Three Essays on Inventory Management

THREE ESSAYS ON INVENTORY MANAGEMENT

JIANG ZHANG

Submitted in partial fulfillment of the requirements

for the Degree of Doctor of Philosophy

Thesis Advisor: Dr. Matthew J. Sobel

Department of Operations

CASE WESTERN RESERVE UNIVERSITY

August 2004 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

______

candidate for the Ph.D. degree *.

(signed)______(chair of the committee)

______

(date) ______

*We also certify that written approval has been obtained for any proprietary material contained therein. I grant to Case Western Reserve University the right to use this work, irrespective of any copyright, for the University’s own purposes without cost to the University or to its students, agents and employees. I further agree that the University may reproduce and provide single copies of the work, in any format other than in or from microforms, to the public for the cost of reproduction.

JIANG ZHANG To my mother: Huishu Jiang

my father: Shuqing Zhang

my wife: Dr. Yan Cao Contents

ListofTables...... viii

ListofFigures...... ix

Acknowledgements...... x

Abstract...... xii

1 InventoryReplenishmentwitha FinancialCriterion...... 1

1.1Introduction...... 1

1.2ModelFormulation...... 7

1.3DynamicProgrammingAnalysis...... 11

1.4 Optimality of (sn, Sn)ReplenishmentPolicies...... 14

1.5 Inﬁnite Horizon Convergence ...... 18

1.6ModelswithSmoothingCosts...... 19

1.7ConcludingRemarks...... 23

v 2 Fill Rate of General Review Supply Systems...... 25

2.1Introduction...... 25

2.2GeneralPeriodicReviewSystem...... 29

2.3 Uncapacitated Single-stage Systems ...... 32

2.4 Gamma and Normal Demand in Single-stage Systems ...... 35

2.4.1 GammaDemandDistribution...... 35

2.4.2 NormalDemandDistribution...... 36

2.4.3 Fill Rate Approximation for Normal Demand Distribution 39

2.5Multi-StageGeneralReviewSystems...... 40

2.5.1 FillRateinTwo-StageSystems...... 42

2.5.2 Fill Rate in Two-Stage Systems with General Leadtime . . 48

2.5.3 NumericalExample...... 51

2.6FillRateinaThree-StageSystem...... 51

2.7Conclusion...... 56

3 Interchangeability of Fill Rate Constraints and Backorder Costs inInventoryModels...... 60

3.1Introduction...... 60

3.2ModelandProblemFormulations...... 66

3.3ContinuousDemand...... 72

3.4DiscreteDemand...... 76

3.5 Interchangeability ...... 79

vi 3.6Examples...... 82

3.6.1 StrictlyPositiveDemandDensity...... 83

3.6.2 NonStrictlyPositiveDemandDensity...... 84

3.6.3 DiscreteDemand...... 86

3.7GeneralizationsandSummary...... 89

3.8Appendix...... 91

Bibliography ...... 94

vii List of Tables

2.1 Fill Rate and its Approximation for Normal Demand ...... 57

2.2 Fill Rate of Two-stage Systems for Normal Demand (a) ...... 58

2.3 Fill Rate of Two-stage Systems for Normal Demand (b) ...... 59

3.4 Distribution Function and Expected Number of Backorders ...... 87

3.5 Values of G−1(·)andB−1(·) ...... 88

3.6 S-optimal Base-Stock Levels and Fill Rates at which they are F-optimal 88

3.7 F-optimal Base-Stock Levels and Unit Stockout Costs at which they are

S-optimal ...... 89

viii List of Figures

2.1 The standard N-stage serial inventory system ...... 30

2.2 The Fill Rate Integral for a system with Normal Demand . . . . 38

3.3 Dependence of S-Optimal Base-Stock Level on Stockout Cost: Non-

negativeDensity...... 84

3.4 Locus of {(b, f)} with the Same Optimal Base-Stock Level: Non-

negativeDensity...... 86

ix Acknowledgements

I would like to express my sincere gratitude to my mentor, Professor Matthew

J. Sobel, who encouraged and guided me through various phases of my doctoral studies with patience. I would also like to thank him for his incredible eﬀort and willingness to help me at any time and any where.

I would specially like to thank my dissertation committee members, Profes- sors Lisa Maillart, Peter Ritchken, and Yunzeng Wang for their generous insight, comments, and support on this work. In addition, my thanks are also owed to Pro- fessors Apostolos Burnetas, Hamilton Emmons, Kamlesh Mathur, Daniel Solow, and George Vairaktarakis for their help throughout my doctoral studies.

I would like to express my appreciation to the Department of Operations, Case

Western Reserve University, for their generous ﬁnancial support. Special thanks to department’s staﬀ, Elaine Iannicelli, Sue Rischar, and Emily Anderson for their help throughout my study in the department.

I had a fabulous time at Case which would not have been possible without the company of friends like Junze Lin, Zhiqiang Sun, Yuanjie He, Huichen Chiang,

x Wei Wei, Xiang Fang, Qiaohai Hu, Will Millhiser, Ant Printezis, Halim Hans, and

Kang-hua Li who have always helped and cheered me up in every possible way.

Finally, I would like to thank my family for their unconditional love, support and encouragement. My special thanks are for my wife Yan who is always there for me through everything.

xi Three Essays on Inventory Management

Abstract

JIANG ZHANG

This dissertation consists of three essays that are related to inventory management.

The first essay models a single-product equity-owned firm which orders products from an outside supplier, borrows short-term capital for solvency, and issues dividends to its shareholders while facing financial risks and demand uncertainty.

The firm maximizes the expected present value of the time stream of dividends. If there is a setup cost in this model, we show that an (s, S) replenishment policy is optimal by jointly optimizing the firm’s operational and financial decisions. The analysis is not a straightforward copy of Scarf’s argument. The second part of this essay studies the same model with a smoothing cost (instead of a setup cost) and shows that the optimal policy has the same form as the traditional smoothing

xii cost model. Although operational decisions and ﬁnancial decisions interact with

each other in these models, the optimal inventory policies have standard forms.

The second essay obtains ﬁll rate formulas for general review inventory models

with base-stock-level policies. Ordering decisions in a general review model are

made every R (R ≥ 1) periods but demand arises every period. We provide exact ﬁll rate formulas for single-stage model with a general demand distribution.

A simple fill rate expression is derived for the model with normally distributed demand. For multi-stage models, we first discuss a general review procedure at each stage and then provide exact fill rate formulas for two-stage and three-stage models.

There are parallel streams of literature which analyze identical models except that one stream has backorder costs and the other has fill rate constraints. The third essay clarifies redundancy in the two streams of dynamic inventory models with linear purchase costs. We show that optimal policies for either kind of model can be inferred from the other. That is, inventory fill rate constraints and backorder costs are interchangeable in dynamic newsvendor models.

xiii Chapter 1

Inventory Replenishment with a

Financial Criterion

1.1 Introduction

Nearly all the literature on optimal inventory management uses criteria of cost minimization or profit maximization. An inventory managers’ goal for example, is modeled as minimizing cost or maximizing profit while satisfying customers’ demands. If inventory decisions do not affect the revenue stream, these two criteria result in the same optimal replenishment policy. Most of this literature treats

firms’ inventory decisions and financial decisions separately. This dichotomy is perhaps due to the perception that inventory managers in a large firm cannot in-

fluence the firm’s financial policy and that financial officers are usually detached

1 2

from the inventory decisions. This separate consideration of financial and operational decisions simplifies management and has its foundation in corporate finance.

The pathbreaking papers, Modigliani and Miller (1958) and Modigliani and Miller

(1963) (hereafter referred to as M-M), show that the firm’s capital structure and its financial decisions should be independent of the firm’s investment and operational decisions if capital markets are perfect.

However, when market imperfections such as taxes and transaction fees are in- troduced, the results characterized from these separate treatments may no longer hold. “Treating real and financial decisions of the firm as independent may not be justified.”(Dammon and Senbet 1988). Other sources of market imperfections include asymmetric information between supplier and retailer, asymmetric information between shareholders and managers, and differential access to financial resources by different firms. For example, many small and medium-sized firms are cash constrained, and their operational decisions are heavily dependent on their financial decisions (such as short-term borrowing). Although the assumed independence of operations and finance has led to the development of intuitively appealing and insightful results, there remains the question of whether joint optimization of both the operational and financial decisions of a firm will generate new insights regarding firm behavior and perhaps overturn or modify existing results.

The M-M theorem not only allows separation of operations and ﬁnance, but also establishes that the ﬁrm’s optimal decisions are the same no matter if it 3

optimizes the value of the firm, dividends, or retained earnings. However, when capital markets are imperfect, the equivalence of different objectives is no longer valid. The literature on agency theory finds that corporate managers, the agents of shareholders, have conflicting interests with the shareholders. Those conflicts are primarily caused by dividends paid to shareholders. “Payouts to shareholders reduce the resources under managers’ control, thereby reducing managers’ control, and making it more likely they will incur the monitoring of the capital markets which occurs when firm must obtain new capital”(Jensen 1986, p. 1). So managers have to seek external funds to finance their projects. Since the external funds are usually unavailable for certain firms or available only at high prices, it somewhat reduces the profitability (or increases the operating costs) of the projects and may affect the performance evaluation of the managers. This conflict may discourage managers from disgorging the cash to shareholders and cause organizational inef-

ﬁciencies.

There is a literature on finance that recognizes the interdependence issues among a firm’s decisions. Most of this literature focuses on the effects of market imperfections on financial structures and decisions. Miller and Rock (1985) extend the standard finance model of the firm’s dividend/investment/financing decisions by allowing asymmetric information between the firm’s managers and outside investors and they show that there exists an equilibrium investment policy which leads to lower levels of investment than the optimum achievable under 4

full information. Myers (1974) shows how investment decisions, i.e., acceptance or rejection of projects, affect the optimal financial structure of a firm and why investment in turn, should be affected by leverage. Long and Racette (1974) shows that the cost of capital of a competitive firm facing stochastic demand is affected by the level of production. Hite (1977) examines the impact of leverage on the optimal stock of capital by the firm and its capital-to-labor ratio. Datan and Ravid (1958) analyze the interaction between the optimal level investment and debt financing. In their model, a firm faces an uncertain price and has to decide on its optimal level investment and debt simultaneously. They show that a negative relationship exists between investment and debt. Dammon and Senbet

(1988) extend capital structure model in DeAngelo and Masulis (1980) and study the eﬀect of corporate and personal taxes on the ﬁrm’s optimal investment and

ﬁnancing decisions under uncertainty. However, this literature does not address how a ﬁrm operates (quantitatively) by considering the interrelationships.

This paper considers interactions between operational and financial decisions and uses a dividend criterion. The primary purpose of this paper is to study how operational decisions are affected by jointly considering financial and operational decisions and using a non-traditional operations objective. We consider a single product equity-owned retail firm which periodically reviews its inventory and retained earnings. Every period the firm faces random demand and replenishes 5

stocks to satisfy customer demand. In addition, every R (a positive integer) periods, the ﬁrm issues a dividend to its shareholders. The ﬁrm seeks to maximize the expected present value of the time stream of dividends (also called shareholders’ wealth).

Shareholders’ wealth maximization is a widely accepted objective of the ﬁrm in the literature (cf. Hojgaard and Taksar 2000, Milne and Robertson 1996, Moyer,

McGuigan, and Kretlow 1990, Sethi 1996, and Taksar and Zhou 1998). This goal states that management should seek to maximize the present value of the expected future returns to the shareholders of the ﬁrm. These returns can take the form of periodic dividend payments or proceeds from the sale of stock.

If the objective of the ﬁrm is to minimize the expected present value of ordering, holding, and shortage costs and the capital market is perfect, the optimal replenishment policies have been characterized for a broad range of conditions. See

Porteus (1990), Graves et al. (1993), and Zipkin (2000) for details and references to the literature.

Some recent research addresses the coordination of ﬁnancial and operational decisions. Li, Shubik, and Sobel (2003) examine the relationship between decisions on production, dividends, and short-term loans in dynamic newsvendor inventory models. They show that there are myopic optimal base-stock policies associated with production decisions and dividend decisions. The present paper proceeds directly from Li, Shubik, and Sobel (2003) and augments their model 6

with a setup cost and smoothing costs. Buzacott and Zhang (1998) look at the interface of finance and production for small firms with limited borrowing. They maximize profit over a finite horizon using a mathematical programming model to optimize inventory and borrowing decisions, and they assume that the demand for the product is known. Archibald, Thomas, Betts, and Johnston (2002) assert that start-up firms are more concerned with the probability of survival than with profitability. They present a sequential decision model of a firm which faces an uncertain bounded demand and whose inventory replenishment decisions are constrained by working capital.

Buzacott and Zhang (2003) incorporate financial capacity into production decisions using an asset-based constraint on the available working capital in a single- period newsvendor model. They model the available cash as a function of assets and liabilities that will be updated according to the dynamics of the production activities. They analyze a leader-follower game between the bank and the retailer, and illustrate the importance of jointly considering production and financial decisions. Babich and Sobel (2002) consider capacity expansion and financial decisions to maximize the expected present value of a firm’s IPO. They treat the IPO event as a stopping time in an infinite-horizon Markov decision process, characterize an optimal capacity-expansion policy, and provide sufficient conditions for a monotone threshold rule to yield an optimal IPO decision. 7

The rest of the paper is organized as follows. Section 1.2 formulates the financial inventory model and §1.3 analyzes the corresponding dynamic program. The structure of the finite-horizon optimal replenishment policy is explored in §1.4 and extended to the infinite horizon optimal policy is discussed in §1.5. Section

1.6 studies the ﬁnancial inventory model with smoothing costs and characterizes optimal replenishment policies. Section 1.7 concludes the paper.

1.2 Model Formulation

We consider an equity-owned retail ﬁrm that sells a single product to meet uncer-

tain periodic demand and orders the product from a supplier with an ample supply.

The ﬁrm can make short-term loans, if necessary, to obtain working capital. As

discussed in the introduction, every R periods, dividends are issued to the shareholders and the objective of the firm is to maximize the expected present value of the time stream of dividends. Negative dividends are interpreted as capital subscriptions, a common phenomenon for young firms. The following chronology occurs in each period. The firm observes the level of retained earnings, wn,and the current physical inventory level, xn. A default penalty (or bankruptcy) p(wn) is assessed if wn < 0, but it is convenient to define p(·) as a function on .We assume that p(·) is convex nonincreasing on . Then the firm chooses the level of its short-term loan, bn, and the order quantity, zn. The restriction R =1sim- plifies the presentation and Section 6 substantiates that R = 1 is without loss of 8

generality.

At the beginning of each period, the ﬁrm also decides on the amount of dividend to declare, vn.ifvn ≥ 0 then it is a dividend issued to the shareholders; if vn < 0 then it is a capital subscription. Also at the beginning of the period, the

+ loan interest ρ(bn)(whereweassumeρ(·) is a convex increasing function on )is paid, and the ordering decision is implemented at a cost of Kδ(zn)+czn,whereK is an ordering setup cost, δ(zn)=1ifzn > 0, and δ(zn) = 0 otherwise. Then demand Dn in period n is realized, and sales revenue net of inventory cost, denoted

+ + g(yn,Dn), is received. For speciﬁty, let g(y,d)=r min{y,d}−h(y−d) −π(d−y)

where yn, r, h and π denote available goods in period n afterdelivery,unitsale price, holding cost and shortage penalty cost, respectively. However, we only use convexity of g(·,d)foreachd ≥ 0. Finally, the loan principal bn is repaid. We assume that the demands D1, D2, ···are independent nonnegative random variables and unmet demands are backlogged.

For convenience but without loss of generality, we assume that the order leadtime is 0. Therefore, the amount of goods that is available to satisfy demand in period n is

yn = xn + zn (1.1)

Let In be the internally generated working capital in period n:

In = wn − p(wn) − vn − czn − ρ(bn) − Kδ(zn) (1.2) 9

That is, In is working capital after the dividend is issued, loan interest and replenishment costs are paid, and before the loan is made and revenue and inventory costs are realized. The total working capital available at the beginning of period n is bn + wn and the residual cash left in the ﬁrm before sales is bn + In. We assume that there is an interest rate γ associated with In, that is, if working capital is positive, the ﬁrm will gain γIn if In ≥ 0orpayapenalty−γIn if In < 0.

Since excess demand is backlogged, the dynamics are as follows:

xn+1 = xn + zn − Dn

wn+1 =(1+γ)[wn − p(wn) − vn − czn − ρ(bn) − Kδ(zn)] + g(yn,Dn)

The first equation balances the flow of physical goods and the second equation balances the cash flow. Using (1.1) and (1.2), the balance equations become

xn+1 = yn − Dn (1.3)

wn+1 =(1+γ)In + g(yn,Dn) (1.4)

We assume that the loan and replenishment quantities are nonnegative:

bn ≥ 0andzn ≥ 0 (1.5)

The following liquidity constraint prevents the expenditures in period n from exceeding the sum of retained earnings and the loan proceeds:

wn + bn − ρ(bn) ≥ p(wn)+vn + czn + Kδ(zn) (1.6) 10

Given xn and wn, from (1.1) and (1.2) the decision variables in period n can be

speciﬁed as yn, In and bn instead of zn, vn and bn.

Let B denote the present value of the time stream of dividends and let β denote the single period discount factor :

∞ n−1 B = β vn (1.7) n=1

Remark Discount factor β can be regarded as the risk neutral discount rate for the shareholders and need not to be a constant every period. Letting β(xn,wn) be an endogenous random variable which depends on the levels of inventory and retained earnings will not alter the results.

