zhang jiang

8/8/2019 ZHANG JIANG

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THREE ESSAYS ON INVENTORY MANAGEMENT

By

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


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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.

JIANG ZHANG

Lisa M. Maillart

Yunzeng Wang

Peter H. Ritchken

06/16/2004

Matthew J. Sobel


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To my mother: Huishu Jiang

my father: Shuqing Zhang

my wife: Dr. Yan Cao


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Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

1 Inventory Replenishment with a Financial Criterion . . . . . . . . . . . 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Dynamic Programming Analysis . . . . . . . . . . . . . . . . . . . 11

1.4 Optimality of (sn, Sn) Replenishment Policies . . . . . . . . . . . 14

1.5 Infinite Horizon Convergence . . . . . . . . . . . . . . . . . . . . . 18

1.6 Models with Smoothing Costs . . . . . . . . . . . . . . . . . . . . 19

1.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 23

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

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 General Periodic Review System . . . . . . . . . . . . . . . . . . . 29

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

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

2.4.1 Gamma Demand Distribution . . . . . . . . . . . . . . . . 35

2.4.2 Normal Demand Distribution . . . . . . . . . . . . . . . . 36

2.4.3 Fill Rate Approximation for Normal Demand Distribution 39

2.5 Multi-Stage General Review Systems . . . . . . . . . . . . . . . . 40

2.5.1 Fill Rate in Two-Stage Systems . . . . . . . . . . . . . . . 42

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

2.5.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . 51

2.6 Fill Rate in a Three-Stage System . . . . . . . . . . . . . . . . . . 51

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Interchangeability of Fill Rate Constraints and Backorder Costs

in Inventory Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2 Model and Problem Formulations . . . . . . . . . . . . . . . . . . 66

3.3 Continuous Demand . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.4 Discrete Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.5 Interchangeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.6.1 Strictly Positive Demand Density . . . . . . . . . . . . . . 83

3.6.2 Non Strictly Positive Demand Density . . . . . . . . . . . 84

3.6.3 Discrete Demand . . . . . . . . . . . . . . . . . . . . . . . 86

3.7 Generalizations and Summary . . . . . . . . . . . . . . . . . . . . 89

3.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

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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 ofG1() and B1() . . . . . . . . . . . . . . . . . . . . . . 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

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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-

negative Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

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

negative Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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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 departments 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,

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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.

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Three Essays on Inventory Management

Abstract

by

JIANG ZHANG

This dissertation consists of three essays that are related to inventory man-

agement.

The first essay models a single-product equity-owned firm which orders prod-

ucts 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 firms operational and financial decisions. The

analysis is not a straightforward copy of Scarfs 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

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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.

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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 crite-

ria 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 firms financial policy and that financial officers are usually detached

1


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from the inventory decisions. This separate consideration of financial and opera-

tional 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 firms capital structure and

its financial decisions should be independent of the firms investment and opera-

tional 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 infor-

mation 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 opti-

mization 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 firms optimal decisions are the same no matter if it


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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 firms decisions. Most of this literature focuses on the effects of mar-

ket imperfections on financial structures and decisions. Miller and Rock (1985)

extend the standard finance model of the firms dividend/investment/financing

decisions by allowing asymmetric information between the firms managers and

outside investors and they show that there exists an equilibrium investment pol-

icy which leads to lower levels of investment than the optimum achievable under


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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 firms 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 sin-

gle product equity-owned retail firm which periodically reviews its inventory and

retained earnings. Every period the firm faces random demand and replenishes


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stocks to satisfy customer demand. In addition, every R (a positive integer) peri-

ods, 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 or-

dering, 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 de-

cisions 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


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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) as-

sert 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 de-

cisions 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 deci-

sions. Babich and Sobel (2002) consider capacity expansion and financial decisions

to maximize the expected present value of a firms 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 mono-

tone threshold rule to yield an optimal IPO decision.


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The rest of the paper is organized as follows. Section 1.2 formulates the finan-

cial 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 share-

holders 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 . Weassume 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 = 1 sim-

plifies the presentation and Section 6 substantiates that R = 1 is without loss of


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generality.

At the beginning of each period, the firm also decides on the amount of divi-

dend to declare, vn. if vn 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 (bn) (where we assume () is a convex increasing function on +) is

paid, and the ordering decision is implemented at a cost of K (zn) + czn, where K

is an ordering setup cost, (zn) = 1 ifzn > 0, and (zn) = 0 otherwise. Then de-

mand 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(yd)+(dy)+

where yn, r, h and denote available goods in period n after delivery, unit sale

price, holding cost and shortage penalty cost, respectively. However, we only use

convexity of g(, d) for each d 0. Finally, the loan principal bn is repaid. We as-

sume that the demands D1, D2, are independent nonnegative random variablesand unmet demands are backlogged.

For convenience but without loss of generality, we assume that the order lead-

time 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 (bn) K(zn) (1.2)


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That is, In is working capital after the dividend is issued, loan interest and replen-

ishment 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 associated with In, that is, if working capital is

positive, the firm will gain In if In 0 or pay a penalty In if In < 0.

