Inventory Management, Pricing and Risk Hedging
in the Presence of Price Fluctuations
by
Caner Canyakmaz
A Dissertation Submitted to the
Graduate School of Sciences and Engineering
in Partial Ful llment of the Requirements for
the Degree of
Doctor of Philosophy
in
Industrial Engineering and Operations Management
July, 2017
Inventory Management, Pricing and Risk Hedging in the Presence of
Price Fluctuations
Ko c University
Graduate School of Sciences and Engineering
This is to certify that I have examined this copy of a doctoral dissertation by
Caner Canyakmaz
and have found that it is complete and satisfactory in all respects,
and that any and all revisions required by the nal
examining committee have been made.
Committee Members:
Prof. S uleyman Ozekici
Asst. Prof. U gur C elikyurt
Asst. Prof. Pelin G ul sah Canbolat
Prof. Re k G ull u
Assoc. Prof. Semih Onur Sezer
Date:
Dedicated to my grandfather...
iii
ABSTRACT
Price uncertainties are among the most critical challenges that retailers and man-
ufacturers have to face. For instance, companies whose operations require procuring
from commodity markets are exposed to commodity price
uctuations which experi-
ence sharp movements frequently. Besides random nature of customer demand, due
to this input and/or selling price volatility, there might be considerable variability in
rms' pro ts. It is vital for these rms to consider price
uctuations in adjusting in-
ventory control and pricing policies, and take a variety of risk management measures.
In this dissertation, we consider such a rm where continuous price changes during
the planning horizon a ect both unit payo from sales as well as customer arrivals. In
a multi-period setting, we rst investigate optimal price-dependent inventory control
policies and numerically illustrate how continuous price
uctuations a ect optimal
controls and resulting payo s. Then, we analyze optimal pricing policies assuming
that selling prices are determined both by market-driven random prices and rm's
markup decision. We show that level of price variability has a negative e ect on rms'
nal pro ts. Finally, in a minimum-variance framework, we explore nancial hedg-
ing strategies of the risk-sensitive rm. We assume inherent price dynamics of the
inventory item is correlated with prices of various products which are freely traded in
nancial markets. This presents an opportunity for the rm to invest in a nancial
portfolio of these products to manage its exposure to price and demand uncertainties
by observing the current inventory, wealth and price levels. In this environment, we
explicitly characterize dynamic variance-minimizing investment decisions of the rm
using dynamic programming. We then explore the risk reduction e ects of minimum-
variance nancial hedges through numerical examples and show that signi cant risk
reductions may be possible by using the right hedge.
iv
OZETC E
Uretici ve perakendecilerin onlem almas gereken en kritik zorluklardan biri yat
belirsizlikleridir. Orne gin operasyonlar emtia piyasalar ndan al m yap lmas n gerek-
tiren sirketler, s urekli de gi sen emtia yatlar na maruz kalmaktad rlar. Talepteki be-
lirsizliklerin d s nda al s/sat s yatlar ndaki belirsizlikler, sirketlerin nakit ak slar nda
onemli derecede de gi skenli ge neden olmaktad r. Bu nedenle sirketler yatlardaki oy-
nakl klar g oz on une alan envanter ve yatlama politikalar geli stirmeli ve riskini kon-
trol edebilece gi ad mlar atmal d rlar. Bu cal smada, planlama d onemindeki s urekli
yat oynakl klar y uz unden hem birim sat s getirisi hem de m u steri geli s zaman-
lar etkilenen bir sirketin end ustriyel ve nansal operasyonlar ele al nmaktad r. _Ilk
k s mda, ce sitli durumlarda en iyi coklu-d onem envanter kontrol u politikalar analiz
edilip say sal orneklerle yat de gi simlerinin bu politikalar ve getirilerini nas l etk-
iledikleri g osterilmektedir. _Ikinci k s mda, sat s yatlar n n piyasa-bazl yatlardan
ve sirketin kar pay kararlar ndan etkilendi gi durumda sirketin en iyi yatlama strate-
jileri incelenmekte ve yat oynakl klar n n sirketin nihai getirilerine negatif etkisi
teorik olarak g osterilmektedir. Son k s mda ise, riske-duyarl bu sirketin nansal
risk azalt m politikalar analiz edilmektedir. Uretilen ve/veya sat s yap lan ur un un
piyasa bazl i csel yat hareketlerinin nansal piyasalarda al m-sat m yap lan ce sitli
ur unlerin yatlar yla kar s l kl ili skilerinin oldu gu kabul edilmektedir. Bu da riske du-
yarl sirketin nansal ur unlerden bir yat r m portf oy u olu sturup, yat ve talep risk-
lerini y onetebilmesi i cin f rsat sa glamaktad r. Bu durumda sirketin envanter, varl k
ve yat seviyelerini g ozlemleyip dinamik bir sekilde olu sturdu gu minimum-varyans
yat r m kararlar , dinamik programlama kullan larak a c k bir bi cimde karakterize
edilmektedir. Do gru nansal yat r m kararlar n n sirketin nakit ak slar nda dikkate
de ger risk azalt mlar sa glad klar ce sitli say sal orneklemelerle g osterilmektedir.
v
ACKNOWLEDGMENTS
First and foremost, I am truly indebted to two great persons, my supervisors,
Prof. S uleyman Ozekici and Prof. Fikri Karaesmen for their inspiring mentorship
and guidance. They have continuously supported me by sharing their wisdom, not
only on this dissertation, but also on every part of my life here and always encouraged
me to be better. They are my ultimate role models and I am very proud to be their
student.
I also would like to thank Asst. Prof. Pelin Canbolat and Asst. Prof. U gur
C elikyurt for taking part in my thesis committee, for patiently reading my reports
and for their valuable suggestions and guidance which led this work to improve.
I also would like to thank to my o ce mates for bringing all the joy and laughter
to our work environment.