For n =1, 2, ···,letHn denote the history up to the beginning of period n, namely,

Hn =(x1,w1,b1,I1,y1,D1, ···,xn−1,wn−1,bn−1,In−1,yn−1,Dn−1,xn,wn)

Let Ωn be the set of all possible Hn sequences. A policy is a nonanticipative rule for choosing y1,I1,b1,y2,I2,b2,....That is, a policy is a rule that, for each n,chooses

yn, In,andbn as a function of Hn.Anoptimal policy maximizes E(B|Hn = η) for each η ∈ Ωn, for all n =1, 2,.... Since a policy speciﬁes the three decisions each period (the amount of supply level, yn, the amount of internally generated working capital, In, and the short-term loan, bn), the ﬁrm can easily determine the order size and the amount of dividends to issue to shareholders by using (1.1) 11

and (1.2). The theme of §3and§4 is the characterization of an optimal policy, i.e., one that maximizes the expected value of (1.7) subject to (1.3)-(1.6).

1.3 Dynamic Programming Analysis

This section gives the dynamic programming equations which correspond to the

dynamics in §2. We then give a proposition that reduces the dimensionality of the decision space from three to two. From (1.2) and (1.3), vn = wn − p(wn) −

In − cyn + cxn − ρ(bn) − Kδ(yn − xn)andxn = yn−1 − Dn−1 (n>1), and using standard procedures (Veinott and Wagner 1965), substituting vn and xn in (1.7), then inserting (1.4) and rearranging terms yields

∞ n B = cx1 − β cDn n=1 ∞ n−1 + β wn − p(wn) − In − (1 − β)cyn − ρ(bn) − Kδ(yn − xn) n=1 ∞ ∞ n n−1 = cx1 + w1 − p(w1) − β cDn − β Kδ(yn − xn) n=1 n=1 ∞ n−1 + β −(1 − β)cyn − In + β (1 + γ)In + g(yn,Dn) n=1 −βp (1 + γ)In + g(yn,Dn) − ρ(bn)

For (b, I, y) ∈3 let

L(b, I, y)=−(1 − β)(I + cy)+βγI + βE g(y,D) − p (1 + γ)I + g(y,D) − ρ(b)

(1.8) 12

Then the expected present value of the dividends can be stated as

∞ ∞ n n−1 E(B)=cx1+w1−p(w1)−E β cDn +E β L(bn,In,yn)−Kδ(yn−xn) n=1 n=1 (1.9)

As in Veinott and Wagner (1965) and in many other references since then, we interpret L(·, ·, ·) as a generalized inventory reward function (revenue minus costs and penalties). Because the ﬁrst four terms in (1.9) do not depend on the decision variables, a policy maximizes (1.9) if and only if it maximizes the last term of (1.9).

So we utilize (1.8) and (1.9) and pursue the following objective:

∞ n−1 sup EH1 β [L(bn,In,yn) − Kδ(yn − xn)] n=1

subject to yn ≥ xn,bn + In ≥ 0, and bn ≥ 0. (1.10)

where the supremum is over the set of all policies, H1 =(x1,w1) is the initial state, and the constraints in (1.10) follow from (1.1), (1.2), (1.5), and (1.6).

It is convenient to analyze the following ﬁnite horizon counterpart of (1.10) and then let N →∞:

N n−1 sup EH1 β [L(bn,In,yn) − Kδ(yn − xn)] n=1

subject to yn ≥ xn,bn + In ≥ 0, and bn ≥ 0 (1.11)

A dynamic recursion that corresponds to (1.11) is ϕ0(·) ≡ 0andforeachx ∈ and n =1, 2,...,

ϕn(x)=sup{Jn(b, I, y) − Kδ(y − x):y ≥ x, I + b ≥ 0,b≥ 0} (1.12) 13

Jn(b, I, y)=L(b, I, y)+βE[ϕn−1(y − D)]. (1.13)

This dynamic program has one state variable, x, and three decision variables (b,

I, y), but the original problem has two state variables, x and w. This reduction occurs because equation (1.4) allows embedding state variable w into the decision variable I, which is also a surrogate decision variable for v.

Let bn(x), In(x)andyn(x) be optimal values of b, I and y in (1.12). The

following proposition states that borrowing should not exceed the amount needed

+ to cover current expenses. That is, bn(x)=[−In(x)] . The proof is similar to that of Proposition 3.2 in (Li, Shubik, and Sobel 2003).

Proposition 1.3.1 For all n =1, 2, ··· and x ∈, if the supremum in (1.12) is achieved, then b =(−I)+ without loss of optimality in (1.12).

Proof From (1.8), (1.12) and (1.13),

ϕn(x)=sup−(1 − β)(I + cy)+βγI + βE g(y,D) − p[(1 + γ)I + g(y,D)] b,I,y

−ρ(b)+βE[ϕn−1(y − D)] − Kδ(y − x):y ≥ x, I + b ≥ 0,b≥ 0

=sup−c(1 − β)y + βE[g(y,D)] + βE[ϕn−1(y − D)] − Kδ(y − x) y +sup −(1 − β)I + βγI − E p[(1 + γ)I + g(y,D)] I +sup −ρ(b):I + b ≥ 0,b≥ 0 : I ∈ : y ≥ x b

The last supremum is achieved by b =0ifI ≥ 0andbyb = −I if I<0 because 14

ρ(·) is monotone increasing. Therefore, ϕn(x)=sup − c(1 − β)y + βE g(y,D) + βE ϕn−1(y − D) − Kδ(y − x) y +sup−(1 − β)I + βγI − βE p((1 + γ)I + g(y,D)) I −ρ[(−I)+]:I ∈ : y ≥ x .

The next section uses Proposition 1.3.1, (1.12), and (1.13) to analyze the dynamic problem and establish conditions that guarantee the optimality of (s, S) policies.

1.4 Optimality of (sn, Sn) Replenishment Poli-

cies

This section establishes conditions under which the optimal ordering policy turns

out to be an (s, S)-type policy for the dividend criterion inventory model. An (s,

S) policy brings the level of inventory after ordering up to S if the initial inventory

level x is below s (where s ≤ S), and orders nothing otherwise. For a ﬁnite horizon

dynamic inventory problem in which the ordering cost is linear plus a ﬁxed setup

cost and the other one-period costs are convex, Scarf (1959) and Zabel (1962)

show that the optimal ordering policy is (sn, Sn). Iglehart (1963) shows that the limiting (s, S) policy characterizes the optimal policy for the inﬁnite horizon problem. Scarf’s proof uses the important concept of K-convexity. In this paper,

we use K-concavity. 15

Deﬁnition A real-valued function f(·)on is K − concave (K ≥ 0) if for all x ∈,λ≥ 0, and θ>0,

λ f(x + λ) − f(x) ≤ K + [f(x) − f(x − θ)] (1.14) θ

Properties of K-concave functions are analogous to those of K-convex functions.

Lemma 1.4.1 (a) f(·) is 0-concave ⇔ f(·) is concave on ;

(b) fi(·) is K-concave, i=1,2,...⇒ α1f1 +α2f2 is α1K1 +α2K2 concave (α1 >

0,α2 > 0);

(d) f(·) is K-concave ⇒ f(·) is continuous on .

Proof f(·)isK-concave if and only if −f(·)isK − convex. So (a) through (d)

follow from properties of K-convex functions (Scarf 1959).

For dynamic program (1.12) and (1.13), we use the following results to show that there is an optimal (s, S) policy.

Let µ and Q(·) be the mean and distribution function of D.LetG(y)=

E[g(y,D)].

Lemma 1.4.2 (a) p[(1 + γ)I + g(y,d)] is convex with respect to (I,y) ∈2 (for each d ≥ 0);

(b) L(·, ·, ·) is a concave function on its domain 3. 16

Proof (a) Since g(·,d) is a convex on for each d ≥ 0, (1+γ)I +g(y,d)isjointly convex in (I,d). The conclusion in (a) follows from monotonicity and convexity of p(·). p[(1 + γ)I + g(y,d)] is convex on I ∈,soisE{p[(1 + γ)I + g(y,D)]}.

(b) Concavity of L(·, ·, ·) follows from deﬁnition (1.8) and (a).

Theorem 1.4.3 If L(b, I, y) →−∞as |y|→∞for all b ≥ 0,andb + I ≥ 0, then there is an optimal (s, S)policy.

Note that the hypothesis uses concavity of L(·, ·, ·) and ensures the existence of maxima of L(b, I, ·). The proof of this theorem is not a paraphrasing of Scarf

(1959). Indeed, it exploits Proposition 1.3.1 to reduce the decision space from 3

to 2, and establishes the K-concave properties of embedded functions.

Proof The concavity of ϕ0(·) ≡ 0 initiates an inductive proof of K-concavity for each n.

For any n ≥ 1, if ϕn−1(·)isK-concave, then

Jn(b, I, y)=L(b, I, y)+βE[ϕn−1(y − D)]

is βK-concave ( 0 <β<1)iny because L(·, ·, ·) is concave and E[ϕn−1(y − D)] is βK-concave due to Lemma 1.4.1 (b).

To prove that ϕn(·)isK-concave, we show that Tn(·)isβK-concave where

+ Tn(y)=sup{Jn(b, I, y):b ≥ 0,b+I ≥ 0}; Proposition 1.3.1 asserts that b =(−I)

is optimal. So

Tn(y)=sup{L(b, I, y)+βE[ϕn−1(y − D)] : b ≥ 0,b+ I ≥ 0} 17

=sup{L(0,I,y)+βE[ϕn−1(y − D)] − ρ(b):b ≥ 0,b+ I ≥ 0}

+ = βE[ϕn−1(y − D)] + sup{L(0,I,y) − ρ[(−I) ]:I ∈}.

+ Let ζ(y)=supI∈{L(0,I,y) − ρ[(−I) ]}.Sinceζ(·):C →and C = {(I,y):

I ∈,y ≥ x} is a convex set, a slight modiﬁcation of Heyman and Sobel (1984)

(Proposition B-4) shows that ζ(·) is concave on by proving that supI∈{L(0,I,y)−

ρ[(−I)+]} is concave in y.

So Tn(·)isβK-concave because βE[ϕn−1(y − D)] is βK-concave by applying

Lemma 1.4.1 (b).

Therefore, ϕn(·)isβK-concave; hence it is K-concave.

The proof that an (s, S) policy is optimal for period n follows the next lemma

which is proved in lemma 7-3 (Heyman and Sobel 1984) (p.314).

Lemma 1.4.4 Suppose T (·) is K-concave, attains its global maximum at S,and

there is a smallest number s ≤ S such that

T (s) ≥−V + T (S) (1.15)

where V ≥ K.Thenϕ(·) is V -concave where

ϕ(x)=sup{T (y) − Vδ(y − x):y ≥ x} x ∈ (1.16)

Since Tn(y) →−∞as |y|→∞and Tn(·) is continuous due to βK-concavity, the global maximum of Tn(·) is attained, say at Sn. Also, let sn be the smallest number x such that x ≤ Sn and Tn(x) ≥−βK + Tn(Sn); so sn is well deﬁned. 18

Finally, K>βKimplies K-concavity of

ϕn(x)=sup{Tn(y) − Kδ(y − x):y ≥ x} x ∈ (1.17)

Therefore, a policy that utilizes (sn, Sn) policy for each n =1, 2, ···,Nis optimal.

1.5 Inﬁnite Horizon Convergence

The analysis in §1.3 replaces the inﬁnite planning horizon in (1.10) with a ﬁnite

planning horizon (n<∞) in (1.11). This section shows that the earlier conclu-

sions regarding the qualitative properties of an optimal policy remain valid for

(1.10). We draw on Iglehart (1963) and Heyman and Sobel (1984, sections 8-5,

8-6), and proves that (a) there are upper and lower bounds for the sequences

{sn} and {Sn} of the ﬁnite horizon optimal policy, (b) the value function of the

finite horizon dynamic program converges as n →∞, (c) the limit value function satisfies the functional equation of dynamic programming, (d) as n →∞, the finite horizon optimal policy converges to a policy that is optimal in the functional equation, and (e) the limit policy inherits the qualitative properties of the finite optimal policies.

We rewrite function Tn(·):

Tn(y)=βE[ϕn−1(y − D)] + ζ(y). (1.18) 19

Assuming that sup(I,b)∈2 {L(b, I, y)}≥0leadstoζ(·) > 0on. Then the

dynamic program in (1.17) and (1.18) with ϕ0(·) ≡ 0 is analogous to (1) in

Iglehart (1963). We modify (1.17) as follows:

ϕn(x)=inf{−Tn(y)+Kδ(y − x):y ≥ x} x ∈. (1.19)

Then −Tn(·)and−ϕ(·)areK-convex, and −ζ(·) is convex. The dynamic program in (1.19) and (1.18) preserves all the properties (convex one-step cost function and

K-convex value functions ) of the one in Iglehart (1963). So Theorems 1 and 2

(Iglehart 1963) hold in our inﬁnite horizon model. Therefore, an (s, S) policy is optimal in (1.10).

1.6 Models with Smoothing Costs

In this section, we replace the setup cost in previous sections with smoothing

costs and provide another test of the following conjecture. For a broad class of

cost structures in inventory models, replacing traditional criteria with the dividend

criterion does not change the form of the optimal replenishment policy. Smoothing

costs discourage intertemporal volatility of replenishment quantities and have a

literature dating at least from Beckmann (1961).

Let e+ be the unit cost of increasing replenishment and e− be the unit cost of decreasing replenishment. So the smoothing cost in period n is

+ + − + e · (zn − zn−1) + e · (zn−1 − zn) . 20

The present value of the time stream of dividends can be expressed as

∞ ∞ n n−1 B = cx1 − β cDn + β [wn − p(wn) − In − (1 − β)cyn n=1 n=1 + + − + − ρ(bn) − e (zn − zn−1) − e (zn−1 − zn) ] (1.20)

Deﬁne e =(e+ + e−)/2 and observe that

+ − + + − + e − e e · (zn − zn−1) + e · (zn−1 − zn) = e|zn − zn−1| + (zn − zn−1) 2

Proceeding as in Sobel (1969), rearranging and collecting terms in (1.20) yields

∞ n−1 B = β −(1 − β)(In + cyn)+βγIn + βg(yn,Dn) n=1 + − e − e 2 − βp (1 + γ)In + g(yn,Dn) − ρ(bn) − e|zn − zn−1|− (1 − β) yn 2 ∞ n + cx1 + w1 − p(w1) − β cDn n=1 + − ∞ e − e n + (1 − β)x1 − β (1 − β)Dn − z0 2 n=1

Let

M(b, I, y)=−(1 − β)(I + cy)+βγI + βE g(y,D) − p[(1 + γ)I + g(y,D)] e+ − e− −ρ(b) − (1 − β)2y 2

Therefore,

∞ n−1 E(B)= β E M(bn,In,yn) − e|zn − zn−1| n=1 ∞ n +E cx1 + w1 − p(w1) − β cDn (1.21) n=1 + − N e − e n + (1 − β)x1 − β (1 − β)Dn − z0 2 n=1 21

Since the last two rows of (1.21) depend only on the distribution of demand and the initial state, we proceed to optimize the ﬁrst row of (1.21).

As in §1.4, we analyze a ﬁnite-horizon counterpart of the inﬁnite-horizon problem and the former converges to the latter. Henceforth, we optimize the following objective:

N n−1 sup EH1 { β [M(bn,In,yn) − e|zn − zn−1|]} n=1

subject to yn ≥ xn,bn + In ≥ 0, and bn ≥ 0 (1.22) where the supremum is over the set of all policies.

A dynamic programming recursion that corresponds to (1.22) is φ0(·, ·) ≡ 0 and for each n =1, 2, ···,N, x ∈,andz ≥ 0,

φn(x, z) = sup[Jn(b, I, y) − e|y − x − z| : y ≥ x, I + b ≥ 0,b≥ 0] (1.23)

Jn(b, I, y)=M(b, I, y)+βE[φn−1(y − D, y − x)] (1.24)

Let bn(x),In(x)andyn(x) be optimal values of b, I and y, respectively, in

+ (1.23). It can be shown that bn(x)=[−In(x)] is optimal as in Proposition 1 in the setup cost model.

Corollary 1.6.1 For all n =1, 2, ··· and x ∈, if the supremum in (1.23) is achieved, then b =(−I)+ is without loss of optimality.

Using Corollary 1.6.1, (1.23) and (1.24) can be written as follows:

φn(x, z)=sup−e|y − x − z| +sup{Jn(b, I, y):I + b ≥ 0,b≥ 0} y≥x 22

=sup−e|y − x − z| + hn(y) (1.25) y≥x

where hn(y)=sup{Jn(b, I, y):I + b ≥ 0,b≥ 0}.

The assumption that M(·, ·, ·) is a concave function on its domain 3 leads to the concavity of hn(·)ony.

Theorem 1.6.2 If M(b, I, y) →−∞as |y|→∞for all (b, y) such that b ≥ 0, and b + I ≥ 0, then for each n and x, there are numbers un(x) and Un(x) with un(x) ≤ Un(x) for each x ∈, such that an optimal policy in (1.25) is ⎧ ⎪ un(x) if x + z

Proof Sketch: It is straightforward to prove inductively that φn(·, ·), Jn(·, ·) and hn(·) are concave functions on their respective domains because M(·, ·, ·)is concave. Since concave functions have one-sided derivatives (except possibly at

their boundaries), let hn(·) denote the left-hand derivative of hn(·). Concavity and a modiﬁcation of Sobel (1969) leads to the structural result in Theorem 1.6.2.

