Adaptive Control of a Reverse Logistic Inventory Model with Uncertain Deteriorations and Disposal Rates

Abstract

An adaptive control of a reverse logistic inventory system with unknown deterioration and disposal rates is considered. An adaptive control approach with a feedback is applied to track the inventory levels toward their goal levels. Also, the updating rules of both deterioration and disposal rates are derived from the conditions of asymptotic stability of the reference model. Important characteristics of the adaptive inventory system are discussed. The adaptive controlled system is modeled by a nonlinear system of differential equations. Finally, the numerical solution of the controlled system is discussed and displayed graphically.

1. Introduction

Applications of optimal control theory to management science, and especially to production planning, are proving to be quite fruitful; see Sethi and Thompson [1] and El-Gohary et al. [2]. The optimal control problem characterizes the system optimal trajectory in time. The application of optimal control theory supplies nice information about the system optimal path and decision rules in time, as for the following examples.(i)Tadj et al. [3] applied the optimal control theory to an inventory system with ameliorating and deteriorating items. They derived the optimal inventory level and optimal production rate for different cases of both amelioration and deterioration rates.(ii)El-Gohary and Elsyed [4] discussed the problem of optimal control of multi-item inventory models with different types of deterioration. They derived inventory levels which minimize the total holding cost of the system.(iii)Alshamrani and El-Gohary [5] applied Pontryagin principle to a two-item inventory model with different types of deteriorating items. They minimized the total cost which includes the sum of the holding costs of inventory items, the holding costs of one item due to the presence of the other, and the production cost.(iv)Foul et al. [6] studied the adaptive control of a continuous-time model of a production inventory system in which a manufacturing firm produces a single product, selling some, and stocking the remaining. The model reference adaptive control with feedback is applied to track the output of the system toward the inventory goal level.(v)El-Gohary and Yassen [7] used the adaptive control and synchronization procedures to the coupled dynamo system with unknown parameters. Based on the Liapunov stability technique, an adaptive control laws are derived so that the coupled dynamo system is asymptotically stable and the two identical dynamo systems are asymptotically synchronized. Also the updating rules of the unknown parameters are derived.

The reverse logistic model is a method for manufacturing of materials and remanufacturing of market returned reusable materials. In this model, the demand is satisfied with the new produced products and with the remanufactured used products. Therefore, there is no difference between manufactured and remanufactured items for satisfying the demand [8, 9].

The aim of this paper is to introduce the mathematical model of a reverse logistic system with two different types of natural deteriorations. We apply an adaptive control technique to derive the updating rules of both of deteriorations and disposal rates from the conditions of asymptotic stability of the reference model. Also, the manufacturing and remanufacturing rates are derived using a suitable feedback technique. Numerical examples for different sets of the system parameters and initial inventory levels are introduced.

The motivation of this paper is to extend and generalize the reverse logistic system and applying an adaptive control approach to stabilize the system about the steady states of this system.

2. The Reverse Logistic Model

In this section, we derive the mathematical model of a reverse logistic model with two different types of deterioration. Reverse logistic is a term for manufacturing new products and remanufacturing used products that is returned from market for future reuse. The demand is to be satisfied with new manufactured products and the remanufactured used products.

2.1. The Mathematical Model

In this subsection, we derive the mathematical model and the perturbed system of a reverse logistic inventory model with two different types of deteriorating items about the inventory goal levels. In what follows, we formulate a two inventory stores reverse logistic model with a continuous disposal rate. The manufactured items go to the first inventory store and then are used to meet the market demand. Returned used items from market go to the second inventory store. Some of the returned items will be disposed while the others will be remanufactured and then go to the first inventory store where they will be used to meet the market demand. It is assumed that items might deteriorate in both inventory stores. The problem can be considered as an adaptive control problem with two state variables 𝐼_𝑖(𝑡), (𝑖=1,2) and two control variables 𝑃_𝑖(𝑡), (𝑖=1,2), and three unknown functions which are the two deterioration rates and the rate of disposal. The aim is to use the adaptive control technique to derive both of manufacturing and remanufacturing rates and the updating rules for the disposal and deterioration rates.