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

xn+1 = xn + zn Dn

wn+1 = (1 + )[wn p(wn) vn czn (bn) K(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 + )In + g(yn, Dn) (1.4)

We assume that the loan and replenishment quantities are nonnegative:

bn 0 and zn 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 (bn) p(wn) + vn + czn + K (zn) (1.6)


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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

denote the single period discount factor :

B =

n=1

n1vn (1.7)

Remark Discount factor can be regarded as the risk neutral discount rate for

the shareholders and need not to be a constant every period. Letting (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, , let Hn denote the history up to the beginning of period n,

namely,

Hn = (x1, w1, b1, I1, y1, D1, , xn1, wn1, bn1, In1, yn1, Dn1, xn, wn)

Let n be the set of all possible Hn sequences. A policyis 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, and bn as a function of Hn. An optimal policy maximizes E(B|Hn = )

for each

n, 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)


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and (1.2). The theme of 3 and 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 (bn) K (yn xn) and xn = yn1 Dn1 (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

B = cx1

n=1

ncDn

+

n=1

n1wn p(wn) In (1 )cyn (bn) K(yn xn)

= cx1 + w1 p(w1)

n=1

ncDn

n=1

n1K(yn xn)

+

n=1

n1(1 )cyn In +

(1 + )In + g(yn, Dn)

p(1 + )In + g(yn, Dn)

(bn)

For (b , I , y) 3 let

L(b , I , y) = (1 )(I+ cy) + I + E

g(y, D) p(1 + )I+ g(y, D)

(b)

(1.8)


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

E(B) = cx1+w1p(w1)E

n=1

n

cDn

+E

n=1

n1L(bn, In, yn)K(ynxn)

(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:

sup EH1

n=1

n1[L(bn, In, yn) K(yn xn)]

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 :

sup EH1 N

n=1

n1[L(bn, In, yn) K (yn xn)]

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

A dynamic recursion that corresponds to (1.11) is 0() 0 and for each x

and n = 1, 2, . . .,

n(x) = sup{Jn(b , I , y) K(y x) : y x, I+ b 0, b 0} (1.12)


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Jn(b , I , y) = L(b , I , y) + E[n1(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) and yn(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),

n(x) = supb,I,y

(1 )(I+ cy) + I + E

g(y, D) p[(1 + )I+ g(y, D)]

(b) + E[n1(y D)] K(y x) : y x, I+ b 0, b 0

= supy

c(1 )y + E[g(y, D)] + E[n1(y D)] K(y x)

+supI

(1 )I+ I E

p[(1 + )I + g(y, D)]

+sup

b

(b) : I + b 0, b 0

: I

: y x

The last supremum is achieved by b = 0 ifI 0 and by b = I if I < 0 because


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() is monotone increasing. Therefore,

n(x) = supy

c(1

)y + Eg(y, D) + En1(y

D) K (y

x)

+ supI

(1 )I+ I E

p((1 + )I+ g(y, D))

[(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. Scarfs proof uses the important concept of K-convexity. In this paper,

we use K-concavity.


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15

Definition A real-valued function f() on is K concave (K 0) if for all

x , 0, and > 0,

f(x + ) f(x) K+

[f(x) f(x )] (1.14)

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

tions.

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

(b) fi() is K-concave, i=1,2,. . . 1f1 + 2f2 is 1K1 + 2K2 concave (1 >

0, 2 > 0);

(c) f() is K-concave f() is V-concave for all V K;

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

Proof f() is K-concave if and only if f() is K 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 and Q() be the mean and distribution function of D. Let G(y) =

E[g(y, D)].

Lemma 1.4.2 (a) p[(1 + )I + g(y, d)] is convex with respect to (I, y) 2 (for

each d 0);

(b) L(, , ) is a concave function on its domain3.


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16

Proof (a) Since g(, d) is a convex on for each d 0, (1+ )I+ g(y, d) is jointly

convex in (I, d). The conclusion in (a) follows from monotonicity and convexity

of p(). p[(1 + )I + g(y, d)] is convex on I , so is E{p[(1 + )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, and b + 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 0() 0 initiates an inductive proof of K-concavity for

each n.

For any n 1, if n1() is K-concave, then

Jn(b ,I,y) = L(b , I , y) + E[n1(y D)]

is K-concave ( 0 < < 1 ) in y because L(, , ) is concave and E[n1(y D)]

is K-concave due to Lemma 1.4.1 (b).

To prove that n() is K-concave, we show that Tn() is K-concave whereTn(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) + E[n1(y D)] : b 0, b + I 0}


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17

= sup{L(0, I , y) + E[n1(y D)] (b) : b 0, b + I 0}

= E[n1(y

D)] + sup

{L(0, I , y)

[(

I)+] : I

}.

Let (y) = supI{L(0, I , y) [(I)+]}. Since () : 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 () is concave on by proving that supI{L(0, I , y)

[(I)+]} is concave in y.

So Tn(

) is K-concave because E[n1(y

D)] is K-concave by applying

Lemma 1.4.1 (b).