Moreover,

un(x)=sup{y : hn(y) ≤ e}

Un(x)=sup{y : hn(y) ≤−e} 23

and (1.26) corresponds to the following optimal replenishment quantity ⎧ ⎪ un(xn) − xn, if xn + zn−1

His method is based on the convexity of the value functions, and can be directly applied in our model. So the inﬁnite horizon counterparts of (1.22), (1.23), (1.24), and (1.26) are valid and the following stationary policy is optimal: ⎧ ⎪ u(x)ifx + z

1.7 Concluding Remarks

In this paper, we consider periodic review inventory systems where the objective

is to maximize the expected present value of the time stream of dividends. We

consider joint ﬁnancial and replenishment decisions and examine models with a

setup cost and with smoothing costs by embedding the state variable for working 24

capital in the replenishment decision variable. This embedding simpliﬁes the analysis and allows the use of existing stochastic optimization techniques to obtain qualitative results. In the setup cost case, the proof is not straightforward. In both cases, the same form of replenishment policy is optimal as when the criterion is cost minimization.

A generalization that permits dividends to be issued every R periods, where

R is a positive integer, would lead to similar results. The only change would be to constrain vn = 0 for non-dividend-paying periods n. With these constraints added to the model, Proposition 1.3.1 and Corollary 1.6.1 remain valid. So Theorems

1.4.3 and 1.6.2 remain essentially unchanged with only minor changes in their proofs. Chapter 2

Fill Rate of General Review

Supply Systems

2.1 Introduction

All inventory systems face a difficult tradeoff between inventory costs and customer service. The fill rate, the long-run average fraction of demand which is sat- isfied immediately from on-hand inventory, is perhaps the most important measure of customer service in professional practice.

There is a literature on formulas for the fill rate under different inventory replenishment policies. Most of it concerns the fill rate in a single-stage system with a demand process consisting of independent and identically distributed normal random variables. Johnson et al. (1995) review the literature on approximations

25 26

for the item ﬁll rate in an uncapacitated single-stage system with normally distributed demand, develop a new approximation, and evaluate the accuracy via simulations of several approximations to estimate the exact ﬁll rate.

Sobel (2004), Glasserman and Tayur (1994), and Glasserman and Liu (1997) consider the fill rate of capacitated periodic review multi-stage supply systems in which each stage reviews its inventory periodically, and there is a constant transportation leadtime between stages. Glasserman et al. develop asymptotic bound and approximations, including diffusion approximations with higher order correction terms, for fill rate and optimal base-stock levels of multi-stage systems, whereas the fill rate formulas in Sobel (2004) are exact and the bounds are valid without asymptotics.

Research on fill rate of periodic review inventory systems usually assumes that the system reviews its inventory every period. In practice, although customer demand may arise every period, a firm may not review its inventory and make ordering (replenishment) decisions every period. For example, consider a retailer who is supplied by a wholesaler who ships products to the retailer by truck once a week. Although the retailer might prefer to replenish her inventory daily, she should reorder goods only shortly before the truck leaves. One may argue that in this situation, if we define the unit period as one week, the system becomes the usual periodic review system. Indeed, this is a widely held perception. However, our results show that the fill rate computed via a rescaled periodic review system 27

diﬀers from the actual ﬁll rate of the system.

Fill rate expressions are sometimes used to optimize the parameters of an inventory policy subject to a lower bound on the ﬁll rate induced by the policy.

Many authors consider optimization problems with service level constraints and most of this literature consists of heuristics and approximations.

Tijms and Groenevelt (1984) consider both periodic review and continuous review (s, S) inventory systems and present a practical approximation for the reorder point s subject to a fill rate constraint and find that the normal approximation gives good results for required service levels when the coefficient of variation of the demand during lead time and review periods does not exceed 0.5.

Silver (1970), Yano (1985), and Platt, Robinson, and Freund (1997) propose heuristic solutions to ﬁll-rate constrained models using (R, Q) policies. Axs¨ater

(2003) considers a continuous-review ﬁll-rate constrained serial system with batch ordering. The system faces a discrete compound Poisson demand process in which the leadtime demand has a negative binomial distribution. He shows that an optimal policy consists of a mixed multistage echelon stock (R, nQ) policy with one of the reorder points varying over time.

Schneider (1978) and Schneider and Ringuest (1990) study service-constrained models with setup costs, focus on (s, S) policies where the order quantities are predetermined, and present several approximations to estimate the reorder point s such that the required service level is achieved. Schneider and Ringuest consider 28

a periodic review system with a ﬁxed leadtime.

Boyaci and Gallego (2001) and Shang and Song (2003) study a periodic review service-constrained serial inventory system where the leadtime demand for the end product is Poisson distributed. Their service measure, the limiting probability of having positive on-hand inventory at the last stage, diﬀers from the ﬁll rate. Boyaci and Gallego focus on base-stock policies, develop heuristic solutions, and discuss the relationship between stockout cost and service-constrained models.

Shang and Song study the same model, and develop closed-form heuristics to ap- proximate optimal base-stock policies for serial service-constrained systems. Our paper concerns exactly optimal policies for periodic review single-stage models with general demand distributions.

An important purpose of modeling is to analyze the sensitivity of system performance to various parameters. So fill rate equations are used to analyze the sensitivity of inventory levels to alternative fill rate goals. In this sense, the role of fill rate targets is similar to that of stockout costs, but practitioners seem to prefer fill rate targets. Van Houtum and Zijm (2000) discuss the possible relations between backorder cost and several types of service contraints. In particular, they establish the one to one correspondence between backorder cost and modified fill rate (one minus the ratio of the average backlog at the end of a period and the mean demand per period) constraint. Chapter 3 of this dissertation show that fill rate constraints and backorder costs are interchangeable in dynamic newsvendor 29

models and establish monotone mappings between the set of optimal polices with backorder costs and the set of optimal policies with ﬁll rate constraints.

The rest of this paper is organized as follows. Section 2.2 introduces the general review inventory model. A single-stage general review model is considered in §2.3 and §2.4. Section 2.3 provides ﬁll rate formulas for general demand distribution.

Specific fill rate formulas are developed for Gamma and Normal distributions of demand in §2.4. Section 2.5 discusses the review mechanisms of a multi-stage system and has the fill rate formulas for general review two-stage systems. Section

2.6 has a formula for the ﬁll rate of a three-stage system. We conclude the paper in §2.7.

2.2 General Periodic Review System

The following model describes a periodic review N-stage serial system which is displayed in Figure 2.1. Materials, parts or products can be ordered from any stage and are then shipped to the next downstream stage. The inventory level at each stage n is reviewed every Rn periods at which time an order is placed for additional items, if any. An order for material placed at the beginning of a review period t with destination stage n arrives at that stage at the beginning of period t+Ln (Ln and Rn are positive integers.), if suﬃcient materials are available at stage n + 1. The outside supplier preceding stage N has ample supplies and can deliver any order that is placed by stage N. Customer demand for the end 30

Figure 2.1: The standard N-stage serial inventory system product arises solely at stage 1 and any excess demand is backlogged. If Rn =1

(1 ≤ n ≤ N), this general review model is a standard serial multi-stage model.

At the beginning of period t,letxnt denote the number of items that are in storage at stage n (n =2, ···,N; t =1, 2, ···). Let x1t be the analogous quantity at stage 1 minus the number of items backlogged, if any, at the beginning of period t.Thatis,x1t is the on-hand physical inventory if x1t ≥ 0, and −x1t is the amount of backordered demand if x1t < 0. Let zNt be the number of items purchased from an outside supplier in period t,andforn

Let Dt be the demand in period t,andletD1,D2, ··· be independent, identically distributed, and nonnegative random variables with distribution function G and ﬁnite mean µ. To avoid trivialities, it is assumed that G(0) < 1. Let G(k)(·)

k denote the k-fold convolution of G(·), i.e., the distribution function of j=1 Dj, and let G0(a)=1(0)ifa ≥ (<)0.

The following chronology of events occurs during each time period t:

• The inventory vector (x1t, ···,xNt) is observed;

• ··· The previously ordered items (z1,t−L1, ,zN,t−LN ) arrive at their respective 31

stages;

• The order size vector (z1t, ···,zNt) is chosen;

• Finally, demand at stage 1 occurs.

Let ynt be the inventory at stage n in period t after deliveries of previously ordered goods but before demand occurs:

y1t = x1t + zt−L1 ; ynt = xnt + zn,t−Ln if n>1. (2.28)

Because an order cannot exceed the upstream inventory,

0 ≤ znt ≤ yn+1,t if 1 ≤ n ≤ N;0≤ zNt. (2.29)

Notice that znt =0ifperiodt is not a review period for stage n. For expository convenience, let period t with t|R = 0 (where a|b denotes a modulo b when a and b are integers) be a review period at stage 1. Then an order decision is made at stage 1 with order size zt units which will be delivered at period t+L1.Ift|R =0, z1t = 0. The on-hand inventory that is available to satisfy demand in period t is

+ (y1t) . Because excess demand (if any) is backlogged, the inventory dynamic are as follows:

x1,t+1 = y1t − Dt; xn,t+1 = ynt − zn−1,t (1

The ﬁll rate, β, is the long run average fraction of demand that can be satisﬁed

immediately from on-hand inventory. So, T + min{(y1t) ,Dt} β = lim E t=1 (2.31) T →∞ T t=1 Dt 32

where (u)+ denote max{u, 0}. The expectation and limit exist in (2.31) for the base-stock policies that are analyzed in subsequent sections.

2.3 Uncapacitated Single-stage Systems

This section considers an uncapacitated single stage general review system in

which products are ordered from an outside supplier every ﬁxed R periods and

are available to satisfy demand in period t + L (where both R and L are positive

integers). We assume that the system uses a base-stock-level policy. Base-stock

policies have been proved to be optimal for periodic review single-stage system

under general conditions and are very easy to implement in practice (cf. Zipkin

2000 and Porteus 2002). Let τ be the base-stock level, so

+ z1t =(τ − y1t) if t|R =0andz1t = 0 otherwise for t =1, 2, ···.

It follows from Lemma 1 in Sobel (2004) that there is no loss of generality in assuming that initial inventory is never higher than τ, i.e., x11 ≤ τ.Asa consequence, for every t ≥ L,

L+[(t−L)|R] y1t = τ − Dt−k, for (t − L)|R =0, 1, ···,R− 1 (2.32) k=1

In Sobel (2004), there is an inconsistency between the fill rate definition and the proof of Theorem 1 . He uses y1t in the fill rate definition [(3) on page 43], but uses x1t to derive the fill rate formula in the proof of Theorem 1 (line 9, page 44).

However, because of the chronology diﬀerences between his model and the present 33

model, substituting x1t in the proof by y1t of our model results in the same ﬁll rate formulas. Formulas (4) and (5) in Sobel (2004) remain valid here if R =1.

The following theorem gives an exact ﬁll rate formula for a general review single-stage inventory system.

Theorem 2.3.1 1 τ β = [G(L)(b) − G(L+R)(b)]db (2.33) Rµ 0

Proof From (2.31), the ﬁll rate is the long run average fraction of demand that is met directly from on-hand inventory. So,

T T + β = lim E min{(y1t) ,Dt}/ Dt T →∞ t=1 t=1 T T + = lim E [ min{(y1t) ,Dt}/T ]/[ Dt/T ] T →∞ t=1 t=1 T 1 + = lim E min{(y1t) ,Dt} /T T →∞ µ t=1 T 1 + = lim E min{(y1t) ,Dt} /(T/R) Rµ T →∞ t=1 R−1 1 + = lim E min{(y1t) ,Dt} /(T/R) T →∞ Rµ i=0 t∈{t:(t−L)|R=i} R−1 1 + = lim E min{(y1t) ,Dt}/(T/R) T →∞ Rµ i=0 t∈{t:(t−L)|R=i}

+ Let Hj = t∈{t:(t−L)|R=j} min{(y1t) ,Dt}, j =0, 1, ···,R− 1. Using (2.32), for all j ∈{0, 1, ···,R− 1},itcanbeshownthat

L+j + lim E[Hj /(T/R)] = E min (τ − Dk) ,DL+j+1 T →∞ k=1 L+j + + = E DL+j+1 − [DL+j+1 − (τ − Dk) ] k=1 34

L+j + + = µ − E [DL+j+1 − (τ − Dk) ] k=1

Let β(τ) make explicit the dependence of β on τ.Then

R−1 R−1 L+j 1 1 + + β(τ)= Hj = µ − E [DL+j+1 − (τ − Dk) ] Rµ j=0 Rµ j=0 k=1 R−1 L+j 1 + + =1− E [DL+j+1 − (τ − Dk) ] Rµ j=0 k=1 and let K(τ)=µ[1 − β(τ,L,R)]. So,

R−1 L+j 1 + + K(τ)= E [DL+j+1 − (τ − Dk) ] R j=0 k=1 1 R−1 ∞ τ = (a + b − τ)dG(L+j)(b)dG(a) R j=0 0 τ −a ∞ ∞ + a dG(L+j)(b)dG(a) 0 τ 1 R−1 ∞ τ = µ 1 − G(L+j)(τ) + (a + b − τ)dG(L+j)(b)dG(a) R j=0 0 τ −a

Leibnitz’ Rule yields

1 R−1 ∞ τ K(τ)= − dG(L+j)(b)dG(a) R j=0 0 τ −a 1 R−1 τ ∞ = − dG(a)dG(L+j)(b) R j=0 0 τ −b 1 R−1 τ = G(τ − b) − 1 dG(L+j)(b) R j=0 0 1 R−1 = G(L+j+1)(τ) − G(L+j)(τ) R j=0 1 = G(L+R)(τ) − G(L)(τ) R

Since β(0) = 0, K(0) = µ[1 − β(0)]. Therefore,

β(τ)=1− K(τ)/µ 35

τ =1− [K(0) + K(a)da]/µ 0 1 τ = [G(L)(b) − G(L+R)(b)]db. Rµ 0

Theorem 2.3.1 characterizes the dependence of the system ﬁll rate on the base-

stock level (τ), demand distribution (G), review period (R), and leadtime (L).

2.4 Gamma and Normal Demand in Single-stage

Systems

This subsection specializes (2.33) for the gamma distribution and the normal

distribution.

2.4.1 Gamma Demand Distribution

Let Γ(j, λ)denotethesumofj independent, identically distributed random variables, each one exponential with parameter λ, i.e.,Γ(j, λ) is a gamma random variable with parameters j and λ.IfD is Γ(γ,λ)whereγ is a positive integer, then µ = E(D)=γ/λ, Var(D)=γ/λ2, the probability density function of D is

λe−λa(λa)γ−1 Γ(γ)

and the distribution function of D is

γ−1 G(a)=P (D ≤ a)=1− e−λa(λa)j/j! j=0 36

Consequently, G(L) and G(L+R) are Γ(Lγ, λ)andΓ[(L + R)γ,λ], respectively. So,

Lγ−1 G(L)(a)=1− e−λa(λa)j/j! j=0

(L+R)γ−1 G(L+R)(a)=1− e−λa(λa)j/j! j=0

It follows from (2.33) that

(L+R)γ−1 Lγ−1 1 τ β = e−λa(λa)j/j! − e−λa(λa)j/j! da Rµ 0 j=0 j=0 (L+R)γ−1 λ τ = e−λa(λa)j/j! da Rγ 0 j=Lγ (L+R)γ−1 1 τ = λe−λa(λa)j/j! da Rγ j=Lγ 0

Therefore, (L+R)γ 1 β = P {Γ(j, λ) ≤ τ} (2.34) Rγ j=Lγ+1

2.4.2 Normal Demand Distribution

Many researchers investigate the ﬁll rate of an inventory system with normally

distributed demands. In order to compare our results with others, we analyze the

general review inventory systems when demand (D) is normally distributed with mean µ and variance σ2 > 0.

Let Φ(·)andφ(·) denote the distribution and density function, respectively, of a standard normal random variable (with mean 0 and variance 1), and let √ b(a, j)=(a − jµ)/(σ j). The normality and independence of demand imply 37

that sums of demands are normally distributed; that is, G(S)(a)=Φ[b(a, S)] for

S ∈ I +. So, formula (2.33) yields

1 τ β = Φ[b(a, L)] − Φ[b(a, L + R)] da 0 Rµ 1 √ b(τ,L) √ b(τ,L+R) = σ L Φ(x)dx − σ L + R Φ(x)dx (2.35) Rµ b(0,L) b(0,L+R)

The evaluation of the integral in (2.35) exploits the following equation (Hadley and Whitin 1963; Sobel 2004; Zipkin 2000).

