INTRODUCTION
In general, in inventory models, two factors of the problem have been of growing interest to the researchers, one being the deterioration of items and the other being the variation in the demand rate with time. We are concerned in this study with dynamic problem that can be represented as an optimal control problem with one state variable (inventory level) and one control variable (rate of manufacturing) subject to time of deterioration. The novelty here is that the time of deterioration is a random variable (lifetime of the commodity) followed by generalized Pareto distribution and we consider the problem of controlling the production rate of a continuous review manufacturing system.
Many real problems and applications involve the control of dynamic systems,
i.e., systems that evolve over time. Thus optimal control theory is a branch
of mathematics developed to find optimal ways to control a dynamic systems either
continuoustime systems or discretetime systems. The optimal control theory
has been applied to different inventoryproduction control problems where researchers
are involved to analyze the effect of deterioration and the variations in the
demand rate with time in logistics. Items deterioration is of great importance
in inventory theory, as shown by the surveys of Raafat (1991),
Shah and Shah (2000) and Goyal and
Giri (2001). Inventory model with Weibull distribution for the lifetime
of a commodity has been studied by many researchers Devi (2000),
Wu et al. (2000), Chen and
Lin (2003), Ghosh and Chaudhuri (2004), Alkhedhairi
and Tadj (2007) and Baten and Kamil (2009). Two
production systems with inventoryleveldependent demand are considered and
Pontryagin maximum principle is used to determine the optimal control by Bounkhel
and Tadj (2005). The optimal control of continuousreview models with deteriorating
items has been addressed by Bounkhel and Tadj (2005),
Tadj et al. (2006) and Benhadid
et al. (2008). Srlnivasa Rao et al. (2005,
2007) studied the inventory models with Pareto distribution
deterioration rate to derive optimal order quantity with total cost minimized.
But no attempt has been made to develop the inventory model as an optimal control
problem and derive an explicit solution of an inventory model with generalized
Pareto distribution deterioration using Pontryagin maximum principle. The continuous
review policy of optimal control approach is to be novel in this framework.
There seems to be no literature on the optimal control of continuous review
manufacturing systems with generalized Pareto distribution deterioration items
rate.
We are especially interested in the application of optimal control theory to
the production planning problem. Various authors attacked their research in
the application of optimal control theory to the production planning problem.
Some of them are: Sethi and Thompson (2000), Salama
(2000), Riddalls and Bennett (2001), Zhang
et al. (2001), Khemlnitsky and Gerchak (2002),
Hedjar et al. (2004, 2007)
and Bounkhel and Tadj (2005). Recently, ElGohary
et al. (2009) contributed to the application of optimal control theory
with the assumptions of constant deterioration rate in production inventory
systems. In the present study, we assume that the demand rate is timedependent
and the time of deterioration rate is assumed to follow a generalized Pareto
distribution as well as a nonnegative discount rate is considered for the inventory
systems.
This study develops an optimal control model and utilizes Pontryagin maximum
principle by Pontrygin et al. (1962) in case of
continuous review policy to derive the necessary and sufficient optimality conditions
for inventory systems where, the novelty we take into consideration in our research
is that the time of deterioration is a random variable followed by the threeparameter
generalized Pareto distribution. The probability density function of a generalized
Pareto distribution having probability distribution of the form:
Where,
is the location parameter and σε(∞, ∞) is the scale
parameter and ξε(∞, ∞) is the shape parameter. The probability
distribution function is:
The instantaneous rate of deterioration of the onhand inventory is given by:
This study deals with continuous review policy to solve the optimal control
models by applying Pontryagin maximum principle. We derive explicit optimal
policies for the inventory models where items are deteriorating with generalized
Pareto distribution that can be used in the decision making process.
MODEL AND NOTATIONS
Consider a system where items are subject to generalized Pareto distributed
deterioration. The fixed length of the planning horizon is T. The following
notations will be used to describe the dynamics of the system:
x(t) 
: 
Inventory level function at any instant of time tε[0,
T] 
u(t) 
: 
Production rate at any instant of time tε[0, T] 
y(t) 
: 
Demand rate at any instant of time tε[0, T] 
The dynamics of the inventory level of the state equation which says that the
inventory at time t is increased by the production rate u(t) and decreased by
the demand rate y(t) and the rate of deterioration 1/(σ(1+ξt)) of
generalized Pareto distribution can be written as according to:
with initial condition x(T) = 0.
Now to build the objective function, we assume that an inventory goal level
and a production goal rate are set and penalties are incurred when the inventory
level, the production rate and the deterioration rate deviate from these goals.
To explicitly write the objective function, we introduce the following additional
notation:
q 
: 
Inventory holding cost incurred for the inventory level to
deviate from its goal 
r 
: 
Production unit cost incurred for the production rate to deviate from
its goal. 

