Adaptive Control of a Production-Inventory Model with Uncertain Deterioration Rate

doi:10.4236/am.2011.29162

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Applied Mathematics, 2011, 2, 1170-1174

doi:10.4236/am.2011.29162 Published Online September 2011 (http://www.SciRP.org/journal/am)

Adaptive Control of a Production-Inventory Model

with Uncertain Deterioration Rate

Fawzy Bukhari

Department of St ati s ti c s and Oper at i o ns Research, College of Science, King Saud University, Riyadh, Saudi Arabia

E-mail: [email protected]

Received July 6, 2011; revised July 22, 2011; accepted July 29, 2011

Abstract

This paper studied a continuous-time model of a production maintenance system in which a manufacturing

firm produces a single product selling some and stocking the remaining. The problem of adaptive control of

a production-maintenance system with unknown deterioration has been presented. Using Liapunov technique,

the production rate and updating rule of deterioration rate are derived as non-linear functions of inventory

level perturbation. Numerical analysis for the system has been presented for a set of parameter values and

demand rate.

Keywords: Production Inventory System, Adaptive Control, Liapunov Technique

1. Introduction

Applications of optimal control theory to management

science, in general, and to production planning, in par-

ticular, are proving to be quite fruitful; see for example

Sethi and Thompson [1]. Naturally, with the optimal

control theory, optimal control techniques came to be

applied to production planning problems and others. For

example, Dobos [2] have discussed the problem of a re-

verse logistics system with special structure. The prob-

lem was presented as an optimal control problem with

two state variables and with three control variables with

an objective to minimize the sum of the quadratic devia-

tion from the goal values. Nahmias [3], Porter and Taylor

[4], Raafat [5], and Khemlnitsky and Gerchak [6] have

discussed the problems of production inventory systems

with inventory levels dependant on demand rate, the

problem of modal control of production inventory sys-

tems, and inventory systems with deteriorating items.

Caldwell [7], Aseltine, Mancini and Sartune [8], Landau

[9], and Narendra and Annaswamy [10] have solved

many problems by using the adaptive control of linear

and non linear dynamical systems using a feedback ap-

proach. El-Gohary and Al-Ruzaiza [11] have studied the

adaptive control of a continuous time three species prey

predator populations. They have derived the nonlinear

feedback control inputs asymptotically stabilized the

system about its steady states. El-Gohary and Yassen [12]

have used the adaptive control and synchronization pro-

cedures to the coupled dynamo system with unknown

parameters. Based on the Liapunov stability technique,

an adaptive control laws are derived such that the cou-

pled dynamo system is asymptotically stable and the two

identical dynamo systems are asymptotically synchro-

nized. Also, the update rules of the unknown parameters

are derived. Tadj, Sarhan and El-Gohary [13] have dis-

cussed the optimal control of an inventory system with

ameliorating and deteriorating items. Different cases for

the difference between the ameliorating and deteriorating

items are considered. Foul, Tadj, and Hedjar [14] have

studied the problem of adaptive control of a production

inventory system in which a manufacturing firm pro-

duces a single product selling some and stocking the re-

maining. Model reference adap tive co ntrol with feedb ack

is applied to track the output of the system toward the

inventory goal level.

The paper studies a production-inventory model which

produces a single item with a certain production rate and

seeks an adaptive actual production and inventory rates.

This model has a dynamic nature and an adaptive control

approach seems particularly well suited to achieve its

goal level and production rate. Also, the unknown dete-

rioration rate will be derived from the conditions of as-

ymptotic stabilization about the steady-state.

The rest of the paper is organized as follows. Follow-

ing this introduction, the mathematical model is intro-

duced in Section 2. In Section 3, the problem of adaptive

control of the production inventory system with un-

F. BUKHARI1171

known deterioration rate is discussed. Also the desired

production rate and updating rule of the deterioration rate

are derived from the conditions of the asymptotic stabil-

ity of the system. In Section 4, we present some illustra-

tive examples. The numerical solution of the controlled

model is presented. Section 5 presents a conclusion of

the paper.

2. Mathematical Model

Consider a manufacturing system that produces a single

product, selling some units and adding others to inven-

tory. We start by writing the differential equation mod-

els. These can be conveniently formulated in terms of

Moreover, the following variables and parameters are

used:

()

t()Pt

: the inventory level at time t,

a()Dt : the actual production rate at time t,

: the demand rate at time t,

()ut : the desired production rate at time t,



: the deterioration rate,



: the inverse of exponential delay time,

I: the inventory level at the steady state,

P: the actual production rate at steady state,

D: the demand rate at steady state.

All functions are assumed to be nonnegative, continu-

ous and differentiable. Assume that at the time instant t

the demand occurs at rate , the actual production at

rate , and the actual production rate responds to

the desired production rate with an exponential

time-delay of

()Dt

()



, it follows that the inventory level

()

t and the actual production rate evolve ac-

cording to the following first order differential equa-

tions:

()

()() ()()(),

()() (),

()

tPt tItDt

PtutPt







 





 



(1)

where, the “dot” means the differentiation with respect to

time and the initial state is0

(0)

I, a

. This

balance equations were also used by Porter and Taylor

(1972), Bradshaw and Porter (1975), and Riddalls and

Bennett (2001) without deterioration rate

(0)

PP



. The

steady-states of this system can be derive by setting both

()

 and equal to zero. That is: ()











pu (2)

Next, in what follows we will discuss the adaptive

control of the produ ction inventory model with unknown

deterioration rate using Liapunov technique.