Last, but not least, I am grateful to my family for their warmest love, patience
and encouragement throughout my doctoral study. Finally, my special thanks are
due to my late granfather Sabri, who insistedly supported me to follow this path. His
memory will always be with me.
vi
TABLE OF CONTENTS
List of Figures ix
Nomenclature x
Chapter 1: Introduction 1
Chapter 2: Literature Review 5
2.1 Inventory Models with Random Prices . . . . . . . . . . . . . . . . . 5
2.2 Joint Inventory Management and Pricing . . . . . . . . . . . . . . . . 8
2.3 Risk-Sensitive Inventory Management . . . . . . . . . . . . . . . . . . 9
2.4 Financial Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Chapter 3: Inventory Models with Randomly Fluctuating Prices 16
3.1 Model Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Backorder Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Model with Lost-Sales . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Partially Backorder Setting . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Compound-Poisson Demand Case . . . . . . . . . . . . . . . . . . . . 35
3.6 Fixed Ordering Cost Case . . . . . . . . . . . . . . . . . . . . . . . . 39
3.7 Some Relevant Price Processes . . . . . . . . . . . . . . . . . . . . . . 41
3.8 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Chapter 4: Markup Pricing in the Presence of Price Fluctuations 51
4.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
vii
4.2 Optimal Inventory Control & Markup Pricing . . . . . . . . . . . . . 55
4.3 The E ect of Price Variability on Expected Pro t . . . . . . . . . . . 60
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Chapter 5: Financial Hedging of Systems with Randomly Fluctuat-
ing Prices 66
5.1 Minimum-Variance Hedging . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Minimum-Variance Hedging for Inventory Models with Demand and
Price Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2.1 Single-Period Model with Dynamic Hedging . . . . . . . . . . 71
5.2.2 Static Hedging Model . . . . . . . . . . . . . . . . . . . . . . . 74
5.2.3 Dynamic Hedging Model . . . . . . . . . . . . . . . . . . . . . 81
5.3 Multi-period Inventory Model with Dynamic Hedging . . . . . . . . . 85
5.4 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Chapter 6: Concluding Remarks 98
Chapter 7: Appendix 101
viii
LIST OF FIGURES
3.1 An overview of the inventory system. . . . . . . . . . . . . . . . . . . 19
3.2 A price process that leads to a non-base-stock system. . . . . . . . . . 32
3.3 E ect of price volatility on optimal expected pro ts. . . . . . . . . . . 47
3.4 Deviation from optimal results when approximate model is used. . . . 48
3.5 E ect of period length on the gap with the approximate model. . . . 49
3.6 E ect of number of periods on the gap with the approximate model. . 50
5.1 Mean-variance e cient frontier. . . . . . . . . . . . . . . . . . . . . . 90
5.2 E ect of number of trading periods on risk reduction. . . . . . . . . . 91
5.3 Histogram of unhedged and hedged cash
ows. . . . . . . . . . . . . . 93
5.4 E ect of strike price for the call option on risk reduction. . . . . . . . 94
5.5 E ect of price sensitivity on risk reduction. . . . . . . . . . . . . . . . 95
5.6 E ect of price volatility on risk reduction. . . . . . . . . . . . . . . . 96
ix
NOMENCLATURE
P : Market price process, P = fPt; t 0g
: Intensity process, = f t; t 0g
N : Demand process, N = fNt; t 0g
(:) : Intensity function
f(:) : Selling price function
T : Customer arrival times, T = fTn; n 1g
T : Length of a single sales period
M : Number of sales periods
x : Inventory level at the beginning of a period
y : Inventory level after ordering
p : Observed price at the beginning of a period
r : Interest rate per unit time
: Discount factor for a period,
= eô€€€rT
h (p) : Unit inventory holding cost
b (p) : Unit backorder (lost-sale) cost
Rt : Total discounted revenue until time t
rt(p) : Expected total discounted revenue until time t
c(y; p) : Expected one-period backorder and holding cost function
g(y; p) : Expected one-period pro t function
Vk(x; p) : Maximum expected total discounted pro t from period k
Gk(y; p) : Expected total discounted pro t from period k
k(y; p) : Expected discounted future pro ts from period k
x
Sk(p) : Base-stock level in period k
sk(p) : Reorder point in period k
K : Fixed ordering cost
4f : Forward di erence operator
zt(p) : Expected discounted price at time t
W : Wiener process, W = fWt; t 0g
r(y; p) : Expected total discounted revenue function
: Proportional sales markup
X : Individual customer demand process, X = fXn; n 1g
Dn : Cumulative demand until nth customer
N(y) : Order of the last customer who makes a purchase
X : Standard deviation of investment return X
X;Y : Correlation coe cient between X and Y
CF(y;N; P) : Unhedged operational cash
ow
: Financial investment strategy
S : Price process vector for nancial securities
G( ; S) : Payo from nancial investment strategy
HCF(y; ;N; P; S) : Hedged cash
ow
T : Set of trading times, T = ft0; :::; tng
C : Covariance matrix of security prices, C = fCijg
(y) : Covariance vector of cash
ow and security prices
(y) : Optimal hedge as a function of operational decision
R[s;t] : Total revenue between times s and t
N[s;t] : Total demand between times s and t
Wk : Wealth level at the beginning of period k
Xk : Inventory level at the beginning of period k
xi
Chapter 1
INTRODUCTION
Today's global economy leaves the rms with many critical challenges to deal
with in terms of uncertainties. Besides the random nature of customer demand, man-
ufacturers ad retailers usually have to consider the
uctuations in commodity prices,
exchange rates etc. as well for their supply chain operations. For instance, com-
panies whose operations require procuring from commodity markets are exposed to
commodity price
uctuations which experience sharp movements frequently. Unsta-
ble economies, supply disruptions due to uncontrolled factors such as earthquakes,
strikes,
uctuating exchange rates are all contributing factors to volatile commodity
prices or input materials. Considering the fact that a signi cant portion of manufac-
turers' expenses are due to raw material costs, it is vital for rms to take a variety of
risk management measures against undesired price movements. Moreover, it is clear
that in this sort of price-
uctuating environments, rms' operational policies such as
ordering and pricing as well as nancial strategies are greatly a ected.
Successful inventory management is an e ective approach to mitigate risks due
to input or selling price
uctuations. Besides its importance in managing the usual
trade-o between holding, shortage and purchase costs, it can create additional value
in
uctuating price environments by adjusting the order sizes in response to the price.
