Table of Contents
Journal of Applied Mathematics and Stochastic Analysis
VolumeΒ 2008, Article IDΒ 158193, 9 pages
Research Article

On the Optimality of (𝑠,𝑆) 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.


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 (𝑠,𝑆) 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 𝐷1,𝐷2,… for the product in periods 1,2,… are independently and identically distributed (i.i.d) random variables with density function πœ“, and finite mean πœ‡<∞.

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 𝑓 be a real-valued function representing the holding and shortage cost with 𝑓(0)=0 and 𝑓(π‘₯)>0 for π‘₯β‰ 0. The cost 𝑐(π‘₯) of ordering an amount π‘₯ is given by 𝑐(π‘₯)=π‘˜+𝑐π‘₯,π‘₯>0,0,π‘₯=0,(1.1) where 𝑐 is the unit cost of the item and π‘˜ is the set-up cost (𝑐>0,π‘˜>0). Costs are assumed to be additive and geometrically discounted at a rate 𝛼,0<𝛼<1, and that unmet demand is completely backlogged.

An admissible replenishment policy consists of a sequence (𝑑𝑖,πœ‰π‘–),𝑖=1,…, where 𝑑𝑖 represents the 𝑖th time of ordering and πœ‰π‘–>0 represents the quantity ordered at time 𝑑𝑖. Write𝒱𝑛={(𝑑𝑖,πœ‰π‘–π’±),𝑖=1,…𝑛},∞=limπ‘›β†’βˆžπ’±π‘›=𝒱.(1.2)

Let π‘₯(𝑛) denote the level of stock at time 𝑛,𝑛=0,1,…, and let ℱ𝑛=𝜎{π‘₯(𝑠),𝑠≀𝑛} be the 𝜎-algebra generated by the history of the inventory level up to time 𝑛. Assume that for each π‘›βˆˆβ„•,𝒱𝑛 is ℱ𝑛-measurable. Then for a given initial inventory level π‘₯ and an ordering policy 𝒱, the infinite horizon discounted cost is defined by𝑦(π‘₯,𝒱)∢=πΈπ’±ξ‚»βˆžξ“π‘‘=0𝛼𝑑𝑓(π‘₯(𝑑))+𝑛𝑖=1𝛼𝑑𝑖𝑐(πœ‰π‘–)ξ‚Ό,(1.3)where the expectation is taken with respect to all possible realizations of the process π‘₯(𝑑) under policy 𝒱. Set𝑦(π‘₯)∢=inf𝒱𝑦(π‘₯,𝒱).(1.4)The objective is to find an admissible policy π’±βˆ— such that 𝑦(π‘₯,π’±βˆ—)=𝑦(π‘₯).

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 𝑛 is an (𝑠𝑛,𝑆𝑛) policy. The principal tool used by Scarf was a concept of 𝐾-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 𝐾-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 (𝑠,𝑆) 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 𝐾-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 (𝑠,𝑆) 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 [5–8].

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

Recall that π‘₯(𝑑) refers to the level of stock at time 𝑑, and consider all possible actions at time 𝑑.

(i) If no order is made, then it follows from (1.3) and (1.4) that𝑦(π‘₯(𝑑))≀𝐸[𝑓(π‘₯(𝑑)βˆ’π·)]+𝛼𝐸[𝑦(π‘₯(𝑑)βˆ’π·)](1.5)or𝑦(π‘₯(𝑑))βˆ’π›ΌπΈ[𝑦(π‘₯(𝑑)βˆ’π·)]≀𝐸[𝑓(π‘₯(𝑑)βˆ’π·)],(1.6)where 𝐷 refers to the demand in a period.