Therefore, n() is K-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. Then () is V-concave where

(x) = sup

{T(y)

V (y

x) : y

x

}x

(1.16)

Since Tn(y) as |y| and Tn() is continuous due to K-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) K + Tn(Sn); so sn is well defined.


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18

Finally, K > K implies K-concavity of

n(x) = sup{Tn(y) K(y x) : y x} x (1.17)

Therefore, a policy that utilizes (sn, Sn) policy for each n = 1, 2, , N is

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 fi-

nite 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) = E[n1(y D)] + (y). (1.18)


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20

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

B = cx1

n=1

n

cDn +

n=1

n1

[wn p(wn) In (1 )cyn (bn) e+(zn zn1)+ e(zn1 zn)+] (1.20)

Define e = (e+ + e)/2 and observe that

e+ (zn zn1)+ + e (zn1 zn)+ = e|zn zn1| + e+ e

2(zn zn1)

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

B =

n=1

n1(1 )(In + cyn) + In + g(yn, Dn)

p(1 + )In + g(yn, Dn)

(bn) e|zn zn1| e

+ e2

(1 )2yn

+

cx1 + w1 p(w1)

n=1

ncDn

+e+ e

2

(1 )x1

n=1

n(1 )Dn z0

Let

M(b , I , y) = (1 )(I+ cy) + I + E

g(y, D) p[(1 + )I+ g(y, D)]

(b) e+ e

2(1 )2y

Therefore,

E(B) =

n=1

n1E

M(bn, In, yn) e|zn zn1|

+E

cx1 + w1 p(w1)

n=1

ncDn (1.21)

+e+ e

2

(1 )x1

Nn=1

n(1 )Dn z0


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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 prob-

lem and the former converges to the latter. Henceforth, we optimize the following

objective:

sup EH1{N

n=1

n1[M(bn, In, yn) e|zn zn1|]}

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 0(, ) 0

and for each n = 1, 2, , N, x , and z 0,

n(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) + E[n1(y D, y x)] (1.24)

Let bn(x), In(x) and yn(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:

n(x, z) = supyx

e|y x z| + sup{Jn(b , I , y) : I+ b 0, b 0}


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22

= supyx

e|y x z| + hn(y)

(1.25)

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() on y.

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

y =

un(x) if x + z < un(x)

x + z if un(x) x + z < Un(x)

Un(x) if x Un(x) x + z

x if Un(x) < x.

(1.26)

Proof Sketch: It is straightforward to prove inductively that n(

,

), 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}


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23

and (1.26) corresponds to the following optimal replenishment quantity

zn =

un(xn) xn, if xn + zn1 < un(xn)

zn1, if un(xn) xn + zn1 < Un(xn)

Un(xn) xn, if xn Un(xn) xn + zn1

0, if Un(xn) < xn.

Sobel (1971) shows in great detail that finite horizon nonstationary policies

converges to infinite horizon stationary policies in the smoothing costs model.

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:

y =

u(x) if x + z < u(x)

x + z if u(x) x + z < U(x)

U(x) if x U(x) x + z

x if U(x) < x

(1.27)

where u(x) = sup{y : h(y) e}, U(x) = sup{y : h(y) e}, and h(y) is the

limit of hn(y) as n .

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


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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.


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Chapter 2

Fill Rate of General Review

Supply Systems

2.1 Introduction

All inventory systems face a difficult tradeoff between inventory costs and cus-

tomer 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 re-

plenishment 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


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26

for the item fill rate in an uncapacitated single-stage system with normally dis-

tributed 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


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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 re-

view (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. Axsater

(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


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28

a periodic review system with a fixed leadtime.

Boyaci and Gallego (2001) and Shang and Song (2003) study a periodic re-

view service-constrained serial inventory system where the leadtime demand for

the end product is Poisson distributed. Their service measure, the limiting prob-

ability 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 per-

formance 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


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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


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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, let xnt 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. That is, x1t is the on-hand physical inventory ifx1t 0, and x1t is the amount

of backordered demand ifx1t < 0. Let zNt be the number of items purchased from

an outside supplier in period t, and for n < N, let znt be the number of items

that are ordered by stage n and removed from stage n + 1 in period t.

Let Dt be the demand in period t, and let D1, D2, be independent, identi-

cally distributed, and nonnegative random variables with distribution function G

and finite mean . To avoid trivialities, it is assumed that G(0) < 1. Let G(k)()

denote the k-fold convolution of G(), i.e., the distribution function ofkj=1 Dj,and let G0(a) = 1 (0) if a

(


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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 + ztL1; ynt = xnt + zn,tLn 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 = 0 if period t 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 zn1,t (1 < n N). (2.30)

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

immediately from on-hand inventory. So,

= limT

E

Tt=1 min{(y1t)+, Dt}T

t=1 Dt

(2.31)


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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 be the base-stock level, so

z1t = ( y1t)+ if t|R = 0 and z1t = 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 , i.e., x11 . As a

consequence, for every t L,

y1t = L+[(tL)|R]

k=1

Dtk, for (t L)|R = 0, 1, , R 1 (2.32)

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


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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

R

0

[G(L)(b) G(L+R)(b)]db (2.33)