∞ [1 − Φ(x)]dx = φ(t)+tΦ(t) − t t

This equation implies

s Φ(x)dx = φ(s) − φ(t)+sΦ(s) − tΦ(t). (2.36) t

Using (2.36) in (2.35) yields the following equation for the ﬁll rate that uses only the standard normal density and tabulated standard normal distribution function: √ β 1 σ L φ b τ,L − φ b ,L b τ,L b τ, L − b ,L b ,L = Rµ [ ( )] [ (0 )] + ( )Φ[ ( )] (0 )Φ[ (0 )] √ − σ L + R φ[b(τ,L+ R)] − φ[b(0,L+ R)] + b(τ,L+ R)Φ[b(τ,L+ R)] − b(0,L+ R)Φ[b(0,L+ R)] √ √ 1 σ L φ b τ,L − b ,L − σ L R φ b τ,L R − φ b ,L R = Rµ [ ( )] Φ[ (0 )] + [ ( + )] [ (0 + )] +(τ − Lµ) Φ[b(τ,L)] − Φ[b(τ,L+ R)] + R Φ[b(τ,L+ R)] + L Φ[b(0,L)] − (L + R)Φ[b(0,L+ R)] . (2.37)

Another deviation of 2.37 uses the following formula:

b xφ(x)dx = φ(a) − φ(b). (2.38) a 38

Figure 2.2: The Fill Rate Integral for a system with Normal Demand

Write the integral in (2.35) as

1 τ b(a,L) b(a,L+R) β = φ(x)dx − φ(x)dx da Rµ 0 −∞ −∞ whose integrand in the a − x plane covers the area in the northeast and southeast quadrants depicted in Figure 2.2 and bounded by the lines a =0,a = τ, x = b(a, L), and x = b(a, L + R). As shown in Figure 2.2 , this area is the union of three sets in the a − x plane. Interchanging the order of integrations, employing formula (2.38), and integrating A1, A2,andA3 individually leads to the same ﬁll rate expression as (2.37). 39

2.4.3 Fill Rate Approximation for Normal Demand Dis-

tribution

Although (2.37) can be calculated easily, the following simple approximation yields

valuable insights:

1 √ βA = − σ L + Rφ[b(τ,L + R)] Rµ +(τ − Lµ) 1 − Φ[b(τ,L + R)] + RΦ[b(τ,L + R)] (2.39)

Approximation (2.39) is derived from (2.37) by observing from Figure 2.2 that

φ[b(0,L)] 0, φ[b(τ,L)] 0, φ[b(0,L+R)] 0, Φ[b(τ,L)] 1, Φ[b(0,L+R)] 0, and Φ[b(0,L)] 0. Our numerical results show that the ﬁll rate approximation

(2.39) is very accurate when τ ≥ (L+R)µ. The numerical comparisons are shown

in Table 2.1.

Table 2.1 use the same normal demand distribution as Table 1 in Sobel (2004),

compute the exact value and approximation of the ﬁll rate in a general review

single-stage system with µ = 2000 and L + R = 5. The last columns of the tables

report the absolute error (100%) of ﬁll rate and its approximation.

In summary, we have the following observations:

• The ﬁll rate increases as the variance (σ2) goes down or base-stock level (τ)

goes up.

• In general, shorter leadtime yields higher ﬁll rate if L + R is ﬁxed. 40

• The approximation performs better when the demand coeﬃcient of variation

(σ/µ) decreases. There are three cases (with high variance, long leadtime) in

our example that cause large errors, but the errors diminish when base-stock

level increases.

2.5 Multi-Stage General Review Systems

An echelon base-stock policy is optimal for a periodic-review multi-stage systems

with linear inventory holding costs at all stages and linear backorder costs at stage

one (Clark and Scarf 1960, Federgruen and Zipkin 1984). It is quite natural to use

echelon base-stock policies in our general review systems. To clearly understand an

echelon base-stock policy, it is useful to deﬁne the following echelon variables(cf.

Clark and Scarf 1960):

• echelon inventory level of a stage is the inventory on hand at this stage plus

inventories at or in transit to all its downstream successor stages minus total

customer backorder at the lowest stage.

• echelon inventory position of stage is the sum of echelon inventory level at

this stage and inventory in transit to the stage.

b snt = the (beginning) echelon inventory level at stage n before any order is received 41

b ant = the (beginning) echelon inventory position at stage n before any order is placed

snt = echelon inventory level at stage n

snt = the echelon inventory position at stage n before demand occurs.

The evolution of the system and dynamics of these echelon variables can be speciﬁed as:

Ln b b b b ant = snt + zn,t−k ant = ant + znt snt = sn,t + zn,t−Ln (2.40) k=1 and

b b an,t+1 = ant − Dt sn,t+1 = snt − Dt (2.41)

The ﬁrst expression in (2.40) can be written

Ln−1 ant = snt + zn,t−k k=0

The formulation in installation variables xnt, ynt,andznt is equivalent to a formulation in echelon variables because xnt = ynt − znt, ynt = snt − an−1,t (let Ln−1 s0t = a0t =0),andznt = ant − k=1 zn,t−k − snt. Because s1t = y1t, (2.31) can be written T + min{(s1t) ,Dt} β = lim E t=1 (2.42) T →∞ T t=1 Dt

The constraints on the order quantities (2.29) correspond to

s1t ≤ a1t ≤ s2t ≤···≤sNt ≤ aNt. (2.43) 42

An echelon base-stock policy depends on echelon base-stock levels τ1, ···,τN

b and can be expressed as: If period t is a review period, aNt =max{aNt,τN } and for n

b + b + speciﬁed by (2.44) is znt =min{(τn −ant) , (sn+1,t −ant) } if t is a review period, and znt =0otherwise.

The subsequent sections discuss the ﬁll rate of general review two-stage and three-stage supply systems.

2.5.1 Fill Rate in Two-Stage Systems

This subsection considers a two-stage general review system in which both stages

have the same review intervals (R1 = R2 = R) and order leadtimes are L1 = L and L2 = 1. Based on the review procedure, if t is a review period, then t − 1isa review period for stage two. Both stages have the same length of review intervals.

We are interested in this system because it is very similar to the single-stage system; in essence, the only diﬀerence is that the order quantity in stage one at a review period is constrained by the inventory at stage two. We shall explicitly characterize the relationship between the echelon inventory level at stage one and base-stock levels τ1 and τ2 for stage one and 2, respectively. It is without loss of generality to assume that the initial echelon inventory positions are no higher 43

than the echelon base-stock levels (cf. Sobel 2004, page 47):

b b At the beginning of each period t, ant,snt for n =1, 2, z2,t−1,andz1,t−k for k =1, ···,Lare known. In fact, there are at most Ln/R ( a/b is the minimum integer that is greater than or equal to a/b ) outstanding orders that in transit to stage n for any period t.

Suppose t is a review period at stage two and R>1. Since R>1, stage

b b one does not place an order in period t.Thena2t = τ2, s2t = s2t, a1t = a1t,and

b s1t = s1t +zt−L.WhendemandDt occurs at stage one, period t+1 isnota review

b b period for stage two, so a2,t+1 = s2,t+1 = τ2 −Dt = s2,t+1 (order has been received).

b b b Period t + 1 is a review period for stage one, a1,t+1 = a1,t − Dt, s1,t+1 = s1t − Dt, and because of ordering, a1,t+1 =min{τ1,s2,t+1}. Order quantity will be delivered

L period later, until then the changes of echelon inventory level (s1t) depend only on the period demand and the orders in transit to stage one.

The following result is consistent with comments in several papers [e.g. Cheng and Zheng (1994, §3.1), Federgruen and Zipkin (1984)]

Lemma 2.5.1

L s1t = a1,t−L − Dt−k k=1

Proof

L−1 s1t = a1t − zt−k k=0 L−1 b = a1t − (a1,t−k − a1,t−k−1) k=0 44

L−1 = a1t − a1,t−k − (a1,t−k−1 − Dt−k−1) k=0 L = a1,t−L − Dt−k k=1

Lemma 2.5.1 states that the echelon inventory level at stage one in period t is

equal to its echelon inventory position in period t − L minus the preceding L

periods’ demands.

As a consequence [similar to the derivation of (2.32)], s1t for any period t can be expressed explicitly. It is without loss of generality to assume that stage one reviews its inventory at period t when t|R =0.Let∆=τ2 − τ1.

Lemma 2.5.2 For any t ≥ L,

L+(t−L)|R { − }− s1t = τ1 +min 0, ∆ Dt−[L+(t−L)|R]−1 Dt−k (2.45) k=1

Proof At t = L,then(t − L)|R =0,anda2,L−1 = τ2 because stage two reviews its inventory one period ahead of stage one. Lemma 2.5.1 assures

L s1L = a1,L − Dt−k k=1 L =min{τ1,τ2 − DL−1}− Dt−k k=1 L+0 = τ1 +min{0, ∆ − DL−0−1}− Dt−k k=1

So (2.45) is valid at t = L and initiates an inductive proof of (2.45).

Assume that (2.45) is valid at t and note that

s1,t+1 = s1,t − Dt + zt+1−L 45

If (t +1− L)|R = 0, then an order placed at period t +1− L is delivered to stage one at period t + 1, applying Lemma 2.5.1 again,

L s1,t+1 = a1,t+1−L − Dt+1−k k=1 L =min{τ1,τ2 − Dt+1−L−1}− Dt+1−L−k k=1 L+(t+1−R)|R { − }− = τ1 +min 0, ∆ Dt+1−[L+(t+1−L)|R]−1 Dt+1−k k=1

If (t +1− L)|R =0,then zt+1−L =0,(t +1− L)|R =(t − L)|R +1,so

s1,t+1 = s1,t − Dt L+(t−L)|R { − }− − = τ1 +min 0, ∆ Dt−[L+(t−L)|R]−1 Dt−k Dt k=1 L+(t+1−R)|R−1 { − }− − = τ1 +min 0, ∆ Dt−[L+(t+1−L)|R−1]−1 Dt−k Dt k=1 L+(t+1−R)|R { − }− = τ1 +min 0, ∆ Dt+1−[L+(t+1−L)|R]−1 Dt+1−k k=1

Lemma 2.5.2 leads to a relatively simple ﬁll rate formula. Notice that expression (2.32) is a special case of (2.45) by letting ∆ = ∞.

Theorem 2.5.3

1 τ1 β = G(∆) G(L)(a) − G(L+R)(a) Rµ 0 ∞ + G(L)(a +∆− c) − G(L+R)(a +∆− c) dG(c) da (2.46) ∆

Proof From (2.31), the ﬁll rate is the long run average fraction of demand that is met directly from on-hand inventory. So,

T T + β = lim E min{(s1t) ,Dt}/ Dt T →∞ t=1 t=1 46

T T + = lim E [ min{(s1t) ,Dt}/T ]/[ Dt/T ] T →∞ t=1 t=1 T 1 + = lim E min{(s1t) ,Dt} /T T →∞ µ t=1 T 1 + = lim E min{(s1t) ,Dt} /(T/R) Rµ T →∞ t=1 R−1 1 + = lim E min{(s1t) ,Dt} /(T/R) T →∞ Rµ i=0 t∈{t:(t−L)|R=i} R−1 1 + = lim E min{(s1t) ,Dt}/(T/R) T →∞ Rµ i=0 t∈{t:(t−L)|R=i}

+ Let Hj = t∈{t:(t−L)|R=j} min{(s1t) ,Dt}, j =0, 1, ···,R−1. Using (2.45) and noticing that the demand variables are independent,

L+j+1 + lim E[Hj/(T/R)] = E min τ1 +min{0, ∆ − D1}− Dk ,DL+j+2 T →∞ k=2

For all j ∈{0, 1, ···,R− 1},itcanbeshownthat

L+j+1 + β(Hj)=E min τ1 +min{0, ∆ − D1}− Dk ,DL+j+2 k=2 L+j+1 + + = E DL+j+1 − DL+j+2 − [τ1 +min{0, ∆ − D1}− Dk] k=2 L+j+1 + + + =1− E DL+j+2 − [τ1 − (D1 − ∆) − Dk] k=2

Let β(τ1, ∆) make explicit the dependence of β on τ1 and ∆. Then

1 R−1 β(τ1, ∆) = β(Hj) Rµ j=0 R−1 L+j+1 1 + + + = 1 − E DL+j+2 − [τ1 − (D1 − ∆) − Dk] Rµ j=0 k=2

Let K(τ1, ∆) = µ[1 − β(τ1, ∆)]. So,

R−1 L+j+1 1 + + + K(τ1, ∆) = E DL+j+2 − [τ1 − (D1 − ∆) − Dk] R j=0 k=2 47

R−1 1 ∞ τ1+∆−c ∞ = R j=0 ∆ 0 τ1+∆−b−c (L+j) (a + b + c − τ1 − ∆)dG(L+j+2)(a)dG (b)dG(c) ∞ ∞ ∞ (L+j) + adG(L+j+2)(a)dG (b)dG(c) ∆ τ1+∆−c 0 ∆ τ ∞ 1 (L+j) + (a + b − τ1) dG(L+j+2)(a)dG (b)dG(c) 0 0 τ1−b ∆ ∞ ∞ (L+j) + adG(L+j+2)(a)dG (b)dG(c) 0 τ1 0

Leibnitz’ Rule yields

R−1 ∞ τ +∆−c ∞ ∂K(τ1, ∆) 1 1 (L+j) = − dG(L+j+2)(a)dG (b)dG(c) ∂τ1 R j=0 ∆ 0 τ1+∆−b−c ∞ ∞ (L+j) + adG(L+j+2)(a)g (τ1 +∆− c)dG(c) ∆ 0 ∞ ∞ (L+j) + −adG(L+j+2)(a)g (τ1 +∆− c)dG(c) ∆ 0 ∆ τ ∞ 1 (L+j) + − dG(L+j+2)(a)dG (b)dG(c) 0 0 τ1−b ∆ ∞ (L+j) + adG(L+j+2)(a)g (τ1)dG(c) 0 0 ∆ ∞ (L+j) + −adG(L+j+2)(a)g (τ1)dG(c) (2.47) 0 0

Notice that the sum of the second and third lines in (2.47) is zero and the sum of the ﬁfth and sixth lines is zero. Further simplifying (2.47),

R−1 ∞ τ +∆−c ∞ ∂K(τ1, ∆) 1 1 (L+j) = −dG(L+j+2)(a)dG (b)dG(c) ∂τ1 R j=0 ∆ 0 τ1+∆−b−c ∆ τ ∞ 1 (L+j) + − dG(L+j+2)(a)dG (b)dG(c) 0 0 τ1−b R−1 ∞ τ +∆−c 1 1 (L+j) = G(L+j+2)(τ1 +∆− b − c) − 1 dG (b)dG(c) R j=0 ∆ 0 ∆ τ 1 (L+j) + G(L+j+2)(τ1 − b) − 1 dG (b)dG(c) 0 0 R−1 ∞ 1 (L+j+1) (L+j) = G (τ1 +∆− c) − G (τ1 +∆− c) dG(c) R j=0 ∆ 48

∆ (L+j+1) (L+j) + G (τ1) − G (τ1) dG(c) 0 R−1 ∞ 1 (L+j+1) (L+j) = G (τ1 +∆− c) − G (τ1 +∆− c) dG(c) R ∆ j=0 (L+j+1) (L+j) + G (τ1) − G (τ1) G(∆) ∞ 1 (L+R) (L) = G (τ1 +∆− c) − G (τ1 +∆− c) dG(c) R ∆ (L+R) (L) + G (τ1) − G (τ1) G(∆) (2.48)

Similarly, it is straightforward to show that for all ∆ ≥ 0, K(0, ∆) = µ.

Therefore,

β(τ1, ∆) = 1 − K(τ1, ∆)/µ τ1 ∂K(a, ∆) =1− K(0, ∆) + da /µ 0 ∂a τ1 ∂K(a, ∆) = − da/µ 0 ∂a 1 τ1 = G(∆) G(L)(a) − G(L+R)(a) da 0 Rµ τ1 ∞ + G(L)(a +∆− c) − G(L+R)(a +∆− c) dG(c)da 0 ∆

2.5.2 Fill Rate in Two-Stage Systems with General Lead-

time

The formula for a two-stage general review system with a single period leadtime

in stage two can be easily extended to a two-stage general review system in which

the review intervals are R1 = R2 = R and order leadtimes are L1 and L2.Weuse the following review procedure which is bases on complete information-sharing between stages 1 and 2: if t is a review period, then t − L2 is a review period for 49

stage two. Both stages have the same length of review intervals.

Lemma 2.5.1 still holds for this general two-stage system. We give the following lemma to express the echelon inventory level at stage one.

Lemma 2.5.4 For any t ≥ L1 + L2,

L +(t−L )|R L2 1 1 s1t = τ1 +min 0, ∆ − D − Dt−k (2.49) t−[L1+(t−L1)|R]−k k=1 k=1

The proof of Lemma 2.5.4 is similar to that of Lemma 2.5.2 except that the order placed by stage two has L2 delay before it can be used to satisfy stage one’s order. L2 (L2) Replacing D1 and G(c)by k=1 Dk and G (c), respectively, in the proof of

Theorem 2.5.3 yields the next result.