: 
Inventory goal level 

: 
Production goal rate 
ρ≥0 
: 
Constant nonnegative discount rate 
We want to keep the inventory x(t) as close as possible to its goal and
also keep the production rate u(t) as close to its goal level .
The quadratic terms and
impose
'penalties' for having either x or u not being close to its corresponding goal
level.
The objective function can be expressed as the quadratic form that we need to minimize:
subject to Eq. 1 and the nonnegativity constraint:
Development of the optimal control model: To build our optimal control model, we consider that a firm can manufactures a certain product, selling some and stocking the rest in a warehouse. We assume that the demand rate varies with time and the firm has set an inventory goal level and production goal rate. We also assume that the firm has no shortage, the instantaneous rate of deterioration of the onhand inventory follows the three parameters generalized Pareto distribution and the production is continuous.
By the virtue of Eq. 1 the instantaneous state of the inventory level x(t) at any time t is governed by the differential equation:
This is a linear ordinary differential equation of first order and its integrating factor is:
Solving the differential equation the onhand inventory at time t is obtained
as:
Assuming that x(0) = x is known and note that the production goal rate can
be computed using the state Eq. 4 as:
We start by defining the variables z(t), k(t) and v(t) such that:
Adding and subtracting the last term:
from the right hand side of Eq. 9 to the Eq.
1 and rearranging the terms we have:
Hence by Eq. 7:
Now substituting Eq. 8 and 9 in Eq.
10 yields:
The optimal control model (2) becomes:
subject to an ordinary differential Eq. 11 and the nonnegativity constraint k(t)≥0, for all tε[0, T].
SOLUTION TO THE OPTIMAL CONTROL PROBLEM
Continuousreview policy: The following theorem is the main result of
this section for the development control (Eq. 12) subject
to revised state Eq. 11. We first assume the firm adopts
a continuousreview policy. We adopt the widely used quadratic objective function
of Holt et al. (1960).
Theorem: Necessary conditions for the pair (k, z) to be an optimal solution of Eq. 11 are:
and
Proof By Pontryagin maximum principle, there exists adjoint function λ(t) such that the Hamiltonian function:
Assume (k, z) is an optimal solution to the objective function (Eq. 11), then:
which is equivalent to:
and
Now, combining (Eq. 14 and 15) yields:
Since all the goal rates must satisfy the state equation, so we have:
which yields with the state Eq. 1:
Combining this Eq. 19 with the Eq. 17
yields the second order differential Eq. 12. From the Eq.
14 and 15 follows
directly. So the proof is complete.
Now we solve the original objection function (2) subject to state Eq.
1 and derive the necessary optimality conditions using Pontryagin maximum
principle (Sethi and Thompson, 2000).
Analytical solution of the original problem: The optimal control approach
consists in determining the optimal control that
minimizes the objective function (2) subject to the state Eq.
1. By the maximum principle of (Pontrygin, 1962),
there exists adjoint function λ(t) such that the Hamiltonian functional
form as:
satisfies the control equation:
the adjoint equation
and the state equation:
Then the control Eq. 21 is equivalent to:
The adjoint Eq. 22 is equivalent to:
and the state Eq. 23 is similar to the Eq. 1.
Substitution expression (Eq. 24) into the state Eq. 1 yields:
From (26) we have:
By differentiating Eq. 26, we obtain:
Substitution expression (Eq. 25) into the Eq. 28 yields:
Finally, substituting expression (Eq. 27 into 29)
to obtain:
Since a closed form solution is not possible, so this boundary value problem can be solved numerically together with initial condition x(0) = 0 and the terminal condition λ(T) = 0.
ILLUSTRATIVE EXAMPLES
In this section, we present some numerical examples. Numerical examples are given for three different cases of demand rates.
•  Demand
rate is constant: y(t) = y = 20 
•  Demand
rate is linear function of time: y (t) = y_{1} (t) t+y_{2}
(t) = t+15 
•  Demand
is sinusoidal function of time: y(t) = 1+sint 
In order to present illustrative examples of the results obtained we use the
following parameters where the planning horizon has length T=12 months, ρ
= 0.001, the inventory holding cost coefficient q = 5 the production cost coefficient
r = 5 The goal inventory level is considered as The
shape and scale parameters of the generalized Pareto distribution rate are considered
as σ = 1 and ξ = 1, respectively. Then the deterioration rate of Pareto
distribution becomes:
The inventory level x(t) interms of the firstorder differential equation
from (Eq. 4) and the secondorder differential Eq.
20 considering the sinusoidal demand function are solved numerically using
the version 6.5 of the mathematical package MATLAB displayed by Fig.
1 and 3, respectively. In particular, whenever the goal
inventory level is considered as the sinusoidal function of time t i.e., then
Fig. 1 does not show the convergence of the optimal level.
Whenever if we take the inventory goal level is as then
Fig. 2 shows the convergence of the optimal inventory towards
inventory goal level. The solution of the secondorder differential equation
is represented by Fig. 2 and shows the state of optimal inventory
level is increasing but converges initially for the certain period of time.

