Periodic review inventory systems with fixed order cost and uniform random yield

Accepted Manuscript

Periodic Review Inventory Systems with Fixed Order Cost and Uniform Random Yield Yuyue Song, Yunzeng Wang PII: DOI: Reference:

S0377-2217(16)30542-2 10.1016/j.ejor.2016.07.005 EOR 13831

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

4 August 2015 24 June 2016 4 July 2016

Please cite this article as: Yuyue Song, Yunzeng Wang, Periodic Review Inventory Systems with Fixed Order Cost and Uniform Random Yield, European Journal of Operational Research (2016), doi: 10.1016/j.ejor.2016.07.005

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Highlights • We consider periodic review inventory control problems with both fixed order cost and uniform random yield. • A lower-and-upper bound structure for the optimal policy at the beginning of any period

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is provided.

• A monotone property about the optimal order quantity in the initial stock level in any period is also provided.

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• A sufficient condition is also provided such that the threshold policy is optimal in each period.

• A few interesting phenomena about the behavior of an optimal policy have been illustrated

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in the paper.

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Periodic Review Inventory Systems with Fixed Order Cost and Uniform Random Yield Yuyue Song1 Faculty of Business Administration, Memorial University of Newfoundland, NL Canada A1B 3X5

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Yunzeng Wang2 A. Gary Anderson Graduate School of Management, University of California, Riverside, CA 92521

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Abstract

We consider periodic review stochastic inventory control problems with both fixed order cost and uniform random yield. Our objective is to characterize some structural properties of the optimal policies so that efficient approaches can then be established towards finding an optimal policy. In particular, we provide a lower-and-upper bound structure for the optimal policy at the beginning of any period,

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such that it is optimal not to order anything if the initial stock level is above the upper bound, and it is optimal to order a positive quantity if the initial stock level is below the lower bound, where the

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optimal quantity can be found efficiently. For any initial stock level in-between the upper and lower bounds in each period, a partial characterization of the optimal policy is provided. Furthermore, we

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show that the optimal order quantity is monotonically decreasing in the initial stock level in any period and this will help the search for the optimal order quantity at any initial stock level. In addition, we

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illustrate a few interesting phenomena about the behavior of an optimal policy. These phenomena show that the structure of the optimal policy for this problem is in general significantly different from that

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for the traditional periodic review inventory control problems with certain yield and fixed order cost. Key Words: Periodic Review, Fixed Order Cost, Stochastic Demand, Inventory Control, Random

Yield.

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Corresponding author, Email: [email protected],

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Email: [email protected]

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Glossary of Notation the planning horizon (number of periods) and the period index, the initial stock level and the stock level after receiving the order in any period, the random demand and its realization in one period, the density function and the expected value of the random demand V , the fixed order cost, the per unit holding, shortage and purchase costs, respectively, the discount factor, the expected inventory holding/shortage cost in one period at y, c(1 − γ)y + L(y) + cγE(V ) with unique minimizer S, the optimal total expected discounted cost from period t until the end of the planning horizon given initial stock level x, Ft (x) = ft (x) + cx, ¯ ¯ = constants defined in Assumption 1, A, A, B, B γN ¯ B} ¯ < B , defined before Theorem 2, N = the smallest integer such that 1−γ max{A, 2

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= = = = = = = = = =

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T, t x, y V, v g(v), E(V ) K h, b, c γ L(y) H(y) ft (x)

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H(y) + γ 0+∞ Ft−1 (y − v)g(v)dv, R1 0 Jt (x + uQ)du for any Q ∈ [0, +∞) and any initial stock level x in period t, the global minimizer of ρt (Q, x) given initial stock level x in period t, the highest initial stock at the beginning of period t such that it is optimal to order a positive quantity, 0 0 Yt , St = the largest and the smallest local minimizers of Jt (y), Yt = the largest real value ∈ (−∞, Yt0 ] such that Jt (Yt ) ≥ K + Jt (Yˆt ) where Yˆt is the global minimizer of Jt (y) over [Yt , Yt0 ], Mt0 = the maximal value of Jt (y) over [St0 , Yt0 ], Yt1 = the stock level larger than Yt0 such that Jt (Yt1 ) = Mt0 (refer to Figure 2), R X 1 = the largest real value X 1 ∈ (−∞, S] such that H(X 1 ) ≥ K + 01 H(X 1 + u(S − X 1 ))du, Xt0 = the initial stock level such that z = Mt0 is the balance level for Jt (y) with y from this initial stock level to Yt1 , Xt = min{Xt0 , X 1 }, Y = the non-period dependent Y −bound defined in Theorem 2, U (y) = an upper bound convex function of Jt (y) over (−∞, +∞), Y 1 = the stock level larger than Y such that H(Y 1 ) = U (Y ) (refer to Figure 3), X 0 = the stock level X 0 is chosen such that z = U (Y ) is the balance level of H(y) with y from X 0 to Y 1 (refer to Figure 3), and X = min{X 0 , X 1 }, the non-period dependent X−bound. = = = =

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Jt (y) ρt (Q, x) Qt (x) st

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Introduction

We consider periodic review stochastic inventory control problems for a single product at a single stage with fixed order cost where the excess demand in each period is completely backlogged. This kind of inventory control problems has been well studied in the inventory management

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literature. One main underlying assumption is that for each replenishment, the amount of stock received from the supplier or the production process matches with the ordered quantity. This means that the supplier or the production process is completely reliable and there is no yield loss. Under this assumption of certain yield, the optimal inventory control policies have very

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nice structures. That is, they are the well-known base-stock policy when the fixed order cost is zero, and the (s, S) policy when there is a non-zero fixed order cost. Furthermore, it is also relatively easy to find the optimal policy parameters numerically. The literature on traditional inventory control problems with certain yield is broad but relatively well-known to the field, and we thus will omit a review of this literature in this paper. Interested readers are referred to

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the foundational work by Scarf (1960), Veinott and Wagner (1965), and Zheng and Fredergruen

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(1991), Zipkin (2000) and references therein.

On the other hand, yield uncertainty in replenishment is a common occurrence, especially in

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many production scenarios. In the semiconductor manufacturing industry, for example, defective outputs are found in many cases because of process uncertainties, poor quality control, and

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other environmental factors (see Nahmias (2005), P385). It is well known that inventory control systems with random yield can behave quite differently from the ones with certain yield and

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the importance of yield randomness on the lot sizing policy is well recognized in the literature, see, for instance, Yano and Lee (1995) and Panagiotidou et al. (2012). In order to understand the impact of random yield on the control of inventory/production systems, one fundamental question is how to efficiently find the optimal order quantity at any initial stock level in each period. In order to answer this question, one needs to characterize the structure of the optimal inventory control policy.

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Under the assumptions of zero fixed order cost and stochastic-proportional yield, Henig and Gerchak (1990) made a significant theoretical contribution for the periodic review inventory control systems with random yield. They showed that a single threshold policy structure is optimal in each period. That is, it is optimal to order when the initial inventory level falls below a threshold level and not to order otherwise. The key observation in Henig and Gerchak

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(1990) is that the convexity property of the cost functions can be preserved from period to period if the fixed order cost is zero. Because the optimal order quantity non-linearly depends on the initial stock level in any period, several research papers have also been published to study heuristic approaches for finding good and easily computable approximations of the optimal order

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quantity. These include the work by Bollapragada and Morton (1999), Inderfurth and Transchel (2007), Inderfurth and Kiesmuller (2015), Inderfurth and Vogelgesang (2013), Li et al. (2008), Huh and Nagarajan (2010) and references therein. There are other research works done for the periodic review inventory systems under different settings, including Yao (1988), Lee and Yano

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(1988), Bassok and Akella (1991), Wang and Gerchak (1996), Chen et al. (2001) and Li and Zheng (2006). All of them assume zero fixed order cost in each period over the planning horizon.

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Yano and Lee (1995) provide a comprehensive review of more general lot-sizing problems with random yield. It is not our intention to provide a comprehensive review about the random yield

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in inventory control and our focus in this paper is about periodic review inventory systems with both fixed order cost and random yield.

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If the fixed order cost in any replenishment is positive, then the nice convexity property of the cost functions cannot be preserved from period to period, and the optimal policy structure is

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not known. To characterize the structure of the optimal policy for such a problem is obviously of interest to the research community. It, however, is also a notoriously hard problem that attracts the attention of many researchers in the filed. For example, not knowing the optimal policy structure, Mazzola et al. (1987) and Zipkin (2000) propose some heuristic approaches for the search of the optimal policy. If the random yield is either 1 or 0 (an order is either supplied in full or is not supplied at all), Ozekici and Parlar (1999) can show that the (s, S) policy is 2

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optimal under reasonable conditions. In the following, if the random yield is uniform, we can provide some partial characterization of the optimal inventory control policy. In particular, an (X, Y ) structure is provided in any period: it is optimal to order nothing if the initial stock level is above Y ; it is optimal to order a positive quantity if the initial stock level is below X, where the optimal quantity can be found efficiently. We further show that the optimal order

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quantity is monotonically decreasing in the initial stock level in any period and this will help the search for the optimal order quantity at any initial stock level. A partial characterization of the optimal policy is also provided for initial stock levels in-between X and Y . Moreover, we also discuss a few interesting phenomena about the behavior of the optimal inventory control policy.