Proof From (2.31), the fill rate is the long run average fraction of demand that

is met directly from on-hand inventory. So,

= limT

E Tt=1

min{(y1t)+, Dt}/T

t=1

Dt

= limT

E

[T

t=1

min{(y1t)+, Dt}/T]/[T

t=1

Dt/T]

=1

lim

TE

T

t=1min{(y1t)+, Dt}

/T

=1

Rlim

TE Tt=1

min{(y1t)+, Dt}

/(T /R)

=1

Rlim

TE

R1i=0

t{t:(tL)|R=i}

min{(y1t)+, Dt}

/(T /R)

=1

Rlim

TE

R1i=0

t{t:(tL)|R=i}

min{(y1t)+, Dt}/(T /R)

Let Hj =

t{t:(tL)|R=j} min{(y1t)+, Dt}, j = 0, 1, , R 1. Using (2.32), for all

j {0, 1, , R 1}, it can be shown that

limT

E[Hj/(T /R)] = E

min

(L+jk=1

Dk)+, DL+j+1

= E

DL+j+1 [DL+j+1 ( L+jk=1

Dk)+]+


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34

= E

[DL+j+1 (L+jk=1

Dk)+]+

Let () make explicit the dependence of on . Then

() =1

R

R1j=0

Hj =1

R

R1j=0

E

[DL+j+1 (

L+jk=1

Dk)+]+

= 1 1R

R1j=0

E

[DL+j+1 ( L+jk=1

Dk)+]+

and let K() = [1 ( ,L,R)]. So,

K() = 1R

R1j=0

E

[DL+j+1 (L+jk=1

Dk)+]+

=1

R

R1j=0

0

a

(a + b )dG(L+j)(b)dG(a)

+0

a

dG(L+j)(b)dG(a)

=1

R

R1j=0

1 G(L+j)()

+0

a

(a + b )dG(L+j)(b)dG(a)

Leibnitz Rule yields

K() =1

R

R1j=0

0

a

dG(L+j)(b)dG(a)

=1

R

R1j=0

0

b

dG(a)dG(L+j)(b)

=1

R

R1j=0

0

G( b) 1

dG(L+j)(b)

=1

R

R1

j=0G(L+j+1)() G(L+j)()

= 1R

G(L+R)() G(L)()

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

() = 1 K()/


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35

= 1 [K(0) +0

K(a)da]/

=1

R

0

[G(L)(b)

G(L+R)(b)]db.

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

stock level (), 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 (j,) denote the sum of j independent, identically distributed random vari-

ables, each one exponential with parameter , i.e., (j,) is a gamma random

variable with parameters j and . If D is (, ) where is a positive integer,

then = E(D) = /, V ar(D) = /2, the probability density function of D is

ea(a)1

()

and the distribution function of D is

G(a) = P(D a) = 1 1j=0

ea(a)j/j!


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36

Consequently, G(L) and G(L+R) are (L,) and [(L + R), ], respectively. So,

G

(L)

(a) = 1 L1

j=0 e

a

(a)

j

/j!

G(L+R)(a) = 1 (L+R)1

j=0

ea(a)j/j!

It follows from (2.33) that

=1

R

0

(L+R)1j=0

ea(a)j/j! L1j=0

ea(a)j/j!

da

= R0

(L+R)1j=L

ea(a)j/j!

da

=1

R

(L+R)1j=L

0

ea(a)j/j!

da

Therefore,

=1

R

(L+R)j=L+1

P{(j,) } (2.34)

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 and variance 2 > 0.

Let () and () 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 j)/(j). The normality and independence of demand imply


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

S I+. So, formula (2.33) yields

=1

R

0

[b(a, L)] [b(a, L + R)]

da

=1

R

Lb(,L)b(0,L)

(x)dx

L + Rb(,L+R)b(0,L+R)

(x)dx

(2.35)

The evaluation of the integral in (2.35) exploits the following equation (Hadley

and Whitin 1963; Sobel 2004; Zipkin 2000).

t

[1 (x)]dx = (t) + t(t) t

This equation implies

st

(x)dx = (s) (t) + s(s) t(t). (2.36)

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:

=1

R

L

[b(, L)] [b(0, L)] + b(, L)[b(, L)] b(0, L)[b(0, L)]

L + R

[b(, L+ R)] [b(0, L+ R)]

+ b(, L+ R)[b(, L+ R)] b(0, L+ R)[b(0, L+ R)]

=1

R

L[b(, L)] [b(0, L)]

L + R

[b(, L+ R)] [b(0, L+ R)]

+ ( L)[b(, L)] [b(, L+ R)]+ R

[b(, L+ R)]

+ L

[b(0, L)]

(L + R)[b(0, L+ R)]

. (2.37)

Another deviation of 2.37 uses the following formula:

ba

x(x)dx = (a) (b). (2.38)


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

Write the integral in (2.35) as

=1

R

0

b(a,L)

(x)dx b(a,L+R)

(x)dxda

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 = , 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, employingformula (2.38), and integrating A1, A2, and A3 individually leads to the same fill

rate expression as (2.37).


