Impact of Control System Structure and Performance of Inventory Goods Flow System with Long-Variable Delay

1Abstract—In this paper, we use a mathematical model of a inventory system with time-varying delivery delay and three control systems in order to compare their properties for the goods flow optimization problem in the inventory systems. Structures of the chosen control systems are based on mathematical discrete equations: a periodic inventory system with adaptive maximal inventory level and perpetual inventory system with adaptive order quantity level as well as the methods proposed by the authors in previous works based on the Smith predictor. The selection of the control systems parameters is done by solving optimization tasks for a specific scenario of time-varying market demand using a genetic algorithm in Matlab/Simulink. In this article, we mainly want to compare which of the control structures is able to achieve a high service level with maintaining of a given inventory maximal level in response to assumed consumer demand scenario.


I. INTRODUCTION
Supply-chain risks can become full-fledged supplychain problemsproblem, causing unanticipated changes in flow due to disruptions or delays.The proper control system is essential.InventoryDecisions concerning inventory control decisions are associated with order batching and when to order.Acting as a buffer to smooth production in response to demand fluctuations is determined as the main role of an inventory.In fact, there is aare plenty of reasons for keeping inventories.The most significant is to satisfy the demand during the replenishment period in order to prevent loss of orders.More important reasons for obtaining and holding inventory can be find in [1]: predictability, fluctuations in demand, unreliability of supply, price protection, quantity discounts, lower ordering costs.It is utmost important to maintain the right balance between demand and orders with a view to minimizing costs.What is more, inventory is a protection against fluctuations in demandfacilitateit facilitates satisfying customer demands.Inventory costs generally fall into ordering costs and holding costs [1].In the worst-case scenario, customer service goes down, sales are lost, lead times lengthen, costs go up, maximal level of inventory is increasing.
Over the years models of inventory systems hashave been being created in a variety of areas, in.In this paper it is considered, the model of inventory proposed by the authors in [2].is considered.We want to extend our research [3], [4][1], [3], [4] over the control systems taken from literature and compare it with originally created system which havehas a better or similar performance.
There are more and more methods of improving the flow of goods in the inventory systems, which in turn use more and more advanced control techniques [3], [4], [5], [6], [7], [8].In [5] considered is [3][4][5][6][7][8][9][10].In [7] linear stationary discrete system with a fixed delay to the effective control of storage systems with perishable goods, using methods based on sliding-mode control. is considered.In [8] [6] it is used a linear-quadratic control in order to reduce the risk of bullwhip effect.
The paper presents aan comparative analysis of the impact of the control system structure and the performance of the inventory goods flow system.Control systems structures based on mathematical discrete equations are given for optimization: a periodic inventory system with adaptive maximal inventory level and perpetual inventory system with adaptive order quantity level as well as the methods proposed by the authors in previous works based on the Smith predictor.Parameters were selected for each of the control system structures through solving optimization tasks for a specific scenario of variable market demand using a genetic algorithm.

II. THE MATHEMATICAL MODEL
Number of products that could potentially be sold from the store is modelled as a certain, unknown in advance limited function of time: If the quantity of products in stock at moment k is sufficiently large, it means that:     From the standpoint of controlling the flow of goods, it is important to maintain certain stocks in the inventory, regardless of transient changes in customer demand, so as.It is necessary to avoid a situation in which the magazine is empty or the quantity of the stored products will beis excessive, or even exceeds the storage capacity ymax.In order to take into account variable time delays in the model associated with The product quantity awaiting shipment at time k is defined by the following relationship: The product quantity stored in the inventory at moment k, called the stock, is therefore given as follows: where:      -production delay -related to the time required to pro- duce or complete the orders, s  -forwarding delay -the time interval indispensable to transport the ordered products to the inventory without waiting time for transport.
For the needs of this work, it is assumed that the inventory model has one supplier.The results can be easily extended to the case with more suppliers with the similar delivery time, i.e. total production, shipping and forwarding delays from each supplier.Moreover, on the basis of the balance of products in the inventory, it is clear that products only have to cumulate or be sold to the customers.

