DYNAMICAL ANALYSIS AND DECISION SUPPORT SYSTEM OF PRODUCTION MANAGEMENT

. Nonlinear system dynamics and feedback control theory are presented for management optimization of supply chain systems. Linearization and simplification methods are widely used in analyzing the system dynamics of supply chains because actual production models are highly complex and nonlinear systems. With advanced system dynamics, it is possible to deal directly with nonlinear dynamical problems without linear approximate methods so that the decision-makers can obtain more accurate results for systematic management strategies. This paper proposes a nonlinear system theory to explore dynamical behavior and control synthesis of production-distribution systems using Forrester’s model. A novel super-twisting sliding mode control (SWT-SMC) algorithm has been presented based on adaptation law, ensuring management optimization against disruptions. The closed-loop system stability has been guaranteed by using the Lyapunov theory. Extensive numerical simulations have been conducted to validate the efficacy and reliability of the adaptive super-twisting sliding mode control (ASWT-SMC) algorithm. Four types of decision criteria have been employed to compare system performance between control strategies. With a superb decision scheme powered by a control algorithm, novel supply chain software can learn an ever-fluctuating production flow and anticipate the need for changes in a real market.


Introduction
Supply chain management controls the entire production flow of a good or service, starting from raw materials to delivering final products to consumers.Recently, more and more business enterprises are investing in management optimization software to get things done, predict risks, and improve overall efficiency.Supply chain management is typically a complex dynamical system due to the large mesh of interlinked participants and their nonlinearities having different objectives.The dynamical analysis plays a vital role in developing an optimization algorithm for the supply chain system.It will provide an effective tool to describe large and complex systems' structure and internal behaviors [1].The system complexity and a lack of integration among supply chain network constituents can lead to potential problems that should be appropriately addressed.It is noted that modern supply chains are about effectively managing data, services, and products bundled into solutions.An efficient management system can be designed if dynamical behaviors are properly understood.The nonlinear control theory can be introduced to balance the risk of stock-out and the costs of supply chain network fluctuations.Even better, an intelligent decision algorithm would respond much faster to market moves than existing processes, requiring minimal human intervention by empowering autonomy.Towill [2] introduced an inventory and order-based production and control system (IOBPCS).Then, system analysis was presented under the dynamic effects of feedforward and feedback paths using third-order coefficient plane models.This work has been used as a basic framework for studying dynamic analysis.Many researchers have conducted the related researches described as follows: Mason-Jones et al. [3] conducted dynamic analysis using feedback control by comparing IOBPCS and APIOBPCS (automatic pipeline, inventory and order based production control system) to explore how bullwhip effect affecting to each echelon in the supply chain networks; Disney and Towill [4] evaluated the performance of a production or distribution-scheduling algorithm termed APIOBPCS which embedded within a vendor managed inventory (VMI) supply chain based on causal loop diagrams and difference equations; Wang et al. [5] introduced a new approach for analyzing system behaviors by using system stability for EPVIOBPCS (estimated pipeline, variable inventory and order based production control system); Li et al. [6] analyzed system dynamics of VMI-TPL (vendor-managed inventory and third-party logistics) model based on VMI-APIOBPCS to observe effectively the replenishment and delivery product quantity between suppliers and retailers when the model is turned from a two-stage into a three-stage structure.
Moreover, the supply chain network is analyzed as a chaotic system illustrating complex dynamical and nonlinear behaviors accompanied by bullwhip effects impacting various system network stages.Many dynamical systems have been presented to investigate chaotic supply chain networks, such as two-dimensional chaotic analysis [7] and three or four-stage networks using beer distribution dynamics [8,9].Moreover, some articles focus on hyperchaotic and higher-order chaotic phenomena.Hwarng and Yuan [10] studied supply chain dynamics through a quasi-chaos analysis.Thomsen et al. [11] introduced the decision-making behavior caused by chaotic and higher-order hyperchaotic phenomena.On the other hand, the supply chain model by Forrester [12] introduced simulation modeling in analyzing complex dynamical behaviors of organizations.This model has become the benchmark for other related studies in deep analysis of systems as well as for expansion and optimization of system components.Specifically, Wikner et al. [13] endeavored to gain more insights into this model by analyzing the system before conducting simplification with linearization; Berry et al. [14] re-designed Forrester's system for smoother and faster responses, and simulation techniques have been presented to obtain more characteristics of system