(ii) If an order of size πœ‰ is made, then the level of stock jumps from π‘₯(𝑑) to π‘₯(𝑑)+πœ‰, and𝑦(π‘₯(𝑑))β‰€π‘˜+infπœ‰>0[π‘πœ‰+𝑦(π‘₯(𝑑)+πœ‰)].(1.7)For π‘₯βˆˆβ„, define the operators 𝐴 and 𝑀 by(𝐴𝑦)(π‘₯)∢=𝑦(π‘₯)βˆ’π›ΌπΈ[𝑦(π‘₯βˆ’π·)],(𝑀𝑦)(π‘₯)∢=π‘˜+infπœ‰>0[π‘πœ‰+𝑦(π‘₯+πœ‰)].(1.8)It follows that the problem of finding the optimal solution to (1.3) reduces to solving the following QVI problem:𝐴𝑦≀𝐹,𝑦≀𝑀𝑦,(π΄π‘¦βˆ’πΉ)(π‘¦βˆ’π‘€π‘¦)=0,(1.9)where𝐹(π‘₯)∢=𝐸[𝑓(π‘₯βˆ’π·)].(1.10)To 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 (𝑠,𝑆) policy in Section 3.

2. An Integral Equation Problem

Consider the space of continuous functions 𝐢(ℝ). Assume that we are given a nonnegative function β„Ž in 𝐢(ℝ).

Further, suppose that

(A1)there exists π›Ύβ„Ž,βˆ’βˆž<π›Ύβ„Ž<∞, such that β„Ž is decreasing on (βˆ’βˆž,π›Ύβ„Ž] and nondecreasing on [π›Ύβ„Ž,∞);(A2)β„Ž(π‘₯)β†’βˆž, as |π‘₯|β†’βˆž.

For 𝐿 in 𝐢(ℝ), define the convolution operator βˆ— byξ€œ(πœ“βˆ—πΏ)(π‘₯)=∞0𝐿(π‘₯βˆ’π‘‘)πœ“(𝑑)𝑑𝑑.(2.1)Now, consider the integral equation1𝐿(π‘₯)βˆ’π›Ό(πœ“βˆ—πΏ)(π‘₯)=β„Ž(π‘₯),π‘₯>𝑠,𝐿(π‘₯)=1βˆ’π›Όβ„Ž(𝑠),π‘₯≀𝑠.(2.2)Here, 𝑠<π›Ύβ„Ž, and it is a free parameter.

Under assumption that β„Ž is in 𝐢(ℝ), the integral equation (2.2) has a unique solution in 𝐢(ℝ) (see [9]). Let 𝐿𝑠 denote this solution. In what follows, there is a list of properties of 𝐿𝑠 which will prove useful in showing the optimality of the (𝑠,𝑆) policy: 𝐿𝑠1(π‘₯)=1βˆ’π›Όβ„Ž(𝑠)βˆ€π‘₯≀𝑠,(2.3)𝐿𝑠isdecreasingon(𝑠,π›Ύβ„Ž],(2.4)𝐿𝑠1(π‘₯)β‰₯1βˆ’π›Όβ„Ž(π›Ύβ„Ž)βˆ€π‘₯inℝ,(2.5)𝐿𝑠(π‘₯)⟢∞asπ‘₯⟢∞.(2.6)

