This paper considers a multiitem inventory model with unknown demand rate coefficients. An adaptive control approach with a nonlinear feedback is applied to track the output of the system toward the inventory goal level. The Lyapunov technique is used to prove the asymptotic stability of the adaptive controlled system. Also, the updating rules of the unknown demand rate coefficients are derived from the conditions of the asymptotic stability of the perturbed system. The linear stability analysis of the model is discussed. The adaptive controlled system is modeled by a system of nonlinear differential equations, and its solution is discussed numerically.
The area of adaptive control has grown rapidly to be one of the richest fields in the control theory. Many books and research monographs already exist on the topics of parameter estimation and adaptive control. Adaptive control theory is found to be very useful in solving many problems in different fields, such as management science, dynamic systems, and inventory systems [
ElGohary and Yassen [
ElGohary and AlRuzaiza [
Tadj et al. [
Foul et al. [
Alshamrani and ElGohary [
Many other studies which are concerned with the production and inventory systems, multiitem inventory control, and inventory analysis can be found in references [
This paper is concerned with a twoitem inventory system with different types of deteriorating items subjected to unknown demand rate coefficients. We derive the controlled inventory levels and continuous rates of supply. Further, the updating rules of the unknown demand rate coefficients are derived from the conditions of asymptotic stability of the reference model. The resulting controlled system is modeled by a system of nonlinear differential equations, and its solution is discussed numerically for different sets of parameters and initial states.
The motivation of this study is to extend and generalize the twoitem inventory system with different types of deterioration and applying an adaptive control approach to this system in order to get an asymptotic controlled system. This paper generalizes some of the models available in the literature, see for example, [
The rest of this paper is organized as follows. In Section
This section uses the mathematical methods to formulate the twoitem inventory system with two different type of deteriorations. In this model, we consider a factory producing two items and having a finished goods warehouse.
This subsection is devoted to introduce the model assumptions and its formulation. It is assumed that the inventory supply rates are equal to the production rates, while the demand rates may adopt two different types. Throughout this paper we use
The main problem of this paper is to present the adaptive control problem for the twoitem inventory system as a control problem with two state variables and two control variables which are the inventory levels
Also, since an analytical solution of the resulting control system is nonlinear and its analytical solution is not available, we solve it numerically and display the solution graphically. We show that the solution of the adaptive controlled system covers different modes of demand rates.
In this subsection, we present a suitable mathematical form for a twoitem inventory system with two types of deteriorations. This mathematical form must be simple to deal with any response of the twoitem inventory model with deterioration to any given input. The differential equations system that governs the time evolution of the twoitem inventory system is found to be as follows [
In this paper, we consider the inventory goal levels
Next, we will derive the steady state solution of (
In what follows, we discuss the numerical solution for the (
in this example, we discuss the numerical solution of (
in this example, we discuss the numerical solution of (
in this example, we discuss the numerical solution of (
Model parameters  Steady states 






Model parameters 















Figure
Parameter 












 
Value 












(a) and (b) are the first and the second inventory levels, respectively, of the uncontrolled system, with quadratic rates of supply. (c) is the trajectory of the inventory system in
First inventory level
Second inventory level
The two items inventory limit cycle
Figure
Parameter 