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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:

A =1

R

L + R[b(, L + R)]

+ ( L)

1 [b(, L + R)]

+ R[b(, L + R)]

(2.39)

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

[b(0, L)] 0, [b(, L)] 0, [b(0, L+R)] 0, [b(, L)] 1, [b(0, L+R)] 0,

and [b(0, L)] 0. Our numerical results show that the fill rate approximation

(2.39) is very accurate when (L + R). 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 = 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 (2) goes down or base-stock level ()

goes up.

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


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40

The approximation performs better when the demand coefficient of variation

(/) 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.

sbnt = the (beginning) echelon inventory level at stage n before any order is

received


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41

abnt = 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:

abnt = sbnt +

Ln

k=1

zn,tk ant = abnt + znt snt = s

bn,t + zn,tLn (2.40)

and

abn,t+1 = ant Dt sbn,t+1 = snt Dt (2.41)

The first expression in (2.40) can be written

ant = snt +Ln1

k=0zn,tk

The formulation in installation variables xnt, ynt, and znt is equivalent to a

formulation in echelon variables because xnt = ynt znt, ynt = snt an1,t (let

s0t = a0t = 0), and znt = ant Ln1k=1 zn,tk snt. Because s1t = y1t, (2.31) canbe written

= limT

E

Tt=1 min{(s1t)+, Dt}

Tt=1

Dt

(2.42)

The constraints on the order quantities (2.29) correspond to

s1t a1t s2t sNt aNt. (2.43)


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An echelon base-stock policy depends on echelon base-stock levels 1, , N

and can be expressed as: If period t is a review period, aNt = max{abNt, N} and

for n < N,

ant =

abnt if n abntmin{n, sn+1,t} otherwise,

(2.44)

and ant = abnt if period t is not a review period for all n N. The order quantity

specified by (2.44) is znt = min{(nabnt)+, (sn+1,t abnt)+} ift is a review period,

and znt = 0 otherwise.

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 1 is a

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 1 and 2 for stage one and 2, respectively. It is without loss

of generality to assume that the initial echelon inventory positions are no higher


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44

= a1t L1k=0

a1,tk (a1,tk1 Dtk1)

= a1,tL L

k=1 Dtk

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 = 2 1.

Lemma 2.5.2 For any t L,

s1t = 1 + min{0, Dt[L+(tL)|R]1} L+(tL)|R

k=1

Dtk (2.45)

Proof At t = L, then (t L)|R = 0, and a2,L1 = 2 because stage two reviewsits inventory one period ahead of stage one. Lemma 2.5.1 assures

s1L = a1,L L

k=1

Dtk

= min{1, 2 DL1} L

k=1

Dtk

= 1 + min{0, DL01} L+0

k=1Dtk

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+1L


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46

= limT

E

[T

t=1

min{(s1t)+, Dt}/T]/[T

t=1

Dt/T]

=

1

limTE

T

t=1 min{(s1t)

+

, Dt}/T=

1

Rlim

TE Tt=1

min{(s1t)+, Dt}/(T /R)

=1

Rlim

TE

R1i=0

t{t:(tL)|R=i}

min{(s1t)+, Dt}

/(T /R)

=1

Rlim

TE

R1i=0

t{t:(tL)|R=i}


Let Hj = t{t:(tL)|R=j}min{(s1t)+, Dt}, j = 0, 1, , R1. Using (2.45) and

noticing that the demand variables are independent,

limT

E[Hj/(T /R)] = E

min

1 + min{0, D1} L+j+1k=2

Dk+

, DL+j+2

For all j {0, 1, , R 1}, it can be shown that

(Hj) = E

min

1 + min{0, D1} L+j+1k=2

Dk+

, DL+j+2

= E

DL+j+1

DL+j+2 [1 + min{0, D1} L+j+1k=2

Dk]++

= 1 E

DL+j+2 [1 (D1 )+ L+j+1k=2

Dk]++

Let (1, ) make explicit the dependence of on 1 and . Then

(1, ) =1

R

R1j=0

(Hj)

= 1R

R1j=0

1 EDL+j+2 [1 (D1 )+

L+j+1k=2

Dk]++

Let K(1, ) = [1 (1, )]. So,

K(1, ) =1

R

R1j=0

E

DL+j+2 [1 (D1 )+ L+j+1k=2

Dk]++


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47

=1

R

R1j=0

1+c0

1+bc

(a + b + c

1

)dG(L+j+2)(a)dG

(L+j)(b)dG(c)

+

1+c

0

a dG(L+j+2)(a)dG(L+j)(b)dG(c)

+0

10

1b

(a + b 1) dG(L+j+2)(a)dG(L+j)(b)dG(c)

+0

1

0

a dG(L+j+2)(a)dG(L+j)(b)dG(c)

Leibnitz Rule yields

K(1, )1

= 1R

R1j=0

1

+c

0

1+bc dG(L+j+2)(a)dG(L+j)(b)dG(c)

+

0

a dG(L+j+2)(a)g(L+j)(1 + c)dG(c)

+

0

a dG(L+j+2)(a)g(L+j)(1 + c)dG(c)