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

Proof of Property (2.4). To show (2.4) argue by contradiction. Assume that 𝐿𝑠 initially does not decrease. In other words, there exists Ξ”, 𝑠<Ξ”<π›Ύβ„Ž such that 𝐿𝑠 is nondecreasing on [𝑠,Ξ”). It follows that for π‘₯ and 𝑑 satisfying 𝑠≀π‘₯≀Δ and 𝑑β‰₯0, 𝐿𝑠(π‘₯)βˆ’πΏπ‘ (π‘₯βˆ’π‘‘)β‰₯0; but (2.2) gives(1βˆ’π›Ό)πΏπ‘ ξ€œ(Ξ”)+π›Όβˆž0(𝐿𝑠(Ξ”)βˆ’πΏπ‘ (Ξ”βˆ’π‘‘))πœ“(𝑑)𝑑𝑑=β„Ž(Ξ”).(2.7)Therefore, (1βˆ’π›Ό)𝐿𝑠(Ξ”)β‰€β„Ž(Ξ”) or 𝐿𝑠(Ξ”)≀(1/(1βˆ’π›Ό))β„Ž(Ξ”). This leads to 𝐿𝑠(Ξ”)<(1/(1βˆ’π›Ό))β„Ž(𝑠) by Assumption (A1). Property (2.3) then implies that 𝐿𝑠(Ξ”)<𝐿𝑠(𝑠), which leads to a contradiction.
To complete the proof, we again argue by contradiction and assume that there exists πœ‚, and Ξ”, 𝑠<πœ‚<Ξ”<π›Ύβ„Ž, such that 𝐿𝑠 is decreasing on [𝑠,πœ‚] and nondecreasing on [πœ‚,Ξ”). Let π‘₯ be such that πœ‚<π‘₯<Ξ”, and 𝐿𝑠(π‘₯)<𝐿𝑠(𝑠). We claim that for 𝑑β‰₯0, 𝐿𝑠(π‘₯)βˆ’πΏπ‘ (π‘₯βˆ’π‘‘)β‰₯𝐿𝑠(πœ‚)βˆ’πΏπ‘ (πœ‚βˆ’π‘‘).(2.8)We have by (2.2)(1βˆ’π›Ό)πΏπ‘ ξ€œ(π‘₯)+π›Όβˆž0(𝐿𝑠(π‘₯)βˆ’πΏπ‘ (π‘₯βˆ’π‘‘))πœ“(𝑑)𝑑𝑑=β„Ž(π‘₯).(2.9)Now, use (2.8) and the fact that 𝐿𝑠(π‘₯)β‰₯𝐿𝑠(πœ‚) to get from (2.9) thatβ„Ž(π‘₯)β‰₯(1βˆ’π›Ό)πΏπ‘ ξ€œ(πœ‚)+π›Όβˆž0(𝐿𝑠(πœ‚)βˆ’πΏπ‘ (πœ‚βˆ’π‘‘))πœ“(𝑑)𝑑𝑑=β„Ž(πœ‚),(2.10)but β„Ž(π‘₯)<β„Ž(πœ‚) 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 𝐿𝑠 is continuous, let π‘₯βˆ— be the first (smallest) solution of 𝐿𝑠(π‘₯βˆ—)=(1/(1βˆ’π›Ό))β„Ž(π›Ύβ„Ž). Clearly, π‘₯βˆ—>𝑠, and 𝐿𝑠 attains its minimum at π‘₯βˆ— on (βˆ’βˆž,π‘₯βˆ—]. Using (2.2), Assumption (A1), and recalling that πœ“ is a density function, we get𝐿𝑠(π‘₯βˆ—)=β„Ž(π‘₯βˆ—ξ€œ)+π›Όβˆž0𝐿𝑠(π‘₯βˆ—βˆ’π‘‘)πœ“(𝑑)𝑑𝑑>β„Ž(π‘₯βˆ—)+𝛼𝐿𝑠(π‘₯βˆ—).(2.11)Therefore, 𝐿𝑠(π‘₯βˆ—)>(1/(1βˆ’π›Ό))β„Ž(π›Ύβ„Ž). This leads to a contradiction. Whence Property (2.4) holds.

Proof of Property (2.6). Using (2.2), we getπΏπ‘ ξ€œ(π‘₯)=β„Ž(π‘₯)+π›Όβˆž0𝐿𝑠𝛼(π‘₯βˆ’π‘‘)πœ“(𝑑)𝑑𝑑β‰₯β„Ž(π‘₯)+1βˆ’π›Όβ„Ž(π›Ύβ„Ž).(2.12)The last inequality follows from Property (2.5). The result is then immediate from Assumption (A2) by taking the limit as π‘₯β†’βˆž. This completes the proof.

We will next present further properties of 𝐿𝑠.

Theorem 2.1. For a given 𝑠<π›Ύβ„Ž, there exists an 𝑆(𝑠),π›Ύβ„Ž<𝑆(𝑠)<∞, which minimizes 𝐿𝑠(π‘₯) for π‘₯ in ℝ.

Proof. The proof follows from Properties (2.3)–(2.6) and the continuity of 𝐿𝑠.