Although it is common to design contracts with suppliers or use nancial derivatives
to hedge against possible price risks, a successful inventory policy should also take the
evolution of prices into account in order to avoid downside risk or sometimes bene t
from advantageous variations (Chod et al. (2010), Caldentey and Haugh (2006)). For
instance a manufacturer, in anticipation of raw material price increases, can invest
2 Chapter 1: Introduction
in inventory to avoid high purchase costs and thus bene t from high selling prices in
future.
Classical inventory models usually take purchase and selling prices as constants.
However, it is clear that for some products, input prices are volatile. Moreover, it
is common that selling prices in some industries may be di cult to predict as well.
For instance, a wholesaler that sells in a di erent currency will bear an exchange
rate risk in selling prices. Some industries such as apparel and high technology face
the problem of variable selling price due to rapid product substitution and short
life cycles. Jewelry retailers which buy and sell products that are made of precious
metals or stones such as gold, silver and diamond may re
ect the
uctuation in input
prices to customers and may charge di erent prices to customers arriving at di erent
times. For inputs that are traded in commodity markets, there is a rich literature on
modeling commodity price processes. Sophisticated models are proposed to take into
account both long-term and short-term price volatilities. On the other hand, most
inventory models ignore the full e ects of such input price volatilities.
This dissertation addresses the inventory management and pricing policies of a rm
where the inventory item faces price volatilities, especially within the ordering cycles.
Due to the nature of the item, input and selling prices as well as customer demand
are all a ected from such price changes. After analyzing operational strategies, risk
hedging policies are investigated. Next, we summarize the main topics studied in the
subsequent chapters.
In Chapter 3 of this dissertation, we propose an inventory model that can po-
tentially integrate sophisticated input price processes with price dependent demand.
In particular, we propose and investigate a multi-period, single-item, periodic review
inventory control model where we explicitly model a continuous-time stochastic mar-
ket price process which determines both purchase and selling price and in
uences the
customer demand. In this environment, the arrival times of customers and the value
of the price process is important as they determine the sales revenue. The model,
which includes both price and demand uncertainties in continuous time, covers many
Chapter 1: Introduction 3
important simpler models as special cases. Our main contributions to the literature
are as follows. First, we capture the e ects of continuous stochastic input and selling
price
uctuations and their e ects on the continuous demand process in a tractable
model. The resulting model has both continuous time and discrete time components
and non-trivial within-period dynamics between xed time points. Second, for this
model, we characterize the optimal ordering policy using dynamic programming. Fur-
ther, our characterization leads to numerically implementable solutions for practically
relevant price processes. Using these solutions, we also generate insights on the ef-
fect of price volatilities on optimal expected pro ts and inventory decisions. To our
knowledge, this has not been addressed before for such general price processes that
are consistent with the nance literature.
In Chapter 4, we examine how such a rm coordinates its inventory and pricing
decisions in a
uctuating-price environment. We assume that the selling price of the
item consists of a market-driven random price which constantly changes during sales
period and rm's operational markup. Rather than determining selling price on its
own, the rm determines a proportional markup on the market price to re
ect the
e ect of prevailing commodity prices on the retail price. For this setting, we explic-
itly characterize the optimal inventory-markup strategy and theoretically analyze the
e ect of price volatilities in rm's optimal controls and pro tability.
In Chapter 5, we analyze the nancial hedging problem of a rm which is exposed
to commodity price
uctuations in its inventory operations. At prespeci ed trading
times throughout the sales season, we assume that the rm has the opportunity to
invest in available nancial securities which are correlated with the commodity price
process. The risk-averse rm then aims to minimize the variance of the cash
ow at the
end of the sales season for any inventory policy by exploiting these correlations. For
this setting, we characterize optimal static and dynamic variance-minimizing trading
policies which use the available information at each trading time. In a numerical
setting, we also investigate the e ect of the number of trading periods and price
volatility on the e ectiveness of nancial hedging by using several derivative securities
4 Chapter 1: Introduction
which are written on the underlying market price of the inventory item.
Apart from simple inventory models and special cases, this problem is known to be
challenging and few general results exist. Our contributions to the literature in this
part are two-fold. First, we apply a variance minimization approach to a rather general
multi-period inventory model where there are both selling price and demand risks that
are driven by a continuous stochastic price process. Second, we explicitly characterize
the minimum-variance hedge at each period by solving a stochastic dynamic program.
Despite the complexity of this dynamic program, its solution turns out to be relatively
simple. This leads to useful characterizations in some important special cases and
to implementable numerical solutions in general. Using this approach, we can then
characterize the bene ts of nancial hedging for di erent plausible nancial portfolios.
The rest of this dissertation is organized as follows. In Chapter 2, we review
some of the related papers in inventory management and pricing literature that deal
with price uncertainties. Furthermore, several risk management approaches including
nancial hedging are reviewed in the context of inventory operations. In Chapter
3, we introduce the problem formulation and analyze the optimal ordering policies
of the inventory system that involves price
uctuations. In Chapters 4 and 5, we
investigate pricing and dynamic nancial hedging strategies for the models outlined
before, respectively. Last, in Chapter 6, we give our concluding remarks and share
our ideas for future studies. An appendix is provided at the end for some of the
derivations and lengthy proofs.
Chapter 2
LITERATURE REVIEW
In this chapter, we review some of the available operations models in literature
which mostly deal with volatile prices. Then, we outline several risk-sensitive models
involving nancial hedging framework.
2.1 Inventory Models with Random Prices
A number of papers explore the e ect of volatile purchase prices on inventory control
problems. Kalymon (1971) extends the classical inventory model of Scarf (1960) that
involves xed ordering costs by incorporating random purchase price which is governed
by a Markov process. For such a system, he proves that a price dependent (s; S) policy
is optimal. Golabi (1985) considers a single-item and deterministic-demand inventory
system. He assumes that at the beginning of each cycle, the ordering price is a random
variable with a known distribution function. He derives a policy where it is optimal to
order for a number of next periods if the price falls into a certain interval. Gavirneni
(2004) studies a periodic review inventory problem where unit purchase cost at each
stage takes values from a discrete set according to a Markovian transition matrix.