To derived the mathematical model we use the following notations:𝐼_𝑖(𝑡): the inventory level in the 𝑖th store at time 𝑡, (𝑖=1,2), 𝑃₁(𝑡): the manufacturing rate at time 𝑡, 𝑃₂(𝑡): the remanufacturing rate at time 𝑡, 𝑃₃(𝑡): the disposal rate at time 𝑡, 𝑆(𝑡): the continuous differentiable demand rate, 𝑅(𝑡): the continuous differentiable return rate, 𝐼_𝑖: the required inventory goal level in the 𝑖th store, 𝑃₁: the required manufacturing goal rate, 𝑃₂: the required remanufacturing goal rate, 𝑃₃: the required disposal goal rate, 𝑅: the return value at the inventory goal level, 𝑆: the demand value at the inventory goal level, 𝜃_𝑖: the deterioration rate in the 𝑖th store.

The time evolution of the inventory levels at time 𝑡 in both the first and second stores can be described by the following set of differential equations:𝐼₁̇(𝑡)=𝑃₁(𝑡)+𝑃₂(𝑡)−𝑆(𝑡)−𝜃₁𝐼₁(𝑡),𝐼₂̇(𝑡)=𝑅(𝑡)−𝑃₂(𝑡)−𝑃₃(𝑡)−𝜃₂𝐼₂(𝑡), (2.1) with initial values 𝐼₁(0)=𝐼₁₀, 𝐼₂(0)=𝐼₂₀, where,𝐼_𝑖(𝑡)≥0,(𝑖=1,2),𝑃_𝑖(𝑡)≥0,(𝑖=1,2,3). (2.2)

The above inequalities represent the conditions of the nonnegativity of the inventory levels, the rates of manufacturing, remanufacturing, and disposal.

Next, we discuss the adaptive control of the reverse logistic inventory system with different types of deteriorating items using Liapunov technique.

3. Adaptive Control Procedure

In this section, we discuss, in details, the adaptive control problem of the reverse logistic inventory system. The rates of manufacturing and remanufacturing will be derived using a feedback control approach. Also, the updating rules of disposal and deterioration rates will be derived from the conditions of asymptotic stability of the perturbed system.

To study the adaptive control problem, we first start by considering the required inventory goal levels and the manufacturing and remanufacturing goal rates as a possible special solution of the modified system.𝐼₁̇(𝑡)=𝑃₁(𝑡)+𝑃₂(𝑡)−𝑆(𝑡)−𝜃₁(𝑡)𝐼₁(𝑡),𝐼₂̇(𝑡)=𝑅(𝑡)−𝑃₂(𝑡)−𝑃₃(𝑡)−𝜃₂(𝑡)𝐼₂(𝑡), (3.1) where, 𝜃₁(𝑡),𝜃₂(𝑡), and 𝑃₃(𝑡) are dynamic estimators of the unknown deterioration rates 𝜃_𝑖, 𝑖=1,2 and disposal rate 𝑃₃. Following, we assume that 𝐼_𝑖(𝑡)=𝐼_𝑖, 𝑃_𝑖(𝑡)=𝑃_𝑖, 𝑅(𝑡)=𝑅, 𝑆(𝑡)=𝑆 and 𝜃_𝑖=𝜃_𝑖, 𝑃₃=𝑃₃ is a special solution for the system (3.1). Therefore, the equations of perturbed state about this solution can be derived by introducing the following new variables:𝜉_𝑖(𝑡)=𝐼_𝑖(𝑡)−𝐼_𝑖,𝜂_𝑖(𝑡)=𝜃_𝑖(𝑡)−𝜃_𝑖,𝑣_𝑖(𝑡)=𝑃_𝑖(𝑡)−𝑃_𝑖,(),𝑑(𝑡)(𝑡)−𝑖=1,2=𝑆𝑆,𝜈(𝑡)=𝑃₃(𝑡)−𝑃₃,𝑟(𝑡)=𝑅(𝑡)−𝑅. (3.2) Substituting from (3.2) into (3.1), we get the following perturbed system:𝜉̇₁(𝑡)=𝑣₁(𝑡)+𝑣₂(𝑡)−𝜃₁𝜉₁(𝑡)−𝐼₁𝜂₁(𝑡)−𝜂₁(𝑡)𝜉₁(𝑡)(𝑡),𝜉̇−𝑑₂(𝑡)=𝑟(𝑡)−𝑣₂(𝑡)−𝜈(𝑡)−𝐼₂𝜂₂(𝑡)−𝜂₂(𝑡)𝜉₂(𝑡)−𝜃₂𝜉₂(𝑡). (3.3)

The system (3.3) will be used to study the adaptive control problem using the Liapunov technique. This technique uses the Liapunov function for the system (3.3).