behaviors.Minegishi and Thiel [15] used Forrester's theoretical paradigm to explain the complex behaviors of a particular type of food industry model.The optimized production system is one of the vital issues for efficient supply chain management in an unpredictable real market.The dynamic performance of the manufacturing operation directly affects an organization's overall efficiency of supply chain networks.Therefore, the independent production system, separated from the complete supply chain, has been discussed by many articles to identify how structure and decision policies can generate system behaviors of manufacturers.Sagawa and Mušič [16] conducted bond graph/mathematical models to describe multi-workstation production systems and proposed state feedback controllers using linear quadratic regulator (LQR).AL-Khazraji et al. [17] simulated the dynamical behaviors of a production-inventory system and then applied control system theory to improve system performance.Spiegler et al. [18] proposed a methodology to establish the occurrence of sustained oscillations in production and inventory models.Sarkar and Moon citebib19 analyzed the system behaviors of EPQ (economic production quantity) models in an imperfect production network.Ferney [20] introduced an approach to the modeling and control phases involved in the manufacturing systems' cooperative design and management process.Especially, Spiegler et al. [21] proposed a new approach to analyzing the Forrester production-distribution model by employing Wikner's inverse method [22].The dynamical model has been simplified first, then linearized to observe the analysis conveniently.In order to obtain a competitive advantage, the production system is continually faced with new challenges to improve its process and to adapt to erratic customer demands.The active controller can be designed to respond to management optimization after system analysis has been completed.The selection of appropriate control methods for decision-making must be in accordance with specific system characteristics.Some control strategies have been well-recognized in supply chain management, such as synchronization and suppression of chaotic supply chain systems by feedback controllers and Lyapunov stability theory [8,16,23].A robust intelligent controller is employed based on sliding mode control theory, and radial basis functions neural networks [24].A robust H-∞ control technique has been utilized to manage a linearized supply chain model [25].The integrated error with state feedback and filtering control scheme is applied to the multi-echelon production distribution system [26].Fuzzy and genetic algorithms have been implemented for managing supply chain risks [27].Recently, adaptive controllers have been widely used for monitoring and optimizing nonlinear dynamical systems because they provide appropriate control actions and achieve better performance than conventional controllers.In addition, adaptive control schemes can be utilized to handle a broad range of parameter variations and uncertainty, where the controller can modify itself according to the parameter changes to realize robust stability and performance [28].Implementing this control method increases system response speed and minimizes processing time in supply chain management, optimizing operation costs for profits [29,30].It depends on many factors, such as the business size, needs, tasks, and processes that need to be automated in a business.Meanwhile, modern adaptive controllers have been applied successfully in many fields, especially adaptive super twisting sliding mode, considered one of the best choices to control nonlinear systems with disturbances/uncertainties.The advantage of achieving robustness and insensitivity to disturbances and uncertainty is that it can improve control actions for business decision-making and optimize operation costs [31].Many other areas have been successfully applied to this controller, such as wind energy conversion system control [32], micro gyroscope control optimization under model uncertainties and external disturbances [33], and synchronization for two chaotic systems [34].However, few studies have been conducted in supply chain management based on this control theory.This paper deals with a methodological framework for dynamically analyzing production systems to gain more insights into nonlinear system behaviors.Instead of the linearized approach of Wikner et al. [12] or Wikner's inverse method by Spiegler et al. [21], the production system is directly analyzed by nonlinear system methods.This will provide more accurate results than the above methods because the linearized approximation always produces inaccuracy biases [35], resulting in increased costs and poor service for supply chain management.This paper synchronizes and optimizes the nonlinear production system using the ASWT-SMC scheme.The production system's closed-loop stability is guaranteed using the Lyapunov stability theory.The performance indices are presented to evaluate the active control algorithms.The rest of this paper is organized as follows.A low-order approximation of Forrester's production model is presented in Section 2. The nonlinear system behaviors are analyzed in Section 3. Section 4 implements the nonlinear control synthesis with complete stability analysis.Extensive numerical simulations are carried out to demonstrate the effectiveness of the designed control algorithm to the supply chains in Section 5. Finally, conclusions are successfully made in Section 6.

