, 2009), Vilnius (Kocamaz & Uyaroglu, 2014a) and many others. To control chaotic systems, linear feedback control method is widely preferred due to configuration simplicity. It has been successfully applied to control many chaotic systems such as Lorenz (Jianzu & Vincent, 1997), Rossler (Hegazi et al., 2001), Chen (Gambino et al., 2006), Liu (Wang & Li, 2010), Rucklidge (Kocamaz & Uyaroglu, 2014b) and many others. In recent years, many researchers have investigated the topics of supply chain modeling, planning, analysis and management. Hou et al. (2009) studied an integrated model of production, inventory and distribution in a two-stage supply chain. Then, Glock (2011) used it together with integrated inventory model. But Amorim et al. (2012) applied it to perishable products. Zhang & Zhou (2012) developed a novel nonlinear complementarity formulation for a supply chain network equilibrium model and some qualitative properties of the model regarding the existence and uniqueness were established under weaker conditions. Yuan & Hwarng (2012) analyzed the impact of customers’ behavior and purchasing decisions on stability with a chaos perspective. Kumar & Tiwari (2013) investigated risk pooling effects of safety stock and running inventory in a supply chain system to minimize the cost along with determining facility location and capacity. In number of studies, supply chain management systems have some unpredictable factors in their dynamics and they have resulted in nonlinearity and chaos (Lu et al., 2004; Donner et al., 2008; Fawcett & Waller, 2011; Ramirez & Pena, 2011). For instance, inventory system, planning and scheduling system may cause chaotic behaviours in system components and inventory levels under different stages. During the supply chain stages, information sharing process between the elements of chain distortions, incorrect inventory policies followed by sub-level suppliers and unsuitable demand forecasting methods based on high level of demand and supply to increase the process variability may cause significant problems for companies. It was firstly expressed by Forrester in 1961 (Towill et al., 2007). Then, a negative effect of aforementioned problems was detected by some researchers during the stages with different perspectives when examining the process variability (Burbidge, 1989; Houlihan, 1987). Afterwards, all of these studies have combined and the negative status is named as “Bullwhip Effect” (Lee et al., 1997). Many studies have been carried out to measure, control and find the evidence of
bullwhip effect in real life business environments, which has great importance both theoretical and empirical purposes. Five major causes of the bullwhip effect are defined as: the usage of demand signal processing, non-zero lead times, order batching, supply shortages and price fluctuations (Lee et al., 1997; Hwarng & Xie, 2008). Chaos at inventory levels and production strategies causes some undesirable problems which could be controlled. The control and synchronization of nonlinear behaviors in supply chain have great importance from the management point of view to avoid undesirable behaviors such as bullwhip effect. Recently, the control and synchronization of chaos in supply chain have been investigated in some studies. The benefits of sharing information about endcustomer demand throughout a multi-level supply chain were shown and a control engineering based measure was proposed to quantify the variance amplification (Dejonckheere et al., 2004). The chaos synchronization of bullwhip effect in a supply chain system was implemented by using radial basis function neural networks (Zhang et al., 2006). Then, the bullwhip effect on the supply chain was counteracted by the linear control theory (Wang et al., 2006). Afterwards, H-∞ control technique was proposed for the management of a supply chain model linearized with nominal operating conditions (Boccadoro et al., 2008). A robust-intelligent controller based on sliding mode control theory and radial basis function neural network was presented to reduce the bullwhip effect in supply chain management (Ghane et al., 2010). In recent years, robust control technique has been proposed with the aim of reducing the bullwhip effect in periodic-review inventory systems with variable lead-time (Ignaciuk & Bartoszewicz, 2011). The control of bullwhip effect in a supply chain management system has significant importance due to eliminating undesirable oscillations and decreasing uncertainties. Thus, it increases the effectiveness of the system. This study carries on further investigations on the synchronization and control of chaotic behavior in supply chain management system. For this purpose, active controllers and linear feedback controllers are employed to achieve the synchronization and control, respectively. Stability analyses of the proposed methods are provided by using the Lyapunov stability theory. Numerical simulations are also given to verify the effectiveness of synchronization and control results. This paper is organized as follows: In Section 2, the chaotic behavior in supply chain management system is described in detail. Active control method is applied to chaotic supply chain system to achieve the synchronization and the simulation results of synchronization are demonstrated in Section 3. In Section 4, the control of chaos in supply chain system via linear feedback control method is applied and demonstrated. Finally, concluding remarks are given in Section 5.
2. Description of Chaotic Supply Chain System The chaotic behavior in supply chain is described by a set of three autonomous differential equations as (Zhang et al., 2006):
x = (m + δm) y − (n + 1 + δn) x + d 1 , y = (r + δr ) x − y − xz + d 2 , z = xy + (k − 1 − δk ) z + d 3
(1)
where x, y, z are state variables, m, n, r, k are constant parameters, δm, δn, δr, δk denote linear perturbation of system parameter m, n, r, k respectively when the system is perturbed, and d 1, d 2, d 3 are nonlinear perturbation in three different states which are in general from the system outside. As seen in Fig. 1, the supply chain system includes three levels: end-customers, distributors and producers. x, y and z represent demand, inventory and produced quantities, respectively. m is delivery efficiency of distributors and n is ratio of customer demand. r and k denote distortion and safety stock coefficients. δm, δn, δr and δk indicate linear distortions and disorders in these three levels. d1, d 2 and d 3 are nonlinear changes in different levels of the supply chain system. When the parameter values of supply chain system are considered as in Table 1, it exhibits chaotic behavior (Zhang et al., 2006). Information Flow Dist. 1 Producer 1 Customers
Dist. 2 Producer 2 Dist. 3 Producer 3 Dist. 4 Material Flow
Fig. 1. A model of supply chain system. Table 1. Values of the chaotic supply chain system parameters Parameters Values
m 10
n 9
r 28
k –5/3
δm 0.1
δn 0.1
δr 0.2
δk 0.3
d1 d2 d3 0.2sin(t) 0.1cos(5t) 0.3sin(t)
Under the initial conditions x(0) = 0, y(0) = –0.11 and z(0) = 9, the time series, 2D phase portraits and 3D phase plane of chaotic supply chain system are shown in Fig. 2, Fig. 3 and Fig. 4, respectively.