The variables 𝜉_𝑖(𝑡),𝑖=1,2 represent the deviations of the inventory levels 𝐼_𝑖(𝑡) from the inventory goal levels 𝐼_𝑖, 𝑖=1,2, while the variables 𝜂_𝑖(𝑡), 𝑖=1,2 represent the deviations of the estimators 𝜃_𝑖(𝑡) of the deterioration rates from the real deterioration rates 𝜃_𝑖, 𝑖=1,2.

The following theorem gives the manufacturing and remanufacturing rates and the updating rules of disposal rate and deterioration rates which ensure the asymptotic stability of the reverse logistic inventory model with uncertain deterioration and disposal rates.

Theorem 3.1. If the manufacturing and remanufacturing rates are given by: 𝑣₁(𝑡)=𝑑(𝑡)−𝑟(𝑡)−𝑘₁𝜉₁(𝑡)−𝑘₂𝜉₂(𝑡),𝑣₂(𝑡)=𝑟(𝑡)−𝑘₁𝜉₁(𝑡), (3.4) and the updating rules of deterioration rates and disposal rate are given by: 𝜂̇₁(𝑡)=𝐼₁𝜉₁(𝑡)+𝜂₁(𝑡)𝜉₁(𝑡)−𝑙₁𝜉₁(𝑡),𝜂̇₂(𝑡)=𝐼₂𝜉₂(𝑡)+𝜂₂(𝑡)𝜉₂(𝑡)−𝑙₂𝜉₂(𝑡),𝜈̇(𝑡)=𝜉₂(𝑡)−𝜇𝜈(𝑡), (3.5) where, 𝑘_𝑖, 𝑙_𝑖, and 𝜇 are positive real control gain parameters that can be selected by the firm.
Then the special solution 𝜉_𝑖(𝑡)=0, 𝜂_𝑖(𝑡)=0, (𝑖=1,2), 𝜈(𝑡)=0, 𝑟(𝑡)=0, 𝑑(𝑡)=0 of the system composed of the two systems (3.3) and (3.5) are asymptotically stable in Liapunov sense.

Proof. The proof of this theorem can be reached by choosing a suitable Liapunov function for the system consisting of (3.3) and (3.5).
We assume this function has the form: Φ𝜉₁,𝜉₂,𝜂₁,𝜂₂,𝜈=²_𝑖=1𝜉²_𝑖(𝑡)+𝜂²_𝑖(𝑡)+𝜈²(𝑡). (3.6) This function is a positive definite function of the variables 𝜉_𝑖(𝑡), 𝜂_𝑖(𝑡), (𝑖=1,2), and 𝜈(𝑡).The total time derivative of the Liapunov function (3.6) along the trajectory of the system composed of the systems (3.3) and (3.5) gives Φ̇=−𝜃₁+𝑘₁𝜉²₁+𝜃₂+𝑘₂𝜉²₂+𝑙₁𝜂²₁+𝑙₂𝜂²₂+𝜇𝜈². (3.7) Since 𝜃₁+𝑘₁>0,𝜃₂+𝑘₂>0, and 𝜇>0, then Φ̇ is a negative definite function of the variables 𝜉_𝑖(𝑡), 𝜂_𝑖(𝑡), (𝑖=1,2), and 𝜈(𝑡), so the special solution 𝜉_𝑖(𝑡)=0, 𝜂_𝑖(𝑡)=0, (𝑖=1,2), 𝜈(𝑡)=0, 𝑟(𝑡)=0, and 𝑑(𝑡)=0 is asymptotically stable in the Liapunov sense. which completes the proof.