Model approximation of Forrester's production system
Forrester introduced the well-known production-distribution model in the 1960s by analyzing complex nonlinear behaviors [12].The system was first described in DYNAMO language, which imposes some limits to achieve a deeper understanding of system behavior and the difficulty of implementing the control system for optimization purposes.In order to obtain more internal characteristics of Forrester's model, the original DYNAMO equation is replaced by an equivalent block diagram that is realized by Wikner et al. [13].A new system structure may provide greater flexibility in analyzing the performance of the supply chain networks through a robust control system.As illustrated in Figure 1, the block diagram represents the factory stage of the original Forrester model via Wikner's translation approach.Through the block diagram, the Forrester model is a highly complex and nonlinear system, with many feedback paths and high-order models, which causes many limitations for system analysis and optimization.Many articles deal with simplification and linearization to facilitate in-depth local analysis of the relationships between the functional clusters in the system as well as easier to connect to the linear control systems [13,21,26].Spiegler [21] simplified a system by removing redundancies.This method calculates the output variable in the summing comparators, and some redundancy variables are combined or removed.All system variables and constants are completely described in Tables 1 and 2.More precisely, there are three summing comparators called SUM 1 , SUM 2 , and SUM 3 (see Fig. 1), simplified as follows.First, collecting all signals in SUM 1 leads to where Next, removing signal redundancies (SR and SS) in SUM 2 leads to In addition, by removing redundancy (MO) in SUM 3 , it will result in the following relation: After all redundancies are removed with some algebraic manipulations, the block diagram for the simplified Forrester's model is shown in Figure 2a.
The pipeline is practically demonstrated by a sixth-order system illustrated in Figure 2a.There are several methods to reduce to a low-order system in the supply chain model.Towill [2]'s works were successfully applied to the reduced-order method proposed by Hsia [36] in the supply chain system.The approximations are made from higher-order to lower-order models based on frequency responses.Other methods were developed by Matsubara [37] and introduced by Jeong et al. [26] to obtain a low-order system from high-order delays.According to the analysis performed by Spiegler et al. [21], the Matsubara approach provides a better result in getting a low-order system of the Forrester pipeline.Based on the Matsubara's method, the low-order pipeline system in Figure 2a is approximated by 3. System modelling and dynamical analysis

Nonlinear mathematical model
The simplified Forrester model shown in Figure 2a includes the discontinuous nonlinearities illustrated by two CLIP functions, which provide merely practical upper bounds for signal rates (blocks A and B).For Assuming that production operates under normal conditions, the manufacturing rate never reaches the capacity limit (AL) and equals the manufacturing rate decision, or MW = MD.On the other hand, if the shipment sent equals the shipping rate tried (SS = ST) and never disrupts actual inventory, the CLIP functions in blocks A and B of Figure 2a can be eliminated.The production model under low-order continuous approximation is shown in Figure 2b, in which  can be utilized to manage the production system realized by active control actions in Section 4 later.