(a)
(b)
(c) Fig. 2. Time series of chaotic supply chain system for (a) x signals, (b) y signals, (c) z signals.
(a)
(b)
(c) Fig. 3. 2D phase portraits of chaotic supply chain system in (a) x–y phase plot, (b) x–z phase plot, (c) y–z phase plot.
Fig. 4. x–y–z phase plane of chaotic supply chain system.
3. Synchronization of Chaos in Supply Chain System
3.1. Synchronization
It is assumed that two supply chain systems are taken where the initial positions are different so as to observe the synchronization of chaos in supply chain. The drive system which is denoted by subscript 1 is to control the response system which is denoted by subscript 2. They are defined as follows: x1 = (m + δm) y1 − (n + 1 + δn) x1 + 0.2 sin( t ), y 1 = (r + δr ) x1 − y1 − x1 z1 + 0.1 cos(5t ),
(2)
z1 = x1 y1 + (k − 1 − δk ) z1 + 0.3 sin( t ) and x 2 = (m + δm) y 2 − (n + 1 + δn) x 2 + 0.2 sin(t ) + u1 (t ), y 2 = (r + δr ) x 2 − y 2 − x 2 z 2 + 0.1 cos(5t ) + u 2 (t ), z 2 = x 2 y 2 + (k − 1 − δk ) z 2 + 0.3 sin( t ) + u 3 (t )
(3)
where u1(t), u2(t) and u 3(t) in Eq. (3) are the control functions to be determined. In order to obtain the control functions for the synchronization, the drive system is subtracted from the response system. e1, e2 and e3 are the state errors between response and drive supply chain systems, they are defined as e1 = x 2 − x1 , e 2 = y 2 − y1 ,
(4)
e 3 = z 2 − z1 .
This leads to e1 = (m + δm)e 2 − (n + 1 + δn)e1 + u 1 (t ), e 2 = (r + δr )e1 − e 2 − x 2 z 2 + x1 z 1 + u 2 (t ),
(5)
e3 = x 2 y 2 − x1 y1 + (k − 1 − δk )e3 + u 3 (t ). Eq. (5) is called the error system. The synchronization problem is to ensure the error system asymptotically stable at the origin. In that, the active control functions u1(t), u2(t) and u3(t) are redefined as follows so as to eliminate the nonlinear terms in error system (5): u1 (t ) = v1 (t ), u 2 (t ) = x 2 z 2 − x1 z1 + v 2 (t ), u 3 (t ) = − x 2 y 2 + x1 y1 + v3 (t ). So, using this notation implies that
(6)
e1 = (m + δm)e 2 − (n + 1 + δn)e1 + v1 (t ), e2 = (r + δr )e1 − e 2 + v 2 (t ),
(7)
e3 = (k − 1 − δk )e3 + v3 (t ).
Eq. (7) is the error dynamics. It is linear and convergent solution can be found under appropriate control inputs v1(t), v2(t) and v3(t) as function of e1, e2 and e3. As long as the solutions of system (7) converge to zero as time t goes to infinity, the synchronization of two identical supply chain systems is realized. There are many possible choices for v1(t), v2(t) and v3(t) control functions. They are considered as follows: v1 (t ) = − (m + δm)e 2 + (n + δn)e1, v 2 (t ) = − (r + δr )e1 ,
(8)
v 3 (t ) = −(k − δk )e3 . Then, error system (7) becomes e1 = − e1 , e2 = −e 2 ,
(9)
e3 = −e3 . Now, the Lyapunov function for system (9) is selected as follows:
1 2 (10) (e1 + e22 + e32 ). 2 It is clear that Lyapunov function (10) has a positive definite function and it is equal to zero at the equilibrium of system (9). Furthermore, the time derivative of Lyapunov function V is V (e1 , e2 , e3 ) =
V = e1e1 + e2 e2 + e3 e3 2
2
2
= −(e1 + e2 + e3 )
(11)
which has negative definite. As a consequence, from Lyapunov direct method, the zero solution of system (11) is asymptotically stable with this choice. This implies that the synchronization of two identical supply chain systems is achieved.
3.2. Numerical Simulations
In this section, computer simulations are performed to illustrate the synchronization of two identical chaotic supply chain systems. The fourth-order Runge–Kutta method with fixed step size being equal to 0.001 is used to simulate the system. The parameter values of supply chain system are selected as m = 10, n = 9, r = 28, k = –5/3, δm = 0.1, δn = 0.1, δr = 0.2 and δk = 0.3 to ensure the chaotic behavior of supply chain system. The initial values are x1(0) = 0, y1(0) = –0.11, z1(0) = 9, x2(0) = –10, y2(0) = –5.11 and z2(0) = 29. When the active controllers are started at t = 20, the observed simulation results for the synchronization of two identical supply chain systems are shown in Fig. 5: (a) displays the x signals, (b) displays the y signals, and (c) displays the z signals. The error signals of synchronization are illustrated in Fig. 6.
(a)
(b)
(c) Fig. 5. The time response of states for the synchronization of supply chain systems with active controllers are activated at t = 20 (a) x signals, (b) y signals, (c) z signals.
Fig. 6. The time response of the error signals for the synchronization of supply chain systems with active controllers are activated at t = 20.