He shows the optimality of order-up-to policies and presents conditions that lead
to monotone order-up-to levels. Chen et al. (2007) consider a multi-period pricing
and inventory management problem in discrete time and show that price dependent
base-stock policies are also optimal when additive exponential utility functions are
used to represent the risk sensitivity of the decision maker. Their results extend
to cases where demand and cost parameters are Markov-modulated. Berling and
Mart nez-de Alb eniz (2011) investigate a Poisson demand system where the purchase
price is a Markov process to study the e ect of price evolution on the optimal policy
6 Chapter 2: Literature Review
and its parameters. Speci cally, they consider both geometric Brownian motion and
Ornstein-Uhlenbeck processes for the price. Utilizing the decomposition method of
Muharremoglu and Tsitsiklis (2008) they characterize optimal base-stock levels as a
series of thresholds and provide an algorithm to calculate them. Following their work,
Berling and Xie (2014) present simple heuristics to calculate those threshold levels
e ciently. Chen et al. (2014) study the impact of purchase price volatility in a multi-
period stochastic inventory system where input prices at each period are random and
independent of the demand distribution. They establish that higher price variability
results in lower costs.
Another related stream of literature is on inventory models withMarkov-modulated
demand. In these papers typically demand distributions change over time, usually
dependent on an external stochastic process (see Song and Zipkin (1993), Ozekici and
Parlar (1999), Cheng and Sethi (1999), Erdem and Ozekici (2002), Gallego and Hu
(2004) and Gayon et al. (2009)). In our model, an internal price process modulates
the demand arrivals.
Our study di ers from the above papers in following points. First, in addition to
input price
uctuations, we also model selling price
uctuations and their impact on
the demand process. In addition, in our case, the demand and the revenue within a
period depend on the continuous price process which connects the optimal ordering
policy to the properties of the price process.
A few authors study inventory systems with changing selling prices. Available
models usually consider an in
ation rate or a deterministic continuous price decrease,
mostly in the context of the Economic Order Quantity (EOQ) model (Erel (1992),
Hariga (1995) and Khouja and Park (2003), and Yang et al. (2011)). Banerjee and
Meitei (2009) speci cally dealt with the e ect of changing selling price. Referring to
the price history of Nokia's two mobile phone models, they consider a single period
stochastic demand inventory model with random lead time and continuously decreas-
ing selling price. They assume uniform demand over the selling season and investigate
retailer pro tability by proving the existence of optimal solutions. In contrast to the
Chapter 2: Literature Review 7
above papers, we model the case of random demand modulated by a randomly
uctu-
ating price process and assume that the selling price is a general function of
uctuating
input price.
There are also some papers that consider rms that can buy from or sell in spot
markets where prices are constantly changing. Goel and Gutierrez (2006) consider
a multi-period stochastic inventory model where a rm may purchase from both
spot and future markets. Haks oz and Seshadri (2007) review existing models that
incorporate spot market procurements with volatile prices in several supply chain
operations. Katariya et al. (2014) consider a multi-period problem of a supplier that
has access to a volatile spot market and more stable long-term contractual customers
to sell items. In each period, the supplier decides on a production quantity and how
much to liquidate in the spot market. They show that the optimal policy consists of
two parameters and provide bounds on these. Guo et al. (2011), on the other hand,
consider a rm that can purchase at any time from a spot market and faces a random
demand at a later random time. The rm meets demand as much as possible and
salvages the leftovers, if any. They prove that optimal policy is a price-dependent
two-threshold policy. Secomandi (2010) studies the warehouse problem of a merchant
that is involved in commodity-trading activities. He assumes that in each period, the
spot price of the commodity evolves as a Markov process. In the presence of both
space and buy-sell limits, he shows that operational and nancial decisions can not
be separated and the optimal policy is characterized by two-stage price-dependent
base-stock targets.
One key di erence of our models is that we explicitly model the customer demand
process in detail and the rm do not have access to an ample spot market. Our model
in that sense is more appropriate for rms who may use commodities to manufacture
specialized products which are not easily traded in spot markets.
Overall, the models presented in this dissertation di er from existing models in the
sense that they incorporate the e ect of random selling prices by explicitly modeling
a continuous-state stochastic price process. They also relate random purchase and
8 Chapter 2: Literature Review
selling prices through a multiplicative retail markup unlike classical models that take
selling price as constants or as a random variable unrelated to the purchase costs.
This case is applicable to situations where a rm is selling in a foreign currency
or selling a commodity-based item such that any price
uctuations in the material
cost also pass to customers as in the jewelry industry. In addition, instead of having
random demands that are realized at the end of sales periods, we model the individual
customer demand also as a stochastic process that depends on the prevailing stochastic
prices. Finally, to our knowledge, there are few results for the challenging lost sales
problem when demand is price dependent.
2.2 Joint Inventory Management and Pricing
A signi cant portion of operations management literature focuses on coordinated
inventory and pricing decisions. In all of these models, whether stochastic or deter-
ministic, customer demand is a function of selling price. Whitin (1955) was the rst
to allow selling price to be set simultaneously with order quantity in the newsvendor
model. His method is to rst determine the optimal ordering quantity as a function of
price and then nd the corresponding optimal price. Reviews by Petruzzi and Dada
(1999), Chan et al. (2004) and Yano and Gilbert (2005) provide the current models in
joint inventory-pricing literature. Focusing on the newsvendor model, Petruzzi and
Dada (1999) investigate both additive and multiplicative demand models and provide
conditions that are su cient to ensure unimodality of the pro t function. With the
introduction of a base price, they are able show that the optimal price can be inter-
preted as the sum of base price and a premium. Federgruen and Heching (1999), on
the other hand, consider a multi-period inventory model with backorders where the
distributions of independent demand functions of each period depend on the price
charged at the current period. They show that optimal inventory pricing strategy
is a base-stock list-price policy which suggests to order up to the optimal base-stock
level and charge the optimal price of a given period if the state of inventory level is
less than that optimal base-stock level. Otherwise, it is optimal to order nothing and
Chapter 2: Literature Review 9
charge the unique optimal corresponding price for the current inventory level. Some
of available models incorporate risk-aversion in the problem as well. Agrawal and Se-
shadri (2000) decide both order quantity and selling price to maximize the expected
utility in a newsvendor setting. They show that a risk-averse retailer will set a higher
selling price and order less compared to a risk-neutral retailer. Chen et al. (2007) con-
sider joint pricing and ordering in a multi-period model with an exponential utility
objective.