Due to the algebraic operations described by Pi (product)
∏︀ , the production model will be a nonlinear dynamical system, even if each individual is linear.The Forrester's model in Figure 2b can be described by a set of differential equations as follows: In turn, the state-space representation for the production system can be described in the vector-matrix form by ẋ =  +  () +  (11) where  = [,   ,   , , , ]  ∈ ℜ 6 is the measurable state vector, and the system matrix  with the fixed nominal parameter is given by In addition,  () is a vector of nonlinear terms in the production system given by For this system, the orders received (rr ) are considered as an external disturbance ( = ).In fact, the information flow is passed from the customer through an intermediate stage of supply chain networks before placing an order to the factory.In this process, the bullwhip effect rises due to the demand fluctuations at each echelon in the supply chain.The disturbance gain matrix is given by  = [1/ DR , 0, 1/ DP , 0, 1, 0]  .
The Jacobian matrix () of the system evaluated at equilibrium point is given by: Based on eigenvalue analysis, local system behaviors are described as follows.With the equilibrium point  1 , the Jacobian matrix () has six eigenvalues:  1 = 0,  2 = −0.5,  3 = −0.25, 4 = −0.125, 5 = 0, and  6 = 0.002.Similarly, there are six eigenvalues in case of  2 :  1 = 0,  2 = −0.5,  3 = −0.25, 4 = 0.125,  5 = 0, and  6 = −0.002.The equilibrium points ( 1 ,  2 ) are unstable based on the Routh-Hurwitz criterion since at least one eigenvalue (  ) has a positive real part [38].By extending with different values of ia, the equilibrium point will be a stable node if the value of ia is positive; otherwise, it will be unstable with negative values of ia.For the system dynamical equations, two terms in equations ( 9) and ( 10) contain nonlinear variables.Bifurcation analysis is performed to clarify the system behaviors for influence by these terms.Because the two nonlinear terms are almost similar, the analysis is only conducted for the case of equation (9). Figure 3 illustrates the bifurcation diagram of the state variables (uo) according to the time constant ( DU ) for the unfilled orders of the production system.Particularly, the system parameters are selected as given in Table 2.There would always be several business activities going on every time.It could be challenging to handle too many activities at once and not let work become chaotic.For the time-series simulation, the base units of time are selected as weeks, and simulation time is set as a total of 52 weeks.It is noted that many control problems are involved in parametric perturbations in the model parameters.The bifurcation diagram in Figure 3 is considered by changing the system control parameter  DU in the range  DU = [0, 3], while keeping other parameters constants given in Table 2.When the bifurcation parameter is increased from  DU = 0 to 3, the chaotic orbits of () is illustrated in periodic one motion in the range of  DU = [0, 1.5].The system showing instability tends to be chaotic and randomly distributed.After this period, the periodic behavior occurs.The system exhibits the limit cycle for  DU = [1.5, 1.75] before becoming stable for the rest of the simulated period.More precisely, the dynamical behavior can be understood as follows.The system state is unstable during the period of a small unfilled order of time constant.If distributed in wide ranges of values, it tends to be stable by increasing the time constant.In reality, the stability of unfilled orders could be a signal of capability for the production network where shipment sent can satisfy customer demand through the number of goods released downstream timely.
Next, the time evolutions of state variables of the supply chains are illustrated in Figure 4. System behaviors are simulated in two cases: production system without external disturbance (requisition orders received) and with external disturbances.It is observed that without disruptions, after a finite time, all states of the production distribution system converge to zero since no customer orders are placed in the factory.On the other hand, requisition orders or external disturbances are imposed by step-type functions.At the beginning of the simulation, the requisition orders (RR = ) are increased by 100 units per week, giving a 10% upward step in the initial value of demand, at  = 1 (week) from the initial steady-state value given as 1000 units/week.Under external disturbances, the production system becomes unpredictable with nonlinearity.The active controller will be introduced later to the Forrester production system to minimize erratic behaviors related to the nonlinear terms and optimize overall production operations.

Production management using system dynamics
Business management software is constantly improving things for enterprises involved in some sort of business.The manufacturer is considered the heart of a supply chain network, so maintaining an optimized manufacturing rate based on customer order information is essential for supply chain management.This will offer other echelons stability and efficient operations in the supply chain networks.Therefore, management optimization for a manufacturer is one of the main topics discussed in many articles [8,16,24,26].With the increasing complexity of supply chain networks, management software for decision-making assists in eliminating errors, completing business tasks, and reporting activities.It increases overall efficiency and effectiveness for system performance.In this study, the controller is designed to optimize the production system by adding the active action () in block SUM 1 of Figure 2b.After implementing the active controller into the production system, the state variable in equation ( 7) is rewritten as follows: For this supply chain model, the critical outputs that need to be optimally managed include manufacturing rate (md ), shipment sent (ss), and inventory actual (ia).Based on the block diagram in Figure 2b, the output variables are described as follows: In fact, there are two sources of main uncertainty impacts on supply chain systems: variations of system parameters and uncertainties from external disturbances.As mentioned, the requisition order received (rr ) in the Forrester system is considered an external disturbance.On the other hand, the system parametric uncertainty always represents the variations caused by changing parameter values.The robust control synthesis for the optimized production system should fully consider these uncertainty factors in reality.The state-space representation of the perturbed production system can be rewritten below, System output: where the perturbed system matrix is given by Ã =  + ∆  ;  represents the fixed nominal matrix, and parametric uncertainty, ∆  , of the state variables.Specifically, all system matrices and vectors are given by Similarly, the perturbed parameter (τ ) is described by a fixed normal value ( ) plus perturbation (∆ ).For instance, each perturbed constant is denoted by τ =   + ∆  ;   represents the nominal parameter, and ∆  denotes parametric perturbation of  parameter mentioned in Table 2.The gain vector associated with active control input () is given by  = [0, 0, 3/ DP , 0, 0, 0]  .The perturbed nonlinear term is given by f (x) =  ( + ∆  ), which also includes the parameter uncertainty described by In this paper, the specific output vector is selected by  = [, , ]  .The perturbed output matrix is given by C =  + ∆  ; the matrix with fixed nominal parameter, , and the matrix with the parametric uncertainty, ∆  .Specifically, it is given by In addition, f ′ (x) =  ′ ( + ∆  ′ ) is a vector of nonlinear terms of the output system given by Other vectors related to the system outputs are described as follows:  ′ = [1, 0, 0]  is the disturbance gain vector of the output system;  = [1, 0, 0]  is the control gain vector of the system output.