Theorem 2.5.5

1 τ1 β = G(L2)(∆) G(L1)(a) − G(L1+R)(a) Rµ 0 ∞ + G(L1)(a +∆− c) − G(L1+R)(a +∆− c) dG(L2)(c) da (2.50) ∆

In what follows, we present an alterative formula for (2.50) using the single stage ﬁll rate formulas. Let β1(τ,L) be the ﬁll rate of a single stage system with review period R, base-stock level τ, and order leadtime L. We introduce incomplete convolutions (cf. van Houtum et al. 1996). Let G(k) be the distribution

k functions of j=1.Letδ>0 and deﬁne ⎧ ⎨⎪ G(k)(x + δ)ifx ≥ 0 (k) Gδ (x)= ⎩⎪ 0ifx<0, 50

[m,n] The incomplete convolution, denoted Gδ ,is

δ [m,n] (m) (n) Gδ (x)= G (δ + x − u)dG (u). 0

[m,n] (m+n) note that G0 (x)=G (x)

Corollary 2.5.6

1 (L2) β = G (∆)β1(τ1,L1)+β1(τ1 +∆,L1 + L2) − β1(∆,L1 + L2) Rµ τ1 [L1,L2] [L1+R,L2] − G∆ (a) − G∆ (a) da (2.51) 0

Proof Rewriting (2.50) and using (2.33),

1 τ1 ∞ β = G(L1)(a +∆− c) − G(L1+R)(a +∆− c) dG(L2)(c)da 0 ∆ Rµ τ1 +G(L2)(∆) G(L1)(a) − G(L1+R)(a) da 0 1 τ1 a+∆ = G(L1)(a +∆− c) − G(L1+R)(a +∆− c) dG(L2)(c)da Rµ 0 ∆ (L2) +G (∆)β1(τ1,L1) 1 τ1 a+∆ = G(L1)(a +∆− c) − G(L1+R)(a +∆− c) dG(L2)(c)da 0 0 Rµ τ1 ∆ − G(L1)(a +∆− c) − G(L1+R)(a +∆− c) dG(L2)(c)da 0 0 (L2) +G (∆)β1(τ1,L1) 1 τ1 = G(L1+L2)(a +∆)− G(L1+L2+R)(a +∆) da 0 Rµ τ1 [L1,L2] [L1+R,L2] (L2) − [G∆ (a) − G∆ (a) da + G (∆)β1(τ1,L1) 0 1 τ1+∆ = G(L1+L2)(u) − G(L1+L2+R)(u) du ∆ Rµ τ1 [L1,L2] [L1+R,L2] (L2) − [G∆ (a) − G∆ (a) da + G (∆)β1(τ1,L1) 0 1 τ1+∆ = G(L1+L2)(u) − G(L1+L2+R)(u) du Rµ 0 51

∆ − G(L1+L2)(u) − G(L1+L2+R)(u) du 0 τ1 [L1,L2] [L1+R,L2] (L2) − [G∆ (a) − G∆ (a) da + G (∆)β1(τ1,L1) 0 1 (L2) = G (∆)β1(τ1,L1)+β1(τ1 +∆,L1 + L2) − β1(∆,L1 + L2) Rµ τ1 [L1,L2] [L1+R,L2] − G∆ (a) − G∆ (a) da (2.52) 0

2.5.3 Numerical Example

It is clear from (2.50) that the ﬁll rate of a two-stage general review system depends

on R, L, τ1,∆(τ2), and G(·). However, it is not easy to reveal the dependences.

Therefore, we perform another set of numerical study on (2.50) to illustrate their relationships. In this study, demand is normally distributed with µ = 10.

We make the following observations from Tables 2.2 and 2.3:

• The ﬁll rate increases as variance goes down, and as echelon base-stock level

at stage one goes up.

• The ﬁll rate decreases as leadtime at stage one and stage two increase.

• The ﬁll rate increases as echelon base-stock level at stage two increases.

2.6 Fill Rate in a Three-Stage System

This section considers a three-stage general review system in which the review

interval is the same at each stage (R1 = R2 = R3 = R) and order leadtimes are 52

L1, L2,andL3. Based on the review procedure, if t is a review period at stage

one, then t − L2 is a review period at stage two, and t − L2 − L3 is a review period at stage three. We assume that an echelon base-stock policy with echelon base-stock levels τ1, τ2,andτ3 is used. Let ∆1 = τ2 − τ1 and ∆2 = τ3 − τ2.

Lemma 2.6.1 For any t ≥ L1 + L2 + L3,

L3 s1t = τ1 +min 0, ∆1 +min 0, ∆2 − D t−[L1+(t−L1)|R]−L2−k k=1 L +(t−L )|R L2 1 1 − D − Dt−k (2.53) t−[L1+(t−L1)|R]−k k=1 k=1

Proof sketch: Suppose t is a review period at stage one. Then the echelon position is

a1t =min{τ1,s2t} =min{τ1,a2,t−L2 − Dt−k} k=1 L2

=minτ1, min{τ2,s3,t−L2 }− Dt−k k=1 L3 L2

=minτ1, min{τ2,a3,t−L2−L3 − Dt−L2−k}− Dt−k k=1 k=1 L3 L2

=minτ1, min{τ2,τ3 − Dt−L2−k}− Dt−k k=1 k=1 L3 L2

=minτ1,τ2 +min{0, ∆2 − Dt−L2−k}− Dt−k k=1 k=1 L3 L2

= τ1 +min 0, ∆1 +min{0, ∆2 − Dt−L2−k}− Dt−k . k=1 k=1

Lemma 2.5.1 yields,

s1t = a1,t−L1 − Dt−k. k=1

An inductive proof similar to that of Lemma 2.5.2 results in (2.53). 53

Theorem 2.6.2

τ1 ∞ ∞ 1 (L1) β = G (a +∆1 +∆2 − c − e) Rµ 0 ∆2 ∆1+∆2−e (L1+R) (L2) (L3) −G (a +∆1 +∆2 − c − e) dG (c)dG (e) ∞ (L1) (L1+R) (L2) (L3) + G (a) − G (a) G (∆1 +∆2 − e)dG (e) ∆2 ∞ (L3) (L1) (L1+R) (L2) +G (∆2) G (a +∆1 − c) − G (a +∆1 − c) dG (c) ∆1 (L1) (L1+R) (L2) (L3) + G (a) − G (a) G (∆1)G (∆2) da. (2.54)

Proof From (2.31),

T T + β = lim E min{(s1t) ,Dt}/ Dt T →∞ t=1 t=1 R−1 1 + = lim E min{(s1t) ,Dt}/(T/R) T →∞ Rµ i=0 t∈{t:(t−L)|R=i}

+ Let Hj = t∈{t:(t−L)|R=j} min{(s1t) ,Dt}, j =0, 1, ···,R − 1, use (2.53), and notice that the demand random variables are independent. Let D(j) have the

j distribution of k=1 Dk. Then limT →∞ E[Hj /(T/R)] is equal to

+ (L3) (L2) (L1+j) E min τ1 +min 0, ∆1 +min{0, ∆2 − D }−D − D ,D

For all j ∈{0, 1, ···,R− 1},itcanbeshownthat

+ + + (L3) + (L2) (L1+j) β(Hj)=1− E D − τ1 − [D − ∆2] + D − ∆1 − D

Let β(τ1, ∆1, ∆2) make explicit the dependence of β on τ1,∆1,and∆2.Then

1 R−1 β(τ1, ∆1, ∆2)= β(Hj) Rµ j=0 54

Let K(τ1, ∆1, ∆2)=µ[1 − β(τ1, ∆1, ∆2)]. So,

K(τ1, ∆1, ∆2) R−1 + 1 + + E D − τ − D(L3) − + D(L2) − − D(L1+j) = R 1 [ ∆2] + ∆1 j=0 R−1 1 ∞ ∞ τ1+∆1+∆2−e−c ∞ = R j=0 ∆2 ∆1+∆2−e 0 τ1+∆1+∆2 −e−c−b (L+j) (L2) (L3) (a + b + c + e − τ1 − ∆1 − ∆2)dG(a)dG (b)dG (c)dG (e) ∞ ∞ ∞ ∞ = adG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e) ∆2 ∆1+∆2−e τ1+∆1 +∆2−e−c 0 ∞ ∆1+∆2−e τ1 ∞ (L1+j) (L2) (L3) + (a + b − τ1)dG(a)dG (b)dG (c)dG (e) ∆2 0 0 τ1−b ∞ ∆1+∆2−e ∞ ∞ + adG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e) ∆2 0 τ1 0 ∆2 ∞ τ1+∆1−c ∞ + 0 ∆1 0 τ1+∆1−c−b

(L1+j) (L2) (L3) (a + b + c − τ1 − ∆1)dG(a)dG (b)dG (c)dG (e) ∆2 ∞ ∞ ∞ + adG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e) 0 ∆1 τ1+∆1−c 0 ∆2 ∆1 τ1 ∞ (L +j) (L ) (L ) + (a + b − τ1)dG(a)dG 1 (b)dG 2 (c)dG 3 (e) 0 0 0 τ −b 1 ∆2 ∆1 ∞ ∞ + adG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e) 0 0 τ1 0

Leibnitz’ Rule yields

∂K(τ1, ∆1, ∆2) ∂τ1 R−1 1 ∞ ∞ τ1+∆1+∆2 −e−c ∞ = − R j=0 ∆2 ∆1+∆2 −e 0 τ1+∆1 +∆2−e−c−b dG(a)dG(L+j)(b)dG(L2)(c)dG(L3)(e) ∞ ∆1+∆2−e τ1 ∞ − dG(a)dG(L1 +j)(b)dG(L2)(c)dG(L3)(e) ∆ 0 0 τ −b 2 1 ∆2 ∞ τ1+∆1−c ∞ − dG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e) 0 ∆ 0 τ +∆ −c−b 1 1 1 ∆2 ∆1 τ1 ∞ − dG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e) 0 0 0 τ1−b ∞ ∞ 1 (L1+R) = G (τ1 +∆1 +∆2 − c − e) R ∆2 ∆1+∆2 −e 55

(L1) (L2) (L3) −G (τ1 +∆1 +∆2 − c − e) dG (c)dG (e) ∞ (L1+R) (L1) (L2) (L3) +[G (τ1) − G (τ1)] G (∆1 +∆2 − e)dG (e) ∆2 ∞ (L3) (L1+R) (L1) (L2) +G (∆2) G (τ1 +∆1 − c) − G (τ1 +∆1 − c) dG (c) ∆1 (L1+R) (L1) (L2) (L3) G (τ1) − G (τ1) G (∆1)G (∆2) (2.55)

It is straightforward to show that K(0, ∆1, ∆2)=µ. Therefore,

β(τ1, ∆1, ∆2)=1− K(τ1, ∆1, ∆2)/µ τ 1 ∂K(a, ∆1, ∆2) =1− K(0, ∆1, ∆2)+ da /µ 0 ∂a τ 1 1 ∂K(a, ∆1, ∆2) = − da. R 0 ∂a

Using (2.55), yields (2.54)

[m,n,q] [m,n,q] An alternate to (2.54) uses (2.33) and (2.50). Let Gδ1,δ1+δ2 and Gδ1,δ2 be the following three-fold incomplete convolutions:

δ δ +δ −v [m,n,q] 1 1 2 (m) (n) (q) Gδ +δ ,δ (x)= G (δ1 + δ2 + x − v − u)dG (u)dG (v); 1 2 1 0 0 δ δ [m,n,q] 1 2 (m) (n) (q) Gδ ,δ (x)= G (δ1 + δ2 + x − v − u)dG (u)dG (v). 2 1 0 0

Corollary 2.6.3 1 (L3) β = G (∆2)β2(τ1, ∆1,L1,L2) Rµ [L ,L (L2+L3) − 2 3 +β1(τ1,L1) G (∆1 +∆2) G∆2 (∆1)

+β1(τ1 +∆1 +∆2,L1 + L2 + L3) − β1(∆1 +∆2,L1 + L2 + L3) τ1 [L1+L2,L3] [L1+L2+R,L3] − G∆ (a +∆1) − G∆ (a +∆1) 0 2 2 − [L1,L3,L2] [L1+R,L3,L2] − [L1,L3,L2] [L1+R,L3,L2] Gδ1+δ2,δ1 (a)+Gδ1+δ2,δ1 (a) Gδ2,δ1 (a)+Gδ2,δ1 (a) da

The proof of Corollary 2.6.3 is similar to that of Corollary 2.5.6 and is omitted. 56

2.7 Conclusion

This paper develops formulas for the ﬁll rates of single-stage and multi-stage supply systems that use base-stock-level policies and have general review intervals.

We provide fill rate formulas for a single-stage general review system and general distributions of demand. When demand is normally distributed, an exact fill rate expression uses only the standard normal distribution function and density function. For the general review multi-stage systems, we first discuss how each stage reviews its inventory and provide a general approach to compute the system

ﬁll rate. 57

σσ/µτ RL β βA %Error 200 0.1 11000 1 4 0.999014 0.999014 0 200 0.1 11000 2 3 0.999507 0.999507 0 200 0.1 11000 3 2 0.999671 0.999671 0 200 0.1 11000 4 1 0.999754 0.999754 0 1000 0.5 11000 1 4 0.789395 0.760095 2.93 1000 0.5 11000 2 3 0.880264 0.880047 0.0217 1000 0.5 11000 3 2 0.919869 0.920032 0.0163 1000 0.5 11000 4 1 0.938963 0.940024 0.1061 2000 1 11000 1 4 0.590938 0.335729 25.5209 2000 1 11000 2 3 0.686992 0.667864 1.9128 2000 1 11000 3 2 0.766131 0.778576 1.2445 2000 1 11000 4 1 0.815568 0.833932 1.8364 200 0.1 12000 1 4 1 1 0 200 0.1 12000 2 3 1 1 0 200 0.1 12000 3 2 1 1 0 200 0.1 12000 4 1 1 1 0 1000 0.5 12000 1 4 0.895048 0.886563 0.8485 1000 0.5 12000 2 3 0.943282 0.943282 0 1000 0.5 12000 3 2 0.962025 0.962188 0.0163 1000 0.5 12000 4 1 0.97058 0.971641 0.1061 2000 1 12000 1 4 0.679695 0.520189 15.9506 2000 1 12000 2 3 0.765023 0.760095 0.4928 2000 1 12000 3 2 0.826923 0.840063 1.314 2000 1 12000 4 1 0.861683 0.880047 1.8364 200 0.1 13000 1 4 1 1 0 200 0.1 13000 2 3 1 1 0 200 0.1 13000 3 2 1 1 0 200 0.1 13000 4 1 1 1 0 1000 0.5 13000 1 4 0.95544 0.953442 0.1998 1000 0.5 13000 2 3 0.976695 0.976721 0.0026 1000 0.5 13000 3 2 0.984318 0.984481 0.0163 1000 0.5 13000 4 1 0.987299 0.98836 0.1061 2000 1 13000 1 4 0.758474 0.664425 9.4049 2000 1 13000 2 3 0.829454 0.832213 0.2759 2000 1 13000 3 2 0.874769 0.888142 1.3373 2000 1 13000 4 1 0.897742 0.916106 1.8364

Table 2.1: Fill Rate and its Approximation for Normal Demand 58

σσ/µτ1 ∆1 RL1 L2 β σσ/µτ1 ∆1 RL1 L2 β 2 0.2 50 0 1 4 1 0.1747 20.2800 1 4 1 1 2 0.2 50 0 1 4 2 0.0037 2 0.2 80 0 1 4 2 0.9940 2 0.2 50 0 2 3 1 0.4985 20.2800 2 3 1 1 2 0.2 50 0 2 3 2 0.0892 2 0.2 80 0 2 3 2 0.9970 2 0.2 50 0 3 2 1 0.6654 20.2800 3 2 1 1 2 0.2 50 0 3 2 2 0.3336 2 0.2 80 0 3 2 2 0.9980 2 0.2 50 0 4 1 1 0.7491 20.2800 4 1 1 1 2 0.2 50 0 4 1 2 0.5 2 0.2 80 0 4 1 2 0.9985 5 0.5 50 0 1 4 1 0.3030 5 0.5 80 0 1 4 1 0.9749 5 0.5 50 0 1 4 2 0.1052 5 0.5 80 0 1 4 2 0.8546 5 0.5 50 0 2 3 1 0.4702 5 0.5 80 0 2 3 1 0.9868 5 0.5 50 0 2 3 2 0.2041 5 0.5 80 0 2 3 2 0.9147 5 0.5 50 0 3 2 1 0.6200 5 0.5 80 0 3 2 1 0.9912 5 0.5 50 0 3 2 2 0.3485 5 0.5 80 0 3 2 2 0.9427 5 0.5 50 0 4 1 1 0.7141 5 0.5 80 0 4 1 1 0.9933 5 0.5 50 0 4 1 2 0.4932 5 0.5 80 0 4 1 2 0.9571 5 0.5 50 20 1 4 1 0.6357 5 0.5 80 20 1 4 1 0.9987 5 0.5 50 20 1 4 2 0.5395 5 0.5 80 20 1 4 2 0.9944 5 0.5 50 20 2 3 1 0.7773 5 0.5 80 20 2 3 1 0.9993 5 0.5 50 20 2 3 2 0.6948 5 0.5 80 20 2 3 2 0.9971 5 0.5 50 20 3 2 1 0.8503 5 0.5 80 20 3 2 1 0.9994 5 0.5 50 20 3 2 2 0.7906 5 0.5 80 20 3 2 2 0.9979 5 0.5 50 20 4 1 1 0.8868 5 0.5 80 20 4 1 1 0.9986 5 0.5 50 20 4 1 2 0.8419 5 0.5 80 20 4 1 2 0.9975

Table 2.2: Fill Rate of Two-stage Systems for Normal Demand (a) 59

σσ/µτ1 ∆1 RL1 L2 β σσ/µτ1 ∆1 RL1 L2 β 5 0.5 50 30 1 4 1 0.6373 5 0.5 50 50 1 4 1 0.6373 5 0.5 50 30 1 4 2 0.6284 5 0.5 50 50 1 4 2 0.6373 5 0.5 50 30 2 3 1 0.7785 5 0.5 50 50 2 3 1 0.7785 5 0.5 50 30 2 3 2 0.7714 5 0.5 50 50 2 3 2 0.7785 5 0.5 50 30 3 2 1 0.8512 5 0.5 50 50 3 2 1 0.8512 5 0.5 50 30 3 2 2 0.8462 5 0.5 50 50 3 2 2 0.8512 5 0.5 50 30 4 1 1 0.8874 5 0.5 50 50 4 1 1 0.8874 5 0.5 50 30 4 1 2 0.8837 5 0.5 50 50 4 1 2 0.8874 5 0.5 80 30 1 4 1 0.9987 5 0.5 80 50 1 4 1 0.9987 5 0.5 80 30 1 4 2 0.9985 5 0.5 80 50 1 4 2 0.9987 5 0.5 80 30 2 3 1 0.9993 5 0.5 80 50 2 3 1 0.9993 5 0.5 80 30 2 3 2 0.9991 5 0.5 80 50 2 3 2 0.9993 5 0.5 80 30 3 2 1 0.9994 5 0.5 80 50 3 2 1 0.9994 5 0.5 80 30 3 2 2 0.9993 5 0.5 80 50 3 2 2 0.9994 5 0.5 80 30 4 1 1 0.9986 5 0.5 80 50 4 1 1 0.9986 5 0.5 80 30 4 1 2 0.9985 5 0.5 80 50 4 1 2 0.9986

Table 2.3: Fill Rate of Two-stage Systems for Normal Demand (b) Chapter 3

Interchangeability of Fill Rate

Constraints and Backorder Costs in Inventory Models

3.1 Introduction

Inventories which encounter uncertain demand lead to risks of both excess supply and unsatisﬁed demand. The associated research on inventory models with stochastic demand studies how best to balance these risks. Initially, the relative importance of the two risks was often parameterized with holding costs and stockout costs. For the past twenty years, research has paid much attention to the service role of inventories and focused on a service system’s ﬁll rate, namely

60 61

the fraction of demand which is immediately met from on-hand inventory. During this latter period, the relative importance of the two risks has often been parameterized with holdings costs and a lower bound on the ﬁll rate. As a result, there are parallel streams of literature which analyze identical models except that one stream has stockout costs and the other has ﬁll rate constraints.