We remark here that 𝑆(𝑠) may not be unique.

For 𝑠<π›Ύβ„Ž, define𝐾(𝑠)=𝐿𝑠(𝑠)βˆ’minπ‘₯βˆˆβ„πΏπ‘ (π‘₯).(2.13)Clearly, 𝐾 is a well-defined function on (βˆ’βˆž,π›Ύβ„Ž) and is nonnegative.

Lemma 2.2. The function 𝐾 is decreasing in 𝑠.

Proof. Let 𝑑<𝑠<π›Ύβ„Ž, and for π‘₯ in ℝ, define𝐷(π‘₯)∢=𝐿𝑑(π‘₯)βˆ’πΏπ‘ (π‘₯).(2.14)It is easy to show that 𝐷 is a solution of (2.2) with the right-hand side changed to ⎧βŽͺ⎨βŽͺβŽ©π‘”(π‘₯)=β„Ž(𝑑)βˆ’β„Ž(𝑠),π‘₯≀𝑑,β„Ž(π‘₯)βˆ’β„Ž(𝑠),𝑑≀π‘₯≀𝑠,0,π‘₯>𝑠.(1) The function 𝑔 is constant on (βˆ’βˆž,𝑑], decreasing on [𝑑,𝑠], and is equal to zero for π‘₯>𝑠. Therefore, a similar argument to that used to show properties (2.4) and (2.5) shows that 𝐷 is decreasing on ℝ and is nonnegative. Since 𝑆(𝑠)>𝑠>𝑑, it follows from Theorem 2.1 that𝐿𝑑(𝑑)βˆ’πΏπ‘ (𝑑)β‰₯𝐿𝑑(𝑆(𝑠))βˆ’πΏπ‘ (𝑆(𝑠))β‰₯𝐿𝑑(𝑆(𝑑))βˆ’πΏπ‘ (𝑆(𝑠)),(2.15)but 𝐿𝑠(𝑑)=𝐿𝑠(𝑠). Therefore, 𝐿𝑑(𝑑)βˆ’πΏπ‘‘(𝑆(𝑑))β‰₯𝐿𝑠(𝑠)βˆ’πΏπ‘ (𝑆(𝑠)), which leads to the required result.

Lemma 2.3. The function 𝐾 is continuous.

Proof. Fix πœ–>0. Since β„Ž is continuous, there exists 𝛿>0 such that |β„Ž(𝑠)βˆ’β„Ž(𝑑)|<((1βˆ’π›Ό)/2)πœ– whenever |π‘ βˆ’π‘‘|<𝛿. Pick 𝑑<π›Ύβ„Ž such that |π‘ βˆ’π‘‘|<𝛿. To make things simple, assume 𝑑<𝑠. It was shown in the proof of Theorem 2.1 that 𝐿𝑑(𝑆(𝑑))βˆ’πΏπ‘ (𝑆(𝑠))≀𝐿𝑑(𝑑)βˆ’πΏπ‘ (𝑠). Now, use the definition of 𝐿𝑑(𝑑) and 𝐿𝑠(𝑠) to get that 𝐿𝑑(𝑆(𝑑))βˆ’πΏπ‘ (𝑆(𝑠))≀(1/(1βˆ’π›Ό))(β„Ž(𝑑)βˆ’β„Ž(𝑠))<πœ–/2; but|𝐾(𝑑)βˆ’πΎ(𝑠)|≀|𝐿𝑑(𝑆(𝑑))βˆ’πΏπ‘ (𝑆(𝑠))|+|𝐿𝑑(𝑑)βˆ’πΏπ‘ πœ–(𝑠)|<2+πœ–2=πœ–.(2.16)Therefore, 𝐾 is continuous.

Lemma 2.4. (i) 𝐾(𝑠)β†’0asπ‘ β†’π›Ύβ„Ž.
(ii) 𝐾(𝑠)β†’βˆžasπ‘ β†’βˆ’βˆž.