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These phenomena, which are unknown in the literature, show that the structure of the optimal inventory control policy for problems with fixed order cost and random yield is significantly different from the one of the traditional periodic review inventory control problems with positive fixed order cost but certain yield.

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In §2, the model framework will be introduced. Then, an (X, Y ) structure of the optimal inventory control policy is provided in §3 and a partial characterization of the optimal policy is

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provided in §4 for initial stock levels in-between X and Y . In §5, we illustrate a few interesting phenomena about the structure of the optimal inventory control policy and we conclude the

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paper in §6. All the proofs are placed in an appendix to avoid interrupting the general flow of

Model Framework

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2

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the paper.

We consider a periodic review inventory control problem with both positive fixed order cost and uniform random yield over T periods, where the last period is indexed by 1 and the first period is indexed by T . For simplicity, we present the model formulation and our analysis only for the stationary environment, i.e., all model parameter values are constants and not perioddependent. The demand V in each period is random with density function g(v)(0 ≤ v < +∞) 3

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and is stochastically independent of the demands in other periods. Let E(V ) be the expected value of V . Like almost all other papers in the literature including Henig and Gerchak (1990), we assume that the yield is stochastically proportional to the order quantity Q in any period, i.e., only U Q is received where U ∈ [0, 1] is a random variable following uniform distribution over [0, 1].

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The ordering cost of any positive quantity Q consists of two components: a positive fixed order cost K(> 0) and a proportional cost cU Q, that is, the per unit cost c is charged only for the received units. In other words, the total ordering cost of Q(≥ 0) units of the product is Kδ(Q) + cU Q, where δ(Q) = 1 if Q > 0 and δ(Q) = 0 otherwise. Following the literature,

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we assume that the order lead time is zero (assuming non-zero lead time would significantly complicate the problem, even for the case with zero fixed order cost). Let L(y) be the expected inventory holding/shortage cost in one period given that the on-hand stock level after receiving the order is y ∈ (−∞, +∞) at the beginning of the period.

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For any given initial stock level x ∈ (−∞, +∞) at the beginning of period t, we wish to characterize the structure of the optimal inventory control policy by assuming that optimal

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policies are followed over all the remaining periods. Let ft (x) be the optimal total expected discounted cost from period t until the end of the planning horizon. Then, we have the following

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dynamic programming (DP) formulation:

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Z 1 Z 1 Z +∞ 1 ft (x) = minQ≥0 {Kδ(Q) + cQ + L(x + uQ)du + γ ft−1 (x + uQ − v)g(v)dvdu}, 2 0 0 0

where γ ∈ (0, 1) is the discount factor and Kδ(Q) + 21 cQ +

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cost for one (the current) period.

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L(x + uQ)du is the expected total

At the end of the planning horizon, there are two possible situations. If there is remaining

stock of x(> 0) units, we assume that it can be returned to the supplier with full credit; and if there is a shortage of x(< 0) units, we assume that it can be satisfied by the supplier. Thus, the cost of x ∈ (−∞, +∞) units at the end of the planning horizon is f0 (x) = −cx. For simplicity,

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let Ft (x) = ft (x) + cx. Then, F0 (x) = 0 and the above DP formulation can be rewritten as Ft (x) = minQ≥0 {Kδ(Q) +

Z 1 0

H(x + uQ)du + γ

Z 1 Z +∞ 0

0

Ft−1 (x + uQ − v)g(v)dvdu},

where H(y) = c(1 − γ)y + L(y) + cγE(V ). Without loss of generality, for the remaining portion of this paper we assume that H(y) is the new expected inventory holding/shortage cost in one

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period given that the on-hand stock level after receiving the order is y ∈ (−∞, +∞). We also assume that H(y) is piecewise differentiable and convex in y over (−∞, +∞) and there exists a unique minimizer S of H(y) over (−∞, +∞). We will also make the following assumption about H(y).

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¯ B, and B ¯ such that −A¯ ≤ H 0 (y) ≤ −A for Assumption 1 There exist positive constants A, A, ¯ for any y ∈ [S + 1, +∞), and −A¯ ≤ H 0 (y) ≤ B ¯ for any any y ∈ (−∞, S − 1], B ≤ H 0 (y) ≤ B y ∈ (S − 1, S + 1).

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For simplicity, in the remaining portion of the paper we define ρt (Q, x) = any Q ∈ [0, +∞) and any initial stock level x in period t where

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Jt (y) = H(y) + γ

Z +∞ 0

Ft−1 (y − v)g(v)dv.

2 R1 0

Jt (x + uQ)du for

(1)

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Thus, Ft (x) can be rewritten through ρt (Q, x) as the following:

An (X, Y ) Structure of the Optimal Inventory Control

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(2)

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Ft (x) = minQ≥0 {Kδ(Q) + ρt (Q, x)}.

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Given that the initial stock level is x ∈ (−∞, +∞) and the order quantity is Q(≥ 0), the corresponding total expected cost for the last period problem is Kδ(Q) + ρ1 (Q, x) where ρ1 (Q, x) = R1 0

H(x+uQ)du. The following results for the last period problem are well known in the literature

(refer to Henig and Gerchak (1990)). 5

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Lemma 1 For any given initial stock level x ∈ (−∞, +∞) at the beginning of the last period, there exists a unique Q1 (x) minimizing ρ1 (Q, x). Furthermore, we have the following properties about Q1 (x), x + Q1 (x), and ρ1 (Q1 (x), x): (1) Q1 (x) = 0 if x ≥ S; and Q1 (x) > 0 if x < S.

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(2) both Q1 (x) and x + Q1 (x) are strictly decreasing in x over (−∞, S]. (3) ρ1 (Q1 (x), x) is convex and strictly decreasing in x over (−∞, S].

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Let s1 ∈ (−∞, S] be the largest real value such that H(s1 ) − ρ1 (Q1 (s1 ), s1 ) ≥ K, i.e., the highest initial stock level in the last period such that it is optimal to order a positive quantity.

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Then, the optimal policy at the beginning of the last period is summarized in the following theorem.

Theorem 1 It is optimal to order a positive quantity for any initial stock level x ≤ s1 ; ordering fixed order cost and random yield.

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nothing otherwise, i.e., the threshold policy is optimal for the last period problem with both positive 2

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For the periodic-review inventory control problems with positive fixed order cost and certain yield , it is well known that S, the unique minimizer of H(y), is a common Y −bound for all

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periods such that it is optimal to order nothing at any initial stock level above Y . But for our problem with uniform random yield, S is not a common Y -bound for all periods and we

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will illustrate this point in §5. In the following, we first provide a Y −bound for high initial stock levels such that it is optimal to order nothing and then an X − bound for low initial stock

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levels such that it is optimal to order a positive quantity. Neither X−bound nor Y −bound is period-dependent.

3.1

Y −Bound

In order to characterize the behavior of Jt (y), we need to provide an estimation about Ft0 (x) and Jt0 (y) and we have the following results. 6

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t Lemma 2 We have − 1−γ A¯ ≤ Jt0 (y) ≤ 1−γ

1−γ t ¯ B 1−γ

1−γ t ¯ B 1−γ

t for any y ∈ (−∞, +∞) and − 1−γ A¯ ≤ Ft0 (x) ≤ 1−γ

for any x ∈ (−∞, +∞). As a consequence, we have |Ft0 (x)| ≤

x ∈ (−∞, +∞) and |Jt0 (y)| ≤

1 ¯ B} ¯ max{A, 1−γ

1 ¯ B} ¯ max{A, 1−γ

for any

for any y ∈ (−∞, +∞).

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Based on the above estimation result in Lemma 2, we will show that Jt (y) is strictly increasing

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when y is high enough. The idea to achieve this objective can be illustrated as follows. For simplicity, suppose that the random demand V is bound by a constant D. By Lemma 2, we 1 ¯ B} ¯ and F 0 (y) ≥ − 1 max{A, ¯ B} ¯ for any y ∈ [S + 1, +∞) in any have Jt0 (y) ≥ − 1−γ max{A, t 1−γ

period t. By combining this fact and the increasing property of H(y) over [S + 1, +∞), from the γ 0 ¯ B} ¯ for any y ∈ [S + 1 + D, +∞) (Note that (y) ≥ − 1−γ expression of Jt+1 (y), we get Jt+1 max{A,

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y − v ∈ [S + 1, +∞) for any y ∈ [S + 1 + D, +∞) in period t + 1). By induction on the period γk 0 ¯ B} ¯ for any y ∈ [S + 1 + kD, +∞) and any positive index, we can get Jt+k (y) ≥ − 1−γ max{A,

integer k. Note that the constant discount factor γ is within (0, 1), let N be the smallest positive integer such that

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γN ¯ B} ¯
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¯ are defined in Assumption 1. Then, we have the following result. where A¯ and B Theorem 2 There exists Y (> S) such that Jt0 (y) ≥

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for any y ∈ [Y, +∞) in any period t.