As expected, the error signals shown in Fig. 6 converge asymptotically to zero. Fig. 5 shows that the synchronization is observed when t ≥ 28, which confirms the validity of proposed active control technique. In order to show the effect of sequential controller scheme, the active controllers are started at different times. The activation times are: tdx = 15, tdy = 20 and tdz = 25. The synchronization simulation results are shown in Fig. 7: (a) displays the x signals, (b) displays the y signals, and (c) displays the z signals. The error signals of synchronization are illustrated in Fig. 8.
(a)
(b)
(c) Fig. 7. The time response of states for the synchronization of supply chain systems with sequential active controllers are activated at tdx = 15, tdy = 20 and tdz = 25 (a) x signals, (b) y signals, (c) z signals.
Fig. 8. The time response of the error signals for the synchronization of supply chain systems with sequential active controllers are activated at tdx = 15, tdy = 20 and tdz = 25.
As expected, the errors shown in Fig. 8 converge asymptotically to zero. Fig. 7 shows that sequential control yields the synchronization which is observed completely when t ≥ 24. It confirms the validity of proposed active control technique. Hence, despite some disruptor factors such as bullwhip effect, the synchronization of supply chains has made the adaptation possible to enterprises on account of obtaining the same demand, inventory, produced quantities in an appropriate time period and has reduced the risks.
4. Control of Chaos in Supply Chain System 4.1. Control
Linear feedback control can be applied to system (1) so as to control the chaos in supply chain system to its equilibrium point. The controlled supply chain system is expressed as follows: x = (m + δm) y − (n + 1 + δn ) x + 0.2 sin(t ) + u 1 , y = (r + δr ) x − y − xz + 0.1 cos(5t ) + u 2 ,
(12)
z = xy + (k − 1 − δk ) z + 0.3 sin( t ) + u 3 where u 1, u 2 and u3 are the single variable feedback controllers. They are described as follows: u1 = −k1 ( x − x ), u 2 = −k 2 ( y − y ),
(13)
u 3 = −k 3 ( z − z ) where ( x , y , z ) is an equilibrium point of the controlled supply chain system (12), and k1, k2, k3 are positive feedback gains. According to Eq. (13), system (12) can be rewritten as follows: x = (m + δm) y − (n + 1 + δn) x + 0.2 sin( t ) − k1 ( x − x ), y = (r + δr ) x − y − xz + 0.1 cos(5t ) − k 2 ( y − y ),
(14)
z = xy + (k − 1 − δk ) z + 0.3 sin( t ) − k 3 ( z − z ).
When the error dynamics in Eq. (14) are asymptotically stable on the zero equilibrium point, the –xz and xy equations equal to zero. Therefore, Eq. (14) can be simplified as
x = (m + δm) y − (n + 1 + δn ) x + 0.2 sin(t ) − k1 ( x − x ), y = (r + δr ) x − y + 0.1 cos(5t ) − k 2 ( y − y ),
(15)
z = (k − 1 − δk ) z + 0.3 sin(t ) − k 3 ( z − z ).
In order to provide the asymptotic stability of the solution ( x , y , z ) for system (15), the direct method of Lyapunov is used. The Lyapunov function is selected as follows: V ( x, y , z ) =
1 2 (x + y 2 + z 2 ) 2
(16)
which has a positive definite function and equal to zero at the equilibrium of system (15). Furthermore, the time derivative of the Lyapunov function V is V = xx + yy + zz = −[(n + 1 + δn + k1 ) x 2 + (1 + k 2 ) y 2 + (−k + 1 + δk + k 3 ) z 2 ]
(17)
Therefore, the derivative of V has negative definite, whenever
n + 1 + δn + k1 ≥ 0,
1 + k 2 ≥ 0,
− k + 1 + δk + k 3 ≥ 0.
(18)
As a result, from Lyapunov direct method, the zero solution of system (15) is asymptotically stable under condition (18). The Jacobi matrix of supply chain system (15) for zero equilibrium point is given by 0 − (n + 1 + δn + k1 ) m + δm J = r + δr − (1 + k 2 ) 0 . 0 0 k − 1 − δk − k 3
(19)
Assuming control parameters k1 = 0 and k3 = 0, the characteristic equation of matrix (19) is ( λ − k + 1 + δk )[λ2 + λ ( n + δn + 2 + k 2 ) + ( n + 1 + δn)(k 2 + 1) − ( m + δm)( r + δr )] = 0. (20)
When the parameters are taken as m = 10, n = 9, r = 28, k = –5/3, δm = 0.1, δn = 0.1, δr = 0.2 and δk = 0.3, Eq. (20) becomes
λ3 + (11.1 + k 2 )λ2 + (−27.2 *10.1 + 89 / 30 *11.1 + (89 / 30 + 10.1)k 2 )λ + 89 / 30 *10.1 * (−27.2 + k 2 ) = 0.
(21)
According to the Routh–Hurwitz criterion, the stability of the controlled system states that all real eigenvalues and all real parts of complex conjugate eigenvalues are negative if and only if the following conditions hold: (i ) c1 > 0, c 2 > 0, c 3 > 0, (ii ) c1c 2 > c 3
(22)
where c1, c2 and c3 are the coefficients of the characteristic equation
λ3 + c1λ2 + c 2 λ + c3 = 0
(23)
of the Jacobi matrix for the linearized system of the controlled system (15) at zero equilibrium point where c1 = 11.1 + k2 > 0, then k2 > –11.1; c2 = –27.2*10.1 + 89/30*11.1 + (89/30 + 10.1)k2 > 0, then k2 > 18.504337; c3 = 89/30*10.1*(–27.2 + k2) > 0, then k2 > 27.2; (11.1 + k2) * (–27.2*10.1 + 89/30*11.1 + (89/30 + 10.1)k2) > 89/30*10.1*(–27.2 + k2), then k2 > 17.4052. As a result, k2 > 27.2. Hence, the controlled supply chain system (14) is asymptotically stable to the zero equilibrium point when the control parameters are taken as k1 = 0, k2 > 27.2 and k3 = 0 which also provide the conditions (18) of Lyapunov stability theory. 4.2. Numerical Simulations
In this section, computer simulations are performed to illustrate the control of chaos in supply chain system. The fourth-order Runge–Kutta method with fixed step size being equal to 0.001 is used to simulate the system. The parameter values of supply chain system are selected as m = 10, n = 9, r = 28, k = –5/3, δm = 0.1, δn = 0.1, δr = 0.2 and δk = 0.3 to ensure the chaotic behavior of supply chain system. The initial values are x(0) = 0, y(0) = –0.11 and z(0) = 9. The controllers are started at t = 20 in all simulations. The observed simulation results for the control of supply chain system with the linear feedback controllers k1 = 0, k2 = 26, k3 = 0 are shown in Fig. 9: (a) displays the x signals, (b) displays the y signals, and (c) displays the z signals.