III. THE CONTROL SYSTEMS DEFINITIONS
The two basic questions any inventory control system must answer are when and how much to order.There are several replenishment techniques attempt to answer this fundamental questions.
Control system determines a certain amount of products to be ordered on the basis of market demand.As a result, a class of inventory models has been designed to cope with situations where demand level fluctuates.
The two classic systems for managing independent demand inventory are periodic review and perpetual review systems [14] [12].
It is essential to create a mathematical description of investigated control systems:

A) Periodic Inventory System with adaptive maximal inventory level
It is a modified version of the classical periodic inventory system which givegives an opportunity to the adaptation of the maximum level of inventory in stock.The block diagram is depicted in fig. 2. It has been put additional factor k1 in order to enable adaptation depending on the market demand.Factor k1 together with a factor k2 make an affine function of maximal inventory capacity depending on market demand, given in the following form: 2 A block diagram of the control system for the periodic inventory system with adaptive maximal inventory level.B) Perpetual Inventory Systeminventory system with adaptive order quantity level.

Sformatowano
A perpetual system keeps records of the amount in storage, and it replenishes when the stock drops to certain level k3.
The reorder pointthreshold, inventory content critical value, is fixed, but review period, order quantity and maximal inventory level are variable (max inventory level depends on demand).The structure of the system is shown in fig. 3. Factor k1 together with a factor k2 make an affine function of maximal inventory capacity depending on market demand, given in the following form: where: k3reorder point.

C) Proportional-derivative Inventory Control Systeminventory control system with Smith predictor and adaptive reference queue length stock level
The structure shown in Figure 4 -theThe control system is based on a classical structure with Smith predictor.It is a kind of a predictive controller, which was developed for control systems, which are characterized by long and inevitable delays.Its structure is based on implementations of the model without delay and with delay.
Based on the control concepts for systems with delays using a Smith predictor it is assumed that an estimated model of the system without delay is given in the form :.: Model of PD controller for error of approximate model and error of approximate model with delay is given in the following form: and a model of delay time-varying form: Model of PD controller for error of approximate model with- and error of approximate model with delay is given in the following form: where: k k  are parameters of the control system.Due to the similarity between the considered class of systems and engineering processes, it is a natural choice to apply control-theoretic methods in the design and analysis of strategies governing the flow of goods.The issue of optimal control requires a mathematical formulation of the process performance index to be optimal.Consider the problem of finding the optimal values of the parameters of a dynamical system with fixed its structure from Figsfig.2-4.In the case of the inventory system indicators can be described by the following relations: where: N is the length of the time horizon.The equation ( 1413) represents a lost opportunity to make sales.In turn, the expression (1514) concerns use of space in the inventory.These indicators are associated with financial costs.Its form is based on the physical interpretation of the problem.
In the present case, it is used scalarization of the objective function is used to form of the weighted sum: w are weighting factors and j can be evalu- ated in arbitrary currency.
For the model described by relationships ( 1) -( 4) and the control systems described by equations ( 5) -( 1312) and a quality indicator in the form of ( 14) -( 1713) -( 15) the optimization problem can be defined in the following form: Where optimization variables and constraints are dependent based on the controller structure: for eq.( 6)(5)   , , , 0, 0, 0 for eq.(8)(7)-(10)(13) The parameters k1 -k3 of controller are non-negative because it is assumed that returns are not taken into consideration.According to this and to the requirements of negative feedback loop, incl.stability issues, the output signal of the controller cannot be negative.The proposed system is aimed at determining the optimal size of deliveries, which.It minimizes the cost ratio consisting of the average weighted total inventory costs including the cost of stocking and maintenance costs and lost benefits, reducing the risk of stoppages.