These streams of literature correspond in mathematical programming to optimization subject to constraints and to optimization of an unconstrained La- grangean. As in nonlinear deterministic optimization, in stochastic optimization the two approaches do not always yield the same results. This paper investigates whether there is redundancy in the two streams of dynamic inventory models with linear purchase costs, namely dynamic newsvendor models. We show the extent to which optimal policies for either kind of model can be inferred from the other.

Here, an inventory replenishment policy is called Stockout-optimal, or S- optimal for short, if it minimizes the long-run average sum of holding and stockout costs per unit time. So, S-optimality corresponds to an unconstrained Lagrangean formulation. A policy is called Fill-Rate-Optimal, or F-optimal for short, if it minimizes the long-run average holding cost per unit time subject to a ﬁll-rate constraint. If demand is continuous, i.e., if the distribution function of demand has a density function, then S-optimality and F-optimality are shown to be equivalent in the following sense:

(a) Corresponding to any unit stockout cost b, there is a base-stock level y and a 62

ﬁll-rate f such that a base-stock policy with parameter y is both S-optimal with stockout cost b and F-optimal with constraint parameter f.

(b) Corresponding to any ﬁll-rate constraint parameter f,thereisabase-stock level y and a stockout cost b such that a base-stock policy with parameter y is both S-optimal with stockout cost b and F-optimal with constraint parameter f.

If demand is an integer-valued random variable, the situation is more com-

plicated. Although a deterministic base-stock level policy is S-optimal for every

stockout cost b, a randomized base-stock level policy is F-optimal for most con-

straint parameters f. Nevertheless, a parametric analysis of either kind of opti-

mality can be accomplished via the other kind in the following sense:

ﬁll-rate f such that a base-stock policy with parameter y is both S-optimal with stockout cost b and F-optimal with constraint parameter f.

(d) There are sequences of ﬁll-rate constraint parameters 0

fk ≤ f

If demand is continuous, the consequence of (a) and (b) is that the structure and parametric analysis of F-optimal inventory policies are implicit in the 63

structure and parametric analysis of S-optimal policies, and conversely. There- fore, it is redundant to analyze one if the other has been solved. If demand is discrete, the consequence of (c) and (d) is that there are discrete sets of ﬁll-rate constraint parameters and stockout costs at which F-optimality and S-optimality are interchangeable.

Our methods apply to the interchangeability of pairs of constraints and costs other than ﬁll rate and stockout cost. For example, in §3.7 we sketch results for balancing (i) inventory turnover ratios versus ﬁll rates, and (ii) inventory costs versus stockout frequency.

We provide a few portals to the large literature that is relevant to these issues.

Porteus (2002) and Zipkin (2000) are treatises on stochastic inventory models.

Also, see Porteus (1990) for a review of dynamic newsvendor models. Most of the literature on ﬁll-rate constrained models consists of heuristics and approximations.

For example, Silver (1970), Yano (1985), and Platt, Robinson, and Freund (1997) propose heuristic solutions to ﬁll-rate constrained models using (R, Q) policies.

Axs¨ater (2003) considers a continuous-review ﬁll-rate constrained serial system with batch ordering. The system faces a discrete compound Poisson demand process in which the leadtime demand has a negative binomial distribution. He shows that an optimal policy consists of a mixed multistage echelon stock (R, nQ) policy with one of the reorder points varying over time.

Schneider (1978) and Schneider and Ringuest (1990) study service-constrained 64

models with setup costs, focus on (s, S) policies where the order quantities are predetermined, and present several approximations to estimate the reorder point s such that the required service level is achieved. Schneider and Ringuest consider a periodic review system with a ﬁxed leadtime.

Van Houtum and Zijm (2000) discuss the extent to which a cost-optimal policy in the dynamic newsvendor model is also optimal for a service-constrained model.

They consider several types of service contraints. When the demand distribution function is strictly increasing on R+, they assert that an S-optimal policy is also F- optimal for an appropriate ﬁll rate lower bound. We conﬁrm their assertion under more general conditions and examine when an F-optimal policy is also S-optimal.

We also show that the fill rate converges to their modified fill rate, characterize 65

the F-optimal policies with continuous and discrete demand distributions, and establish the correspondences between backorder costs and ﬁll rate contraints.

Chen and Zheng (1993) present nonlinear backorder costs in the setup cost inventory model and shows that certain demand distributions guarantee the quasi- convexity of cost rate function. Therefore, results derived from quasi-convex cost functions remain valid. Rosling (2002a, 2002b) extends this inventory model

(Chen and Zheng 1993). Rosling (2002a) gives more general conditions than that of Chen and Zheng (1993) on the demand process for which the cost function is quasi-convex on the inventory position. Rosling (2002b) has square-root algorithm to ﬁnd the optimal inventory policy. It also investigates the inventory problem with various service-contraints and discusses the relationship between shortage costs and service constraints. Our paper focus on the dynamic newsvendor model and assumes one linear shortage cost which is the case in nearly all of the literature with backorder costs.

The ability to calculate the ﬁll rate is a prerequisite to ﬁnding an F-optimal

policy. Johnson et al. (1995) review ﬁll rate approximations in periodic-review

single-stage models with base-stock level policies. Glasserman and Tayur (1994),

Glasserman and Liu (1997), Glasserman (1997), Sobel (2004), and Chapter 2 of

this dissertation have other approximations and exact expressions for ﬁll rates of

single-stage and multi-stage systems with base-stock level policies. 66

The paper has the following organization. §3.2 introduces the model and notation and §3.3 characterizes F-optimal policies when demand has a density function.

§3.4 has the corresponding characterization when demand is an integer-valued random variable. §3.5 relates F-optimality to S-optimality with characterizations and algorithms. §3.6 contains numerical examples and §3.7 generalizes and summa- rizes the results.

3.2 Model and Problem Formulations

We consider a periodic-review inventory model in which an inventory manager

has a single product that faces independent and identically distributed nonneg-

ative demands D1, D2, ··· in successive periods. For any positive integer j let

j j D = n=1 Dn,letD have the same distribution as D1,andletµ = E(D)with

0 <µ<∞. Ordering decisions are made at the beginning of each period, and each order takes L periods to deliver, i.e., items ordered in period n are delivered at the beginning of period n + L (where L is a scalar integer). If a is the order-up-to level in period n, then the expected excess inventory and excess back-

L + + ordered demand in period n+L are E[(a− j=0 Dn+j ) ] (deﬁne u =max{u, 0})

L + L + and E[( j=0 Dn+j − a) ], respectively. Let Bn+L(a)=( j=0 Dn+j − a) and

L + Hn+L(a)=(a− j=0 Dn+j ) ,andletB(a)=E[Bn+L(a)] and H(a)=E[Hn+L(a)].

Given an inventory replenishment policy, initial inventory level, and inbound

N N N shipments due in periods 1, ..., L,letF , KI ,andKS denote, respectively, the 67

ﬁll rate, the average end-of-period inventory level, and the average end-of-period stockout level, all during periods L +1, ···,L+ N.

Let yn denote the inventory position after placing an order in period n,namely

the physical inventory (net of backlog) at the beginning of the period plus the sum

of shipments which will arrive at the beginning of periods n, n+1, ···,n+L.Since

L−1 yn − j=0 Dn+j is the amount of goods that will be available to satisfy Dn+L,

N L−1 N N F = E min {Dn+L,yn − Dn+j } Dn+L n=1 j=0 n=1 N N KI = E Hn+L(yn) /N n=1 N N KS = E Bn+L(yn) /N n=1

We assume that excess demand is backordered and order quantities are nonnegative. Let xn denote the inventory position net of backlog at the start of period n,soxn+1 = yn − Dn and let π be an inventory replenishment policy, namely, a

nonanticipative decision rule that yields nonnegative order quantities, i.e., yn ≥ xn

for all n. We assume that there are no planned backorders, so, yn ≥ max{xn, 0}.

Let c, h,andb be the respective unit costs of replenishment, end-of-period inventory, and end-of-period backorders. Let F denote the inﬁnite-horizon ﬁll

N rate, namely, F = lim infN→∞ F .

These problems are considered in subsequent sections:

N infπ lim inf Eπ|x (hKI ):F ≥ f (3.56) N→∞ 1 N N infπ lim inf Eπ|x (hKI + bKS ) (3.57) N→∞ 1 68

N N infπ lim inf Eπ|x (hKI − λF ) (3.58) N→∞ 1

Problem (3.56) minimizes the average inventory cost subject to a ﬁll rate constraint. Problem (3.57) minimizes the average sum of inventory and stockout costs. Problem (3.58) is a Lagrangean with terms for the average inventory cost and the ﬁll rate. In (3.56) - (3.58) there is no loss of generality in the assumption, henceforth made, that h = 1. In each problem, the primary issues are the form of an optimal policy and the set of optimal policies as a parameter is varied: f in

(3.56), b in (3.57), and λ in (3.58). However, The agenda is reduced by showing that (3.58) is a special case of (3.57). The identity z =(z)+ − (−z)+ implies

L Bn(a)=Hn(a) − a + Dn+j and B(a)=H(a) − a +(L +1)µ j=0

Therefore, for N<∞, the criterion in (3.57) is

N N lim inf Eπ|x (KI + bKS ) N→∞ 1 N = lim inf Eπ|x E Hn+L(yn)+bBn+L(yn) /N N→∞ 1 n=1 N L = lim inf Eπ|x E Bn+L(yn)+yn − Dn+j + bBn+L(yn) /N N→∞ 1 n=1 j=0 N = lim inf Eπ|x B(yn)(1 + b)+yn /N − (L +1)µ N→∞ 1 n=1

Rewriting F N as

N N N F = E [Dn+L −Bn+L(yn)] Dn+L n=1 n=1

the criterion in (3.58) is 69

N N KI − λF N L = E [Bn+L(yn)+yn − Dn+j ]/N n=1 j=0 N N − λE [Dn+L −Bn+L(yn)] Dn+L n=1 n=1 N L = E [Bn+L(yn)+yn − Dn+j ]/N n=1 j=0 N N N − λE Dn+L − Bn+L(yn) Dn+L n=1 n=1 n=1 N L = E [Bn+L(yn)+yn − Dn+j ]/N n=1 j=0 N N − λE 1 − Bn+L(yn) Dn+L n=1 n=1

N The law of large numbers implies that n=1 Dn+L/N → µ with probability one as N →∞.Since0<µ<∞, Slutsky’s theorem (Cram´er 1946, p. 255) yields

N N Bn+L(yn)/N D limN→∞ inf Bn+L(yn)/N lim inf n=1 −→ n=1 N→∞ N n=1 Dn+L/N µ

D where −→ denotes convergence in distribution.

Therefore,

N N n=1 Bn+L(yn) 1 n=1 Bn+L(yn) lim inf Eπ|x = lim inf Eπ|x (3.59) N→∞ 1 N N→∞ 1 n=1 Dn+L µ N if the right-side expectation in (3.59) exists. Henceforth, we restrict attention to policies for which the expectation exists.

Suppose policy π induces yn ≥ 0, for all n. The Helly-Bray Theorem (Lo`eve

1963, p. 182) implies that the right-side expectation in (3.59) exists if demand 70

is a bounded random variable. A less restrictive justiﬁcation begins with the observation N N n+L 0 ≤ Bn+L(yn)]/N ≤ Dj /N n=1 n=1 j=n whose right side has expectation (L +1)µ to which it converges in probability (by the strong law of large numbers). Since 0 <µ<∞, it follows that

N E{ n=1 Bn+L(yn)]/N } exists and is ﬁnite for every N and has a convergent sub-

sequence. That is, the right-side expectation in (3.59) exists.

Employing (3.59),

N N lim inf Eπ|x (KI − λF ) N→∞ 1 N L = lim inf Eπ|x E [Bn+L(yn)+yn − Dn+j ]/N N→∞ 1 n=1 j=0 N −λE 1 − Bn+L(yn) (Nµ) n=1 N = lim inf Eπ|x E Bn+L(yn)+yn N→∞ 1 n=1 L − Dn+j +(λ/µ)Bn+L(yn) N − λ j=0 N = lim inf Eπ|x B(yn)(1 + λ/µ)+yn N − (L +1)µ − λ N→∞ 1 n=1

Therefore, varying b in (3.57) is the same as varying λ in (3.58). The remainder

of the paper concerns only (3.56) and (3.57).

This argument results in the following property.

Proposition 3.2.1

1 N F =1− lim sup Eπ|x B(yn)/N (3.60) N→∞ 1 µ n=1 71

Proof N N N n=1[Dn+L −Bn+L(yn)] n=1 Bn+L(yn)/N F = E N =1− E N n=1 Dn+L n=1 Dn+L/N

which implies (3.60).

We note that Proposition 3.2.1 shows that the modified fill rate of Van Houtum and Zijm (2000) converges to the fill rate.

Let G(·) be the distribution function of DL+1 and let m =sup{a : G(a)=0} and M =inf{a : G(a)=1} be the smallest interval for which P {m ≤ DL+1 ≤

M} =1.LetG−1(θ)=inf{a : G(a) ≥ θ}, 0 ≤ θ ≤ 1. We denote ν as the mean of

L + 1 periods demand, so ν =(L +1)µ. It is well known (cf. Porteus 1990, 2002 and Zipkin 2000) that yn =max{xn,y∗} for all n is an S-optimal policy with

b y∗ = G−1( ) (3.61) b +1

So the parametric analysis of (3.57) consists of solving (3.61) as b spans positive real numbers.

In (3.56), the ﬁll-rate constrained minimization of inventory cost, Proposition

3.2.1 permits replacement of the ﬁll rate constraint with a constraint on the average backorder level. Therefore, an F-optimal policy solves the following problem:

N inf lim inf Eπ|x H(yn)/N : π N→∞ 1 n=1 N lim sup Eπ|x B(yn)/N ≤ µ(1 − f) (3.62) N→∞ 1 n=1

The following properties of H(·)andB(·) are useful in section §3.3. The straightforward proof is omitted. 72

Lemma 3.2.2 H(·) is convex on + and is strictly increasing on (m, ∞),and

B(·) is convex on + and is strictly decreasing on (0,M).Ifm ≥ 0 and M<∞, then H(a)=0for a ≤ m, H(a)=a − ν for a ≥ M, B(a)=ν − a for a ≤ m,and

B(a)=0for a ≥ M.

Deﬁnition A function is injective (or an injection) if it maps at most one point in the domain to each possible value in the range.

Let B−1(θ)=sup{y : B(y) ≥ θ},0≤ θ ≤ ν,andH−1(η)=sup{y : H(y) ≤

η},0≤ η ≤ M − ν.SinceH(·)andB(·) are strictly monotone and continuous

(due to convexity) on (m, ∞)and(0,M), respectively, they are injections; i.e.,

B[B−1(θ)] = θ and H[H−1(η)] = η. In subsequent sections we analyze problem

(3.62) for continuous demand and discrete demand.

3.3 Continuous Demand

First we analyze the following N-period version of (3.62) and then let N →∞:

N N π|x n π|x n ≤ − infπ E 1 H(y ) : E 1 B(y ) Nµ(1 f) (3.63) n=1 n=1

The single-period newsvendor model with a ﬁll rate constraint is the special case of N = 1 in (3.63):

π|x 1 π|x 1 ≤ − infπ E 1 [H(y )] : E 1 [B(y )] µ(1 f) (3.64)

The following result states that a base-stock policy is F-optimal for (3.64). 73

Proposition 3.3.1 If demand has a density function, then y = max{y∗,x} is

F-optimal in (3.64) with

y∗ = B−1[µ(1 − f)]. (3.65)

Proof Since H(·) is convex and increasing and B(·) is covex and decreasing, the smallest y that satisﬁes the constraint achieves the constrained minimum of

H(·).

∗ If x1 ≤ y , then the base-stock policy in (3.65) is optimal in (3.63) regardless

of the value of N.

∗ ∗ Lemma 3.3.2 If demand has a density and x1 ≤ y ,thenyn = y , n =1, 2, ···,N, is F-optimal in (3.63) for all ﬁnite horizons N ∈ I +.