(a)
(b)
(c)
Fig. 9. The time response of states for the control of supply chain system with the linear feedback controller is taken as k2 = 26 and activated at t = 20 (a) x signals, (b) y signals, (c) z signals.
As expected, the signals of the controlled chaotic supply chain system shown in Fig. 9 do not converge asymptotically to zero equilibrium point, because the linear feedback controller is taken as k2 = 26 which is smaller than 27.2. When the linear feedback controllers are supposed as k1 = 0, k2 = 27.2, k3 = 0, and k1 = 0, k2 = 28, k3 = 0, the observed simulation results for the control of them are shown in Fig. 10 and Fig. 11, respectively.
(a)
(b)
(c) Fig. 10. The time response of states for the control of supply chain system with the linear feedback controller is taken as k2 = 27.2 and activated at t = 20 (a) x signals, (b) y signals, (c) z signals.
(a)
(b)
(c) Fig. 11. The time response of states for the control of supply chain system with the linear feedback controller is taken as k2 = 28 and activated at t = 20 (a) x signals, (b) y signals, (c) z signals. As expected, the signals of the controlled chaotic supply chain system shown in Figs. 10 and 11 converge asymptotically to zero equilibrium point. By taking the controller parameter as k2 ≥ 27.2, especially k2 = 28 particular choice, the control is observed when t ≥ 22. It confirms the validity of the proposed linear feedback control technique. Hence, despite some disruptor factors such as bullwhip effect, the control of supply chain has eliminated the periodic trajectories of demand, inventory, produced quantities in an appropriate time period and has made them stable.
5. Conclusions
Nowadays, it is generally desired that supply chain management systems should be effective, but it is difficult to operate these systems especially when there are some distortions and nonlinear behaviors. For this reason, most of the researchers have focused on bullwhip effect which is one of the undesirable behaviors in supply chain systems and can cause chaotic phenomenon. Bullwhip effect is detected under certain circumstances. It is possible that correct demand information is able to be retrieved by enterprise resource planning system on time. So, the effectiveness of supply chain system increases by choosing appropriate synchronization and control mechanisms. In this way, more realistic management models may be determined by means of adding some appropriate variables to the model. The purpose of this paper is to investigate the synchronization and control of chaos in supply chain management system. Based on the Lyapunov stability theory, active controllers have been designed and added to the system for the synchronization of chaos in two identical supply chain management systems. The sequential usage of active controllers in the synchronization is also investigated. Besides, linear feedback controllers have been designed and added to the system for the control of chaos in the supply chain management system by using the Lyapunov stability theory and Routh–Hurwitz criterion. All the theoretical analyses are confirmed with numerical simulations. Computer simulations show that synchronization and control of chaos in supply chain management system are realized in an appropriate time period which verifies the effectiveness of proposed synchronization and control methods.
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Highlights of manuscript “Synchronization and Control of Chaos in Supply Chain Management”, in Special Issues on “Applications of Computational Intelligence and Fuzzy Logic to Manufacturing and Service Systems” of Computers & Industrial Engineering journal:
• Chaos in supply chain management is described. • Active controllers are applied for the synchronization. • Linear feedback controllers are applied for the control. • Lyapunov function is used to realize synchronization and control. • Computer simulations verify the effectiveness of synchronization and control strategies.
S0360-8352(14)00292-7 http://dx.doi.org/10.1016/j.cie.2014.09.025 CAIE 3817
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Please cite this article as: Göksu, A., Kocamaz, U.E., Uyaroğlu, Y., Synchronization and Control of Chaos in Supply Chain Management, Computers & Industrial Engineering (2014), doi: http://dx.doi.org/10.1016/j.cie.2014.09.025
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Synchronization and Control of Chaos in Supply Chain Management Alper Göksu 1*, Uğur Erkin Kocamaz 2, and Yılmaz Uyaroğlu 3 1
Department of Industrial Engineering, Engineering Faculty, Sakarya University, Esentepe Campus, 54187 Serdivan, Sakarya, Turkey [email protected] 2 Department of Computer Technologies, Vocational School of Karacabey, Uludağ University, 16700 Karacabey, Bursa, Turkey [email protected] 3 Department of Electrical & Electronics Engineering, Engineering Faculty, Sakarya University, Esentepe Campus, 54187 Serdivan, Sakarya, Turkey [email protected] *
Corresponding Author: Alper Göksu E-mail: [email protected] Tel: +90 264 295 5492 Fax: +90 264 295 5664 Abstract This paper presents the synchronization and control of a chaotic supply chain management system based on its mathematical model. For this purpose, active controllers are applied for the synchronization of two identical chaotic supply chain management systems. Also, linear feedback controllers are designed and added to the nonlinear supply chain management system to achieve the control of the system. In these methods, synchronization and control are established by using Lyapunov stability theory. As a result, the synchronization and control of chaotic supply chain management system are realized numerically. Computer simulations are performed to verify the robustness of proposed synchronization and control methods. Keywords: Supply chain management, Chaos, Synchronization, Control, Active control, Linear feedback control.