V. SIMULATION RESEARCH AND ANALYSIS
In this section we conduct computer simulations.The structures of control systems in Figuresfig.2, 3, 4 are applied.The main purpose of this section is to compare properties of three different control structures: Periodic Inventory Systemperiodic inventory system with adaptive maximal inventory level and Perpetual Inventory Systemperpetual inventory system with adaptive order quantity level, present in economic environment and Proportionalproportional-derivative Inventory Control Systeminventory control system with Smith predictor and adaptive reference queue lengthstock level proposed by authors.With a view to simulation research of the control systems for a discrete, non-stationary linearhybrid model with signal bound described by equations ( 1) -( 4), the control systems described by equations ( 5) -( 1312) and the quality indicator in the form of ( 14) -( 1713) -( 15), the following values of the system parameters are held: 14 . For the simulations purposes, it is assumed that 1 w is either 60 or 530.The sampling period is 1 day.Specific time-varying market demands functions shown in The sampling period is 1 day.Specific time-varying market demand functions shown in fig.5, 6 are taken into consideration.Demand signal is shown in fig. 5 is the assumed scenario proposed by the authors.The presumed function of market demand consists of step change at k=0.Next, it is constant until k=300.After that, it linearly increases with slope 6 products per day for 100 days, then it is constant for 200 days.From k=600 the presumed demand decrease with slope -1.5 products per day until k=700 and then it is constant until the end of the scenario (k=1000).This demand signal has these different values in order to show the control system performance at different conditions.The signal in fig. 5 is used for verifying optimal controller parameters.As a function that controls the periodic time-variability of the model q(k) it is assumed following periodic function: 0 for rem ,14 0 1 for rem ,14 0 where: the function rem is the remainder of the division.Tuning of the control system are based on a the criterion (1) and trapezoidal demand signal plotted in fig. 5. On the basis of the results we try to evaluate: how does the controller structure impact on the properties of the inventory control system.In order to verify the solution for different scenario rectangular demand test signal plotted in fig.6 are employed.Before developing the theoretical argumentation, it is important to first establish definitions and foundational concepts underlying this research.
To solve the optimization problem (17) in order to determine parameters of controllers, a genetic algorithm with parameters: population size 50, elite count: 2, crossover fraction: 0.8 is used.The genetic algorithm has been receiving great attention and it has successfully been applied to other    In order to compare the different control systems, consequently marked by square, 'x' and triangle some additional evaluations are made.One of them are plot in the objective space for rectangular demand shown in fig.8 with controllers optimized for trapezoidal demand depicted in fig.6.The results do not differ significantly from each other.In order to finalize the comparison we analyse the stock level response for different control systems, weights 1 w and consumerscustomer demands.Let's now move on to the system response which represents inventory levels.On the basis of the inventory stock, we can decide how control systems react for the demand and have aan opportunity to differ them.One the other hand, the same formulation in case of triangular signal shows that the non-dominated set contains subsets of solutions for PDIS-SP-ARQLARSL and PIS-AMIL.So itIt can therefore be concluded that the PDIS-SP-ARQLARSL and PIS-AMIL control system have better robustness for the changes in the assumed scenario of consumercustomer demands.Time responses for systems PIS-AMIL and PDIS-SP-ARQLARSL remain adequate level of orders and stocks, also in a wide range of changes of w1.
Inventory Control Systemcontrol system has to provide weighted balance between this two indicators: 1 j and 2 j .The analysed results for the structures PIS-AMIL and PDIS-SP-ARQLARSL are similar, however.However, by comparing the obtained value of the cost indicator j, presented in figure 7 and 8 it can be noted that for small values of w1 is the little advantage of the system PDIS-SP-ARQLARSL for both trapezoidal and rectangular signal.
Our analysis shows that adaptive reference stock level and adaptive reference queue length has a smoothing effect on the order variability.