Now, by substituting from (3.4) into (3.3) and considering the updating rules (3.5), we get the following controlled system:𝜉̇₁(𝑡)=−𝑘₁𝜉₁(𝑡)−𝐼₁𝜂₁(𝑡)−𝜃₁+𝜂₁(𝑡)𝜉₁(𝑡),𝜉̇₂(𝑡)=−𝑘₁𝜉₁−𝐼₂𝜂₂−𝜃₂+𝜂₂𝜉₂−𝜈(𝑡),𝜂̇₁(𝑡)=𝜉²₁(𝑡)+𝐼₁𝜉₁(𝑡)−𝑙₁𝜂₁(𝑡),𝜂̇₂(𝑡)=𝜉²₂(𝑡)+𝐼₂𝜉₂(𝑡)−𝑙₂𝜂₂(𝑡),𝜈̇(𝑡)=𝜉₂(𝑡)−𝜇𝜈(𝑡). (3.8)

The above system is used to study the time evolution of inventory levels and dynamic estimators of deterioration rates and disposal rate. It appears from (3.8) that the analytical solution of the system is difficult to derive since it is nonlinear and therefore we solve it numerically in the Section 5. In the next section, we discuss some characteristics of the adaptive system.

4. Some Characteristics of the Model

In this section, we discuss some characteristics of the adaptive reverse logistic inventory system with unknown deterioration and disposal rates.

Lemma 4.1. There are always manufacturing, remanufacturing, and disposal activities.

Proof. Using the asymptotic stability conditions of the system (3.3), we find that the manufacturing and remanufacturing rates are: 𝑃₁(𝑡)=𝑃₁+𝑑(𝑡)−𝑟(𝑡)−𝑘₁𝜉₁(𝑡)−𝑘₂𝜉₂(𝑡),𝑃₂(𝑡)=𝑃₂+𝑟(𝑡)−𝑘₁𝜉₁(𝑡). (4.1) Also, note that when the perturbations 𝜉_𝑖(𝑡),𝑑(𝑡), and 𝑟(𝑡) tend to zero, we find that the manufacturing and remanufacturing activity tend to their goal values 𝑃₁ and 𝑃₂, respectively.
The solution of the differential equation 𝑃̇₃(𝑡)−𝑃₃+𝜇𝑃₃(𝑡)−𝑃₃=𝜉₂(𝑡) (4.2) is given by 𝑃₃(𝑡)=𝑃₃+𝑒^𝜇𝑡𝑒^−𝜇𝑡𝜉₂(𝑡)𝑑𝑡+𝑐, (4.3) where 𝑐 is a constant. This gives the estimator of the disposal activity.

Lemma 4.1 can be interpreted as follows: the manufacturing rate is forced continuously to its goal rate and the dynamic estimator of the disposal rate is forced continuously to its goal value.

Lemma 4.2. If the inventory size of goal levels 𝐼₁ and 𝐼₂ satisfy the system (2.1). That is 𝑃₁+𝑃₂−𝜃₁𝐼₁−𝑆=0,𝑃₂+𝑃₃−𝜃₂𝐼₂−𝑅=0, (4.4) then the deviation of the inventory levels from the goal values is equal to zero and the inventory levels, the manufacturing, and remanufacturing rates and the disposal rates are equal to the goal values that is: lim_𝑡→∞𝐼₁(𝑡)=𝐼₁,lim_𝑡→∞𝐼₂(𝑡)=𝐼₂. (4.5)

Proof. Since Φ(𝜉₁,𝜉₂,𝜂₁,𝜂₂,𝜈)>0 and Φ̇(𝜉₁,𝜉₂,𝜂₁,𝜂₂,𝜈)<0 along the trajectory of the system composed of the two systems (3.3) and (3.5), then, we have lim_𝑡→∞𝜉_𝑖(𝑡)=0,(𝑖=1,2), (4.6) then we get lim_𝑡→∞𝐼_𝑖(𝑡)=𝐼_𝑖,(𝑖=1,2). (4.7)

5. Numerical Solution

The objective of this section is to provide numerical solutions of the system (3.8) for different sets of the system parameters and initial conditions. The numerical solution algorithm is based on numerical integration of the system using the Runge-Kutta method. This section displays graphically the numerical solution of the adaptive controlled system (3.8).

5.1. Example 1

This example considerers the case for which both of demand and disposal rates are constants. In this example, In Table 1, a set of parameter values is assumed.

Table 1 

The numerical result of this example are displayed graphically in Figures 1(a) to 1(f). Figure 1 shows the time behavior of the perturbations of the inventory levels and the manufacturing and remanufacturing rates. It is depicted that after some time, the inventory levels and the manufacturing and remanufacturing rates track perfectly their goal levels and rates, respectively.