Proof The claim is valid for N = 1 (Proposition 3.3.1). For any N>1, rewrite

(3.63) as N N π n π n ≤ − infπ E H(y ) : E B(y ) Nµ(1 f) (3.66) n=1 n=1

For any N>1, yn ≥ 0forn =1, ···,N. Then (3.66) is a convex nonlinear program for which the following Karush-Kuhn-Tucker conditions are necessary and suﬃcient when demand has a density:

E[H (yn)] + γ · E[B (yn)] − ρn =0forn =1, 2, ···,N(3.67) N γ · E[B(yn)] − Nµ(1 − f) = ρnyn =0forn =1, 2, ···,N(3.68) n=1 N E[B(yn)] ≤ Nµ(1 − f),yn ≥ 0,γ ≥ 0, and ρn ≥ 0. for n =1, 2, ···,N(3.69) n=1 74

Since H(·) > 0andB(·) < 0on(m, M), if γ = 0, then (3.67) implies that

ρn = E[H (yn)] > 0forn =1, ···,N. Consequently, from (3.68), yn =0for n =1, ···,N.Butthen,

N E[B(yn)] = Nν = N(L +1)µ>Nµ(1 − f) for all f>0 n=1

which contradicts (3.69).

So γ = 0, and (3.68) yields

N E[B(yn)] = Nµ(1 − f). (3.70) n=1

Also, from (3.67),

E[H (y1)] E[H (y2)] ··· E[H (yN )] = = = E[B (y1)] E[B (y2)] E[B (yN )]

So, y1 = y2 = ··· = yN because H(·)andB(·) are convex and monotone. There- fore, with (3.70),

E[B(y1)] = E[B(y2)] = ···= E[B(yN)] = µ(1 − f)

Since B(·) is an injection, it follows that

∗ −1 y1 = y2 = ···= yN = y = B [µ(1 − f)]

∗ ∗ Since x1 ≤ y , xn+1 = yn − Dn and P {Dn ≥ 0} = 1 ensure that yn = y ≥ xn for all n. Therefore, adding the constraints yn ≥ xn for all n reduces the feasibility set

∗ ∗ of (3.66), but does not aﬀect the optimality of yn = y for all n (given x1 ≤ y ). 75

The transition from (3.63) to (3.62) is consistent with the large literature which connects ﬁnite horizon and inﬁnite horizon inventory models. We exploit the fact that eventually the inventory level is at least as low as the back-stock level y∗

(regardless of the initial inventory level). The proof is brief, straightforward, and

omitted.

∗ ∗ Lemma 3.3.3 With probability one there is a period n < ∞, such that xn ≤ y

for all n ≥ n∗.

Proposition 3.3.4 If demand has a density, then the base stock policy yn =

∗ ∗ max{y ,xn} for all n,withy specified in (3.65), is F-optimal for (3.62), the infinite horizon problem with a fill rate constraint.

N ≤ − Proof For all N and for all π such that Eπ|x1 [ n=1 B(yn)] Nµ(1 f), if

∗ x1 ≤ y , N ∗ NH(y ) ≤ Eπ|x1 H(yn) (Lemma 3.3.2) n=1 N ∗ ≤ 1 H(y ) Eπ|x1 H(yn) N n=1

Therefore, if π is feasible in (3.62),

N ∗ 1 H(y ) ≤ lim inf Eπ|x H(yn) N→∞ 1 N n=1

Therefore, π∗ is optimal in (3.62) because it is feasible due to Lemma 3.3.3 and

N n∗−1 N 1 ∗ lim inf Eπ∗|x H(yn) = lim inf Eπ∗|x H(xn)+ H(y ) N→∞ 1 N→∞ 1 n=1 N n=1 n=n∗ 76

N 1 ∗ = lim inf Eπ∗|x H(y ) N→∞ 1 N n=n∗ = H(y∗)

Proposition 3.3.4 states that a nonrandomized base-stock policy, with target base-stock level B−1[µ(1 − f)], is F-optimal when demand has a density. The

optimal policy is randomized when demand is discrete.

3.4 Discrete Demand

L+1 Demand and order quantities are integer-valued in this section. Let ξj = P {D = j},j =0, 1, ..., M. For a target ﬁll rate f<1, identify k

B(k +1) <µ(1 − f) ≤ B(k). If B(k)=µ(1 − f), it is clear from §3that yn = k for all n =1, 2,... is F-optimal. The remainder of this section concerns the typical situation in which B(k +1)<µ(1 − f)

That is, a randomized policy dominates the best unrandomized policy. We note that yn = k +1foralln is F-optimal among unrandomized policies.

Recall that xn denotes the inventory level at the beginning of period n.We prove that the following policy, labeled π∗, is F-optimal among randomized policies. For all n,ifxn = k +1,thenyn = k +1. Ifxn ≤ k then yn = k +1 with 77

probability β and yn = k with probability 1 − β where

γ(1 − ξ0) B(k) − µ(1 − f) β = ,γ= (3.71) 1 − γξ0 B(k) − B(k +1)

We conﬁrm that π∗ is feasible and then prove that it is optimal.

Lemma 3.4.1 The ﬁll rate of π∗ with parameters in (3.71) is f.

Proof Without loss of generality let x1 ≤ k.Then{yn} is a Markov chain with states k and k + 1, transition probabilities pkk =1− β, pk,k+1 = β, pk+1,k =

(1 − β)(1 − ξ0), and pk+1,k+1 =1− (1 − β)(1 − ξ0), and stationary probabilities

∗ (1 − β)(1 − ξ0) ∗ β λk = λk+1 = 1 − (1 − β)ξ0 1 − (1 − β)ξ0

Using (3.71),

µ(1 − f) − B(k +1) B(k) − µ(1 − f) λ∗ = λ∗ = (3.72) k B(k) − B(k +1) k+1 B(k) − B(k +1)

Therefore, the ﬁll rate, using (3.60) and (3.72), is

1 F =1− [λ∗ B(k)+λ∗ B(k +1)] µ k k+1 1 [µ(1 − f) − B(k +1)]B(k)+[B(k) − µ(1 − f)]B(k +1) =1− = f µ B(k) − B(k +1)

Let Π(f) be the set of all randomized stationary policies which result in F ≥ f,

π for π ∈ Π(f)let {λj } be the stationary distribution of {yn} from initial state 0,

π π and write Hλ for j H(j)λj .

Proposition 3.4.2 Policy π∗ is F-optimal among randomized stationary policies. 78

∗ Proof Since π∗ ∈ Π(f) from Lemma 3.4.1, we show that Hλπ =min{Hλπ : π ∈

Π(f)}.

Since there are no planned backorders and P {0 ≤ DL+1 ≤ M} =1,with probability one max{0,xn}≤yn ≤ M and −M ≤ xn ≤ M for all n. Under

any stationary policy π, the inventory position {yn} is a Markov chain with states

{0, ···,M}.Sinceξ0 < 1, state 0 is recurrent. Therefore, let {λa : a =0, ···,M} be the stationary distribution.

The following linear program minimizes the average holding cost subject to a

ﬁll rate constraint. Find {λa : a =0, ···,M} in order to

M minimize H(a) λa a=0 M subject to B(a) λa ≤ µ(1 − f) (3.73) a=0 M λa =1 a=0

λa ≥ 0,a=0, ···,M

The proof is completed with the following lemma which is proved in the Appendix.

∗ ∗ Lemma 3.4.3 The optimal solution of (3.73) is λk = λk, λk+1 = λk+1,and

λa =0for a = k and a = k +1.

Let wja be the probability of ordering a − j units when the inventory level

∗ ∗ is j; j = −M, ···,M; a =0, ···,M. Policy π induces wjk =(1− β)(λkξk−j +

∗ ∗ ∗ ∗ λk+1ξk+1−j ), wj,k+1 = β(λkξk−j + λk+1ξk+1−j )ifj ≤ kwk+1,k+1 = λk+1ξ0,and wja =0,otherwise. 79

We note that π∗ is not uniquely optimal because other policies can induce the same stationary distribution of the inventory position.

3.5 Interchangeability

For b ≥ 0andf ∈ [0, 1], let yS(b)=G−1[b/(b +1)]andyF (f)=B−1[µ(1 − f)].

If b is the unit stockout cost, then yS(b)=G−1[b/(b + 1)] is an S-optimal base- stock level. If f constrains the ﬁll rate and demand is continuous, then yF (f)=

B−1[µ(1 − f)] is an F-optimal base-stock level. If demand is integer-valued, there

is a randomization among yF (f)andyF (f) + 1 that is F-optimal. This section focuses on two questions. First, for a unit stockout cost b,isthereanf such that yF (f) is an S-optimal base stock level? Second, if f constrains the ﬁll rate, is there a b such yS(b) is an F-optimal base stock level (randomization among

yS(b)andyS(b) + 1 if demand is discrete)? Both questions have positive answers

and if demand is continuous, there is a formula for b in terms of f and conversely.

Although the relation between b and f is more complicated with discrete demand,

if B[yF (f)] = µ(1 − f), then an F-optimal policy employs base stock level yF (f)

(with probability one). We say that demand has a strictly positive density g(·)if g(a) > 0,m

Let F (y) be the ﬁll rate induced by the base-stock policy with base-stock level y.ThenF (M)=1.LetbU =inf{b ≥ 0:G−1[b/(b +1)]=M}. 80

Proposition 3.5.1 If demand is continuous: yF (f)=yS(b) if (3.74) (equivalently (3.75)) is valid G B−1[µ(1 − f)] b = (3.74) 1 − G B−1[µ(1 − f)] B G−1[b/(b +1)] f =1− (3.75) µ

Also, yF (·) is an injection on [0, 1],

yF F [yS(b)] = yS(b)=G−1[b/(b + 1)] (3.76)

yS G[yF (f)]/{1 − G[yF (f)]} ≥ yF (f) (3.77) and, if the density is strictly positive, the inequality in (3.77) is satisﬁed as an equality and yS(·) is an injection on [0, bU ].

Proof When the distribution of demand has a density, yF (·)isaninjectionon

[0, 1] because both B(·)andB−1(·) are continuous and strictly monotone. There- fore, equating yS(b)=G−1[b/(b + 1)] with yF (f)=B−1[µ(1 − f)] and applying

B(·) to both sides yields (3.74) and (3.75). Similarly, Proposition 3.2.1 implies

F [yS(b)] = 1 − B{yS(b)}/µ. With the deﬁnitions of yF (·)andyS(·), this implies

(3.76). For (3.77), let a = yF (f)inyS{G(a)/[1 − G(a)]} = G−1[G(a)] ≥ a with equality if G has a strictly positive density. Finally, if demand has a strictly positive density, then yS(·) is an injection because both G(·)andG−1(·) are continuous and strictly monotone. 81

If demand has a strictly positive density function, Proposition 3.5.1 states that

S-optimality and F-optimality are equivalent in the following sense:

(a) Corresponding to any unit stockout cost b, a base-stock policy with base-

stock level yS(b) is F-optimal if f =1− B[yS(b)]/µ.

(b) Corresponding to any ﬁll-rate constraint parameter f, a base-stock policy with base-stock level yF (f) is S-optimal if b = G[yF (f)] 1 − G[yF (f)] .

The properties are weaker if demand is discrete or if it is continuous with a density that is not strictly positive. In both instances, G(·) is not strictly monotone which causes discontinuities in G−1(·). If demand is discrete, recall the

L+1 notation ξj = P {D = j},j=0, 1, ..., M. For expository convenience, in the next result we assume ξj > 0, j =0, 1, ···,M. Only minor modiﬁcations are

needed if ξj =0forsomej.

Proposition 3.5.2 If demand is discrete:

1. If G[yF (f) − 1]/{1 − G[yF (f) − 1]}

yF (f) is S-optimal with parameter b.

2. If f =1− B[yS(b)]/µ,thenyS(b) is F-optimal with parameter f.If1 −

B[yS(b)]/µ ≤ f<1−B[yS(b)+1]/µ, then a randomized policy that employs

base-stock level yS(b) with probability 1 − β and yS(b)+1 with probability β

is F-optimal; β is speciﬁed by (3.71).

F F F Proof Forpart1,letb1 = G[y (f)−1]/{1−G[y (f)−1]} and b2 = G[y (f)]/{1− 82

F G[y (f)]}.SinceG(·) is nondecreasing on [0, M], b1

b1 G−1( )=G−1{G[yF (f) − 1]} = yF (f) − 1 b1 +1 b2 G−1( )=G−1{G[yF (f)]} = yF (f) b2 +1

−1 F Since b1

3.4.2.

The relationship between S-optimality and F-optimality is more complicated when demand is discrete than when it is continuous because F-optimal policies are random only in the former case. Nevertheless, a parametric analysis of either kind of optimality can be accomplished via the other kind in the following sense:

(c) Corresponding to any unit stockout cost b, base-stock level yS(b)isF- optimal with constraint parameter f =1− B[yS(b)]/µ. Also, if 1 − B[yS(b)]/µ < f<1 − B[yS(b)+1]/µ, then a policy which randomizes between base-stock levels

yS(b)andyS(b) + 1 is F-optimal with parameter f.

(d) Corresponding to any ﬁll-rate constraint parameter f, base-stock level

yF (f) is S-optimal for any stockout cost b in the interval G[yF (f) − 1]/{1 −

G[yF (f) − 1]}

3.6 Examples

This section illustrates the results with numerical examples having continuous

demand with a strictly positive density, continuous demand lacking a strictly 83

positive density, and discrete demand. We shall clarify that qualitative results obtained in the previous sections do not depend on leadtime L. For expository convenience, the following numerical examples use L =0.

3.6.1 Strictly Positive Demand Density

In this subsection, the density function is g(a)=1/10 and G(a)=a/10, a ∈

[0, 10]; so µ = 5. Therefore, G−1(θ)=10θ, B(y)=E[(D − y)+]=(10− y)2/20, √ and B−1(θ)=10− 20θ. Using (3.61), an S-optimal policy employs base-stock

level b 10b yS(b)=G−1( )= b +1 b +1

In this example, yS(·) is concave and increasing. Using (3.65), an F-optimal policy has base-stock level

yF (f)=B−1[µ(1 − f)] = 10 − 20µ(1 − f)=10− 10 1 − f

In this example, yF (·) is convex and increasing. Simplifying yS(b)=yF (f) yields

(3.74) and (3.75): √ 1 − 1 − f b2 +2b b = √ f = (3.78) 1 − f (b +1)2

Each equation in (3.78) uniquely deﬁnes the one-to-one mapping between b and f. Considering b as a function of f,inthisexampleb(·) is convex and increasing.

Considering f as a function of b,inthisexamplef(·) is concave and increasing. 84

3.6.2 Non Strictly Positive Demand Density

In this subsection, g(a)=1/10 for a ∈ [0, 5] and a ∈ [6, 11], and otherwise g(a) = 0. Therefore, µ =5.5andG(a)=a/10 if a ∈ [0, 5], G(a)=1/2if a ∈ (5, 6], and G(a)=(a − 1)/10 if a ∈ (6, 11]. So G−1(θ)=10θ if 0 ≤ θ ≤ 1/2 and G−1(θ)=10θ +1 if1/2 <θ≤ 1. That is, G−1(·) is strictly increasing but discontinuous at θ =1/2. Using (3.61) an S-optimal policy employs base-stock level ⎧ ⎪ 10b/(b +1), if 0 ≤ b ≤ 1 b ⎨ yS(b)=G−1( )= b +1 ⎩⎪ (11b +1)/(b +1), if b>1. Although yS(·) is strictly increasing as in Section 3.6.1, it is neither concave nor continuous due to the discontinuity of G−1(·). As Figure 3.3 shows, no stockout cost b induces yS(b) ∈ (5, 6]. However, yS(b)=G−1[b/(b + 1)] is not necessarily uniquely optimal. In this case, if b =1,anybase-stocklevelin[5, 6] is optimal.

Figure 3.3: Dependence of S-Optimal Base-Stock Level on Stockout Cost: Non- negative Density 85

Continuing, ⎧ 2 ⎪ (y − 20y + 110)/20 if 0 ≤ y ≤ 5 ⎪ ⎪ ∞ ⎨ B(y)= (a − y)g(a)da = (17 − 2y)/4if5

Let P = {x :0≤ x ≤ 11} and Q = {x :5

{yF (f):0≤ f ≤ 1} and P − Q = {yS(b):0≤ b}.Fory ∈ P − Q, the one-to-one correspondence between b and f given in (3.74) and (3.75) is the following:

10b =10− 100 − 110f for y ∈ [0, 5] b +1 11b +1 =11− 110 − 110f for y ∈ (6, 11] b +1

Thus, for each b, yF (f) is uniquely S-optimal with

f = 10(b2 +2b)/[11(b +1)2] ∈ [0, 15/22].

Conversely, yS(b) is uniquely F-optimal using

b =[10− 100 − 110f]/ 100 − 11f ∈ [0, 1].

There are similar relationships for b ∈ (0, ∞]andf ∈ (17/22, 1], where

f = [11(b2 +2b)+1]/[11(b +1)2 and b =[10− 110 − 110f]/ 110 − 110f. 86

In this example, there is no one-to-one mapping between b and f.Iff ∈

(15/22, 17/22], yF (f)=11f − 5/2 is an F-optimal base-stock level but there is no

b for which yS(b)=11f − 5/2; cf. Figure 3.4.