3. Adaptive Control Problem

In this section, we will discuss in details the adaptive

control problem of the reference model. The rates of de-

mand and desired production, and updating rule of dete-

rioration rate will be derived from the condition of as-

ymptotic stability of the reference model. To study the

adaptive control p roblem we start by considering the val-

ues of required inventory goal level, and actual produc-

tion goal rate as the steady-state solution of the reference

model Equation (1). We consider the adaptive system as

follows:

()() ()()(),

()() (),

()

tPt tItDt

PtutPt







 





 



(3)

where, ˆ()t



is the dynamic estimator of the unknown

deterioration rate



. Following, we assume that ()



()

Pt P



, ()ut u



, ()Dt D, and ˆ()t



 is a spe-

cial solution for the system Equation (3). The equations

of perturbed states of the reference system about its

steady states can be derived by introducing the following

new variables:

()(), ()(),

()(),()(), ()(),

tItI tPtP

vtutudtDtDtt









 





 



 (4)

Substituting from Equation (4) into Equation (3) we

get the following perturbed system:

121 1

()()() ()()()(),

()()(),

[]

tttt tItdt

tvtt

 

 



 







 



(5)

The system Equation (5) will be used to study the

adaptive control problem using the Liapunov technique

for the system of Equation (4). The following theorem

gives the desired production rate and updating rule for

deterioration rate ensure the asymptotic stability of the

reference production inventory model with uncertain

deterioration rate.

Theorem 1

If the perturbations of both desired production rate and

updating rule o f the deterioration rate are giv e n by :





()() ()vttkt



 



(6)

and

() ()()()ttItmt





 

 (7)

where, k, and m are positive real control gains parameters.

Then the special solution () 0,(1,2)

iti



, () 0vt



and () 0t





of the systems Equations (5) and (7) is

F. BUKHARI

1172

asymptotically stable in Liapunov sense if the demand

rate is linear function of the perturbation of the inventory

level or the actual production rate. Otherwise, the as-

ymptotic stability of the solution requires further mathe-

matical analysis.

Proof: The proof of this theorem based on the selec-

tion of a suitable Liapunov function for the system con-

sists of the two combined systems Equations (5) and

(7).

Assume this function has the form:



12 1

, ,()()

 





 (8)

This function is a positive definite function of the

variables (),( 1,2)

iti



, and ()t



. The total time de-

rivative of the Liapunov function Equation (8) along the

trajectory of the systems Equations (5) and (7) give:

222

()()()()() ()[]tktmtdt

 

 

1

t (9)

Since 0k





() ()

ft t

,, the sign of depends

upon the perturbation of the demand rate. In this paper,

we will consider the following two possibilities which

are: 1

0m 



()dt



 or 2

()() ()dt gtt





. Hence, the

total time derivative of the Liapunov function takes the

form:

222

12 1

(()) ()() ()(),

()()()()() ()(),

fttkt mt

tktmtgtt



 

 





 





(10)

So, the special solution: () 0,(1,2)

iti



, ()0vt



and () 0t



 is asymptotically stable in the Liapunov

sense if: 2( )



 which comp letes th e proof.

Now, substituting the adaptive controlled desired pro-

duction rate and updating rule of the deterioration rate

Equation (5) into Equation (4) we get the following con-

trolled system:

121 1

2122

()()() ()()()(),

()() ()(),

tttt tItdt

ttktt

tIttmt

 



 



 



 







(11)

The above system is used to study the time evolution

of inventory levels and dynamic estimators of deteriora-

tion rates. It appears from Equation (11) that the analyti-

cal solution of the system is difficult to derive since it is

non-linear and therefore we solve it numerically in the

next section.

4. Numerical Solution

The objective of this section is to study the numerical

solution of the problem to determine an adaptive control

strategy for a produ ction inventory model with unknown

deterioration rate. The numerical solution algorithm is

based on numerical integration of the system using the

Runge-Kutta method. This section displays graphically

the numerical integration of system Equation (11) of

differential equations for a set of parameter values.

4.1. Example 1

In this example we will graphically display the system

behavior subjected to linear demand rate of the inventory

perturbation. The following set of parameter values is

assumed:

Parameter kmf





Value 332 0.05 50 0.2

With the following values of initial perturbations of

inventory level, actual production rate, and deterioration

rate: 1(0) 40





, 2(0)15





, (0) 0.02



 respectively.

The numerical results are illustrated in Figures 1(a)-(d)

as follows:

Figures 1(a)-(c) have indicated that both the perturba-

tions of inventory level and deterioration rate are damp-

ing oscillate about their goal values. While Figure 1(b)

indicates that the actual production rate perturbation ex-

ponentially tends to its goal value. Finally, Figure 1(d)

shows that the desired production rate exponentially os-

cillates about its goal value.