(a) Perturbation of the first inventory level

(b) Perturbation of the second inventory level

(c) Perturbation of the first deterioration rate

(d) Perturbation of the second deterioration rate

(e) Perturbation of the manufacturing rate

(f) Perturbation of the remanufacturing rate

Figure 1 

(a): the perturbation of the first inventory level 𝜉1(𝑡), (b): the perturbation of the second inventory level 𝜉2(𝑡), (c): the perturbation of the first deterioration rate 𝜂1(𝑡), (d): the perturbation of the second deterioration rate 𝜂2(𝑡), (e): the perturbation of the manufacturing rate 𝜈1(𝑡), and (f): the perturbation of the remanufacturing rate 𝜈2(𝑡), with the initial perturbations: 𝜉1(0)=5, 𝜉2(0)=1; 𝜂1(0)=0.02, 𝜂2(0)=0.02; 𝜈(0)=5, respectively.

5.2. Example 2

In this example, we discuss the case in which both of demand and disposal rates are time dependent. Namely, the demand rate is a sinusoidal function of time 𝑑(𝑡)=1+sin(𝑡) and the disposal rate is an exponential of time 𝑟(𝑡)=𝑒^−5𝑡.

In Table 2, a set of parameter values is assumed.

Table 2 

The numerical results of this example are illustrated in Figures 2(a) to 2(f). We conclude that both the perturbations of inventory levels and the manufacturing and remanufacturing rates extensively oscillate about their goal levels and rates, respectively, and finally tend to zero.

(a) Perturbation of the first inventory level

(b) Perturbation of the second inventory level

(c) Perturbation of the first deterioration rate

(d) Perturbation of the second deterioration rate

(e) Perturbation of the manufacturing rate

(f) Perturbation of the remanufacturing rate

Figure 2 

(a): the perturbation of the first inventory level 𝜉1(𝑡), (b): the perturbation of the second inventory level 𝜉2(𝑡), (c): the perturbation of the first deterioration rate 𝜂1(𝑡), (d): the perturbation of the second deterioration rate 𝜂2(𝑡), (e): the perturbation of the manufacturing rate 𝜈1(𝑡), and (f): the perturbation of the remanufacturing rate 𝜈2(𝑡), with the initial perturbations: 𝜉1(0)=25, 𝜉2(0)=10; 𝜂1(0)=0.02, 𝜂2(0)=0.02; 𝜈(0)=25, respectively.

5.3. Example 3

In this example, we discuss the case when both of demand rate and disposal rate are solenoidal functions of time, where 𝑟(𝑡)=1+cos(5𝑡), 𝑑(𝑡)=1+sin(5𝑡).

In Table 3, a set of parameter values is assumed.

Table 3 

The numerical results are illustrated in Figures 3(a) to 3(f). We conclude that both of the perturbations of inventory levels and the manufacturing and remanufacturing rates are oscillating about their goal levels and rates, respectively, and tend to zero.

(a) Perturbation of the first inventory level

(b) Perturbation of the second inventory level

(c) Perturbation of the first deterioration rate

(d) Perturbation of the second deterioration rate

(e) Perturbation of the manufacturing rate

(f) Perturbation of the remanufacturing rate

Figure 3 

(a): the perturbation of the first inventory level 𝜉1(𝑡), (b): the perturbation of the second inventory level 𝜉2(𝑡), (c): the perturbation of the first deterioration rate 𝜂1(𝑡), (d): the perturbation of the second deterioration rate 𝜂2(𝑡), (e): the perturbation of the manufacturing rate 𝜈1(𝑡), and (f): the perturbation of the remanufacturing rate 𝜈2(𝑡), with the initial perturbations: 𝜉1(0)=5, 𝜉2(0)=20; 𝜂1(0)=0.2, 𝜂2(0)=0.3; 𝜈(0)=5, respectively.

6. Conclusion

In this paper, the mathematical model of a reverse logistic inventory system with deterioration rates has been studied. We have shown how to use the Liapunov technique to study an adaptive control with feedback to solve a reverse logistic inventory model. Some characteristics of the adaptive control system have been discussed. Numerical simulations have been conducted to validate the results obtained. The updating rules of dynamic estimators of deteriorations and disposal rates have been derived from the conditions of the asymptotic stability.