Figure 3.4: Locus of {(b, f)} with the Same Optimal Base-Stock Level: Nonneg- ative Density

3.6.3 Discrete Demand

a In this subsection, ξk =1/10, k =0, 1, ···, 9; so µ =4.5, G(a)= a=0 1/10

(0 ≤ a ≤ 9), and G−1(θ)=inf{a : G(a) ≥ θ}(0 ≤ θ ≤ 1). Deﬁne G−1(0) = 0 to avoid a triviality. Discrete demand causes G(·) to be piece-wise constant, so

(3.61) speciﬁes a many-to-one correspondence between unit stockout costs and

S-optimal base stock levels. That is, yS(·) is piece-wise constant. Here, using

(3.65),

B(y)=E[(D − y)+]=(9−y)(10 − 2y + y)/20 87

9 yF (f)=B−1[µ(1 − f)] = sup{y : (a − y) ≥ 45(1 − f)} a= y

That is, yF (·) is piece-wise constant (and discontinuous) although B(·)iscon- tinuous. Table 3.4 speciﬁes G(·)andB(·) and Table 3.5 tabulates G−1(·)and

B−1(·). Tables 3.6 and 3.7 illustrate Proposition 3.5.2. Table 3.6 gives yS(b)as b varies, together with the values of f for which yS(b)=yF (f). Table 3.7 gives yF (f)asf varies, together with the values of b which cause yS(b)=yF (f). It is apparent from Tables 3.6 and 3.7 that the parametric analysis of optimal policies can be accomplished either with b or f.

y G(y) B(y)=E[(D − y)+] [0, 1) 0.1 4.5-0.9y [1, 2) 0.2 4.4-0.8y [2, 3) 0.3 4.2-0.7y [3, 4) 0.4 3.6-0.6y [4, 5) 0.5 3.5-0.5y [5, 6) 0.6 3.0-0.4y [6, 7) 0.7 2.4-0.3y [7, 8) 0.8 1.7-0.2y [8, 9) 0.9 0.9-0.1y [9, ∞) 1 0

Table 3.4: Distribution Function and Expected Number of Backorders

For example, when 7/3

θ G−1(θ) θ B−1(θ) [0, 0.1] 0 (3.6, 4.5] 0 (0.1, 0.2 ] 1 (2.8, 3.6] 1 (0.2, 0.3 ] 2 (2.1, 2.8] 2 (0.3, 0.4 ] 3 (1.5, 2.1] 3 (0.4, 0.5 ] 4 (1.0, 1.5] 4 (0.5, 0.6 ] 5 (0.6, 1.0] 5 (0.6, 0.7 ] 6 (0.3, 0.6] 6 (0.7, 0.8 ] 7 (0.1, 0.3] 7 (0.8, 0.9 ] 8 (0, 0.1] 8 (0.9, 1.0 ] 9 0 9

Table 3.5: Values of G−1(·)andB−1(·)

S − B[yS(b)] − B[yS(b)] − B[yS(b)+1] b y (b) 1 µ [1 µ , 1 µ ) [0, 1/9] 0 0 [0, 0.2) (1/9, 1/4] 1 0.2 [0.2, 0.377) (1/4, 3/7] 2 0.377 [0.377, 0.533) (3/7, 2/3] 3 0.533 [0.533, 0.667) (2/3, 1] 4 0.667 [0.667, 0.778) (1, 3/2] 5 0.778 [0.778, 0.867) (3/2, 7/3] 6 0.867 [0.867, 0.933) (7/3, 4] 7 0.933 [0.933, 0.978) (4, 9] 8 0.978 [0.978, 1) (9, ∞) 9 1 1

Table 3.6: S-optimal Base-Stock Levels and Fill Rates at which they are F-optimal

optimal among deterministic policies, a randomized policy is better. For example,

if f =0.95, then (cf. Proposition 3.5.2), the following randomized policy strictly

dominates all deterministic policies. Order nothing when the inventory level is at

least 8, and otherwise order up to 7 and 8 with respective probabilities 1 − β and

β. Using (3.71) and Table 3.4, γ =[B(7) − 4.5(1 − 0.95)]/[B(7) − B(8)] = 0.375 and β =[0.375(1 − 0.1)]/[1 − 0.375(0.1)] = 0.35. 89

F G[yF (f)−1] G[yF (f)] f y (f) ( 1−G[yF (f)−1], 1−G[yF (f)]] [0, 0.2) 0 [0, 1/9] [0.2, 0.378) 1 (1/9, 1/4] [0.378, 0.533) 2 (1/4, 3/7] [0.533, 0.667) 3 (3/7, 2/3] [0.667, 0.778) 4 (2/3, 1] [0.778, 0.867) 5 (1, 3/2] [0.867, 0.933) 6 (3/2, 7/3] [0.933, 0.978) 7 (7/3, 4] [0.978, 1.0) 8 (4, 9] 1.0 9 (9, ∞)

Table 3.7: F-optimal Base-Stock Levels and Unit Stockout Costs at which they are S-optimal

3.7 Generalizations and Summary

Numerous pairs of constraints and costs - not only ﬁll rate constraints and stockout

costs - are interchangeable in the sense of §3.5. The key prerequisite is convexity and monotonicity as in Lemma 3.2.2. As a ﬁrst example, consider balancing inventory turnover ratio versus ﬁll rate. We model the turnover ratio as the ratio of the demand to the average of the inventory levels at the beginning and ending of the period. Accordingly, if τ denotes the constraining upper bound on turnover ratio, then the argument that leads to Proposition 1 shows that the constraint corresponds to

N lim sup E{ [yn + H(yn)]/N }/µ ≤ 2µ/τ N→∞ n=1

We consider (i) maximization of the ﬁll rate subject to this constraint, and (ii)

minimization of the long-run average value of h[H(yn)+yn]/2+bB(yn). Let h =1 90

without loss of generality, and let Q(y)=H(y)+y. Obvious analogues of the results in Sections 3 through 5 are valid for these problems because Q(·)isconvex and strictly increasing on (m, M). For example, if demand has a density, then the

H following base-stock policy is optimal for problem (i): yn = max{y ,xn} for all n, with yH = Q−1(2µ/τ). The following base-stock policy is optimal for problem (ii):

T T −1 yn = max{y ,xn} for all n,withy = G [(b − 1)/(b + 1)]. Interchangeability follows from the fact that yH and yT span the same set as τ and b span the nonnegative numbers.

A second example of the broader applicability of the methods used in earlier sections is the task of balancing inventory costs and stockout frequency. The task can be formalized in several ways including (iii) minimizing the long-run average inventory cost subject to an upper bound on the long-run relative frequency of stockout, and (iv) minimizing a weighted linear combination of inventory cost and probability of stockout. If demand is continuous with a concave distribution function and m is the upper bound on stockout probability, then the following

u base-stock policy is optimal for problem (iii): yn = max{y ,xn} for all n,with

yu = G−1(1 − m). In problem (iv), if the weight on inventory cost is normalized to unity and c is the weight on stockout probability, then the following base-stock

c c policy is optimal for problem (iv): yn = max{y ,xn} for all n,wherey minimizes

H(y)+c[1 − G(y)]. Interchangeability follows from the fact that yu and yc span thesamesetasm spans [0, 1] and c spans the nonnegative numbers. 91

Generally, in this paper we make precise the extent to which Stockout-optimality

(S-optimality) and Fill-rate-optimality (F-optimality) are interchangeable in dynamic newsvendor models. We also show that F-optimality is achieved by a base- stock policy that is unrandomized if demand is continuous. If demand is discrete, it is achieved by a stationary policy that randomizes between two adjacent base- stock levels. Also, there is a weakly monotone mapping from the set of S-optimal policies to the set of F-optimal policies. For practical purposes, the parametric analysis of either kind of problem can be performed via an algorithm directed at the other case. So it is unnecessary to study both cases separately.

3.8 Appendix

Proof of Lemma 3.4.3

Proof Expression (3.72) is feasible in (3.73). In order to prove optimality, let δa

be the reduced cost of λa in the simplex algorithm; if δa ≥ 0 for all a = k, k +1, then (3.73) is optimal. The corresponding basis matrix of (3.73) is

B(k) B(k +1) B = 11

Since det(B)=B(k) − B(k +1)> 0, for each λa the reduced cost is

H(a)[B(k) − B(k +1)]− H(k)[B(a) − B(k +1)]+H(k +1)[B(a) − B(k)] δa = B(k) − B(k +1) (3.79) 92

Let ∆a be the numerator of (3.79); since B(k) − B(k +1)> 0, δa and ∆a have thesamesigns.

If a

∆a = H(a)[B(k) − B(k +1)]− H(k)[B(a) − B(k +1)]+H(k +1)[B(a) − B(k)]

= H(a)[B(k) − B(k +1)]− H(k)[B(a) − B(k)+B(k) − B(k +1)]

+H(k +1)[B(a) − B(k)]

=[H(k +1)− H(k)][B(a) − B(k)] − [H(k) − H(a)][B(k) − B(k +1)]

We use an induction starting with ∆k = 0 to show that ∆a ≥ 0. If ∆k−n ≥ 0, for a = k − n − 1andn =0, ···,k− 2

∆k−n−1 =[H(k +1)− H(k)][B(k − n − 1) − B(k)]

−[H(k) − H(k − n − 1)][B(k) − B(k +1)]

=[H(k +1)− H(k)][B(k − n − 1) − B(k − n)+B(k − n) − B(k)]

−[H(k) − H(k − n)+H(k − n) − H(k − n − 1)][B(k) − B(k +1)]

=[H(k +1)− H(k)][B(k − n − 1) − B(k − n)]

−[H(k − n) − H(k − n − 1)][B(k) − B(k +1)]+∆k−n. (3.80)

Since H(·)andB(·) are convex and monotone, ∆k−n−1 ≥ 0 because

H(k +1)− H(k) ≥ H(k − n) − H(k − n − 1) ≥ 0

B(k − n − 1) − B(k − n) ≥ B(k) − B(k +1)≥ 0.

If a>k+1,

∆a = H(a)[B(k) − B(k +1)]− H(k)[B(a) − B(k +1)] 93

+H(k +1)[B(a) − B(k +1)+B(k +1)− B(k)]

=[H(a) − H(k +1)][B(k) − B(k +1)]− [H(k +1)− H(k)][B(k +1)− B(a)]

Another induction concludes that ∆a ≥ 0 for all a>k+1. Bibliography

Archibald, T. W., L. C. Thomas, J. M. Betts, and R. B. Johnston (2002). Should start-up companies be cautious? inventory policies which maximise survival probabilities. Management Science 48, 1161–1174. Axsäter, S. (2003). Note: Optimal policies for serial inventory systems under fill rate constraints. Management Science 49, 247–253. Babich, V. and M. J. Sobel (2002). Pre-IPO operational and financial decisions. Technical Report: TM-760R, Department of Operations, Weather- head School of Management, Case Western Reserve University. Beckmann, M. J. (1961). Production smoothing and inventory control. Opera- tions Research 9, 456–467. Boyaci, T. and G. Gallego (2001). Serial production/distribution systems under service constraints. Manufacturing & Service Operations Management 3, 43– 50. Buzacott, J. A. and R. Q. Zhang (1998). Production and financial decisions in a start-up firm. Working paper, York University, Toronto, Canada. Buzacott, J. A. and R. Q. Zhang (2003). Inventory management with asset- based financing. Working paper, York University, Toronto, Canada. Chen, F. and Y. Zheng (1993). Inventory models with general backorder costs. European Journal of Operational Research 65, 175–186. Chen, F. and Y. Zheng (1994). Lower bound for multi-echelon stochastic inventory systems. Management Science 45, 1426–1443. Clark, A. J. and H. Scarf (1960). Optimal policies for a multi-echelon inventory problem. Management Science 6, 475–490. Cramér, H. (1946). Mathematical Methods of Statistics. Princeton University Press, Princeton, NJ. Dammon, R. and L. Senbet (1988). The effect of taxes and depreciation on corporate investment and financial leverage. Journal of finance 43, 357–373. Datan, A. and S. Ravid (1958). On the interaction of real and financial decisions of the firm under uncertainty. The American Economic Review 48, 261–297.

94 95

DeAngelo, H. and R. Masulis (1980). Optimal capital structure under corporate and personal taxation. Journal of financial economics 7, 3–26. Federgruen, A. and P. Zipkin (1984). Computational issues in an infinite horizon multi-echelon inventory problem with stochastic demand. Operations Research 32, 818–836. Glasserman, P. (1997). Bounds and asymptotics for planning critical safety stocks. Operations Research 45, 244–257. Glasserman, P. and T. Liu (1997). Corrected diffusion approximations for a multistage production-inventory system. Mathematics of Operations Re- search 22, 186–201. Glasserman, P. and S. Tayur (1994). The stability of a capacitated, multi- echelon production-inventory system under a base-stock policy. Operations Research 42, 913–925. Graves, S. C., A. H. G. Rinnoy Kan, and P. H. Zipkin (1993). Logistics of replenishment and inventory. In Handbooks in Operations research and Man- agement Science, Volume 4. North-Holland. Hadley, G. J. and T. M. Whitin (1963). Analysis of Inventory Systems. Prentice Hall, Englewood Cliffs, NJ. Heyman, D. P. and M. J. Sobel (1984). Stochastic Models in Operations Re- search Volume II: Stochastic Optimization. McGraw-Hill, New York, NY. Hite, G. (1977). Leverage, output effects, and the m-m theorems. Journal of financial economics 4, 177–202. Hojgaard, B. and M. Taksar (2000). Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy. Working Paper, SUNY-Stony Brook, NY. Iglehart, D. L. (1963). Optimality of (s, S) policies in the infinite horizon dynamic inventory problem. Management Science 9, 259–267. Jensen, M. C. (1986). Agency costs of free cost flow, corporate finance and takeovers. American Economic Review, Papers and Proceedings 76, 323– 329. Johnson, M. E., H. Lee, T. Davis, and R. Hall (1995). Expressions for item fill rates in periodic inventory systems. Naval Research Logistics 42, 57–80. Li, L., M. Shubik, and M. J. Sobel (2003). Control of dividends, capital subscriptions, and physical inventories. Working paper, TM-763, Department of Operations, Weatherhead School of Management, Case Western Reserve University. 96

Loève, M. (1963). Probability Theory. D. Van Nostrand Company, Inc. Prince- ton, NJ. Long, M. and G. Racette (1974). Stochastic demand, output and the cost of capital. Journal of finance 29, 499–506. Miller, M. and K. Rock (1985). Dividend policy under asymmetric information. Journal of finance 40, 1031–1051. Milne, A. and D. Robertson (1996). Firm behaviour under the threat of liqui- dation. Journal of economic dynamics and control 20, 1427–1449. Modigliani, F. and M. H. Miller (1958). The cost of capital, corporate finance and the theory of investment. The American Economic Review 48, 261–297. Modigliani, F. and M. H. Miller (1963). Corporate income taxes and the cost of capital: a correction. The American Economic Review 53, 433–443. Moyer, F. C., J. R. McGuigan, and W. J. Kretlow (1990). Contemporary Fi- nancial Management, Fourth Edition. West Publishing company, St. Paul, MN. Myers, S. (1974). Interaction of corporate financing and investment decisions - implications of capital budgeting. Journal of finance 29, 1–25. Platt, D. E., L. W. Robinson, and R. B. Freund (1997). Tractable (Q, R) heuristic models for constrained service levels. Management Science 43, 951– 965. Porteus, E. L. (1990). Stochastic inventory theory. In Handbooks in OR and MS, Chapter 12. Editors: Heyman, D. P. and Sobel, M. J.,Volume2.Elsevier Science Publishers. Porteus, E. L. (2002). Foundations of Stochastic Inventory Theory. Stanford University Press, Stanford, CA. Rosling, K. (2002a). Inventory cost rate functions with nonlinear shortage costs. Operations Research 50, 1007–1017. Rosling, K. (2002b). The square-root algorithm for single-item inventory optimization. Working paper, School of Industrial Engineering, VäxjöUniver- sity, SE-351 95 Växjö, Sweden. Scarf, H. (1959). The optimality of (S, s) policies in the dynamic inventory problem. In Stanford University, Mathematical methods in the social sciences., pp. 196–202. Stanford university press. Schneider, H. (1978). Methods for determining the re-order point of an (s, S) ordering policy when a service level is specified. J. Oper. Res. Soc. 29, 1181– 1193. 97

Schneider, H. and J. L. Ringuest (1990). Power approximation for computing (s, S) policies using service level. Management Science 36, 822–834. Sethi, S. P. (1996). When does the share price equal the present value of future dividends. Economic Theory 8, 307–319. Shang, H. K. and J. Song (2003). Analysis of serial supply chain with a service constraint. Working paper, The Fuqua School of Business, Duke University, Durham, NC 27708. Silver, E. A. (1970). A modified formula for calculating customer service under continuous inventory reviews. AIIE Transactions 2, 241–245. Sobel, M. J. (1969). Production smoothing with stochastic demand I: finite horizon case. Management Science 16, 195–207. Sobel, M. J. (1971). Production smoothing with stochastic demand II: infinite horizon case. Management Science 17, 724–735. Sobel, M. J. (2004). Fill rates of single-stage and multi-stage supply systems. Manufacturing & Service Operations Management 6, 41–52. Taksar, M. and X. Zhou (1998). Optimal risk and dividend control for a company with a debt liability. Insurance: mathematics and economics 22, 105– 122. Tijms, H. C. and H. Groenevelt (1984). Simple approximations for the reorder point in periodic and continuous review (s, s) inventory systems with service level constraints. European Journal of Operational Research 17, 175–190. Van Houtum, G. J., K. Inderfurth, and W. H. M. Zijm (1996). Material coordination in stochastic multi-echelon systems. European Journal of Operational Research 95, 1–23. Van Houtum, G. J. and W. H. M. Zijm (2000). On the relation between cost and service model for general inventory systems. Statistical Neerlandia 54, 127–147. Veinott, A. F. and H. Wagner (1965). Computing optimal (s, S) policies. Man- agement Science 11, 525–552. Yano, C. A. (1985). New algorithm for (Q, r) systems with complete back- ordering using a fill-rate criterion. Naval Research Logistics Quarterly 32, 675–688. Zabel, E. (1962). A note on the optimality of (s, S) policies in inventory theory. Management Science 9, 123–125. Zipkin, P. (2000). Foundations of Inventory Management. McGraw-Hill, New York, NY.