Journal of Applied Mathematics and Stochastic Analysis

VolumeΒ 2008, Article IDΒ 158193, 9 pages

http://dx.doi.org/10.1155/2008/158193

## On the Optimality of Inventory Policies: A Quasivariational Approach

Department of Statistics and Operations Research, College of Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

Received 22 February 2008; Revised 9 June 2008; Accepted 17 July 2008

Academic Editor: HoΒ Lee

Copyright Β© 2008 Lakdere Benkherouf. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper revisits the classical discrete-time stationary inventory model. A new proof, based on the theory of quasivariational inequality (QVI), of the optimality of policy is presented. This proof reveals a number of interesting properties of the optimal cost function. Further, the proof could be used as a tutorial for applications of QVI to inventory control.

#### 1. Introduction

Consider an inventory model which consists in controlling the level of stock of a single product where the demands for the product in periods are independently and identically distributed (i.i.d) random variables with density function , and finite mean .

Assume that at the beginning of each period the system is reviewed and we are allowed to increase the level of stock to any level we wish. Orders are assumed to be delivered immediately.

Let be a real-valued function representing the holding and shortage cost with and for . The cost of ordering an amount is given by where is the unit cost of the item and is the set-up cost . Costs are assumed to be additive and geometrically discounted at a rate , and that unmet demand is completely backlogged.

An admissible replenishment policy consists of a sequence where represents the th time of ordering and represents the quantity ordered at time . Write

Let denote the level of stock at time and let be the -algebra generated by the history of the inventory level up to time . Assume that for each is -measurable. Then for a given initial inventory level and an ordering policy , the infinite horizon discounted cost is defined bywhere the expectation is taken with respect to all possible realizations of the process under policy . SetThe objective is to find an admissible policy such that .

Scarf [1] considered a finite horizon version of the problem described in (1.3). He showed using dynamic programming that if the one period expected holding plus shortage cost function is convex, then the optimal policy for period is an policy. The principal tool used by Scarf was a concept of -convexity which he introduced in the same paper. Subsequently, Iglehart [2] extended Scarf's result to the infinite horizon case by showing that the property of -convexity holds for the infinite period stationary model. Veinott [3] replaced the requirement of the convexity of the one period expected holding plus shortage cost by a quasiconvexity requirement and added other conditions. Again using dynamic programming, he showed the optimality of an policy.

In this paper, we approach the problem of determining the optimal inventory policy as an impulse control problem, the theory which has been developed by Bensoussan and Lions [4]. Under this theory, the Bellman equation of dynamic programming for the inventory problem leads to a set of quasivariational inequalities (QVIs) whose solution leads to the optimal inventory policy. This approach leads to a new proof of the result which does not use -convexity and is based on the examination of some properties of an integral equation. Previous applications of QVI to inventory control revolved around diffusion processes from which the machinery needed to prove optimality of policy was not simple. This paper we hope can serve as a tutorial of applications of QVI to inventory control. Readers interested in applications of QVI to inventory control may consult [5β8].

Before we embark on the proof we will first formulate the problem described in (1.3) as a QVI.

Recall that refers to the level of stock at time , and consider all possible actions at time .

(i) If no order is made, then it follows from (1.3) and (1.4) thatorwhere refers to the demand in a period.

(ii) If an order of size is made, then the level of stock jumps from to , andFor , define the operators and byIt follows that the problem of finding the optimal solution to (1.3) reduces to solving the following QVI problem:whereTo solve the QVI given in (1.9), we examine an integral equation problem related to the QVI. This is done in Section 2. The properties obtained of the integral equation are then used to show the optimality of policy in Section 3.

#### 2. An Integral Equation Problem

Consider the space of continuous functions . Assume that we are given a nonnegative function in .

Further, suppose that

(A1)there exists , such that is decreasing on and nondecreasing on ;(A2), as .

For in , define the convolution operator byNow, consider the integral equationHere, , and it is a free parameter.

Under assumption that is in , the integral equation (2.2) has a unique solution in (see [9]). Let denote this solution. In what follows, there is a list of properties of which will prove useful in showing the optimality of the policy:

Property (2.3) is the boundary condition of (2.2).

*Proof of Property (2.4). *To show
(2.4) argue by contradiction. Assume that initially does not decrease. In other words,
there exists , such that is nondecreasing on .
It follows that for and satisfying and , ;
but (2.2) givesTherefore, or .
This leads to by Assumption (A1).
Property (2.3) then
implies that ,
which leads to a contradiction.

To complete the
proof, we again argue by contradiction and assume that there exists ,
and , ,
such that is decreasing on and nondecreasing on .
Let be such that ,
and .
We claim that for , We have by (2.2)Now, use (2.8) and the fact that to get from (2.9) thatbut by Assumption (A1). This leads to a
contradiction. This ends the proof.

*Proof of Property (2.5). *Assume that Property (2.5) is not true. Using Property
(2.4)
and the fact that is continuous, let be the first (smallest) solution of .
Clearly, ,
and attains its minimum at on .
Using (2.2), Assumption (A1), and recalling that is a density function, we getTherefore, .
This leads to a contradiction. Whence Property (2.4) holds.