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Hence, Jt (y) is strictly increasing over [Y, +∞) and it is optimal to order nothing at any initial stock level above Y in any period t.

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Remark:

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We can propose a better period-dependent Yt −bound such that it is optimal to

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order nothing at any initial stock level x ∈ (Yt , +∞) in any period t. Let Yt0 be the largest local minimizer of Jt (y) and Yt ∈ (−∞, Yt0 ] be the largest real value such that Jt (Yt ) ≥ K+Jt (Yˆt ) where Yˆt is the global minimizer of Jt (y) over [Yt , Yt0 ]. Then, it is obvious that Jt (x) < K + ρt (Q, x) for any x ∈ (Yt , +∞) and Q ∈ (0, +∞).

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Next, through characterizing the behavior of Jt (y) when y is low, we will provide an X−bound such that −ρt (Q, x) is unimodal in Q for any given x ∈ (−∞, X]. Hence, it is easy to compute the optimal order quantity Qt (x) for any initial stock level x ∈ (−∞, X] in any period t. 7

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3.2

X−Bound

For any given initial stock level x in period t, note that Qt (x) is the global minimizer of ρt (Q, x) = 0

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Jt (x + uQ)du in Q over [0, +∞). If Qt (x) > 0, then Qt (x) satisfies the first order condition of Jt0 (x + uQt (x))udu = 0 and this can be rewritten as the following through the integration by

parts: Qt (x)Jt (x + Qt (x)) =

Z x+Qt (x) x

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Jt (y)dy.

(3)

The right hand-side of the above equation is the size of the area below z = Jt (y) but above z = 0 with y from x to x + Qt (x) and the left hand-side is the size of a rectangle with length

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Qt (x) with y from x to x + Qt (x) and width Jt (x + Qt (x)). Based on this understanding, we can provide a very intuitive explanation about the above first order condition in eqn (3): The size of the area above the level of z = Jt (x + Qt (x)) but below z = Jt (y) is equal to the size of the area below the level of z = Jt (x + Qt (x)) but above z = Jt (y) over [x, x + Qt (x)]. Thus, we call

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z = Jt (x + Qt (x)) the balance level for z = Jt (y) over [x, x + Qt (x)] and this is illustrated in the

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following Figure 1.

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Figure 1: Illustration of the First Order Condition in Eqn (3)

If it is optimal to order at the initial stock level x in period t, then the following result will

characterize how Jt (y) behaves near x + Qt (x). Lemma 3 If it is optimal to order a positive quantity at the initial stock level x in period t, then Jt (x) − Jt (x + Qt (x)) ≥ K and Jt (y) is strictly increasing at x + Qt (x). 8

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Next, we will investigate the behavior of the order-up-to level x + Qt (x) in x and this will further reduce the searching effort for Qt (x). Lemma 4 For any given initial stock levels x < xˆ in period t, if x + Qt (x) ≥ xˆ and Jt (y) ≥ Jt (x + Q(x)) for any y ∈ [x, xˆ], then we have x + Qt (x) > xˆ + Qt (ˆ x).

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The above requirement in Lemma 4 means that if the two initial stock levels x and xˆ are near to each other and the value of Jt (y) over [x, xˆ] is not small, then the order-up-to level at x should be higher than the one at xˆ. In each period t, by Lemma 4, we can get the following property about x + Qt (x) in x, which is very similar to the one in Henig and Gerchat (1990) under the

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assumption of zero fixed order cost.

Proposition 1 For any initial stock levels x < xˆ, if it is optimal to order a positive quantity at any initial stock level y ∈ [x, xˆ], then y + Qt (y) is strictly decreasing in y over [x, xˆ].

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For the periodic review inventory systems with certain yield, it is well known that the optimal

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order-up-to level is a constant. But this observation does not hold true anymore if there is a

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random yield in the supply process. In order to hedge against the random yield in the supply process, the optimal order-up-to level will be decreasing in terms of the initial stock level and

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this means that the optimal order quantity is much bigger than the situation with certain yield. In the following, we will characterize the behavior of the optimal expected discounted cost

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function Ft (x) in the initial stock level x in period t. 2

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Proposition 2 Both Ft (y) and Jt (y) are strictly decreasing in y over (−∞, S].

Now, we are ready to provide a period dependent bound Xt such that it is optimal to order

a positive quantity at any initial stock level x below Xt and −ρt (Q, x) is unimodal in Q over [0, +∞). Thus, for any x ∈ (−∞, Xt ], the search of Qt (x) is simple. Let Yt0 be the largest local minimizer and St0 be the smallest local minimizer of Jt (y) over (−∞, +∞). By Theorem 2 and Proposition 2, it is obvious that S ≤ St0 ≤ Yt0 < Y . Based on the relationship between St0 and Yt0 , we consider two cases. 9

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(1) If St0 < Yt0 , then we let Mt0 be the maximal value of Jt (y) over [St0 , Yt0 ]. Corresponding to this Mt0 , let Yt1 ∈ [Yt0 , +∞) such that Jt (Yt1 ) = Mt0 . Then, let Xt0 ∈ (−∞, St0 ] be the initial stock level such that z = Mt0 is the balance level for Jt (y) with y from Xt0 to Yt1 (refer to Figure 2).

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(2) If St0 = Yt0 , then Jt (y) has a unique minimizer St0 . In this case, we let Xt0 = St0 . We further define X 1 as the largest real value X 1 ∈ (−∞, S] such that H(X 1 ) ≥ K + H(X 1 + u(S − X 1 ))du where H(y) is the expected holding/shortage cost in one period and S

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is the minimizer of H(y). Let Xt = min{Xt0 , X 1 }. Then, we have the following result.

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Figure 2: Illustration of Xt0 Bound in Period t

Theorem 3 For any initial stock level x ∈ (−∞, Xt ] in period t, it is optimal to order a positive 2

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quantity and −ρt (Q, x) is unimodal in Q over [0, +∞).

The above result means that a binary search will be enough to find the optimal order quantity

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at any initial stock level below Xt . Thus, the search for the optimal order quantity is very efficient. One benefit of this result is that the optimal order quantity can be used to evaluate the performance of different heuristics proposed in the literature. Next, we will provide a non-period dependent bound X such that it is optimal to order a positive quantity Qt (x) at any initial stock level x below X and −ρt (Q, x) is unimodal in terms of Q over [0, +∞) in any period t. In general, X is much lower than Xt in period t. It is obvious 10

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that we have Jt (y) ≥ H(y) for any y ∈ (−∞, +∞) in any period t. We can construct a convex function U (y) as follows:    

U (y) =   

) + b E(V 1−γ

h y, 1−γ

if y ≥ 0,

) − b E(V 1−γ

b y, 1−γ

if y < 0.

have H(y) ≤ Jt (y) ≤ U (y)

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It is easy to show that Jt (y) ≤ U (y) for any y ∈ (−∞, +∞) in any period t. Thus, we always

for any y ∈ (−∞, +∞) in any period t. Based on this relationship, let Y 1 ≥ Y such that

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H(Y 1 ) = U (Y ) where Y is defined in Theorem 2. We choose X 0 ≤ S such that z = U (Y ) is the

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balance level for z = H(y) with y from X 0 to Y 1 (refer to Figure 3).

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Figure 3: Illustration of X 0 Bound for All Periods

Let X = min{X 0 , X 1 } where X 1 is defined before Theorem 3. Then, we have the following

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result (since the proof is very similar to the one for Theorem 3, we omit it). Corollary 1 For any initial stock level x ∈ (−∞, X] in any period, it is optimal to order a positive quantity and −ρt (Q, x) is unimodal in Q over [0, +∞).

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Therefore, we can find a period-independent (X, Y ) bound and also a period-dependent (Xt , Yt ) bound in period t. Intuitively, the (Xt , Yt ) bound in period t should be much better than the (X, Y ) bound and the numerical results in Table 1 clearly illustrate this point, but it 11

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Table 1: (X, Y ) and (Xt , Yt ) Bounds

(X5 , Y5 )

(X4 , Y4 )

(X3 , Y3 )

(X2 , Y2 )

(X1 , Y1 )

(X, Y )

0

(8.998, 14)

(8.998, 14)

(8.998, 14)

(8.998, 14)

(8.998, 9)

(-311, 213)

5

(7.149, 8.995)

(7.149, 9.229)

(7.149, 10.841)

(7.149, 10.825)

(7.149, 8.084)

(-311, 213)

10 (5.298, 8.690)

(5.298, 8.866)

(5.298, 9.704)

(5.298, 8.940)

(5.298, 7.169)

(-311, 213)

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K

is clear that the (X, Y ) bound has theoretical importance. In this numerical experiment, the

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random demand in each period has two possible realizations and the parameter values are T = 5, b = 20, h = 0.2, c = 2, γ = 0.8, v1 = 5 (with probability of 0.7) and v2 = 9 (with probability of 0.3). We have tested three different scenarios by varying the fixed order cost of K. It seems that (Xt , Yt ) bounds are very good and the range in-between Xt and Yt is small. In the following, we

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will investigate the optimal policy structure for all initial stock levels in-between Xt and Yt in

Optimal Policy for Initial Stock Levels in [Xt, Yt]

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4

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period t.