+0

10

1b

dG(L+j+2)(a)dG(L+j)(b)dG(c)

+0

0

a dG(L+j+2)(a)g(L+j)(1)dG(c)

+00 a dG(L+j+2)(a)g

(L+j)

(1)dG(c)

(2.47)

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),

K(1, )

1=

1

R

R1j=0

1+c0

1+bc

dG(L+j+2)(a)dG(L+j)(b)dG(c)

+

0

1

0

1b dG(L+j+2)(a)dG

(L+j)(b)dG(c)

=1

R

R1j=0

1+c0

G(L+j+2)(1 + b c) 1

dG(L+j)(b)dG(c)

+0

10

G(L+j+2)(1 b) 1

dG(L+j)(b)dG(c)

=1

R

R1j=0

G(L+j+1)(1 + c) G(L+j)(1 + c)

dG(c)


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48

+0

G(L+j+1)(1) G(L+j)(1)

dG(c)

=1

R

R1

j=0

G(L+j+1)(1 + c) G

(L+j)(1 +

c)dG(c)+G(L+j+1)(1) G(L+j)(1)

G()

=1

R

G(L+R)(1 + c) G(L)(1 + c)

dG(c)

+G(L+R)(1) G(L)(1)

G()

(2.48)

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

Therefore,

(1, ) = 1 K(1, )/

= 1 K(0, ) +

10

K(a, )

ada/

= 10

K(a, )

ada/

=1

R

G()

10

G(L)(a) G(L+R)(a)

da

+10

G(L)(a + c) G(L+R)(a + c)dG(c)da

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. We use

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


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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,

s1t = 1 + min

0, L2k=1

Dt[L1+(tL1)|R]k

L1+(tL1)|Rk=1

Dtk (2.49)

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 ones order.

Replacing D1 and G(c) byL2

k=1 Dk and G(L2)(c), respectively, in the proof of

Theorem 2.5.3 yields the next result.

Theorem 2.5.5

=1

R10

G(L2)()G(L1)(a) G(L1+R)(a)

+

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 1(, L) be the fill rate of a single stage system

with review period R, base-stock level , and order leadtime L. We introduce

incomplete convolutions (cf. van Houtum et al. 1996). Let G(k)

be the distribution

functions ofk

j=1. Let > 0 and define

G(k) (x) =

G(k)(x + ) if x 0

0 if x < 0,


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

G

[m,n]

(x) =0 G

(m)

( + x u)dG(n)

(u).

note that G[m,n]0 (x) = G

(m+n)(x)

Corollary 2.5.6

=1

R

G(L2)()1(1, L1) + 1(1 + , L1 + L2) 1(, L1 + L2)

1

0G[L1,L2] (a)

G[L1+R,L2] (a)da (2.51)

Proof Rewriting (2.50) and using (2.33),

=1

R

10

G(L1)(a + c) G(L1+R)(a + c)

dG(L2)(c)da

+G(L2)()10

G(L1)(a) G(L1+R)(a)

da

=1

R

10

a+

G(L1)(a + c) G(L1+R)(a + c)

dG(L2)(c)da

+G(L2)()1(1, L1)

=1

R

10

a+0

G(L1)(a + c) G(L1+R)(a + c)

dG(L2)(c)da

10

0

G(L1)(a + c) G(L1+R)(a + c)

dG(L2)(c)da

+G(L2)()1(1, L1)

=1

R

10

G(L1+L2)(a + ) G(L1+L2+R)(a + )

da

1

0

[G[L1,L2]

(a) G[L1+R,L2]

(a)

da + G(L2)

()1(1, L1)

=1

R

1+

G(L1+L2)(u) G(L1+L2+R)(u)

du

10

[G

[L1,L2] (a) G[L1+R,L2] (a)

da + G(L2)()1(1, L1)

=1

R

1+0

G(L1+L2)(u) G(L1+L2+R)(u)

du


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51

0

G(L1+L2)(u) G(L1+L2+R)(u)

du

1

0[G

[L1,L2] (a)

G

[L1+R,L2] (a)da + G(L2)()1(1, L1)

=1

R

G(L2)()1(1, L1) + 1(1 + , L1 + L2) 1(, L1 + L2)

10

G

[L1,L2] (a) G[L1+R,L2] (a)

da

(2.52)

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, 1, (2), 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 = 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


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53

Theorem 2.6.2

=

1

R10

2

1+2e

G

(L1)

(a + 1 + 2 c e)G(L1+R)(a + 1 + 2 c e)

dG(L2)(c)dG(L3)(e)

+G(L1)(a) G(L1+R)(a)

2

G(L2)(1 + 2 e)dG(L3)(e)

+G(L3)(2)1

G(L1)(a + 1 c) G(L1+R)(a + 1 c)

dG(L2)(c)

+G(L1)(a) G(L1+R)(a)

G(L2)(1)G

(L3)(2)

da. (2.54)

Proof From (2.31),

= limT

E Tt=1

min{(s1t)+, Dt}/T

t=1

Dt

=1

Rlim

TE

R1i=0

t{t:(tL)|R=i}


Let Hj =

t{t:(tL)|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

distribution ofj

k=1 Dk. Then limT E[Hj/(T /R)] is equal to

E

min

1 + min

0, 1 + min {0, 2 D(L3)} D(L2)