Control synthesis using ASWT-SMC algorithm
The supply chain management software helps businesses support, improve, and systematize processes.It should provide robust performance and stability, ensuring efficiency and profits against unpredictable real markets.Especially, the shortest lead time while ensuring the best possible quality during the production process with optimal costs can realize an efficient production system.Therefore, selecting an active controller for a nonlinear system must be carefully considered to realize a multi-objective management algorithm for critical business decisions.Sliding mode control (SMC) has been recognized as one of the potentially best control schemes for nonlinear systems due to guaranteeing its quick finite-time convergence, tracking accuracy, and robustness against uncertainty [31].However, in practice, the main drawback of SMC is numerical chattering, which could negatively impact business decision-making and management costs.In this work, an adaptive STW algorithm can cope with these problems by generating a continuous control function (or chattering attenuation) to adapt its control gains to the system perturbations with the unknown boundary [33,34].Thereby, this algorithm can improve the controller's efficiency, reduce operating costs, and speed up the decision process to supply the quantity of goods to the real market in the shortest possible lead time.Based on the block diagram in Figure 2b, the decision law () is activated in the manufacturing state, thereby controlling the manufacturing rate as a target and optimizing the inventory and shipment sent levels.The overall block diagram of the ASWT-SMC technique for production management is illustrated in Figure 5.
By letting the error be  (∈ ℜ), the tracking error can be defined as follows: where the desired value is defined by  *  and the actual output is   .The sliding surface is proposed by where  is the positive constant (∈ ℜ + ), determining the convergence rate.The sliding mode dynamics can be derived as where The control law  is designed by a two-stage structure by combining the equivalent control and the adaptive super twisting control algorithm as follows: In this scheme, the equivalent part   = − 1 / 3 is a continuous control for the nominal dynamics of the supply chain system, to drive the system's state on the sliding surface.In addition,   is a discontinuous control part realized by using adaptive super twisting sliding mode.This discontinuous switching control part is implemented to cope with external disturbance and parametric uncertainty, attenuating the chattering, and improving the control performance against uncertainty, essential for managing supply chain networks.Thus, the sliding mode dynamics in equation ( 28) can be rewritten as The ASTW-SMC algorithm is realized as follows: where  = (, σ, ) and  = (, σ, ) are some adaptive gains to be designed.Theorem 1.Consider the dynamical model of the Forrester system by equations ( 20) and ( 21) and the sliding surface variable proposed by equation (27).If the control law (29) based on the adaptive sliding mode scheme is realized in the production system, the output tracking error in equation ( 26) will converge to zero asymptotically with the following adaptive gains: where the control gains ( 1 ,  1 , , and ) are arbitrary positive constants (∈ ℜ + ).The control systems given by equations ( 30) and (31) are presented in the following form: Proof.First, the sliding mode dynamics (34) could be converted into a convenient form for the Lyapunov stability analysis, which will be used to analyze the stability properties of a designed controller.A new state vector is introduced for this purpose, Then, the system (35) can be deduced as derivatives of the state variables, Equation ( 36) can be rewritten in a vector-matrix form: ż = ( 1 ) + ( 1 ) (37) where Next, the following Lyapunov function candidate is introduced to verify the system's closed-loop stability given by equations ( 36) and (37): where It is worth noting that the matrix  is positive definite if  > 0 and (∈ ℜ).Some constants ( * > 0 and  * > 0) satisfy the bounded conditions, || ≤  * and || ≤  * .The time derivative of the Lyapunov function ( 38) is calculated by The first two terms of equation ( 39) are computed by recalling equation ( 38), The matrix  =    +   can be calculated such that where , and  = 2 +  + 4 2 .
Then,  will be a positive definite matrix with a minimal eigenvalue, ensuring  min () ≥ 2 if  satisfies the following: Therefore, it can be derived from inequality (40) Now, equation ( 35) can be rewritten as Finally, it can be obtained that V0 ≤ − max( ) .Given equations ( 45), ( 39) can be rewritten as follows: By recalling a well-known inequality and combining with equations ( 39) and (45), the following inequality can be obtained: where  = min(,  1 ,  2 ).By utilizing equations ( 48), ( 46) can be further rewritten as Now the adaptive gains () and () calculated by the updated law equations ( 32) and ( 33) can be assumed as bounded.Then there exist positive constants ( * and  * ) satisfying () <  * and () <  * , ∀ ≥ 0. By using this assumption, equation ( 49) can be reduced to the following: This will be rewritten by where )︁ .
For finite time convergence, it should be assuring  = 0 via adaptation of the gains (, ) as follows: By selecting  = 2 21 √︁ 2 1 , equations ( 33) and (52) coincide, since It is noted that the finite-time convergence () should satisfy inequality (42).At that time, () is supposed to increase in accordance with equation (53) in the range of equation (42) met, which guarantees the positive definiteness of the matrix .After that, the finite convergence is guaranteed according to equation (51).Also it does not satisfy to increase the gains () and () by making γ() = 0 as soon as the sliding variable  and its derivative converges to zero.Therefore, adaptation law in equations ( 32) and ( 33) can be effectively realized to complete control synthesis for the production system [31,32].Then, the proof is completed.Now, the boundedness of () and () are provided as follows.
Proof.A solution to equation ( 32) can be constructed as where   is denoted as a finite reaching time.The adaptive gain is bounded, since  = 2 +  + 4 2 and () are bounded.The proposition is proven.By the proposition, it suffices to define the following finite reaching time.
Proposition 2. As soon as inequality (42) is reached in infinite time, the adaptation control law in equations (32) and (33) drives the sliding variable , and its trajectory converges to zero in finite time that is bounded and estimated as follows: where  = min(,  1 ,  2 ).
Proof.The inequality (42) is fulfilled in infinite time since its right-hand side is bounded, and the adaptive gain () is linearly increasing with respect to time following equations ( 32) and (33).Inequality (55) is obtained by directly integrating inequality (51), considering that  = 0 due to the adaptation law in equations ( 32) and (33).The proof is completed.