Proof. (i) Recall that 𝐾(𝑠)β‰₯0 and that 𝐿𝑠(π‘₯)β‰₯(1/(1βˆ’π›Ό))β„Ž(π›Ύβ„Ž) by Property (2.5). In particular, 𝐿𝑠(𝑆(𝑠))β‰₯(1/(1βˆ’π›Ό))β„Ž(π›Ύβ„Ž). It follows that 𝐿𝑠(𝑠)βˆ’(1/(1βˆ’π›Ό))β„Ž(π›Ύβ„Ž)β‰₯𝐾(𝑠)β‰₯0 or (1/(1βˆ’π›Ό))(β„Ž(𝑠)βˆ’β„Ž(π›Ύβ„Ž))β‰₯𝐾(𝑠)β‰₯0. The result is then immediate from the continuity of β„Ž and Assumption (A2) by letting π‘ β†’π›Ύβ„Ž.
(ii) Define𝐿𝑠(π‘₯)∢=𝑔𝑠(π‘₯),(2.17) where π‘”π‘ βŽ§βŽͺ⎨βŽͺβŽ©π›Ό(π‘₯)=β„Ž(π‘₯)+11βˆ’π›Όβ„Ž(𝑠),π‘₯>𝑠,1βˆ’π›Όβ„Ž(𝑠),π‘₯≀𝑠.(2.18) The function 𝐿𝑠 is decreasing on (βˆ’βˆž,π›Ύβ„Ž]. Therefore, for π‘₯ in (βˆ’βˆž,π›Ύβ„Ž],πΏπ‘ ξ€œ(π‘₯)<∞0𝐿𝑠(π‘₯βˆ’π‘‘)πœ“(𝑑)𝑑𝑑.(2.19)WriteπΊπ‘ πΏβˆΆ=π‘ βˆ’πΏπ‘ .(2.20)It is not difficult to show that 𝐺𝑠 satisfies the following:𝐺𝑠(π‘₯)βˆ’π›Ό(πœ“βˆ—πΊπ‘ π›Ό)(π‘₯)=𝐿1βˆ’π›Όβ„Ž(𝑠)βˆ’π›Ό(πœ“βˆ—π‘ πΊ)(π‘₯),π‘₯>𝑠,𝑠(π‘₯)=0,π‘₯≀𝑠.(2.21)
Again, a similar argument used to prove Property (2.4) can be used to show that 𝐺𝑠 is increasing on (βˆ’βˆž,π›Ύβ„Ž). Therefore, 𝐺𝑠(π›Ύβ„Ž)β‰₯𝐺𝑠(𝑠)=0. This in turn leads to 𝐿𝑠(π›Ύβ„Ž)β‰₯𝐿𝑠(π›Ύβ„Ž). Now,𝐾(𝑠)=𝐿𝑠(𝑠)βˆ’πΏπ‘ (𝑆(𝑠))β‰₯𝐿𝑠(𝑠)βˆ’πΏπ‘ (π›Ύβ„ŽπΏ)β‰₯𝑠𝐿(𝑠)βˆ’π‘ (π›Ύβ„Ž).(2.22)The right-hand side of (2.22) is equal to β„Ž(𝑠)βˆ’β„Ž(π›Ύβ„Ž) with limit ∞ as π‘ β†’βˆ’βˆž. This completes the proof of the lemma.

Now consider the problem of finding a solution 𝑠 to the problem𝐾(𝑠)=π‘˜.(2.23)

Theorem 2.5. There exits a unique number 𝑠<π›Ύβ„Ž such that 𝐾(𝑠)=π‘˜.

Proof. The proof is immediate from Theorem 2.1 and Lemmas 2.2–2.4.

3. Optimality of (𝑠,𝑆) Policy

Recall the definitions of the functions 𝑦 and 𝐹 in (1.9) and let𝐿(π‘₯)∢=𝑦(π‘₯)+𝑐π‘₯,β„Ž(π‘₯)∢=(1βˆ’π›Ό)𝑐π‘₯+𝐹(π‘₯)+π›Όπ‘πœ‡.(3.1)It is an easy exercise to see that for π‘₯>𝑠,𝐴𝑦=𝐹 is equivalent to 𝐴𝐿=β„Ž, which is the integral equation (2.2) for π‘₯>𝑠.