Fig. 1:  The
inventory level x(t) interms of the firstorder differential equation.
Optimal inventory level of the product with time. Source: Author compution 

Fig. 2:  The
inventory goal level with time when it is fixed as .
Optimal inventory level of the product with time t. Source: Author compution 
However, in the subsections we present the model to measure the performance
using different demand patterns. The production level with time t given
from the Eq. 6 considering the mentioned above different demand
rates

Fig. 3:  The
inventory level x(t) interms of the secondorder differential equation.
Source: Author compution 

Fig. 4:  Optimal
production policy for constant demand rate with time. Source: Author compution 
and keeping all other parameters unchanged yielded the figures represented
by the Fig. 46. These Fig.
46 the slight variations of the optimal production level
with time with changing the shape of the demand functions. It is observed that
the optimal production rates are not very sensitive to changes in the demand
functions in case of generalized Pareto distribution.
Constant demand function: In this subsection, we present the model with constant demand function. Substituting

Fig. 5:  The
Optimal production policy for linear demand rate with time. Source: Author
compution 

Fig. 6:  The
Optimal production policy for sinusoidal demand rate with time. Source:
Author compution 
y_{1}(t) = y_{1} =20 instead of y(t) in the controlled system (4) we have:
from which the production goal rate can
be computed (assuming x(0) = x) as:
displayed by Fig. 4.
Linear demand function: In this subsection, we present the model with linear demand function. Substituting linear y_{2}(t) = t+15 instead of y(t) in the controlled system (4) we have:
from which the production goal rate can
be computed (assuming x(0) = x) as:
displayed by Fig. 5.
Sinusoidal demand function: In this subsection, we present the model with sinusoidal demand function. Substituting y_{2}(t) = 1+sin(t) instead of y(t) in the controlled system (4) we have:
from which the production goal rate can
be computed (assuming x(0) = x) as:
displayed by Fig. 6.
CONCLUSION
In this study, we developed an optimal control model in inventoryproduction system with generalized Pareto distribution deteriorating items. We derived the explicit solution of the optimal control models of an inventoryproduction system under a continuous reviewpolicy using Pontryagin maximum principle. However, we gave numerical illustrative examples for this optimal control of a productioninventory system with Pareto distribution deteriorating items.
ACKNOWLEDGMENT
The authors wish to acknowledge the support provided by Fundamental Research Grant Scheme, No. 203/PJJAUH/671128, Universiti Sains Malaysia, Penang, Malaysia for conducting this research.