Performance measure for control strategies
When the production system is established, there are standards to measure whether the supply chain management is efficient, delivers value to customers, and meets company goals.Management optimization algorithms of the production system should be assessed through performance criteria, considering the system's accuracy and stability and the system's responses.In the system control theory, several indices are used to evaluate system performance in the time domain.Implementing a specific decision algorithm for a business enterprise isn't always easy, as different types of business management software are used for other purposes.Four kinds of performance criteria for time tracking capabilities are considered for quantitative measure of performance and penalty for executing control actions: the integral of absolute error (IAE), the integral of time absolute error (ITAE), the integral of squared error (ISE) and the integral of time square error (ITSE) [39].The control system parameters, such as SMC and STW-SMC, are manually adjusted, or automatic tuning like ASWT-SMC, so that the performance indices are intended to reach extremum values, typically as small as possible for tracking errors.Performance index is calculated over a time interval.In detail, the control problem works by minimizing the cost functions, which are integrals of error signals and time.A general form of integral performance indices can be defined as where  and  are the fixed real numbers.In the performance criteria, () is the tracking error that can be found by comparing the target production output and the actual response.It is noted that the indices ( 0,1 ,  0,2 ,  1,1 , and  1,2 ), respectively, represent IAE, ISE, ITAE, and ITSE criteria.Each index has specific features for error performance in the time domain.The IAE can put equal weight on small and large errors even if they happen sooner or later under the control actions, in which the system's response speed will not be fully assessed through this indicator.By using ITAE, more weight is put on time than on variation, meaning the error that occurs after a long time is given much more heavily than this at the beginning of the response.On the contrary to the above two indices, ISE and ISTE penalize the square of the weights of the system error.Control systems specified to reduce ISE will tend to eliminate significant errors quickly but will tolerate minor errors persisting for an extended period of time.Often, this leads to a fast initial response and low amplitudes but a slow recovery and sustained oscillation since the system is much underdamped.In addition, the ITSE index continues to put the penalty for significant errors more than small ones.Hence, the system recovery will be quick for high amplitude values but may maintain low amplitude errors for longer.These indicators are key quantitative measures while maintaining the stability and efficiency of the manufacturing system [40].