If the initial stock level is higher than Yt or lower than Xt in period t, we know how to figure out

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the corresponding optimal order quantity. In this section, we shall show how to figure out Qt (x) at x ∈ [Xt , Yt ].

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From both the practical and the computational points of view, we would like to point out two critical issues regarding the search of Qt (x). First, we can only compute the value of Jt (y) over some discrete points (for simplicity, we only focus on integral values of y over (−∞, +∞)). Without loss of generality, we assume that both Xt and Yt are integers and we only need to compute the value of Jt (y) over (−∞, +∞) while y is an integer. For any given integral initial stock level yˆ, let Jt (y) be linear in y over [ˆ y , yˆ +1] passing through the two points of (ˆ y , Jt (ˆ y )) and

12

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(ˆ y +1, Jt (ˆ y +1)). Therefore, we will treat Jt (y) as a piecewise linear function of y over (−∞, +∞). Secondly, for the computation of integrations, we only focus on some discrete realization values of the random yield. We choose a small real value  be the step size such that 1/ is an integer. Let uj = j for j = 0, 1, · · · , 1/. Then, ρt (Q, x) =

R1 0

Jt (x + uQ)du =

P1/

j=0

Jt (x + uj Q) and

number of local maximizers/minimizers of Jt (y) over [Xt , Yt ].

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other integrations can be computed in a similar way. Based on this understanding, there is finite

For simplicity, we denote the left derivative of Jt (y) at y ∈ (−∞, +∞) by Jt0 (y) in this section and also introduce the following two definitions.

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Definition 1 If there exists an initial stock level x0 such that it is optimal to order a positive quantity at x0 but optimal to order nothing for all initial stock levels which are on the right-hand side of x0 and near to x0 , then we call x0 a locally highest order point. Otherwise, it is optimal to order a positive quantity at x0 but optimal to order nothing for all initial stock levels which 2

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are on the left-hand side of x0 and near to x0 , then we call x0 a locally lowest order point.

Definition 2 If K + Jt (x0 + Qt (x0 )) = Jt (x0 ) for some initial stock level x0 in period t, then we 2

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call x0 a critical order point.

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Note that both locally highest order points and locally lowest order points are critical order points and we have K + Jt (x0 + Qt (x0 )) = Jt (x0 ) for any critical order point x0 in period t. At

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a critical order point, we can identify a sufficient condition such that it is optimal to order a positive quantity for all initial stocks which are on the left-hand side but near to this critical

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order point.

Lemma 5 If Jt0 (x0 )Qt (x0 ) + K < 0 for a critical order point x0 ∈ [Xt , Yt ], then there exists δ0 > 0 such that it is optimal to order a positive quantity for all initial stock levels in [x0 − δ0 , x0 ]. 2 Suppose that there are k(t) locally highest order points at the beginning of period t and we denote them by st1 > · · · > sti > · · · > stk(t) . It is obvious that sti ∈ (Xt , Yt ) for any 1 ≤ i ≤ k(t). 13

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If k(t) = 1 in period t, then the threshold policy is optimal at the beginning of period t. The optimal policy for all initial stock levels in-between Xt and Yt is provided in the following result. Theorem 4 If Jt (y) is piecewise linear in y over [Xt , Yt ], then there exists a sequence of initial t stocks levels Xt = rk(t) ≤ stk(t) · · · < rit ≤ sti < · · · < r1t ≤ st1 = st such that it is optimal to order k(t)

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a positive order quantity for any initial stock level x ∈ ∪i=1 [rit , sti ]; optimal to order nothing for k(t)

any initial stock level x ∈ [Xt , Yt ]\ ∪i=1 [rit , sti ].

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We can identify a sufficient condition such that the threshold policy is optimal at the beginning of period t, i.e., k(t) = 1 for period t. Based on the proof of Theorem 4, we get the following

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result.

Corollary 2 If Jt (x) is piecewise linear and Jt0 (x)Qt (x) + K ≤ 0 for any x ≤ st1 = st in period t, then the threshold policy is optimal.

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The requirement in the above corollary is not very restrictive. If Jt (x) is decreasing fast and the optimal order quantity Qt (x) is not small for any x ≤ st1 (note that Qt (x) is decreasing in

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terms of x), then the threshold policy is the optimal inventory control policy in period t. So, if the fixed order cost is zero, then the requirement is automatically satisfied and the threshold

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policy is optimal in all periods. In the remaining portion of this section, we would like to focus

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on the search of Qt (x) for any given initial stock level x ∈ [Xt , Yt ] in period t. By the proof of Lemma 3, we know that Qt (x), the global minimizer of ρt (Q, x) in Q over

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[0, +∞), can only be in a closed interval such that Jt (y) is strictly increasing and −ρt (Q, x) is unimodal in Q over that interval. Note that Jt (y) is strictly decreasing at Xt by Proposition 2 and strictly increasing at Yt by w(t)

the definition of Yt . Let a1t < · · · < ait < · · · < at w(t)−1

[Xt , Yt ] and b1t < · · · < bit < · · · < bt

w(t)

respectively. For convenience, we define bt

be all the local minimizers of Jt (y) over

be all the local maximizers of Jt (y) over (Xt , Yt ), = +∞. Then, Jt (y) is strictly increasing in y over

w(t)

w(t)

∪i=1 [ait , bit ] and strictly decreasing in y over [Xt , Yt ] \ ∪i=1 [ait , bit ]. 14

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For any given x ∈ [Xt , Yt ], if Jt (y) is always increasing over [x, +∞), then Qt (x) = 0; x(t)

Otherwise, let x(t) be the first index ∈ {1, 2, · · · , w(t)} such that at

≥ x. Let Qit (x) be

the minimizer of ρt (Q, x) over [ait , bit ] for any x(t) ≤ i ≤ w(t) and it can be found efficiently through binary search. Then, we have the following result.

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Proposition 3 For any given initial stock level x in period t, if Jt (y) is increasing over [x, +∞), then Qt (x) = 0; Otherwise, Qt (x) can be chosen from a finite discrete set {Qit (x)|x(t) ≤ i ≤ w(t)}.

Some Observations about the Optimal Policy

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5

2

The presence of the random yield in the periodic review inventory control problems has made the structure of the optimal policy very complicated. If the fixed order cost is zero, then it is well known that the threshold policy is optimal. Because the optimal order quantity nonlinearly

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depends on the initial stock level in any period, some heuristic approaches are proposed in the literature to approximate the optimal policy. If there is a positive fixed order cost, to the best

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of our knowledge, no research work has been published in the literature for the periodic review inventory control problems with random yield.

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In the following, we observe a few phenomena about the structure of the optimal inventory

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control policy. These phenomena further show that the structure of the optimal policy is very complicated and quite different from the well known (s, S) policy for the periodic review inventory

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problems with both positive fixed order cost and certain yield. We will illustrate these interesting phenomena by considering a very special setting: uniform random yield over [0, 1], deterministic constant demand d > 0, and linear holding/shortage cost in each period. Let y ∈ (−∞, +∞) be the on-hand stock level after receiving the order at the beginning of one period, then the holding/backlogging cost L(y) in one period is h(y − d) for any y ∈ [d, +∞) and b(d − y) for any y ∈ (−∞, d). For simplicity, let cˆ = c(1 − γ). Thus, H(y) = cˆy + L(y) + cdγ

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can be expressed as: H(y) =

      

cd + (h + cˆ)(y − d), if y ∈ [d, +∞) cd + (b − cˆ)(d − y), if y ∈ (−∞, d).

Without loss of generality, we assume that b > cˆ. Differentiation with respect to x on both sides

we get Q01 (x)Q1 (x) =

b+h (x cˆ+h

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of the first order condition in eqn. (3), which is satisfied by Q1 (x) in the last period problem, − d). Note that Q1 (d) = 0. Thus, for any x ≤ d, we get Q1 (x) =

s

b+h (d − x) cˆ + h

(4)

and it is linearly decreasing in x. Note that we denote the highest initial stock level to order a

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positive quantity at the beginning of the last period by s1 , which can be obtained by solving this equation: (b − cˆ)(d − s1 ) = K + (ˆ c + h)[s1 + Q1 (s1 ) − d]. Hence, we get

As H(x) − H(x + Q1 (x)) = [(b + h) −

K

(b + h) −

q

q

(b + h)(ˆ c + h)

.