Assume that β„Ž satisfies Assumptions (A1) and (A2) and let 𝑠<0 be the unique solution of (2.23). This value of 𝑠 leads to a value of 𝑆(𝑠) which minimizes 𝐿𝑠 (this may not be unique). Further, let 𝑆 denote the generic value of 𝑆(𝑠). We will next show that the policy which asserts that if the level of stock π‘₯<𝑠, order up to level 𝑆: else do not order, solves the QVI given by (1.9). The proof of optimality relies on the concept of non-π‘˜-decreasing functions which may be found in [10, page 137].

Definition 3.1. A function π‘£βˆΆβ„β†’β„ is non-π‘˜-decreasing if π‘₯≀𝑦 implies that𝑣(π‘₯)β‰€π‘˜+𝑣(𝑦).(3.2)

Note that the concept of non-π‘˜-decreasing is weaker than the concept of π‘˜-convexity which is a standard tool for showing optimality of (𝑠,𝑆) policy; see [10] for more details.

Our objective is to show that the function 𝐿𝑠 is non-π‘˜-decreasing. Note from Properties (2.3)–(2.6) that 𝐿𝑠 is constant on (βˆ’βˆž,𝑠], then decreases at least down to π›Ύβ„Ž, reaches its minimum at some 𝑆, and eventually goes to ∞ as π‘₯β†’βˆž. Non-π‘˜-decreasing means that the function 𝐿𝑠 cannot have a drop bigger than π‘˜ beyond 𝑆. LetΞ”β„Ž=min{π‘₯>π›Ύβ„Ž,𝐿𝑠(π‘₯)=𝐿𝑠(𝑠)}.(3.3)

Note that Ξ” exists and is unique. Set𝒦(𝑠)=𝐿𝑠(𝑠)βˆ’minπ‘₯β‰€Ξ”β„ŽπΏπ‘ (π‘₯).(3.4)

Theorem 3.2. For 𝑠<π›Ύβ„Ž, the solution 𝐿𝑠 of (2.2) satisfies 𝐿𝑠(π‘₯)βˆ’πΏπ‘ (𝑦)≀𝒦(𝑠)βˆ€π‘₯≀𝑦.(3.5)

Proof. The proof is by contradiction and only a sketch of the proof will be given. Consider the setβ„›(𝒦(𝑠))={π‘₯β‰₯Ξ”β„Ž,𝐿𝑠(π‘₯)βˆ’πΏπ‘ (𝑦)>𝒦(𝑠),forsome𝑦β‰₯π‘₯}.(3.6)If β„›(𝑠) is empty, there is nothing to prove and theorem is true. Assume that β„›(𝒦(𝑠)) is not empty, in which case it can be shown that there exists a triplet (𝑆1,𝑆2,𝑆3) such that π›Ύβ„Ž<𝑆1<Ξ”<𝑆2<𝑆3 such that on the interval [𝑠,𝑆3],𝐿𝑠 attains its minimum at 𝑆1, and its maximum at 𝑆2 (as shown in Figure 1) with𝐿𝑠(𝑠)βˆ’πΏπ‘ (𝑆1)=𝐿𝑠(𝑆2)βˆ’πΏπ‘ (𝑆3)∢=𝒦(𝑠).(3.7)We will next show that this cannot happen. Using (2.2), we get𝐿𝑠(𝑆2)=β„Ž(𝑆2ξ€œ)+π›Όβˆž0𝐿𝑠(𝑆2βˆ’π‘‘)πœ“(𝑑)𝑑𝑑,(3.8)𝐿𝑠(𝑆3)=β„Ž(𝑆3∫)+π›Όβˆž0𝐿𝑠(𝑆3βˆ’π‘‘)πœ“(𝑑)𝑑𝑑.(3.9)It follows that for 𝑑β‰₯0,𝐿𝑠(𝑆2βˆ’π‘‘)βˆ’πΏπ‘ (𝑆3βˆ’π‘‘)≀𝒦(𝑠).(3.10)Using (3.7)–(3.9), we get𝒦(𝑠)=β„Ž(𝑆2)βˆ’β„Ž(𝑆3ξ€œ)+π›Όβˆž0(𝐿𝑠(𝑆2βˆ’π‘‘)βˆ’πΏπ‘ (𝑆3βˆ’π‘‘))πœ“(𝑑)π‘‘π‘‘β‰€β„Ž(𝑆2)βˆ’β„Ž(𝑆3)+𝛼𝒦(𝑠).(3.11)This leads to (1βˆ’π›Ό)𝒦(𝑠)β‰€β„Ž(𝑆2)βˆ’β„Ž(𝑆3)<0 since β„Ž is increasing on (π›Ύβ„Ž,∞) by Assumption (A1). Therefore, we have a contradiction that 𝒦(𝑠)>0. Therefore, 𝑅(𝑠) is empty. This completes the proof.