Numerical simulation
Numerical simulation tests are conducted to evaluate the effectiveness of the ASTW-SMC controller through comparisons.For extensive performance tests, parametric uncertainties with perturbation values are proposed in Table 3.To evaluate more detailed system performance under active controllers, other variables (system error, control actions, and performance index) of the production system are shown in Figures 7-9, respectively.As shown in Figure 7, system tracking errors are asymptotically converged to zero with almost no overshoot when ASWT-SMC is activated, while other controllers are not performing well.Hence, despite some uncertainties, such as parameter perturbations and external disturbances, the proposed control algorithm can eliminate oscillatory trajectories of the manufacturing rate of the Forrester system in much less time, ensuring robust stability and performance.From graphical analysis, it is observed that with SMC and STW-SMC, the actual responses can track the desired target values.However, the overshoot and undershoot of the response may cause dynamic oscillations in the first eight weeks, leading to a relatively slow response and long settling time.These phenomena will cause severe problems in efficient production management when the production system cannot meet customer lead time and will cause excess inventory.While the system response under ASWT-SMC can converge to desired values in much less time, overshoot is comparatively more minor due to an adaptive mechanism.The transient responses are reasonable compared with those of the other control algorithms.In summary, the timedomain responses demonstrate that the system responses under the ASTW-SMC controller have the fastest rise time, quickest setting time, and the smallest maximum percentage of overshoot.Figure 8 shows the control activities under different control algorithms.Besides the ability to obtain a faster reduction rate in tracking errors, the controller relies on the automatic parametric adjustment, which can make the control action smaller without chattering than other methods.
To have a clearer view of control optimization, some performance evaluations based on cost functions mentioned before are shown in Figure 9. Various objective functions based on error performance indices are used to evaluate the control algorithms.According to the test results, all control performances with similar settings have achieved different values.In detail, ASTW-SMC provides the lowest cost for both cases, giving more weight to response time (IAE and ITAE) and variations (ISE and ISTE) compared to other controllers.This is due to the advantage of applying an adaptive algorithm to automatically adjust control gains based on the actual tracking errors and the sliding surface variables, while the fixed values of the control gains are set for STW-SMC and SMC.Those will cause higher index values, which make the manufacturing operations more expensive and riskier, thus indirectly destabilizing and increasing operation costs to manage the entire supply chain networks.Although STW-SMC is not as effective as ASTW-SMC, it still performs better than conventional SMCs due to its ability to minimize chattering effectively, which is critical for the SMC scheme.In supply chain management, the performance indicators maintained at lower and fewer mutations contribute to reducing operating expenditures and keeping product flow stable.Besides, the numerical results have confirmed the validity and reliability of the ASTW-SMC algorithm for suppressing undesirable behaviors of supply chains.The proposed method is superior to current control algorithms in all aspects of business decision-making.
Next, ASTW-SMC has been implemented to control the Forrester system to optimize production operations through manufacturing rate (md ), shipment sent (ss), and inventory actual (ia), by establishing an objective target function of manufacturing rate.The active controller plays a crucial role in helping the company management system achieve the target goals in the least amount of time.Here, the target function is given by the step function.After the simulation starts, the target value is increased by 140 units per week, offering a 10% upward step in the initial demand value, at  = 1 (week) from the initial, in which the sustained value is given as 1000 units/week.As shown in Figure 10, it can be observed that the proposed controller can guarantee more accurate and effective tracking.It can quickly accomplish the manufacturing rate target and keep the production  system stable until the end of the simulation period.The area marked in Figure 10a represents the optimal level between the original and optimized systems under active controllers.The transient responses of shipment sent (ss) and inventory actual (ia) are illustrated in Figures 10b and 10c, respectively.It is obvious that the controller also helps to restrain parametric uncertainty's effect on these variables effectively.This is particularly important in the supply chain where sustained inventory stability and smooth shipment sent can help make commitments to customers, reduce inventory costs, and increase the efficiency of connectivity between factories and other echelons of the supply chain networks.
In order to build a more efficient production system, many inside factors need to be considered, especially the effective connections between components in the system, thereby changing time constants, which can make the system stagnant and have high system maneuverability.Figure 10 shows that although the manufacturing rate can be optimized by quickly reaching the desired reference trajectory under the optimal controller, the shipment sent can only achieve much lower values than the orders from customers.This abnormality is caused by time delay in processing dynamics or connections among system constituents.In order to see how the time constants affect the system responses, Figure 11 illustrates the dependency of the output variables (ss and ia) to inventory time constant  AI and delay in the unfilled order,  DU , accordingly.Some time constants are not of constant values, in which the base unit for time is the week.Rather, they are functions of the system's operation environments.According to the results from Figure 11a, the longer the inventory time constant value, the more products will be stuck in the inventory.On the contrary, the goods are quickly released from inventory if the time constant is short.It means that the processing speed of activities in this area is high, so the stocks are always kept low.Similarly, Figure 11b shows that the volume of shipment sent is small if the delay time in unfilled orders is large, whereas the goods are quickly shipped downstream in large numbers when the time for the unfilled order is improved.