Synchronization and Control of Chaos in Supply Chain Management Abstract This paper presents the synchronization and control of a chaotic supply chain management system based on its mathematical model. For this purpose, active controllers are applied for the synchronization of two identical chaotic supply chain management systems. Also, linear feedback controllers are designed and added to the nonlinear supply chain management system to achieve the control of the system. In these methods, synchronization and control are established by using Lyapunov stability theory. As a result, the synchronization and control of chaotic supply chain management system are realized numerically. Computer simulations are performed to verify the robustness of proposed synchronization and control methods. Keywords: Supply chain management, Chaos, Synchronization, Control, Active control, Linear feedback control.
1. Introduction A chaotic system is a nonlinear dynamical system which sensitively depends on initial conditions. Chaos has been intensively found and investigated in a variety of fields since Lorenz discovered the first chaotic attractor (Lorenz, 1963). Because of the undesired complex behavior of chaos, the synchronization and control of chaotic systems have been one of the major issues in engineering. Hubler was the first to introduce an adaptive control for chaotic systems (Hubler, 1989) and then, Ott, Grebogi and Yorke developed a method called OGY to control chaotic systems (Ott et al., 1990). Along with these, Pecora and Carroll introduced the idea of synchronizing two identical chaotic systems (Pecora & Carroll, 1990). Since these pioneering studies, various types of chaos control and synchronization methods such as active control, linear feedback control, sliding mode control, impulsive control, passive control and backstepping design have been proposed and applied to the chaotic systems. Among them, the most preferred and effective one for the synchronization is the active control method. It has been used for the synchronization of many chaotic systems such as Lorenz (Bai & Lonngren, 1997; Bai & Lonngren, 2000), Rossler (Agiza & Yassen, 2001), Chen (Agiza & Yassen, 2001), unified (Ucar et al., 2006), Bonhöffer–Van der Pol (Njah & Vincent, 2009), Vilnius (Kocamaz & Uyaroglu, 2014a) and many others. To control chaotic systems, linear feedback control method is widely preferred due to configuration simplicity. It has been successfully applied to control many chaotic systems such as Lorenz (Jianzu & Vincent, 1997), Rossler (Hegazi et al., 2001), Chen (Gambino et al., 2006), Liu (Wang & Li, 2010), Rucklidge (Kocamaz & Uyaroglu, 2014b) and many others. In recent years, many researchers have investigated the topics of supply chain modeling, planning, analysis and management. Hou et al. (2009) studied an integrated model of production, inventory and distribution in a two-stage supply chain. Then, Glock (2011) used it together with integrated inventory model. But Amorim et al. (2012) applied it to perishable products. Zhang & Zhou (2012) developed a novel nonlinear complementarity formulation for a supply chain network equilibrium model and some qualitative properties of the model regarding the existence and uniqueness were established under weaker conditions. Yuan & Hwarng (2012) analyzed the impact of customers’ behavior and purchasing decisions on stability with a chaos perspective. Kumar & Tiwari (2013) investigated risk pooling effects of safety stock and running inventory in a supply chain system to minimize the cost along with determining facility location and capacity. In number of studies, supply chain management systems have some unpredictable factors in their dynamics and they have resulted in nonlinearity and chaos (Lu et al., 2004; Donner et al., 2008; Fawcett & Waller, 2011; Ramirez & Pena, 2011). For instance, inventory system, planning and scheduling system may cause chaotic behaviours in system components and inventory levels under different stages. During the supply chain stages, information sharing process between the elements of chain distortions, incorrect inventory policies followed by sub-level suppliers and unsuitable demand forecasting methods based on high level of demand and supply to increase the process variability may cause significant problems for companies. It was firstly expressed by Forrester in 1961 (Towill et al., 2007). Then, a negative effect of aforementioned problems was detected by some researchers during the stages with different perspectives when examining the process variability (Burbidge, 1989; Houlihan, 1987). Afterwards, all of these studies have combined and the negative status is named as “Bullwhip Effect” (Lee et al., 1997). Many studies have been carried out to measure, control and find the evidence of
bullwhip effect in real life business environments, which has great importance both theoretical and empirical purposes. Five major causes of the bullwhip effect are defined as: the usage of demand signal processing, non-zero lead times, order batching, supply shortages and price fluctuations (Lee et al., 1997; Hwarng & Xie, 2008). Chaos at inventory levels and production strategies causes some undesirable problems which could be controlled. The control and synchronization of nonlinear behaviors in supply chain have great importance from the management point of view to avoid undesirable behaviors such as bullwhip effect. Recently, the control and synchronization of chaos in supply chain have been investigated in some studies. The benefits of sharing information about endcustomer demand throughout a multi-level supply chain were shown and a control engineering based measure was proposed to quantify the variance amplification (Dejonckheere et al., 2004). The chaos synchronization of bullwhip effect in a supply chain system was implemented by using radial basis function neural networks (Zhang et al., 2006). Then, the bullwhip effect on the supply chain was counteracted by the linear control theory (Wang et al., 2006). Afterwards, H-∞ control technique was proposed for the management of a supply chain model linearized with nominal operating conditions (Boccadoro et al., 2008). A robust-intelligent controller based on sliding mode control theory and radial basis function neural network was presented to reduce the bullwhip effect in supply chain management (Ghane et al., 2010). In recent years, robust control technique has been proposed with the aim of reducing the bullwhip effect in periodic-review inventory systems with variable lead-time (Ignaciuk & Bartoszewicz, 2011). The control of bullwhip effect in a supply chain management system has significant importance due to eliminating undesirable oscillations and decreasing uncertainties. Thus, it increases the effectiveness of the system. This study carries on further investigations on the synchronization and control of chaotic behavior in supply chain management system. For this purpose, active controllers and linear feedback controllers are employed to achieve the synchronization and control, respectively. Stability analyses of the proposed methods are provided by using the Lyapunov stability theory. Numerical simulations are also given to verify the effectiveness of synchronization and control results. This paper is organized as follows: In Section 2, the chaotic behavior in supply chain management system is described in detail. Active control method is applied to chaotic supply chain system to achieve the synchronization and the simulation results of synchronization are demonstrated in Section 3. In Section 4, the control of chaos in supply chain system via linear feedback control method is applied and demonstrated. Finally, concluding remarks are given in Section 5.