D(L1+j)

+, D

For all j {0, 1, , R 1}, it can be shown that

(Hj) = 1

ED

1

[D(L3)

2]

+ + D(L2)

1

+

D(L1+j)

+

+

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

(1, 1, 2) =1

R

R1j=0

(Hj)


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Let K(1, 1, 2) = [1 (1, 1, 2)]. So,

K(1,1,2)

=1

R

R1j=0

E

D

1

[D(L3) 2]+ + D(L2) 1

+ D(L1+j)++

=1

R

R1j=0

2

1+2e

1+1+2ec0

1+1+2ecb

(a + b + c + e 1 1 2)dG(a)dG(L+j)(b)dG(L2)(c)dG(L3)(e)

=

2

1+2e

1+1+2ec

0

adG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)

+

2

1+2e0

10

1b

(a+ b 1)dG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)

+2

1+2e0

1

0


+

20

1

1+1c0

1+1cb

(a + b + c 1 1)dG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)

+

20

1

1+1c

0


+

20

10

10

1b

(a + b 1)dG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)

+ 2

0

1

0

1

0

adG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)Leibnitz Rule yields

K(1, 1, 2)

1

=1

R

R1j=0

2

1+2e

1+1+2ec0

1+1+2ecb

dG(a)dG(L+j)(b)dG(L2)(c)dG(L3)(e)

21+2e0

101b dG(a)dG

(L1+j)

(b)dG

(L2)

(c)dG

(L3)

(e)

20

1

1+1c0

1+1cb

dG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)

20

10

10

1b

dG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)

=1

R

2

1+2e

G(L1+R)(1 + 1 + 2 c e)


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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.


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57

/ R L A %Error

200 0.1 11000 1 4 0.999014 0.999014 0200 0.1 11000 2 3 0.999507 0.999507 0200 0.1 11000 3 2 0.999671 0.999671 0200 0.1 11000 4 1 0.999754 0.999754 0

1000 0.5 11000 1 4 0.789395 0.760095 2.931000 0.5 11000 2 3 0.880264 0.880047 0.02171000 0.5 11000 3 2 0.919869 0.920032 0.01631000 0.5 11000 4 1 0.938963 0.940024 0.10612000 1 11000 1 4 0.590938 0.335729 25.52092000 1 11000 2 3 0.686992 0.667864 1.91282000 1 11000 3 2 0.766131 0.778576 1.24452000 1 11000 4 1 0.815568 0.833932 1.8364200 0.1 12000 1 4 1 1 0200 0.1 12000 2 3 1 1 0200 0.1 12000 3 2 1 1 0200 0.1 12000 4 1 1 1 0

1000 0.5 12000 1 4 0.895048 0.886563 0.84851000 0.5 12000 2 3 0.943282 0.943282 01000 0.5 12000 3 2 0.962025 0.962188 0.01631000 0.5 12000 4 1 0.97058 0.971641 0.10612000 1 12000 1 4 0.679695 0.520189 15.9506

2000 1 12000 2 3 0.765023 0.760095 0.49282000 1 12000 3 2 0.826923 0.840063 1.3142000 1 12000 4 1 0.861683 0.880047 1.8364200 0.1 13000 1 4 1 1 0200 0.1 13000 2 3 1 1 0200 0.1 13000 3 2 1 1 0200 0.1 13000 4 1 1 1 0

1000 0.5 13000 1 4 0.95544 0.953442 0.19981000 0.5 13000 2 3 0.976695 0.976721 0.00261000 0.5 13000 3 2 0.984318 0.984481 0.01631000 0.5 13000 4 1 0.987299 0.98836 0.10612000 1 13000 1 4 0.758474 0.664425 9.40492000 1 13000 2 3 0.829454 0.832213 0.27592000 1 13000 3 2 0.874769 0.888142 1.33732000 1 13000 4 1 0.897742 0.916106 1.8364

Table 2.1: Fill Rate and its Approximation for Normal Demand


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58

/ 1 1 R L1 L2 2 0.2 50 0 1 4 1 0.17472 0.2 50 0 1 4 2 0.00372 0.2 50 0 2 3 1 0.49852 0.2 50 0 2 3 2 0.08922 0.2 50 0 3 2 1 0.6654

2 0.2 50 0 3 2 2 0.33362 0.2 50 0 4 1 1 0.74912 0.2 50 0 4 1 2 0.55 0.5 50 0 1 4 1 0.30305 0.5 50 0 1 4 2 0.10525 0.5 50 0 2 3 1 0.47025 0.5 50 0 2 3 2 0.20415 0.5 50 0 3 2 1 0.62005 0.5 50 0 3 2 2 0.34855 0.5 50 0 4 1 1 0.7141

5 0.5 50 0 4 1 2 0.49325 0.5 50 20 1 4 1 0.63575 0.5 50 20 1 4 2 0.53955 0.5 50 20 2 3 1 0.77735 0.5 50 20 2 3 2 0.69485 0.5 50 20 3 2 1 0.85035 0.5 50 20 3 2 2 0.79065 0.5 50 20 4 1 1 0.88685 0.5 50 20 4 1 2 0.8419