 
Value 










(a) and (b) are the first and the second inventory levels of the uncontrolled system, with constant rates of supply. (c) is the trajectory of the inventory system in
First inventory level
Second inventory level
The two items inventory limit cycle
The concept of stability is concerned with the investigation and characterization of the behavior of dynamic systems. Stability analysis plays a crucial role in system theory and control engineering and has been investigated extensively in the past century. Some of the most fundamental concepts of stability were introduced by the Russian mathematician and engineer Alexandr Lyapunov in [
In this section, we discuss the linear stability analysis of the system (
The characteristic equation is given by:
The roots of the characteristic equation will be complex numbers with negative real parts if the following conditions can be satisfied:
Therefore the system (
The roots of the characteristic equation will be negative real numbers if the following conditions can be satisfied:
If the conditions (
Next, we discuss some special cases in which the rates of supply take different functions of the inventory levels:
when the supply rates do not depend on the inventory levels, the linear stability conditions are reduced to
when the supply rates are linear function of the inventory levels,
when the supply rates are quadratic functions of the inventory levels,
In what follows, we study the problem of adaptive control. In order to study this problem, we start by obtaining the perturbed system of the twoitem inventory model about its steady states
The system (
In adaptive control systems, we are concerned with changing the properties of dynamic systems so that they can exhibit acceptable behavior when perturbed from their operating point using a feedback approach.
The problem that we address in this section is the adaptive control of the twoitem inventory system with different types of deterioration which are subjected to unknown demand rate coefficients. In such study, we assume that the demand coefficients
In what follows, we discuss the asymptotic stability of the special solution of the system (
This solution corresponds to the steady states solution of the system (
The following theorem determines both of the perturbations of the continuous rates of supply
If the perturbations of the continuous supply rates and the updating rules of the unknown parameters
The proof of this theorem can be reached by using the Liapunov technique. Assume that the Liapunov function of the system of equations (
Substituting from (
In Section
The objective of this section is to study the numerical solution of the problem of determining an adaptive control strategy for the twoitem inventory system subjected to different types of deterioration and unknown demand rate coefficients. To illustrate the solution procedure, let us consider simple examples in which the system parameters and initial states take different values. In these examples, the numerical solutions of the controlled twoitem inventory system with unknown demand rate coefficients are presented. The numerical solution algorithm is based on the numerical integration of the system using the RungeKutta method.
Substituting from (
In this example, a numerical solution of the adaptive controlled system (
Parameter 














 
Value 














The numerical results are illustrated in Figure
(a) and (b) are the perturbation of the first and second inventory levels about their inventory goal levels as the demand rate is a constant. (c) and (d) are the difference between dynamic estimators of the first and second demand rates and their real values.
Perturbation of first inventory level
Perturbation of second inventory level
Estimator of the first demand coefficient
Estimator of the second demand coefficient
In this example, a numerical solution of the adaptive controlled system (
Parameter 














 
Value 














The numerical results are illustrated in Figure
(a) and (b) are the perturbation of the first and second inventory levels about their inventory goal levels as the demand rate is a linear function of the inventory level. (c) and (d) are the difference between dynamic estimators of the first and second demand rates and their real values.
Perturbation of first inventory level
Perturbation of second inventory level
Estimator of the first demand coefficient
Estimator of the second demand coefficient
In this example, a numerical solution of the adaptive controlled system (
Parameter 














 
Value 














The numerical results are illustrated in Figure
(a) and (b) are the perturbation of the first and second inventory levels about their inventory goal levels as the demand rate is an exponential function of the time. (c) and (d) are the difference between dynamic estimators of the first and second demand rates and their real values.
Perturbation of first inventory level
Perturbation of second inventory level
Estimator of the first demand coefficient
Estimator of the second demand coefficient
In this example, a numerical solution of the adaptive controlled system (
Parameter 














 
Value 














The numerical results are illustrated in Figure
(a) and (b) are the perturbation of the first and second inventory levels about their inventory goal levels as the demand rate is an exponential function of the time. (c) and (d) are the difference between dynamic estimators of the first and second demand rates and their real values.
Perturbation of first inventory level
Perturbation of second inventory level
Estimator of the first demand coefficient
Estimator of the second demand coefficient
We have shown in this paper how to use an adaptive control approach to study the asymptotic stabilization of a twoitem inventory model with unknown demand rate coefficients. A nonlinear feedback approach is used to derive the continuous rate of supply. The Liapunov technique is used to prove the asymptotic stability of the adaptive controlled system. Also, the updating rules of the unknown demand rate coefficients have been derived by using the conditions of the asymptotic stability of the perturbed system. Some numerical examples are presented to:
investigate the asymptotic behavior of both inventory levels and demand rate coefficient at the steady state;
estimate the unknown demand rate coefficients.