2. Description of Chaotic Supply Chain System The chaotic behavior in supply chain is described by a set of three autonomous differential equations as (Zhang et al., 2006):
x = (m + δm) y − (n + 1 + δn) x + d 1 , y = (r + δr ) x − y − xz + d 2 , z = xy + (k − 1 − δk ) z + d 3
(1)
where x, y, z are state variables, m, n, r, k are constant parameters, δm, δn, δr, δk denote linear perturbation of system parameter m, n, r, k respectively when the system is perturbed, and d 1, d 2, d 3 are nonlinear perturbation in three different states which are in general from the system outside. As seen in Fig. 1, the supply chain system includes three levels: end-customers, distributors and producers. x, y and z represent demand, inventory and produced quantities, respectively. m is delivery efficiency of distributors and n is ratio of customer demand. r and k denote distortion and safety stock coefficients. δm, δn, δr and δk indicate linear distortions and disorders in these three levels. d1, d 2 and d 3 are nonlinear changes in different levels of the supply chain system. When the parameter values of supply chain system are considered as in Table 1, it exhibits chaotic behavior (Zhang et al., 2006). Information Flow Dist. 1 Producer 1 Customers
Dist. 2 Producer 2 Dist. 3 Producer 3 Dist. 4 Material Flow
Fig. 1. A model of supply chain system. Table 1. Values of the chaotic supply chain system parameters Parameters Values
m 10
n 9
r 28
k –5/3
δm 0.1
δn 0.1
δr 0.2
δk 0.3
d1 d2 d3 0.2sin(t) 0.1cos(5t) 0.3sin(t)
Under the initial conditions x(0) = 0, y(0) = –0.11 and z(0) = 9, the time series, 2D phase portraits and 3D phase plane of chaotic supply chain system are shown in Fig. 2, Fig. 3 and Fig. 4, respectively.
(a)
(b)
(c) Fig. 2. Time series of chaotic supply chain system for (a) x signals, (b) y signals, (c) z signals.
(a)
(b)
(c) Fig. 3. 2D phase portraits of chaotic supply chain system in (a) x–y phase plot, (b) x–z phase plot, (c) y–z phase plot.
Fig. 4. x–y–z phase plane of chaotic supply chain system.
3. Synchronization of Chaos in Supply Chain System
3.1. Synchronization
It is assumed that two supply chain systems are taken where the initial positions are different so as to observe the synchronization of chaos in supply chain. The drive system which is denoted by subscript 1 is to control the response system which is denoted by subscript 2. They are defined as follows: x1 = (m + δm) y1 − (n + 1 + δn) x1 + 0.2 sin( t ), y 1 = (r + δr ) x1 − y1 − x1 z1 + 0.1 cos(5t ),
(2)
z1 = x1 y1 + (k − 1 − δk ) z1 + 0.3 sin( t ) and x 2 = (m + δm) y 2 − (n + 1 + δn) x 2 + 0.2 sin(t ) + u1 (t ), y 2 = (r + δr ) x 2 − y 2 − x 2 z 2 + 0.1 cos(5t ) + u 2 (t ), z 2 = x 2 y 2 + (k − 1 − δk ) z 2 + 0.3 sin( t ) + u 3 (t )
(3)
where u1(t), u2(t) and u 3(t) in Eq. (3) are the control functions to be determined. In order to obtain the control functions for the synchronization, the drive system is subtracted from the response system. e1, e2 and e3 are the state errors between response and drive supply chain systems, they are defined as e1 = x 2 − x1 , e 2 = y 2 − y1 ,
(4)
e 3 = z 2 − z1 .
This leads to e1 = (m + δm)e 2 − (n + 1 + δn)e1 + u 1 (t ), e 2 = (r + δr )e1 − e 2 − x 2 z 2 + x1 z 1 + u 2 (t ),
(5)
e3 = x 2 y 2 − x1 y1 + (k − 1 − δk )e3 + u 3 (t ). Eq. (5) is called the error system. The synchronization problem is to ensure the error system asymptotically stable at the origin. In that, the active control functions u1(t), u2(t) and u3(t) are redefined as follows so as to eliminate the nonlinear terms in error system (5): u1 (t ) = v1 (t ), u 2 (t ) = x 2 z 2 − x1 z1 + v 2 (t ), u 3 (t ) = − x 2 y 2 + x1 y1 + v3 (t ). So, using this notation implies that
(6)
e1 = (m + δm)e 2 − (n + 1 + δn)e1 + v1 (t ), e2 = (r + δr )e1 − e 2 + v 2 (t ),
(7)
e3 = (k − 1 − δk )e3 + v3 (t ).
Eq. (7) is the error dynamics. It is linear and convergent solution can be found under appropriate control inputs v1(t), v2(t) and v3(t) as function of e1, e2 and e3. As long as the solutions of system (7) converge to zero as time t goes to infinity, the synchronization of two identical supply chain systems is realized. There are many possible choices for v1(t), v2(t) and v3(t) control functions. They are considered as follows: v1 (t ) = − (m + δm)e 2 + (n + δn)e1, v 2 (t ) = − (r + δr )e1 ,
(8)
v 3 (t ) = −(k − δk )e3 . Then, error system (7) becomes e1 = − e1 , e2 = −e 2 ,
(9)
e3 = −e3 . Now, the Lyapunov function for system (9) is selected as follows:
1 2 (10) (e1 + e22 + e32 ). 2 It is clear that Lyapunov function (10) has a positive definite function and it is equal to zero at the equilibrium of system (9). Furthermore, the time derivative of Lyapunov function V is V (e1 , e2 , e3 ) =
V = e1e1 + e2 e2 + e3 e3 2
2
2
= −(e1 + e2 + e3 )
(11)
which has negative definite. As a consequence, from Lyapunov direct method, the zero solution of system (11) is asymptotically stable with this choice. This implies that the synchronization of two identical supply chain systems is achieved.