*Proof of Property (2.6). *Using
(2.2), we getThe last inequality follows from
Property (2.5). The result is then immediate from Assumption (A2) by
taking the limit as .
This completes the proof.

We will next present further properties of .

Theorem 2.1. *For a given ,
there exists an ,
which minimizes for in .*

*Proof. *The proof follows from Properties
(2.3)β(2.6) and the continuity of .

We remark here that may not be unique.

For , defineClearly, is a well-defined function on and is nonnegative.

Lemma 2.2. *The function is decreasing in .*

*Proof. *Let ,
and for in ,
defineIt is easy to show that is a solution of (2.2) with the right-hand
side changed to The function is constant on ,
decreasing on , and is equal to zero for .
Therefore, a similar argument to that used to show properties (2.4)
and (2.5)
shows that is decreasing on and is nonnegative. Since ,
it follows from Theorem 2.1 thatbut .
Therefore, ,
which leads to the required result.

Lemma 2.3. *The function is continuous.*

*Proof. *Fix .
Since is continuous, there exists such that whenever .
Pick such that .
To make things simple, assume .
It was shown in the proof of Theorem 2.1 that .
Now, use the definition of and to get that ;
butTherefore, is continuous.

Lemma 2.4. *(i) **(ii) *

*Proof. *(i) Recall that and that by
Property
(2.5). In particular, .
It follows that or .
The result is then immediate from the continuity of and Assumption (A2) by letting

(ii) Define where
The function is decreasing on .
Therefore, for in ,WriteIt is not difficult to show that satisfies the following:

Again, a similar argument used to prove
Property
(2.4)
can be used to show that is increasing on .
Therefore, .
This in turn leads to .
Now,The right-hand side of (2.22) is
equal to with limit as .
This completes the proof of the lemma.

Now consider the problem of finding a solution to the problem

Theorem 2.5. *There exits a unique number such that .*

*Proof. *The proof is immediate from Theorem
2.1 and Lemmas 2.2β2.4.

#### 3. Optimality of Policy

Recall the definitions of the functions and in (1.9) and letIt is an easy exercise to see that for is equivalent to , which is the integral equation (2.2) for .

Assume that satisfies Assumptions (A1) and (A2) and let be the unique solution of (2.23). This value of leads to a value of which minimizes (this may not be unique). Further, let denote the generic value of . We will next show that the policy which asserts that if the level of stock , order up to level : else do not order, solves the QVI given by (1.9). The proof of optimality relies on the concept of non--decreasing functions which may be found in [10, page 137].

*Definition 3.1. *A function is non--decreasing if implies that

Note that the concept of non--decreasing is weaker than the concept of -convexity which is a standard tool for showing optimality of policy; see [10] for more details.

Our objective is to show that the function is non--decreasing. Note from Properties (2.3)β(2.6) that is constant on , then decreases at least down to , reaches its minimum at some , and eventually goes to as . Non--decreasing means that the function cannot have a drop bigger than beyond . Let

Note that exists and is unique. Set

Theorem 3.2. *For the solution of (2.2) satisfies *

*Proof. *The proof is by contradiction and
only a sketch of the proof will be given. Consider the setIf is empty, there is nothing to prove and
theorem is true. Assume that is not empty, in which case it can be shown
that there exists a triplet such that such that on the interval attains its minimum at , and its maximum at (as shown in Figure 1) withWe will next show that this
cannot happen. Using (2.2), we getIt follows that for ,Using (3.7)β(3.9), we
getThis leads to since is increasing on by Assumption (A1). Therefore, we have a
contradiction that .
Therefore, is empty. This completes the proof.

As a corollary of Theorem 3.2, we have the following.

Corollary 3.3. *For ,
a solution of the equation is a global minimum of the function .*

Note that the proof of Theorem 3.2 revealed that the value of belongs to some interval . Further, the results of the previous section should make a numerical search for the value an easy exercise.

Theorem 3.4. *The function defined from the pair which solves (2.23) solves (1.9).*

*Proof. *We need to show that for and for .
To show for ,
let ;
therefore, ,
and is equivalent to ;
but .
Therefore, is equivalent to ,
which is true since is decreasing for .

To show that for ,
note thatIf ,
then can be written as .
This is true since is non--decreasing. This completes the proof.

It is worth noting that Assumption (A1) is equivalent to saying that is quasiconvex. Also, Assumption (A2) can be weakened by replacing it by . The limit when can be inferred from (2.22) and the limit when can be obtained from the proof of Property (2.6). The optimality of policy remains true.

In this short paper, an alternative proof of the optimality of policy was given. The proof also revealed that finding optimal values of is a simple exercise in numerical analysis. It is hoped that this new proof will lead to new insights in the examination of some stochastic inventory models.

#### Acknowledgments

The author would like to thank Michael Johnson for useful discussions on the topic of quasivariational inequalities. He also benefited from the comments of two anonymous referees.

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