(b + h)(ˆ c + h)](d − x) is decreasing in x, similarly as in

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s1 = d −

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the proof of Theorem 1, the threshold policy is optimal, i.e., it is optimal to order at any initial stock level x ∈ (−∞, s1 ]. Therefore, we get the optimal cost function F1 (x) for the last period: if x ∈ [d, +∞),

cd + (b − cˆ)(d − x),

if x ∈ [s1 , d),

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cd + (ˆ c + h)(x − d),

      

q

K + cd + [ (b + h)(ˆ c + h) − (ˆ c + h)](d − x), if x ∈ (−∞, s1 ).

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F1 (x) =

       

q

Note that the slope value of [ (b + h)(ˆ c + h) − (ˆ c + h)] is strictly less than (b − cˆ) but larger

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than (ˆ c + h) if b > 4ˆ c + 3h. By utilizing the above expression of F1 (x), in the following we can get a few very interesting observations about the optimal policy structure.

5.1

Dependence of the Y-Bound on the Demand Realizations

For the dynamic inventory control systems with certain yield, it is well known that S, the minimizer of the single period expected holding/shortage cost function H(y), is a very good 16

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Y −bound in all periods, i.e., it is optimal to order nothing at any initial stock level x ∈ [Y, +∞) in all periods where Y = S. But this property is gone under random yield situation and we will illustrate this in the following. For t = 1, it is obvious that S = d. For t = 2, note that J2 (y) = H(y) + γF1 (y − d). Based on the above closed form expression of F1 (x), if b > ( γ+1 )2 (ˆ c + h) − h, then J2 (y) is strictly γ

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decreasing over (−∞, 2d] and strictly increasing over [2d, +∞) (refer to Figure 4).

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Figure 4: Illustration of J2 (y)

Let s2 be the highest initial stock level to order a positive quantity at the beginning of period

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2. Please note that the difference of 2d and s2 only depends on the shape of J2 (y) but does not depend on d. When d > 2d − s2 , then the first optimal order point s2 at the beginning of

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period 2 should be higher than S, which is the minimizer of H(y) and is equal to the constant demand d. Even further, it is easy to show that s2 is strictly increasing in d. Therefore, the

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Y −bound depends on the demand realization. The reason behind this result is that F1 (x) is strictly decreasing over (−∞, s1 ], not flat like the situation with fixed order cost and certain

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yield.

5.2

Non-Optimality of the Generalized Linear Inflation Policy

If the fixed order cost is zero, then it is well known that the Linear Inflation (LI) policies are performing very well in the stochastic periodic review inventory control systems with random yield. An LI policy has two parameters θ and β. Under this policy, the optimal order quantity 17

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Qt (x) at the initial stock level x in any period t is given by: Qt (x) =

      

β(θ − x), if x ∈ (−∞, θ], if x ∈ (θ, +∞).

0,

We call that θ(> 0) is the target inventory level and β(≥ 1) is the inflation factor.

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If the fixed order cost is positive, Zipkin (2000) suggests an analogue of LI policies: choose (r, s) and a constant β such that β(s − x) will be ordered if the initial stock level x is below r; Otherwise, nothing is ordered. For convenience, we call this type of policies as Generalized Linear Inflation (GLI) policies and we denote it by (r, s, β). The key observation is that the optimal GLI policy parameter values depend on the given initial stock level x. This is summarized in the

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following proposition by checking the two-period problem.

Proposition 4 Generalized Linear Inflation (GLI) policies are not optimal.

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In general, the GLI policies are not optimal for the multi-period model with random yield.

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For the single period model with deterministic demand and uniform random yield, GLI policies are indeed optimal but the inflation factor is dependent on both the random yield distribution

Non-Continuity of the Optimal Order Quantity

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5.3

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and the cost factors.

For the traditional stochastic periodic review inventory control systems with certain yield, it

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is well known that the optimal order quantity Qt (x) at the initial stock level x in period t is continuous in terms of x. Even further, if the fixed order cost is zero, we know that the

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optimal order quantity Qt (x) is also continuous for the stochastic periodic review inventory control systems with random yield. If there is a positive fixed order cost, then we have the following result.

Proposition 5 For the stochastic periodic review inventory control systems with both random yield and positive fixed order cost, the optimal order quantity Qt (x) can be non-continuous in terms of the initial stock level x ∈ (−∞, +∞) in period t. 18

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Proof. In order to show the non-uniqueness of the optimal order quantity in terms of the initial stock level, let us look at the behavior of the expected cost function J2 (y) with respect to the q

on-hand stock level y in period 2, which is illustrated in Figure 5 if γ < (

b+h cˆ+h

− 1)−1 . Even

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further, under this situation, the difference of J2 (2d) − J2 (d) tends to +∞ when d tends to +∞.

Figure 5: Illustration of J2 (y)

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It is optimal to order Q2 (s1 ) at s1 and we assume that s1 + Q2 (s1 ) < J2 (2d). Note that the balance level for J2 (y) with y from d + s1 to d + s1 + Q2 (d + s1 ) is higher than the value of

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J2 (2d). This means that the balance level of z = J2 (s1 + Q2 (s1 )) is less than the balance level of z = J2 (y2 ). When the initial stock level x is decreasing from s1 , we can see that Q2 (x) is not

Conclusion

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6

2

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continuous.

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We have investigated the inventory control problems with both positive fixed order cost and uniform random yield in a periodic-review stochastic demand environment where the excess demand is completely backlogged. The structure of the optimal inventory control policy has been an open problem in the literature for decades and our paper, to the best of our knowledge, is the first contribution in the literature regarding this issue. We have provided an (X, Y ) structure in the paper and the search for the optimal order quantity at any initial stock level is

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much easier. We also have provided evidence to show that there are some significant differences between the periodic review inventory problems with certain yield and the ones with random yield. Our (X, Y ) structure can provide a clear picture regarding the optimal policy if the initial stock levels are below X or above Y . But for any initial stock level that falls in between X and

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Y , a partial characterization of the optimal policy is provided and more research efforts should be required regarding the search of the optimal order quantity in the future. A sufficient condition is also provided such that the threshold policy is optimal in each period.

One restriction in our paper is that we assume uniform random yield. Even if this is the case,

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the corresponding analysis is very complicated. This paper is the first effort in the literature to address both fixed order cost and random yield in a periodic review stochastic demand environment. Our next step of research is to investigate the periodic review inventory system with both fixed order cost and general random yield. Under the general random yield situation, the

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very intuitive explanation about the first order condition in eqn (3) will disappear. So one basic question is how to rewrite the first order condition such that a geometric explanation will help

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us in the analysis of the optimal policy structure for the periodic review inventory systems with both fixed order cost and general random yield. Song (2016) has made some progress regarding

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this matter and he can show the existence of the (X, Y ) bound structure under a very mild assumption about the density function of the general random yield. But there is no any charac-

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terization about the search of the optimal order quantity at any initial stock level in between X and Y . The methodology in Song (2016) is completely different from the one in this paper and

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the optimal policy characterization for the uniform yield situation in this paper is much stronger than the one in Song (2016) for the general random yield situation. For example, Song (2016) cannot show the monotonically decreasing property of the optimal order quantity in terms of the initial stock over (−∞, +∞) and he can only show this monotonically decreasing property over (−∞, X]. More efforts should be required in the future to characterize the optimal policy structure for the periodic review systems with both fixed order cost and the general random 20

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yield.

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References [1] Bassok, Y., and R. Akella. 1991. Ordering and production decision with supply quality and demand uncertainty. Management Science 37, 1556-1574. [2] Bollapragada, S., and T. E. Morton. 1999. Myopic heuristics for the random yield problem.

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Operations Research 47(5), 713-722.

[3] Chen, J., D. Yao, and S. Zheng. 2001. Optimal replenishment and rework with multiple unreliable supply sources. Operations Research 49, 430-443.

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[4] Henig, M. and Y. Gerchak. 1990. The structure of periodic review policies in the presence of random yield. Operations Research 38(4), 634-643.

[5] Huh, W.T. and M. Nagarajan. 2010. Linear inflation rules for the random yield problem: Analysis and computations. Operations Research 58(1), 244-251.

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[6] Inderfurth, K. and G. P. Kiesmuller. 2015. Exact and heuristic linear-inflation policies for

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an inventory model with random yield and arbitrary lead times. European Journal of Operational Research 245, 109-120.

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[7] Inderfurth, K. and S. Transchel. 2007. Note on ”Myopic heuristics for the random yield

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problem”. Operations Research 55(6), 1183-1186. [8] Inderfurth, K. and S. Vogelgesang. 2013. Concepts for safety stock determination under

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stochastic demand and different types of random yield. European Journal of Operational Research 224, 293-301.

[9] Lee, H.L., and C.A. Yano. 1988. Production control for multistage systems with variable yield loses. Operations Research 36, 269-278. [10] Li, Q., and S. Zheng. 2006. Joint inventory and replenishment and pricing control for systems with uncertain yield and demand. Operations Research 40, 434-444. 22

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[11] Li, Q., H. Xu, and S. Zheng. 2008. Periodic-review inventory systems with random yield and demands: bounds and heuristics. IIE Transactions 54, 696-705. [12] Mazzola, J.B., W.F. McCoy and H.M. Wagner. 1987. Algorithms and heuristics for varaibleyield lot sizing. Naval Research Logistics 34, 67-86.