VI. CONCLUSIONS
The overall objective of inventory management is to achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds.The balancing act between liquidity and profitability is key to good inventory management.
Regarding the objective, a decision maker musthas to make two fundamental decisions: the timing and size of orders (i.e., when to order and how much to order).
The risk of shortage can be reduced by holding safety stock, which are stocksis a stock in excess of average demand to compensate for variability in demand and lead time, but.It is important, that safety stock cannot be high.
The main factorsfactor of the control selection is to minimize costs and losses relating to holding costs.Shortage is extremely disadvantageous situationthe demand is greater than the amount of products, because orders have not keep pace with the time-variable changing needs of consumers.customers.
On the basis of empirical analysis, it was found that the structure of the control system has a great impact on the performance of the inventory goods flow system with long-variable delay.The goal was to minimize the total cost of the system and maximize service level simultaneously.The structure of the PIS-AOQL provides the worst reaction for high varying demands, the reference inventory level is higher almost 3-4 times than in other systems.The system with adaptive reference stock level PIS-AMIL has much better performance.Pareto front for inventory control system for trapezoidal signal in figure 6 shows that PDIS-SP-ARQLARSL has the best values of indicators which represents lost opportunities to make sales and use of space in the inventory.
For future study, we will focus on more complex systems e.g.hybrid systems associated with supply chain and taking into account the specific effects occurring in such objects.

Fig. 1 .
Fig. 1.Block diagram of inventory system with control.Assuming that known are the delays are known , p s  

Fig. 2 A
Fig. 2 A block diagram of the control system for Classical Periodic Inventory System.

Fig. 3 A
Fig.3A block diagram of the control system for Perpetual Inventory Systemthe perpetual inventory system with adaptive order quantity level.

Figure
Figure of the control system is shown in Figurefig.4. The variables 1 3

Fig. 4 A
Fig. 4 A block diagram of the control system for Proportionalthe proportional-derivative Inventory Control Systeminventory control system with Smith predictor and adaptive reference queue lengthstock level. .IV. OPTIMIZATION CRITERION It is extremely important to use the most simplest model of optimization model for a certain problem in numerical optimization issuesas simple as possible.Simplification of optimization model not only contribute to reduce the calculation time but also enables you to find solutions closer to the global minimum repeatedly in the case of multimodal issues.The issue of optimal control requires a mathematical formulation of the process performance index to be optimal.Consider the problem of finding the optimal values of the parameters

Fig. 5
and 6 are taken into consideration.

Fig. 7 .
Fig. 7. Pareto front for inventory control system for trapezoidal signal Pareto front with shortages cost 1 j and holding cost 2 j calculated for 11 logarithmic spaced weights in the range 40,1000   and trapezoidal signal is depicted in fig. 7. Results for PIS-AOQL control system are marked by blue triangles and blue smoothing line.Points of Pareto front for PIS-AMIL are marked by green 'x' and green smoothing line whereas for PDIS-SP-ARQLARSL control system are marked by yellow squares and yellow smoothing line.

Fig. 8 .
Fig. 8. Pareto front for inventory control system for rectangular signal It can be noticed from fig. 7 that solutions for PDIS-SP-ARQLARSL control system are non-dominated although results for 'x' and PDIS-SP-ARQLARSL are close to each other.The solutions for PIS-AOQL are dominated by PDIS-SP-ARQLARSL and PIS-AMIL however they are also relatively near the non-dominated solution of PDIS-SP-ARQL.ARSL.In order to compare the different control systems, consequently marked by square, 'x' and triangle some additional evaluations are made.One of them are plot in the objective space for rectangular demand shown in fig.8with controllers optimized for trapezoidal demand depicted in fig.6.The results do not differ significantly from each other.

Fig. 7 .
Fig. 7. Pareto front for inventory control system for trapezoidal signal

Fig. 9 .
Fig. 9.The level of stocks of three control systems vs. the discrete time for the presumed function of market demand for trapezoidal signal, where w1=60

Fig. 11
Fig.11The level of stocks of three control systems invs.the discrete time function for the presumed function of market demand for trapezoidal signal, where w1=530

Fig. 12 .
Fig.12.The level of stocks of three control systems in the discrete time function for the presumed function of market demand for rectangular signal, where w1=530

Fig. 12 .
Fig.12.The level of stocks of three control systems vs, the discrete time for the presumed function of market demand for rectangular signal, where w1=530

TABLE I .
VARIABLES USED FOR SIMULATIONVariable Value p  production delay 14 days s 