With our model, we contribute to the literature by incorporating the e ect of
uctuating commodity prices into inventory and pricing decisions. Unlike traditional
models that use a demand function that has an error term and a deterministic part
which changes with respect to pricing decision, we decide on a proportional markup
and model a customer arrival process that is modulated by a stochastic price process
as well as the markup decision.
2.3 Risk-Sensitive Inventory Management
In this dissertation, we mainly consider inventory systems where a continuous and
Markovian commodity price process determines both the selling prices and instanta-
neous arrival rates of the customers. These both contribute to the total risk of the
nal cash
ow. Although in the next two chapters we analyze risk-neutral order-
ing and pricing policies, in Chapter 5 we take a risk-sensitive approach and examine
minimum-variance hedging policies. In this part it is worth reviewing some of the
available risk-sensitive inventory management models.
Although there are several papers that incorporate price related risks into inven-
tory models, the majority of the risk-sensitive inventory management literature deals
with alleviating demand related risks. Numerous approaches have been proposed in
which the objective function is adjusted to re
ect risk preferences of the decision
maker. The most notable approaches include expected utility maximization, Mean-
Variance (MV) criterion, satis cing probability maximization, Value-at-Risk (VaR)
and conditional-Value-at-Risk (CVaR).
10 Chapter 2: Literature Review
Due to its analytical tractability, the expected utility criterion is relatively more
frequently used among the approaches outlined. Lau (1980) was the rst to analyze
the newsvendor problem under the expected utility criterion. He uses nth-degree
polynomial approximation to a general utility function and provides a numerical so-
lution mechanism. Similarly, Eeckhoudt et al. (1995) use utility theory and consider a
risk-averse newsvendor to investigate the e ects of changes in riskiness of background
wealth and demand. They conclude that as risk-aversion increases, the resultant op-
timal order quantity decreases. Under decreasing absolute risk aversion, they show
that increases in initial wealth also increase the number of orders. Bouakiz and Sobel
(1992) consider a multi-period inventory control problem where the objective is to
minimize the expected utility of discounted costs in nite and in nite planning hori-
zons. They characterize the optimal replenishment strategy as a base-stock policy
provided that ordering costs are linear. Chen et al. (2007) extend the nite hori-
zon problem studied in Bouakiz and Sobel (1992). They take a di erent economics
perspective and aim to maximize expected utilities from consumption
ows rather
than typical cash
ows to avoid the so called temporal risk problem. As thoroughly
explained in Smith (1998), this problem occurs when decision makers are sensitive
to the time at which uncertainties are resolved. Chen et al. (2007) prove that when
the additive utility is exponential, the optimal replenishment policies have the same
structure as those for risk-neutral cases, i.e., base-stock policies are optimal. When
general utility functions are used, on the other hand, optimal policies become wealth-
dependent.
Although a signi cant portion of available models use the expected utility frame-
work for capturing risk sensitivity of the decision maker, it only helps to reduce the
risks by adjusting the order quantity appropriately. However, these deviations in
these control variables may cause service levels to fall considerably and it is usually
better to have other risk reduction plans which also work for any ordering decision.
Finally, expected utility approach is often criticized being impractical since it is usu-
ally hard to estimate the utility function of an individual and usually they are not
Chapter 2: Literature Review 11
mathematically tractable for arbitrary utility functions.
Another approach that has received considerable attention in risk management is
the use of MV approach that Markowitz (1959) introduced on the portfolio selection
problem. In his study, Markowitz (1959) considers an investor who wants to allocate
his initial wealth among a number of risky assets to minimize the variance of the re-
turn while expecting a predetermined level of return. By utilizing the mean-variance
framework, he determines a set of e cient portfolios o ering appropriate risk and re-
turn levels. Although it originated in nance, the mean-variance method is becoming
useful for any problem involving con
icting objectives. In the context of inventory
management, this approach is very similar to portfolio selection model in which the
returns are now the expected pro ts from inventory operations and the risk is the
variation of pro ts.
The use of MV approach in inventory management began with Lau (1980) who uses
an MV objective function for the newsvendor problem to incorporate risk-aversion. He
shows that the expected pro t maximizer yields an upper bound on the optimal order
quantity. Choi et al. (2008) consider both mean-downside risk and mean-variance on
two cases where the selling price is a decision variable or not. They conclude that
for both cases optimal order quantities are the same regardless of the two objective
functions. Berman and Schnabel (1986), Choi et al. (2008) and Wu et al. (2009) also
utilize MV framework on single-period inventory problems. Few works employ MV
approach in multi-period inventory models. Chen and Federgruen (2000) consider a
base-stock policy for a single item inventory system. Using the MV method, they
construct the e cient frontier for two performance measures: long-run holding costs
and a measure that is a function of both the expectation and the variance of customer
waiting time.
A di erent line of research is the use of satis cing probability maximization
method, which is basically maximizing the probability of achieving a target pro t
level. This method is particularly useful for a decision maker if satisfying a certain
pro t level is more important than the level of extra pro t. Lau (1980) and Sankara-
12 Chapter 2: Literature Review
subramanian and Kumaraswamy (1983) consider satis cing probability maximization
objective in newsvendor models. Lau and Lau (1988) and Li et al. (1991) extend their
works to two-product cases of the newsvendor model. Parlar and Weng (2003) use
probability of achieving the expected pro t instead of a xed target pro t. This
approach is not very popular compared to the expected utility and the MV method
since it does not speci cally deal with the variations in operational pro ts.