York.

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[13] Nahmias, S. 2005. Production and Operations Analysis, 5th edition. McGraw-Hill, New

[14] Ozekici, S. and M. Parlar. 1999. Inventory models with unreliable suppliers in a random

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environment. Annals of Operations Research 91, 123-136.

[15] Panagiotidou, S., G. Nenes, C. Zikopoulos. 2013. Optimal procurement and sampling decisions under stochastic yield of returns in reverse supply chains. OR Spectrum 35, 1-32. [16] Scarf, H. 1960. The Optimality of (s, S) Policies in the Dynamic Inventory Problems. Chap.

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13 in Arrow, K. J., S. Karlin, and P. Suppes(eds), Mathematical Methods in the Social

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Sciences, Stanford University Press.

[17] Song, Y. 2016. Periodic review inventory systems with both fixed order cost and general

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random yield. Working paper, Memorial University of Newfoundland. [18] Veinott, A., and H. M. Wagner. 1965. Computing optimal (s, S) inventory policies. Man-

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agement Science 11(5), 525-552.

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[19] Wang, Y. and Y, Gerchak. 1996. Periodic review production models with variable capacity, random yield, and uncertain demand. Management Science 42, 130-137.

[20] Yano, C. A., and H. L. Lee. 1995. Lot sizing with random yields: a review. Operations Research 43(2), 311-334. [21] Yao, D.D. 1988. Optimal run quantities for an assembly system with random yield. IIE Transactions 20(4), 399-403. 23

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[22] Zheng, Y., and Fredergruen, A. 1991. Finding optimal (s, S) policies is about as simple as evaluating a single policy. Operations Research 39, 654-665. [23] Zipkin, P. H. 2000. Foundations of Inventory Management. The McGraw-Hill Companies,

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Inc.

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Appendix Proof of Theorem 1: Let P (x) = H(x) −

R1 0

H(x + uQ1 (x))du. In order to prove the theorem, it is sufficient to show that

Also note that Q1 (x) satisfies can be simplified as

R1 0

Z 1 0

[H 0 (x) − H 0 (x + uQ1 (x))(1 + uQ01 (x))]du.

H 0 (x + uQ1 (x))udu = 0. By this fact, the above expression of P 0 (x)

P 0 (x) =

Z 1 0

[H 0 (x) − H 0 (x + uQ1 (x))]du.

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P 0 (x) =

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P (x) is decreasing in x over (−∞, S). Differentiation of P (x) with respect to x, we get

Since H(y) is convex, we get [H 0 (x) − H 0 (x + uQ1 (x))] ≤ 0 for any u ∈ [0, 1], i.e., P 0 (x) ≤ 0. Therefore, if it is optimal to order a positive quantity at an initial stock x, then it is also optimal to order a positive quantity at any initial stock level below x, i.e., the threshold policy is optimal.

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Proof of Lemma 2:

2

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¯ for any y ∈ (−∞, +∞) by Assumption 1. For t = 1, we know that J1 (y) = H(y) and −A¯ ≤ J10 (y) ≤ B For any given initial stock level x at the beginning of period 1, let us consider three cases. For the first case, there exists a δ > 0 such that it is optimal to order nothing for any y ∈ (x − δ, x + δ). Then, we

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¯ For have F1 (y) = J1 (y) for any y ∈ (x − δ, x + δ) and F10 (x) = J10 (x). Hence, we have −A¯ ≤ F10 (x) ≤ B. the second case, there exists a δ > 0 such that it is optimal to order for any y ∈ (x − δ, x + δ). Then,

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R1 0

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we have F1 (y) = K +

J1 (y + uQ1 (y))du and F10 (x) = −A¯ ≤ F10 (x) =

Z 1 0

R1 0 0 J1 (x + uQ1 (x))du. Hence, we get

¯ J10 (x + uQ1 (x))du ≤ B.

¯ based on both the right derivative and the For the remaining case, we always have −A¯ ≤ F10 (x) ≤ B left derivative. Therefore, the first part of the lemma always holds true when t = 1. Suppose that the result holds true for period t. Then, we need to investigate period t + 1. Note that Jt+1 (y) = H(y) + γ −

R +∞ 0

0 (y) = H 0 (y) + γ Ft (y − v)g(v)dv, hence we get Jt+1

R +∞ 0

Ft0 (y − v)g(v)dv and

t t+1 1 − γ t+1 ¯ 1 − γt ¯ 0 ¯ + γ1 − γ B ¯ = 1 − γ B. ¯ A = −A¯ − γ A ≤ Jt+1 (y) ≤ B 1−γ 1−γ 1−γ 1−γ

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t+1 0 ¯ Similarly to the above argument for t = 1, from the above inequality, we get − 1−γ 1−γ A ≤ Ft+1 (x) ≤

1−γ t+1 ¯ 1−γ B.

Therefore, the first part of the lemma always holds true for any t. Let t tend to infinity, the

second part of the lemma is obvious and we have completed the proof.

2

Proof of Theorem 2:

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Note that the demand realizations in each period may be either bounded or unbounded. Thus, we need to consider two different cases regarding the demand realizations in each period. • Case 1: Bounded demand realizations in each period.

Suppose that the demand realizations in each period are bounded by D. Under this situation,

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we choose Y = S + 1 + N D. For any t ≤ N , it is obvious that Jt (y) is strictly increasing in terms of y over [Y, +∞). For any t > N , then Jt (y) can be obtained by Jt−N (y) after N 1 ¯ B} ¯ for any y ∈ [S, +∞) in any period iterations. By Lemma 2, we get Jk0 (y) ≥ − 1−γ max{A,

k. For any given period t − N , after N iterations (i.e., at the beginning of period t) we have γN ¯ ¯ 1−γ max{A, B}

>

for any y ∈ [Y, +∞). Hence, Jt (y) is also strictly increasing in

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terms of y over [Y, +∞).

B 2

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Jt0 (y) ≥ B −

• Case 2: Unbounded demand realizations in each period.

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Under this case, for any period t, we can select a non-period dependent bound D0 for the demand realizations such that

Z +∞

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|

D0

0 Ft−1 (y − v)g(v)dv| ≤

Z +∞ D0

1 B ¯ B}g(v)dv ¯ max{A, ≤ . 1−γ 4

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Similarly to the above bounded demand case, we can find a constant m0 such that H 0 (y) + γ

Z D0 0

0 Ft−1 (y − v)g(v)dv ≥

B 2

for any y ∈ [S + 1 + m0 , +∞) in period t. Let Y = S + 1 + m0 , by combining the above two facts, we get Jt0 (y) ≥

B 4

for any y ∈ [Y, +∞) in period t. Thus, in this case, Jt (y) is also strictly

increasing over [Y, +∞) and we have completed the proof.

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Proof of Lemma 3: For simplicity, in this proof we denote Jt (y) and Qt (x) by J(y) and Q(x), respectively. Since it is R1

optimal to order Q(x)(> 0) at the initial stock level x, J(x) ≥ K + 0

R −1 x+Q

J(x + uQ)du = Q

condition:

x

0

J(x + uQ(x))du. Note that

J(y)dy for any Q ∈ (0, +∞) and Q(x) satisfies the following first order

Q(x)J(x + Q(x)) =

Z x+Q(x)

J(y)dy.

x

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R1

R x+Q(x)

Rewriting the above equation, we get K + J(x + Q(x)) = K + Q(x)−1

x

uQ(x))du. As it is optimal to order at the initial stock level x, J(x) ≥ K + combining these two facts together, we get the first result of the lemma.

R1 0

J(y)dy = K +

R1 0

J(x +

J(x + uQ(x))du. Thus,

dJ(y) dy |{y=x+Q(x)}

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Then, we show that J(y) is strictly increasing at x + Q(x). In the remaining of this proof, if does not exist, we still use the same notation to represent both the left and the right

derivatives of J(y) at x + Q(x).

Differentiation with respect to x on both sides of the above first-order condition, we get Q0 J(x +

M

Q) + QJ 0 (x + Q)(1 + Q0 ) = J(x + Q)(1 + Q0 ) − J(x) and this can be simplified as

It is obvious that

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QJ 0 (x + Q)(1 + Q0 ) = J(x + Q) − J(x). dJ(y) dy |{y=x+Q(x)}

6= 0; Otherwise, we will get J(x + Q(x)) − J(x) = 0 and this is dJ(y) dy |{y=x+Q(x)}

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not possible by the first result of the lemma. Suppose that

contradiction. Let us consider the function of Γ(Q) = QJ(x + Q) −

R x+Q x

< 0. Then, we will get a

J(y)dy. Based on the above

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first-order condition, we know that Γ(Q(x)) = 0. Its first derivative value at Q(x) is Q(x)J 0 (x + Q(x)) and it is less than zero. Hence, Γ(Q) is zero at Q = Q(x) and also strictly decreasing at Q(x). Note R1

d(

0

J(x+uQ)du) dQ

=

Γ(Q) . Q2

This means that Q(x) is the local maximizer of

AC

that

a contradiction with the definition of Q(x), the global minimizer of We have completed the proof.