3.2. Numerical Simulations
In this section, computer simulations are performed to illustrate the synchronization of two identical chaotic supply chain systems. The fourth-order Runge–Kutta method with fixed step size being equal to 0.001 is used to simulate the system. The parameter values of supply chain system are selected as m = 10, n = 9, r = 28, k = –5/3, δm = 0.1, δn = 0.1, δr = 0.2 and δk = 0.3 to ensure the chaotic behavior of supply chain system. The initial values are x1(0) = 0, y1(0) = –0.11, z1(0) = 9, x2(0) = –10, y2(0) = –5.11 and z2(0) = 29. When the active controllers are started at t = 20, the observed simulation results for the synchronization of two identical supply chain systems are shown in Fig. 5: (a) displays the x signals, (b) displays the y signals, and (c) displays the z signals. The error signals of synchronization are illustrated in Fig. 6.
(a)
(b)
(c) Fig. 5. The time response of states for the synchronization of supply chain systems with active controllers are activated at t = 20 (a) x signals, (b) y signals, (c) z signals.
Fig. 6. The time response of the error signals for the synchronization of supply chain systems with active controllers are activated at t = 20.
As expected, the error signals shown in Fig. 6 converge asymptotically to zero. Fig. 5 shows that the synchronization is observed when t ≥ 28, which confirms the validity of proposed active control technique. In order to show the effect of sequential controller scheme, the active controllers are started at different times. The activation times are: tdx = 15, tdy = 20 and tdz = 25. The synchronization simulation results are shown in Fig. 7: (a) displays the x signals, (b) displays the y signals, and (c) displays the z signals. The error signals of synchronization are illustrated in Fig. 8.
(a)
(b)
(c) Fig. 7. The time response of states for the synchronization of supply chain systems with sequential active controllers are activated at tdx = 15, tdy = 20 and tdz = 25 (a) x signals, (b) y signals, (c) z signals.
Fig. 8. The time response of the error signals for the synchronization of supply chain systems with sequential active controllers are activated at tdx = 15, tdy = 20 and tdz = 25.
As expected, the errors shown in Fig. 8 converge asymptotically to zero. Fig. 7 shows that sequential control yields the synchronization which is observed completely when t ≥ 24. It confirms the validity of proposed active control technique. Hence, despite some disruptor factors such as bullwhip effect, the synchronization of supply chains has made the adaptation possible to enterprises on account of obtaining the same demand, inventory, produced quantities in an appropriate time period and has reduced the risks.
4. Control of Chaos in Supply Chain System 4.1. Control
Linear feedback control can be applied to system (1) so as to control the chaos in supply chain system to its equilibrium point. The controlled supply chain system is expressed as follows: x = (m + δm) y − (n + 1 + δn ) x + 0.2 sin(t ) + u 1 , y = (r + δr ) x − y − xz + 0.1 cos(5t ) + u 2 ,
(12)
z = xy + (k − 1 − δk ) z + 0.3 sin( t ) + u 3 where u 1, u 2 and u3 are the single variable feedback controllers. They are described as follows: u1 = −k1 ( x − x ), u 2 = −k 2 ( y − y ),
(13)
u 3 = −k 3 ( z − z ) where ( x , y , z ) is an equilibrium point of the controlled supply chain system (12), and k1, k2, k3 are positive feedback gains. According to Eq. (13), system (12) can be rewritten as follows: x = (m + δm) y − (n + 1 + δn) x + 0.2 sin( t ) − k1 ( x − x ), y = (r + δr ) x − y − xz + 0.1 cos(5t ) − k 2 ( y − y ),
(14)
z = xy + (k − 1 − δk ) z + 0.3 sin( t ) − k 3 ( z − z ).
When the error dynamics in Eq. (14) are asymptotically stable on the zero equilibrium point, the –xz and xy equations equal to zero. Therefore, Eq. (14) can be simplified as
x = (m + δm) y − (n + 1 + δn ) x + 0.2 sin(t ) − k1 ( x − x ), y = (r + δr ) x − y + 0.1 cos(5t ) − k 2 ( y − y ),
(15)
z = (k − 1 − δk ) z + 0.3 sin(t ) − k 3 ( z − z ).
In order to provide the asymptotic stability of the solution ( x , y , z ) for system (15), the direct method of Lyapunov is used. The Lyapunov function is selected as follows: V ( x, y , z ) =
1 2 (x + y 2 + z 2 ) 2
(16)
which has a positive definite function and equal to zero at the equilibrium of system (15). Furthermore, the time derivative of the Lyapunov function V is V = xx + yy + zz = −[(n + 1 + δn + k1 ) x 2 + (1 + k 2 ) y 2 + (−k + 1 + δk + k 3 ) z 2 ]
(17)
Therefore, the derivative of V has negative definite, whenever
n + 1 + δn + k1 ≥ 0,
1 + k 2 ≥ 0,
− k + 1 + δk + k 3 ≥ 0.