A more recent approach used in the context of risk-averse inventory management
is value-at-risk (VaR), a downside risk measure that is commonly used in nancial
risk management. By de nition, VaR is the lowest amount that will not be exceeded
with a given probability level (see Jorion (2007)). In the context of inventory manage-
ment, lowest amount refers to the lowest pro ts associated with a speci c inventory
replenishment policy. Luciano et al. (2003) apply VaR on an in nite-horizon problem
with no lead time and constant order quantity at the beginning of each replenishment
cycle. Ozler et al. (2009) investigate a multi-product newsvendor problem using VaR
as a risk measure. They derive exact distribution functions for the two-product case
and develop an approximation for the general N-product case. An alternative to VaR
is the conditional value-at-risk (CVaR) measure which is de ned as the conditional
expectation of losses above the VaR value. Gotoh and Takano (2007) consider the
minimization of CVaR in the context of the newsvendor problem. They show that,
due to its convexity, usage of CVaR leads to tractable problems. Ahmed et al. (2007)
analyze coherent risk measures such as CVaR and mean-absolute deviation in a multi-
period single-item inventory problem. Coherent risk measures are a class of functions
satisfying certain axioms introduced in Artzner et al. (1999). According to these ax-
ioms, although CVaR is coherent, VaR is not, since it does not satisfy convexity and
subadditivity.
2.4 Financial Hedging
There is a growing interest in integrating supply chain operations and risk hedging
with nancial instruments. It is understood that the existence of nancial instruments
Chapter 2: Literature Review 13
whose movements are correlated with the random components in rms' operations,
usually allow decision makers to hedge against possible risks (Gaur and Seshadri
(2005), Caldentey and Haugh (2006)). It is well known that rms often use nancial
products to hedge apparent risks involving price or exchange-rate uncertainty. For
instance, a rm procuring from a commodity market may hedge their risk using
commodity futures. Another example would be a producer selling to a foreign market
in foreign currency units. This also presents an opportunity to invest in derivatives
of the particular foreign currency as it will be correlated with producer's pro ts in
the domestic currency.
Unlike the expected utility and the MV methods which only adjust the order
quantities to manage the inventory risks, nancial hedging approach takes a more
proactive role and promises to reduce the risks for any inventory policy in the presence
of relevant nancial products. Considering the variety of nancial products o ered
in developed economies, nancial hedging may provide a good opportunity for risk-
sensitive rms to control their risks.
The rst study that investigates the e ect of nancial markets on inventory poli-
cies is Anvari (1987). He uses the well-known Capital Asset Pricing Model (CAPM)
in a newsvendor setting with no setup cost. In case of normally distributed demand,
he shows that optimal order quantity varies as the covariance between demand and
market returns change. Chung (1990) enhances Anvari's model by sharpening the
optimality conditions and showing that the optimal strategy can be simpli ed to a
single equation regardless of the sign of covariance. Gaur and Seshadri (2005) con-
sider hedging demand risk in inventory models motivated by the statistical nding
that demand for discretionary purchase items and the S&P 500 index are highly cor-
related. In a newsvendor framework, they analyze both perfect and imperfect hedging
cases and characterize the optimal hedge. Caldentey and Haugh (2006) examine the
problem of dynamically hedging a risk-averse rm's operational pro ts in continuous
time. Their central modeling insight is to view operational pro ts as an asset in
the rm's portfolio and address the hedging problem as one of the most extensively
14 Chapter 2: Literature Review
studied problems in the nance literature: nancial hedging in incomplete markets.
They give a fairly general modeling insight where the decision maker's objective is to
maximize the expected utility from pro ts by simultaneously considering both opera-
tional and trading strategies. Chen et al. (2007) consider the opportunity of nancial
hedging in their multi-period inventory control models where inventory and trading
decisions are made at discrete time points. They further assume that security prices
and cost parameters are world-driven, i.e., they are modulated by an external envi-
ronmental process. They show that under a partially complete nancial market, full
hedging is possible if additive exponential utility functions are used. Ding et al. (2007)
attack the problem of integrating the operational and nancial hedging decisions of
a rm selling to both foreign and domestic markets. Such a rm will obviously face
demand and exchange rate uncertainties. As an operational hedge, the rm may post-
pone their capacity commitment until uncertainties are resolved. This can be done
using capacity allocation option. As a nancial hedge, the rm may use currency op-
tions for protection against exchange-rate risk. Using MV approach in their models,
the authors conclude that expected pro t increase and risk reduction is possible via
operational and nancial hedging. Chod et al. (2010) investigate when operational
and nancial hedging are substitutes and complements. Ni et al. (2016) introduce a
certainty equivalent operator to nd optimal hedging-consistent decisions in presence
of non- nancial random factors that can not be hedged through nancial markets.
In an illustrative example on commodity procurement and storage, they show the
optimality of base-stock policies and characterize the nancial hedging portfolio.
Recent works by Okyay et al. (2014), Say n et al. (2014) and Tekin and Ozekici
(2015) extend Gaur and Seshadri (2005) by incorporating supply uncertainty and
discuss the implication of using expected utility and mean-variance objective functions
in nancial hedging context. In particular, Okyay et al. (2014) nd the variance
minimizing nancial portfolio for a given order quantity. We also take a similar
approach but our models are more general since they capture continuous-time price
uctuations and their in
uence on demand as well as multi-period cases with dynamic
Chapter 2: Literature Review 15
decision making.
Few papers consider both random demand and
uctuating prices in the context
of nancial hedging. Kouvelis et al. (2013) analyze the inventory operations of a risk-
averse rm that procures from both a volatile spot market and a long-term supplier.
Exposed to both demand and price uncertainty, the rm is assumed to have access
to nancial securities written on the commodity price. In a multi-period setting, the
objective of the decision maker is to dynamically maximize the interperiod mean-
variance utility of the rm's cash
ow. In a similar work, Kouvelis et al. (2015) study
the same setting in Kouvelis et al. (2013) without a long-term supplier. Assuming
that the objective is to maximize the mean-variance utility of the terminal wealth,
they characterize optimal time-consistent inventory and nancial hedging policies.
Our work di ers from Kouvelis et al. (2013) and Kouvelis et al. (2015) in two major
ways. First, we analyze a rather di erent operational model where the main ran-
domness is due to within-period price
uctuations. More speci cally, in each decision
interval, a continuous stochastic price process drives both selling prices and demand
arrivals in our model. Moreover, in the multi-period case, leftover inventory is not
liquidated at each period and is carried over to satisfy customer demand which makes
the dynamic program of joint-optimization rather intractable Second, in the context
of nancial hedging, we focus on the speci c objective of variance minimization where
the decision maker seeks a minimum-variance hedge for any given operational policy.