R1 0

R1 0

J(x + uQ)du and this is

J(x + uQ)du in Q over [0, +∞). 2

Proof of Lemma 4: Suppose that the theorem does not hold true and we have x + Qt (x) ≤ x ˆ + Qt (ˆ x). Then, we will get a contradiction. If Jt (y) is increasing over [x + Qt (x), x ˆ + Qt (ˆ x)], then we should have Sˆ1 = Sˆ2 and

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S¯1 = S¯2 (refer to Figure 6) by the intuitive explanation of the first order condition. But it is obvious

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that S¯1 > Sˆ1 and S¯2 ≤ Sˆ2 and this is a contradiction.

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Figure 6: Jt (y) is increasing over [x + Qt (x), xˆ + Qt (ˆ x)]

If Jt (y) is not increasing over [x + Qt (x), x ˆ + Qt (ˆ x)], based on the relationship between Jt (x + Qt (x)) and Jt (ˆ x + Qt (ˆ x)), we divide the remaining discussion into two cases. Before the detailed discussion for each case, note that Jt (y) is strictly increasing near both x + Qt (x) and x ˆ + Qt (ˆ x) by Lemma 3.

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(a) Jt (y) is not increasing over [x + Qt (x), x ˆ + Qt (ˆ x)] and Jt (x + Qt (x)) ≤ Jt (ˆ x + Qt (ˆ x)). Note that Jt (y) ≥ Jt (x + Qt (x)) for any y ∈ [x, x ˆ]. Thus, the area size above the level line of

ED

z = Jt (x+Qt (x)) but below Jt (y) is less than the area size below the level line of z = Jt (x+Qt (x)) but above Jt (y) over [ˆ x, x + Qt (x)] by the intuitive explanation of Qt (x). This means that

PT

ˆ >Q ˆ −1 Jt (x + Qt (x)) = Jt (ˆ x + Q)

Z xˆ+Qˆ x ˆ

ˆ Jt (ˆ x + uQ)du ≥ Jt (ˆ x + Qt (ˆ x)) ≥ Jt (x + Qt (x)),

CE

ˆ = (x + Qt (x)) − x where Q ˆ. The last two inequalities are based on the definition of Qt (ˆ x) and the assumption of Jt (x + Qt (x)) ≤ Jt (ˆ x + Qt (ˆ x)) under this case. The above is obviously a

AC

contradiction and Case (a) is not possible. (b) Jt (y) is not increasing over [x + Qt (x), x ˆ + Qt (ˆ x)] and Jt (x + Qt (x)) > Jt (ˆ x + Qt (ˆ x)). ˆ=x Let Q ˆ + Qt (ˆ x) − (x + Qt (x)). First, based on the definition of Qt (x), we get Z xˆ+Qt (ˆx) x+Qt (x)

ˆ Jt (y)dy ≥ Jt (x + Qt (x))Q.

Otherwise, we consider Jt (y) over the interval [x, x ˆ + Qt (ˆ x)] and we can find a lower value of R1 0

Jt (x + uQ) than Jt (x + Qt (x)), which is not possible by the definition of Qt (x). Secondly, we

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consider Jt (y) over [ˆ x, x + Qt (x)], we must have

R x+Qt (x) x ˆ

Jt (y)dy ≥ Jt (ˆ x + Qt (ˆ x))[x + Qt (x) − x ˆ];

otherwise, there will be a contradiction with the definition of Qt (ˆ x) and we can find a lower value of

R1 0

Jt (ˆ x + uQ)du than Jt (ˆ x + Qt (ˆ x)). As a consequence of this, considering Jt (y) over

[x + Qt (x), x ˆ + Qt (ˆ x)], we get

x+Qt (x)

Jt (y)dy =

Z xˆ+Qt (ˆx) x ˆ

Jt (y)dy −

Z x+Qt (x) x ˆ

Jt (y)dy

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Z xˆ+Qt (ˆx)

≤ Qt (ˆ x)Jt (ˆ x + Qt (ˆ x)) − Jt (ˆ x + Qt (ˆ x))[x + Qt (x) − x ˆ] ˆ = Jt (ˆ x + Qt (ˆ x))Q.

Note that Jt (x + Qt (x)) > Jt (ˆ x + Qt (ˆ x)) under this Case (b), combing this and the above two

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inequalities, we get a contradiction and hence Case (b) is also not possible.

By combining Cases (a) and (b) together, it is also not possible if Jt (y) is not increasing over [x + Qt (x), x ˆ + Qt (ˆ x)]. Thus, our assumption at the beginning of the proof is not true and we must have

Proof of Proposition 1:

2

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x + Qt (x) > x ˆ + Qt (ˆ x). We have completed the proof.

ED

We can choose a very small positive number δ > 0 such that Jt (y) ≥ Jt (ˆ x+Qt (ˆ x)) and |Jt (y)−Jt (ˆ y )| < K for any y, yˆ ∈ [ˆ x − δ, x ˆ] because Jt (y) is continuous in y over (−∞, +∞). We also know that it

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is optimal to order a positive quantity at any initial stock level y ∈ [ˆ x − δ, x ˆ]. From this fact, we get Jt (y)−Jt (y +Qt (y)) ≥ K. Thus, it is not possible to have y +Qt (y) ≤ x ˆ and we must have y +Qt (y) > x ˆ.

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Therefore, all the requirements in Lemma 4 are satisfied regarding the initial stock levels y and x ˆ and we have y + Qt (y) > x ˆ + Qt (ˆ x) by Lemma 4, i.e., y + Qt (y) is strictly decreasing in y over [ˆ x − δ, x ˆ]. 2

AC

Continuing this process, then we can see that y + Qt (y) is strictly decreasing in y over [x, x ˆ].

Proof of Proposition 2: For t = 1, the theorem holds true from the analysis for the single period problem. Suppose that it holds true for t − 1 where t ≥ 2. Then, we will show that it also holds true for t. It is obvious that Jt (y) is decreasing in y on (−∞, S] by the above assumption for t − 1 and the expression of Jt (y) in Eqn. (1). In the following, we shall show that Ft (y) is also decreasing in y on

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(−∞, S]. If it is optimal to order a positive quantity Qt (x) at x (if multiple optimal order quantities are possible, then we always choose the largest one), by Lemma 3, there exists an interval of [x1 , x2 ) such that x + Qt (x) ∈ [x1 , x2 ) and Jt (y) is strictly increasing for any y ∈ [x1 , x2 ). For any small δ > 0, if it is optimal to order at x ˆ = x − δ, we know that x ˆ + Qt (ˆ x) ≥ x + Qt (x) know that x ˆ + Qt (ˆ x) ∈ (x + Qt (x), x2 ). Hence, Ft (ˆ x) = K +

R1 0

R1 0

Jt (x + uQ)du, we

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by Proposition 1. If δ is small enough, as Qt (x) is the largest global minimizer of

Jt (ˆ x + uQt (ˆ x))du = K + Jt (ˆ x + Qt (ˆ x)) ≥

K + Jt (x + Qt (x)) = Ft (x). If it is not optimal to order at x ˆ = x − δ, we still have Ft (ˆ x) = Jt (ˆ x) > Jt (x) ≥ Ft (x). We have completed the proof.

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Proof of Theorem 3:

2

Since H(y) is convex, it is obvious that it is optimal to order a positive quantity at any initial stock level below Xt based on the definition of X 1 where Xt = min{Xt0 , X 1 }.

Next, we want to show that −ρt (Q, x) is unimodal in terms of Q over [0, +∞) for any initial stock R1 0

Jt (x + uQ)du = Q−1

M

level x below Xt0 . Because ρt (Q, x) =

x

R x+Q

QJt (x + Q) − x ∂ρt (Q, x) = ∂Q Q2

ED

R x+Q

Jt (y)dy, then we get

Jt (y)dy

.

For any initial stock level x ≤ Xt0 , the balance level of z = Jt (x+Qt (x)) must be above z = Mt0 . For any

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level below the balance level of z = Jt (x + Qt (x)) (refer to Figure 2), it is obvious that

∂ρt (Q,x) ∂Q

any level above the balance level of z = Jt (x + Qt (x)) (refer to Figure 2), it is obvious that Hence, we complete the proof.

< 0; For

∂ρt (Q,x) ∂Q

> 0.