Conclusions
Modern supply chain management affects productivity, service quality, delivery, customer experience, and cost-effectiveness.Empowering decision algorithms is now as organizationally crucial as empowering people.Decision-makers in business enterprises accept the reality that intelligent management algorithms need greater autonomy to succeed.Production management is one of the vital issues for the optimized supply chain networks.Most current approaches by system dynamics focus on approximation schemes by linearization, which only provides local behaviors of the production system.Instead, the paper deals with the comprehensive dynamical analysis and control synthesis for the production-distribution model by combining simplification and nonlinear system theory.The proposed methods can provide efficient nonlinear analysis tools to gain more significant insights into the underlying operation mechanisms of complex production systems.Explicitly, the ASTW-SMC algorithm is proposed to control the production system against model uncertainties as well as external disturbances in unpredictable real markets.Compared with the conventional control strategies, the top features include the superiority and effectiveness of the proposed control algorithm verified by analytical and numerical tools.The extensive simulation results demonstrate that the proposed method can not only make the system output track the reference trajectory quickly and precisely, but also can effectively restrain the control action chattering, which makes decision policy in the supply chain management smoother.For supply chain management systems, these features can help decision-makers speed up production plans and supply goods faster to other constituents at the factory while saving production costs and ensuring network efficiency through optimal controllers.The future work will expand the dynamical analysis and control synthesis of non-linear Forrester systems across complete supply chain networks.The better and more effective a company's production management is, the better it protects its business reputation and long-term sustainability.Without question, top management would have to trust its computationally excellent management software.

Figure 2 .
Figure 2. Block diagrams for the simplified lower-order Forrester's model: (a) simplified model, and (b) continuous nonlinearity model.

Figure 3 .
Figure 3. Bifurcation diagram of chaotic supply chain system.

Figure 5 .
Figure 5. Block diagram of production system management using active controller.

Figure 6 .
Figure 6.Time histories of state variables for nonlinear suppression where controllers are activated at  = 25 (weeks).

Figure 7 .
Figure 7. Time histories of error signals under active controllers.

Figure 10 .
Figure 10.Time histories for synchronization with active controllers: (a) manufacturing rate, (b) shipment sent, and (c) inventory actual.

Figure 11 .
Figure 11.System responses for difference time constants: (a) inventory actual, and (b) shipment sent.

Table 1 .
System variables of Forrester's model.

Table 2 .
Time constants of Forrester's model.