Figure 1: Plot of the function 𝐿𝑠.

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

Corollary 3.3. For 𝑠<π›Ύβ„Ž, a solution 𝑆(𝑠) of the equation 𝐿𝑠(𝑠)βˆ’πΏπ‘ (𝑆(𝑠))=π‘˜ is a global minimum of the function 𝐿𝑠.

Note that the proof of Theorem 3.2 revealed that the value of 𝑆 belongs to some interval (π›Ύβ„Ž,Ξ”β„Ž). Further, the results of the previous section should make a numerical search for the value (𝑠,𝑆) an easy exercise.

Theorem 3.4. The function 𝐿𝑠 defined from the pair (𝑠,𝑆) which solves (2.23) solves (1.9).

Proof. We need to show that 𝑦<𝑀𝑦 for π‘₯β‰₯𝑠 and 𝐴𝑦≀𝐹 for π‘₯≀𝑠. To show 𝐴𝑦≀𝐹 for π‘₯≀𝑠, let π‘₯≀𝑠; therefore, 𝐿𝑠(π‘₯)=𝐿𝑠(𝑠), and 𝐴𝑦(π‘₯)≀𝐹(π‘₯) is equivalent to π΄πΏπ‘ β‰€β„Ž(π‘₯); but 𝐿𝑠(𝑠)=(1/(1βˆ’π›Ό))β„Ž(𝑠). Therefore, 𝐴𝐿𝑠(𝑠)β‰€β„Ž(π‘₯) is equivalent to β„Ž(𝑠)β‰€β„Ž(π‘₯), which is true since β„Ž is decreasing for π‘₯β‰€π›Ύβ„Ž.
To show that 𝑦≀𝑀𝑦(π‘₯) for π‘₯β‰₯𝑠, note that𝑀𝑦(π‘₯)=π‘˜+𝑐(π‘†βˆ’π‘₯)+𝑦(𝑆)for𝑠≀π‘₯≀𝑆,𝑀𝑦(π‘₯)=π‘˜+𝑦(π‘₯)forπ‘₯β‰₯𝑆.(3.12)If 𝑠≀π‘₯≀𝑆, then 𝑦(π‘₯)≀𝑀𝑦(π‘₯) can be written as 𝐿𝑠(π‘₯)β‰€π‘˜+𝐿𝑠(𝑆). This is true since 𝐿𝑠 is non-π‘˜-decreasing. This completes the proof.

It is worth noting that Assumption (A1) is equivalent to saying that β„Ž is quasiconvex. Also, Assumption (A2) can be weakened by replacing it by lim|π‘₯|β†’βˆžβ„Ž(π‘₯)>β„Ž(π›Ύβ„Ž)+π‘˜. The limit when π‘₯β†’βˆ’βˆž can be inferred from (2.22) and the limit when π‘₯β†’βˆž can be obtained from the proof of Property (2.6). The optimality of (𝑠,𝑆) policy remains true.

In this short paper, an alternative proof of the optimality of (𝑠,𝑆) policy was given. The proof also revealed that finding optimal values of (𝑠,𝑆) 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.


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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