CE

2

AC

Proof of Lemma 5: Differentiation on both sides of the first order condition in eqn (3) with respect to x, we get the following: Jt0 (x + Qt (x))[1 + Q0t (x)] =

Jt (x + Qt (x)) − Jt (x) . Qt (x)

Let ∆(x) = Jt (x) − Jt (x + Qt (x)) and we want to investigate the behavior of ∆(x) in terms of x. Its first derivative can be expressed as the following: ∆0 (x) =

Jt0 (x)Qt (x) − Jt (x + Qt (x)) + Jt (x) . Qt (x)

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Note that we have Jt0 (x0 )Qt (x0 ) + K < 0 and Jt (x0 ) − Jt (x0 + Qt (x0 )) = K. Combining these two facts and the above expression of ∆0 (x) together, we get ∆0 (x0 ) < 0. Thus, for all initial stock levels very near to x0 from the left-hand side, it is optimal to order a positive quantity and we denote the smallest such initial stock level as x0 − δ0 . We complete the proof.

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Proof of Theorem 4:

2

Let x0 ∈ [Xt , Yt ] be the initial stock level that we would like to focus on. First, let x0 = st1 , which is the biggest locally highest order point in period t. Based on the slope of Jt (y) on the left-hand side of x0 , we consider three cases:

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Case 1: The slope of Jt (y) at the left-hand side of x0 is less than − QtK (x0 ) .

Under this case, we have Jt0 (x0 )Qt (x0 ) + K < 0. By Lemma 5, it is optimal to order a positive quantity at initial stock levels which are on the left-hand side of x0 and near to x0 . The interval of such initial stock levels at the left-hand side of x0 can be extended further leftwards until xˆ0

M

such that it is optimal to order a positive order quantity at any initial stock level x ∈ [xˆ0 , x0 ] and xˆ0 is the first critical order point on the left-hand side of x0 , i.e., Jt (x0 ) = Jt (xˆ0 + Qt (xˆ0 )) + K.

ED

Then, we let x0 be replaced by xˆ0 . If Jt0 (x0 )Qt (x0 ) + K < 0 still holds true, then we continue the above process under this case. At the end, we will have either Jt0 (x0 )Qt (x0 ) + K = 0 or

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Jt0 (x0 )Qt (x0 ) + K > 0.

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Case 2: The slope of Jt (y) at the left-hand side of x0 is equal to − QtK (x0 ) . Under this case, we have Jt0 (x0 )Qt (x0 ) + K = 0. Note that Qt (x) is strictly decreasing in terms

AC

of x by Lemma 4. Remember that in this section we assume that Jt (y) is piece-wise linear over (−∞, +∞). Let x ˆ0 be the highest breakpoint of Jt (y) on the left-hand side of x0 . Thus, for any

initial stock level x ∈ [ˆ x0 , x0 ], we have Jt0 (x) = Jt0 (x0 ) < 0 and Jt0 (x)Qt (x) + K < 0. Combining this fact and the expression of ∆0 (x) in the proof of Lemma 5, it is optimal to order a positive

quantity at any initial stock level x ∈ [ˆ x0 , x0 ]. Similarly as in Case 1, the interval of [ˆ x0 , x0 ] can be extended further leftwards until y0 such that it is optimal to order a positive order quantity at any initial stock level x ∈ [y0 , x0 ] and y0 is the first critical order point on the left-hand side of x0 , i.e.,

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Jt (y0 ) = Jt (y0 + Qt (y0 )) + K. Then, we let x0 be replaced by y0 . If either Jt0 (x0 )Qt (x0 ) + K = 0 or Jt0 (x0 )Qt (x0 ) + K < 0, we continue the process under either Case 2 or Case 1, respectively. At the end, we will have Jt0 (x0 )Qt (x0 ) + K > 0. Case 3: The slope of Jt (y) at the left-hand side of x0 is bigger than − QtK (x0 ) .

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Under this case, we have Jt0 (x0 )Qt (x0 ) + K > 0. It is optimal to order nothing on the left-hand side of x0 .

We can continue the process under Case 1 and Case 2 until the situation of Case 3 appears. Then, we get the interval of [r1t , st1 ] such that it is optimal to order a positive quantity at any initial stock level x ∈ [r1t , st1 ] and optimal to order nothing at initial stock levels which are on the left-hand side of r1t and

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near to r1t . Then, let x0 = st2 and repeat the above process and we complete the proof.

Proof of Proposition 4:

2

In order to show the result, we only need to focus on a special case: The two-period problem with

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uniform random yield and deterministic demand. First, we show that the highest ordering initial stock

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level at the beginning of period 2 is less than d + s1 but larger than d when both d and b are large enough. Secondly, we shall show the threshold policy is optimal in period 2. Finally, we will show that the optimal order quantity Q2 (x) is not linear in the initial stock level x. Thus, GLI polices are not

PT

optimal.

Recall that J2 (y) = H(y) + γF1 (y − d) for any y ∈ (−∞, +∞) and it is a piecewise linear function

CE

(please refer to Figure 4). Based on the expressions of both H(y) and F1 (x), it is obvious that −b4 > −b2 > −b3 > b1 if b and d are large enough.

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Note that J2 (y) is a summation of γF1 (y − d) and a strictly linearly increasing function of L(y)

on [d + s1 , +∞). Hence, it is not optimal to order for any initial stock level x ∈ [d + s1 , +∞) at the beginning of period 2. Then, when d is large enough, the highest ordering initial stock level s2 at the beginning of period 2 should be within [d, d + s1 ) (note that d − s1 is a constant and not dependent on d by the expression of s1 ). Secondly, we shall show that the threshold policy is optimal at the beginning of period 2. Differen-

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tiating on both sides of the first order condition satisfied by Q2 (x), we get Q02 (x)Q2 (x) =

J2 (x + Q2 (x)) − J2 (x) − Q2 (x)J20 (x + Q2 (x)) . J20 (x + Q2 (x))

Then, based on the linear assumption of Jt (y) around s2 and s2 + Q2 (s2 ), the above expression of Q02 (x) (a1 −a3 )+(b1 −b3 )x b1

Q2 (x) =

q

< 0. Hence, we get p

(b1 − b3 )x2 + 2(a1 − a3 )x + c0 / b1

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is simplified as Q02 (x)Q2 (x) =

where c0 is a constant. Also note that J2 (x)−J2 (x+Q2 (x)) = a3 +b3 x−a1 −b1 (x+Q2 (x)). Differentiation with respect to x, we get

p

b1 −b3 b1 ,

we get

d[J2 (x)−J2 (x+Q2 (x))] |{x=−∞} dx

= (b3 − b1 ) + b1

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q

Since Q02 (−∞) = −

d[J2 (x) − J2 (x + Q2 (x))] = (b3 − b1 ) − b1 Q02 (x). dx

q

b1 −b3 b1

< (b3 − b1 ) +

(b1 − b3 )2 = 0 (note that b1 > 0 and b3 < 0). Hence, [J2 (x) − J2 (x + Q2 (x))] is strictly decreasing at

any x near −∞. Based on the expression of Q2 (x), we know that Q2 (x) is either concave or convex in x. If Q2 (x) is convex, then [J2 (x) − J2 (x + Q2 (x))] is concave and decreasing in x over [d, s2 ]. If Q2 (x) is concave, then [J2 (x) − J2 (x + Q2 (x))] is strictly convex and we will show that there does not exist d[J2 (x)−J2 (x+Q2 (x))] dx

= 0, i.e., Q02 (x0 ) =

M

x0 ∈ [d, s2 ] such that

b3 −b1 b1 .

Suppose that this x0 does exist, then

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[J2 (x) − J2 (x + Q2 (x))] should be strictly increasing at s2 and it is optimal to order at any initial stock level above s2 , i.e., this is a contradiction with the definition of s2 . Thus, if Q2 (x) is concave, we also have that [J2 (x) − J2 (x + Q2 (x))] is decreasing in x over [d, s2 ]. As a summary, we always have that

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[J2 (x) − J2 (x + Q2 (x))] is strictly decreasing in x over [d, s2 ]. Similarly to the above argument (note

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that a3 and b3 should be replaced by a4 and b4 ), we also have that [J2 (x) − J2 (x + Q2 (x))] is strictly decreasing in x over (−∞, d]. Therefore, the threshold policy is optimal at the beginning of period 2. Finally, we need to show that Q2 (x) in x is not linear over (−∞, s2 ] where s2 ∈ (d, d + s1 ). Suppose

AC

that Q2 (x) is linear, then we will get a contradiction. By the above expression of Q2 (x) if Q2 (x) is linear √ 3 )+(b1 −b3 )x for any x ∈ [d, s2 ]. Similarly, Q2 (x) = − (a1 −a √ 4 )+(b1 −b4 )x for on [d, s2 ], then Q2 (x) = − (a1 −a b1 (b1 −b3 )

b1 (b1 −b4 )

any x ∈ (−∞, d]. Note that J2 (y) is continuous at x = d, hence we get a1 − a3 + (b1 − b3 )d = a1 − a4 + (b1 − b4 )d. Substituting this into the two different expressions of Q2 (x) at x = d, we get b3 = b4 and this is a contradiction because b4 < b3 . Therefore, Q2 (x) is not linear and we have completed the proof.

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