(18)
As a result, from Lyapunov direct method, the zero solution of system (15) is asymptotically stable under condition (18). The Jacobi matrix of supply chain system (15) for zero equilibrium point is given by 0 − (n + 1 + δn + k1 ) m + δm J = r + δr − (1 + k 2 ) 0 . 0 0 k − 1 − δk − k 3
(19)
Assuming control parameters k1 = 0 and k3 = 0, the characteristic equation of matrix (19) is ( λ − k + 1 + δk )[λ2 + λ ( n + δn + 2 + k 2 ) + ( n + 1 + δn)(k 2 + 1) − ( m + δm)( r + δr )] = 0. (20)
When the parameters are taken as m = 10, n = 9, r = 28, k = –5/3, δm = 0.1, δn = 0.1, δr = 0.2 and δk = 0.3, Eq. (20) becomes
λ3 + (11.1 + k 2 )λ2 + (−27.2 *10.1 + 89 / 30 *11.1 + (89 / 30 + 10.1)k 2 )λ + 89 / 30 *10.1 * (−27.2 + k 2 ) = 0.
(21)
According to the Routh–Hurwitz criterion, the stability of the controlled system states that all real eigenvalues and all real parts of complex conjugate eigenvalues are negative if and only if the following conditions hold: (i ) c1 > 0, c 2 > 0, c 3 > 0, (ii ) c1c 2 > c 3
(22)
where c1, c2 and c3 are the coefficients of the characteristic equation
λ3 + c1λ2 + c 2 λ + c3 = 0
(23)
of the Jacobi matrix for the linearized system of the controlled system (15) at zero equilibrium point where c1 = 11.1 + k2 > 0, then k2 > –11.1; c2 = –27.2*10.1 + 89/30*11.1 + (89/30 + 10.1)k2 > 0, then k2 > 18.504337; c3 = 89/30*10.1*(–27.2 + k2) > 0, then k2 > 27.2; (11.1 + k2) * (–27.2*10.1 + 89/30*11.1 + (89/30 + 10.1)k2) > 89/30*10.1*(–27.2 + k2), then k2 > 17.4052. As a result, k2 > 27.2. Hence, the controlled supply chain system (14) is asymptotically stable to the zero equilibrium point when the control parameters are taken as k1 = 0, k2 > 27.2 and k3 = 0 which also provide the conditions (18) of Lyapunov stability theory. 4.2. Numerical Simulations
In this section, computer simulations are performed to illustrate the control of chaos in supply chain system. The fourth-order Runge–Kutta method with fixed step size being equal to 0.001 is used to simulate the system. The parameter values of supply chain system are selected as m = 10, n = 9, r = 28, k = –5/3, δm = 0.1, δn = 0.1, δr = 0.2 and δk = 0.3 to ensure the chaotic behavior of supply chain system. The initial values are x(0) = 0, y(0) = –0.11 and z(0) = 9. The controllers are started at t = 20 in all simulations. The observed simulation results for the control of supply chain system with the linear feedback controllers k1 = 0, k2 = 26, k3 = 0 are shown in Fig. 9: (a) displays the x signals, (b) displays the y signals, and (c) displays the z signals.
(a)
(b)
(c)
Fig. 9. The time response of states for the control of supply chain system with the linear feedback controller is taken as k2 = 26 and activated at t = 20 (a) x signals, (b) y signals, (c) z signals.
As expected, the signals of the controlled chaotic supply chain system shown in Fig. 9 do not converge asymptotically to zero equilibrium point, because the linear feedback controller is taken as k2 = 26 which is smaller than 27.2. When the linear feedback controllers are supposed as k1 = 0, k2 = 27.2, k3 = 0, and k1 = 0, k2 = 28, k3 = 0, the observed simulation results for the control of them are shown in Fig. 10 and Fig. 11, respectively.
(a)
(b)
(c) Fig. 10. The time response of states for the control of supply chain system with the linear feedback controller is taken as k2 = 27.2 and activated at t = 20 (a) x signals, (b) y signals, (c) z signals.
(a)
(b)
(c) Fig. 11. The time response of states for the control of supply chain system with the linear feedback controller is taken as k2 = 28 and activated at t = 20 (a) x signals, (b) y signals, (c) z signals. As expected, the signals of the controlled chaotic supply chain system shown in Figs. 10 and 11 converge asymptotically to zero equilibrium point. By taking the controller parameter as k2 ≥ 27.2, especially k2 = 28 particular choice, the control is observed when t ≥ 22. It confirms the validity of the proposed linear feedback control technique. Hence, despite some disruptor factors such as bullwhip effect, the control of supply chain has eliminated the periodic trajectories of demand, inventory, produced quantities in an appropriate time period and has made them stable.
5. Conclusions
Nowadays, it is generally desired that supply chain management systems should be effective, but it is difficult to operate these systems especially when there are some distortions and nonlinear behaviors. For this reason, most of the researchers have focused on bullwhip effect which is one of the undesirable behaviors in supply chain systems and can cause chaotic phenomenon. Bullwhip effect is detected under certain circumstances. It is possible that correct demand information is able to be retrieved by enterprise resource planning system on time. So, the effectiveness of supply chain system increases by choosing appropriate synchronization and control mechanisms. In this way, more realistic management models may be determined by means of adding some appropriate variables to the model. The purpose of this paper is to investigate the synchronization and control of chaos in supply chain management system. Based on the Lyapunov stability theory, active controllers have been designed and added to the system for the synchronization of chaos in two identical supply chain management systems. The sequential usage of active controllers in the synchronization is also investigated. Besides, linear feedback controllers have been designed and added to the system for the control of chaos in the supply chain management system by using the Lyapunov stability theory and Routh–Hurwitz criterion. All the theoretical analyses are confirmed with numerical simulations. Computer simulations show that synchronization and control of chaos in supply chain management system are realized in an appropriate time period which verifies the effectiveness of proposed synchronization and control methods.
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Highlights of manuscript “Synchronization and Control of Chaos in Supply Chain Management”, in Special Issues on “Applications of Computational Intelligence and Fuzzy Logic to Manufacturing and Service Systems” of Computers & Industrial Engineering journal:
• Chaos in supply chain management is described. • Active controllers are applied for the synchronization. • Linear feedback controllers are applied for the control. • Lyapunov function is used to realize synchronization and control. • Computer simulations verify the effectiveness of synchronization and control strategies.