4.2. Example 2

In this example we will graphically display the system

behavior subjected to linear demand rate of the actual

production rate perturbation. The following set of pa-

rameter values is assumed:

Parameter k m f





Value 0.050.010.2 0.05 200 5

With the following values of initial perturbations of

inventory level, actual production rate, and deterioration

rate: 1(0) 50





, 2(0)15





, (0) 0.02



 respectively.

The numerical results are illustrated in Figures 2(a)-(d)

as follows:

Figures 2(a)-(d) have indicates that the perturbations of

inventory level, deterioration rate, and desired production

rate are damping oscillate about their goal values. While

Figure 2(b) indicates that the actual production rate per-

turbation exponen tially tends to its goal value.

F. BUKHARI

1173

-10

-20

-30

0.5 1 1. 5 2

(a) (b)

Figure 1. (a) Perturbation of inventory level; (b) Perturbation of actual production rate; (c) Perturbation of deterioration

rate; (d) Perturbation of desired production rate.

(a) (b)

F. BUKHARI

1174

Figure 2. (a) Perturbation of inventory level; (b) Perturbation of actual production Rate; (c) Perturbation of deterioration

rate; (d) Perturbation of desired production Rate.

5. Conclusions

In this paper we have discussed the problem of adaptive

control of a production inventory system with unknown

deterioration rate. The desired production rate and up-

dating rule of deterioration rate are derived from the

condition of asymptotic stability about its steady state

using Liapunov technique. The demand rate is consid-

ered as linear function of both inventory level and actual

production rate. Numerical examples are presented to

show the system behavior for a set of parameter values.

It was found that the actual production rate exponentially

tends to its goal value while the inventory level, deterio-

ration rate, and desired production rate are damping os-

cillate about their goal values.

6. References

[1] S. Sethi and G. Thompson, “Optimal Control Theory:

Appli-Cations to Management Science and Economics,”

2nd Edition, Kluwer Academic Publishers, Dordrecht,

2000.

[2] I. Dobos, “Optimal Production-Inventory Strategies for a

HMMS-type Reverse Logistics System,” International

Journal of Production Economics, Vol. 81-82, 2003, pp.

351-360. doi:10.1016/S0925-5273(02)00277-3

[3] S. Nahmias, “Perishable Inventory Theory: A Review,”

Operations Research, Vol. 30, No. 3, 1982, pp. 680-708.

doi:10.1287/opre.30.4.680

[4] B. Porter and F. Taylor, “Modal Control of Produc-

tion-Inventory Systems,” International Journal of Sys-

tems Science, Vol. 3, No. 3, 1972, pp. 325-331.

doi:10.1080/00207727208920270

[5] F. Raafat, “Survey of Literature on Continuously Dete-

riorating Inventory Models,” Journal of the Operational

Research Society, Vol. 42, 1991, pp. 27-37.

doi:10.2307/2582993

[6] E. Khemlnitsky and Y. Gerchak, “Optimal Control Ap-

proach to Production Systems with Inventory Level De-

pendent Demand,” IIE Transactions on Automatic Con-

trol, Vol. 47, No. 3, 2002, pp. 289-292.

doi:10.1109/9.983360

[7] W. Caldwell, “Control System with Automatic Response

Adjustment,” US Patent No. 2,517,081., Filed 25, 1950.

[8] J. Aseltine, A. R. Mancini and C. W. Sartune, “A Survey

of Adaptive Control Systems,” IRE Transactions on Au-

tomatic Control, Vol. 3, No. 6, 1958, pp. 102-108.

doi:10.1109/TAC.1958.1105168

[9] I. D. Landau, “Adaptive Control: The Model Reference

Approach,” Marcel Dekker, New York, 1979.

[10] K. S. Narendra and A. M. Annaswamy, “Stable Adaptive

Systems,” Prentice-Hall, Englewood Cliffs, 1989.

[11] A. El-Gohary and A. Al-Ruzaiza, “Chaos and Adaptive

Control in Two Prey, One Predator System with Nonlin-

ear Feedback,” Chaos, Solitons and Fractals, Vol. 34,

2007, pp. 443-453. doi:10.1016/j.chaos.2006.03.101

[12] A. El-Gohary and R. Yassen, “Adaptive Control and

Synchronization of a Coupled Dynamo System with Un-

certain Parameters,” Chaos, Solitons and Fractals, Vol.

29, 2006, pp. 1085-1094.

doi:10.1016/j.chaos.2005.08.215

[13] L. Tadj, A. Sarhan and A. El-Gohary, “Optimal Control

of an Inventory System with Ameliorating and Deterio-

rating Items,” Journal of Applied Sciences, Vol. 10, 2008,

pp. 243-255.

[14] A. Foul, L. Tadj and R. Hedjar, “Adaptive Control of

Inventory Systems with Unknown Deterioration Rate,”

Journal of King Saud University-Science, in Press, 2011,

doi:10.1016/j.jksus.2011.02.001