This objective has some nice features. First, by de nition, the operational policies are
independent of the nancial portfolio and operational pro ts can be accounted for in-
dependently. Therefore, the nancial hedge supports the operation towards variance
reduction without enforcing operational policy changes. Second, it turns out that
the minimum-variance hedge leads to tractable solutions. This allows experimenting
with di erent operational policies to computationally explore non-dominated mean-
variance policies. Moreover, the formulation also leads to a tractable solution in a
multi-period setting with a dynamic ordering policy where inventory carrying is al-
lowed. This provides a hedging approach for general multi-period inventory problems.
Chapter 3
INVENTORY MODELS WITH RANDOMLY
FLUCTUATING PRICES
In this chapter, we investigate the ordering policies and their implications for a
series of inventory systems where the main randomness is due to randomly
uctuating
prices during sales cycles. More speci cally, we consider systems such that a stochas-
tic price process a ect both input and selling prices as well as customer arrivals.
These models may be relevant for rms which manufacture or procure items which
consists of a market-driven component whose price is constantly changing according
to an external process, and then the rm sells those items considering its own opera-
tional costs and random market price of the component. When price
uctuations are
re
ected in the selling price of the item, then consequently the customer arrivals are
a ected.
The models presented in this chapter belong to the class of periodic-review inven-
tory models where an ordering decision is made at prede ned time points. The main
distinction from the current inventory management literature is that these models
also incorporate the e ect of random selling prices by modeling customer demand as
a process rather than a single random variable. This type of modeling which explicitly
considers within-period price
uctuations proves to be very important as these
uc-
tuations a ect ordering, pricing as well as risk-hedging policies of the rm which will
be examined in the upcoming chapters. In the next section, we present the speci cs
of these models.
Chapter 3: Inventory Models with Randomly Fluctuating Prices 17
3.1 Model Setting
In all of the models considered in upcoming sections, we assume that there exists
a nonnegative Markov process P = fPt; t 0g with state space R+ = [0;1) which
models the input price of the inventory item. We call this the market price of the
item to denote its relation to the market-driven component of the inventory item. We
also assume that items are sold at arriving customers according to a nonnegative and
deterministic selling price function f > 0; that is if a customer arrives at time t; then
the corresponding selling price at that time is f (Pt) : The general selling price function
f may include any potential proportional and/or incremental markups that the rm
sets. Note that unless f is a constant function, the rm passes any
uctuations
in market price of the item to customers. A constant selling price function is the
standard assumption in most basic inventory management models. For now, we do
not assume any particular form for the selling price function f. However in some
special cases in this chapter and throughout Chapter 4, we will speci cally assume
that the rm uses a multiplicative selling price function f (p) = p where 1 is
the proportional retail markup.
For the inventory system, we assume that there are M periods whose lengths are
equal to T units of time where at the beginning of each sales period the rm places
an order. Based on our de nition, the values of the random prices at review times
(i.e., PT ; P2T ; ::) are the random purchase prices for the rm. Unlike most of the
inventory management papers that model customer demand as a random variable
to be realized at the end of each review period, we assume that there is a customer
arrival process and it is modulated by the random price process that we consider. This
also explicitly incorporates the e ect of
uctuating selling prices into the operational
model. More speci cally, we assume that the unit customer demand process is a
modulated Poisson process where stochastic arrival rate at time t is t = (Pt) and
(:) is a deterministic, nonnegative function of random price realizations. Customers
arrive according to this process and at each arrival they demand one unit of the item.
This is relaxed at Section 3.5 as we investigate the compound Poisson case where each
18 Chapter 3: Inventory Models with Randomly Fluctuating Prices
customer demands a random amount of the item.
We do not necessarily assume that (:) is a decreasing function of price. If, for
example, the rm deals with commodity-based items whose prices are constantly
changing and can be freely traded in markets, the demand for these types of products
does not necessarily decrease as price increases. Customers may also be willing to
buy in anticipation of future price increases even if prices have already increased.
In our setting, since arrival rate is a function of stochastic price, we have a stochas-
tic arrival rate process = f t; t 0g which modulates the customer arrival process.
These types of models are referred as doubly stochastic Poisson processes introduced
by Cox (1955), or shortly, Cox processes. If is a deterministic function rather than
a stochastic process, we have a non-stationary Poisson process. Since we assume that
P is a Markov process, is also Markovian.
We denote the customer purchase process by N = fNt; t 0g where Nt denotes
the number of sales by time t and N0 = 0: The arrival times of the customers form
a random sequence T = fTn; n 1g where T and N are related as fTn tg =
fNt ng : Here we remark that equal period length assumption is not a necessity
and the models presented in this chapter can easily be extended to cover variable
period lengths. The structure of the optimal policies remains unchanged.
At the beginning of each period, the decision maker observes the current price
and inventory level to make an ordering decision. We assume that there is no lead
time and the entire order is received immediately at the beginning of each period.
We investigate two distinct cases where, at rst, we allow backordering for unsatis ed
demands. Secondly, we consider the lost-sale case. We specify the basics of these
models in Section 3.2 and Section 3.3, respectively.
Chapter 3: Inventory Models with Randomly Fluctuating Prices 19
Figure 3.1: An overview of the inventory system.
We de ne the following additional notation:
x : Inventory level at the beginning of a period
y : Inventory level after ordering
p : Observed price at the beginning of a period
r : Interest rate per unit time
: Discount factor for a period
ô€€€
= eô€€€rT
h (p) : Inventory holding cost per unit as a function of initial market price p,
b (p) : Backorder (or lost-sale) cost per unit as a function of initial market price p
We assume that for each period, unit inventory holding and unit backorder (or
lost-sale) costs are general nonnegative functions of initial price for that period. The
reasoning for the holding cost to depend on the initial price is clear as it consists of
physical storage costs as well as the opportunity costs for the inventory investments.
Since we assume that market prices represent the purchase prices, it is natural that
unit inventory holding cost for the product is a function of initial price p: For the
backorder or lost-sale costs, the reasoning is not as straightforward as the holding cost.