/ 1 1 R L1 L2 2 0.2 80 0 1 4 1 12 0.2 80 0 1 4 2 0.99402 0.2 80 0 2 3 1 12 0.2 80 0 2 3 2 0.99702 0.2 80 0 3 2 1 1

2 0.2 80 0 3 2 2 0.99802 0.2 80 0 4 1 1 12 0.2 80 0 4 1 2 0.99855 0.5 80 0 1 4 1 0.97495 0.5 80 0 1 4 2 0.85465 0.5 80 0 2 3 1 0.98685 0.5 80 0 2 3 2 0.91475 0.5 80 0 3 2 1 0.99125 0.5 80 0 3 2 2 0.94275 0.5 80 0 4 1 1 0.9933

5 0.5 80 0 4 1 2 0.95715 0.5 80 20 1 4 1 0.99875 0.5 80 20 1 4 2 0.99445 0.5 80 20 2 3 1 0.99935 0.5 80 20 2 3 2 0.99715 0.5 80 20 3 2 1 0.99945 0.5 80 20 3 2 2 0.99795 0.5 80 20 4 1 1 0.99865 0.5 80 20 4 1 2 0.9975

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


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/ 1 1 R L1 L2 5 0.5 50 30 1 4 1 0.6373

5 0.5 50 30 1 4 2 0.62845 0.5 50 30 2 3 1 0.77855 0.5 50 30 2 3 2 0.77145 0.5 50 30 3 2 1 0.85125 0.5 50 30 3 2 2 0.84625 0.5 50 30 4 1 1 0.88745 0.5 50 30 4 1 2 0.88375 0.5 80 30 1 4 1 0.99875 0.5 80 30 1 4 2 0.99855 0.5 80 30 2 3 1 0.9993

5 0.5 80 30 2 3 2 0.99915 0.5 80 30 3 2 1 0.99945 0.5 80 30 3 2 2 0.99935 0.5 80 30 4 1 1 0.99865 0.5 80 30 4 1 2 0.9985

/ 1 1 R L1 L2 5 0.5 50 50 1 4 1 0.6373

5 0.5 50 50 1 4 2 0.63735 0.5 50 50 2 3 1 0.77855 0.5 50 50 2 3 2 0.77855 0.5 50 50 3 2 1 0.85125 0.5 50 50 3 2 2 0.85125 0.5 50 50 4 1 1 0.88745 0.5 50 50 4 1 2 0.88745 0.5 80 50 1 4 1 0.99875 0.5 80 50 1 4 2 0.99875 0.5 80 50 2 3 1 0.9993

5 0.5 80 50 2 3 2 0.99935 0.5 80 50 3 2 1 0.99945 0.5 80 50 3 2 2 0.99945 0.5 80 50 4 1 1 0.99865 0.5 80 50 4 1 2 0.9986

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


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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 param-

eterized 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 op-

timization 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-

optimalfor 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 equiv-

alent in the following sense:

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


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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, there is a base-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:

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

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 < f1 < f2 < < 1,

base-stock level parameters y1 < y2 < , and stockout cost parameters b1 1, rewrite

(3.63) as

inf

E N

n=1

H(yn)

: E N

n=1

B(yn) N(1 f)

(3.66)

For any N > 1, yn 0 for n = 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)] + E[B(yn)] n = 0 for n = 1, 2, , N(3.67)

N

n=1

E[B(yn)] N(1 f)

= nyn = 0 for n = 1, 2, , N(3.68)N

n=1

E[B(yn)] N(1 f), yn 0, 0, and n 0. for n = 1, 2, , N(3.69)


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Since H() > 0 and B() < 0 on (m, M), if = 0, then (3.67) implies that

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

n = 1, , N. But then,

Nn=1

E[B(yn)] = N = N(L + 1) > N(1 f) for all f > 0

which contradicts (3.69).

So = 0, and (3.68) yields

Nn=1

E[B(yn)] = N(1 f). (3.70)

Also, from (3.67),

E[H(y1)]

E[B(y1)]=

E[H(y2)]

E[B(y2)]= = E[H

(yN)]

E[B(yN)]

So, y1 = y2 = = yN because H() and B() are convex and monotone. There-

fore, with (3.70),

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

Since B() is an injection, it follows that

y1 = y2 = = yN = y = B1[(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 ofyn = y for all n (given x1 y).


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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, with y specified in (3.65), is F-optimal for (3.62), the

infinite horizon problem with a fill rate constraint.

Proof For all N and for all such that E|x1[N

n=1 B(yn)] N(1 f), if

x1 y,NH(y) E|x1

Nn=1

H(yn)

(Lemma 3.3.2)

H(y) E|x1 1

N

Nn=1

H(yn)

Therefore, if is feasible in (3.62),

H(y

) limN infE|x11

N

N

n=1 H(yn)

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

limN

infE|x1 N

n=1

H(yn)

= limN

infE|x1

zhang jiang

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