20 Chapter 3: Inventory Models with Randomly Fluctuating Prices
Usually, backorder costs depend on the selling price of the item and the length of the
backorder period for any customer, etc. However, since the distribution of both selling
prices and customer demand processes are determined by the initial market price in
our models, we can simply use an approximate value for the backorder cost b (p) where
it is a general nonnegative function of initial market price. Note that typical constant
unit holding and backorder costs are just special cases where h (p) = h and b (p) = b
for some constants h and b.
3.2 Backorder Case
In this model, we allow backordering customer demand in case of inventory shortage.
We assume that in case of a backorder, the selling price is set and paid at the time
of customer arrival rather than at the time of actual product delivery. In other
words, any unsatis ed customer is charged at the time of arrival. We assume that
the backordered demand is satis ed at the beginning of the next period. At every
period, the decision maker observes the current price p and inventory level x to make
an ordering decision which maximizes the expected total discounted pro ts. Since we
assume that the unit demands arrive according to a doubly stochastic Poisson process
which is modulated by a Markovian price process and the length of intervals are the
same, the probability distribution of number of sales in any interval only depend
on the initial price at the beginning of that period. That is, they are conditionally
independent. In addition, given the initial price P0; the probability distribution of
total demand until time t is a Poisson random variable with random mean measure,
i.e.,
P fNt = k j P0g = E
eô€€€MtMk
t
k!
j P0
where
Mt =
Zt
0
(Ps) ds
is the expected number of arrivals until time t given market prices. Note that in
case each customer demands one unit of the item, expected total demand during time
Chapter 3: Inventory Models with Randomly Fluctuating Prices 21
interval [0; t] is given by
E [NtjP0] = E [MtjP0] =
Zt
0
E [ (Ps) jP0] ds:
Expected Discounted Revenue
For both backorder and lost-sales models, total revenue in each period is calculated
by summing the revenue from each sale. However, based on the particular backorder
setting we investigate, the item is sold to each arriving customer regardless of stock
availability and each backordered customer yields a backorder and repurchase cost.
Let us de ne the total discounted revenue collected in time interval [0; t] for any t > 0
as
Rt =
XNt
n=1
eô€€€rTnf (PTn) : (3.1)
Note that f (PTn) is the selling price for the nth customer who arrived since the
beginning of the period and eô€€€rTn is to discount the unit revenue to the beginning
of the period. Summation is performed until the arrival of last customer, i.e., Ntth
customer where Nt is the total number of customers who arrived by time t: In this
chapter, we assume that the decision maker is risk-neutral and aims to maximize the
expected total discounted pro t. For this, let us rst de ne expected total discounted
revenue during [0; t] as a function of initial price by
rt (p) = E [Rt j P0 = p] : (3.2)
Note that the expectation in (3:1) is taken with respect to the random components;
number of arrivals Nt, arrival time vector T (the former can be obtained from the
latter) and market prices between [0; t], i.e., fPt; t 2 [0; t]g. The expected discounted
revenue function in (3:2) can be computed as follows
rt (p) = E [Rt j P0 = p]
=
Zt
0
eô€€€rsE [f (Ps) (Ps) jP0 = p] ds: (3.3)
22 Chapter 3: Inventory Models with Randomly Fluctuating Prices
The derivation is given in Appendix. Note that in case is constant, i.e., customers
arrive according to a regular Poisson process with constant rate, then
rt (p) = tft(p)
where
ft(p) =
Zt
0
eô€€€rsE [f (Ps) jP0 = p] ds
is the average discounted selling price and t is the expected number of customers
arrived by time t:
A similar approach for the total revenue also appears in Grubbstr om (2010) who
considers a single-period problem where demand is modeled as a compound renewal
process. He assumes that selling price is constant and customers that arrive according
to a renewal process demand a random amount of the product. There is no xed sales
period and items are sold until all inventory is depleted. Although in our model we
sum all individual revenues from each arriving customer, our model construction is
somewhat di erent in the sense that we have a nite sales season and selling price is
a stochastic process that also modulates the customer arrival process.
Model Dynamics
The dynamics of the backorder model is as follows. At the beginning of any period, if
the current inventory level and market price are x and p respectively, and order-up-to
level decision is y x, (x and y are integers) the immediate expected pro t for the
period is
g (y; x; p) = ô€€€p (y ô€€€ x) + rT (p) ô€€€ c (y; p) (3.4)
where
c (y; p) = E
b (p) (NT ô€€€ y)+ + h (p) (y ô€€€ NT )+ j P0 = p
(3.5)
and we de ne x+ = max(0; x). The rst term in (3:4) is the total purchase cost
for y units ordered at the initial price p: Second term is the total revenue collected
until time T and the last term is the one-period backorder and inventory holding cost
Chapter 3: Inventory Models with Randomly Fluctuating Prices 23
function given in (3:5) in which a cost of b (p) 0 is charged for each unit backordered
and a cost of h (p) 0 is charged for every remaining unit. Note that NT denotes the
number of arrivals during the period and one-period expected pro t is independent
of the period. This is due to the fact that conditional random prices
PkT+Tn j PkT
d=
PTn j P0
have the same distribution for any period k since market price process is assumed to
be Markovian and time-homogeneous This in turn implies that N is also Markovian
whose distribution depends only on the initial market price p: Remember that we do
not put any restriction on the price process except the Markov property. Since, for
now, we assume that each demand is of size 1; the state space for inventory level in
backorder case is Z; i.e., set of integers.
In the last period, without loss of generality, we assume that all remaining items
are lost, i.e., there is no salvage value. However, at the current price, the rm is
required to raise the inventory level to zero if it turns out to be negative, meaning
that there are backordered customers.
For now, we assume that the decision maker is risk-neutral and aims to maximize
the expected total discounted pro ts. We use dynamic programming to solve this
problem to optimality. We de ne the value function Vk(x; p) as the maximum expected
total discounted pro t for periods from k to M if the initial inventory is x and price
is p. We also de ne the expected discounted one-period revenue function with initial
price p as
g(y; p) = ô€€€py + rT (p) ô€€€ c (y; p) : (3.6)
Then, the dynamic programming equation (DPE) is
Vk(x; p) = max
y x
Gk(y; p) + px (3.7)
where
k(y; p) = E [Vk+1(y ô€€€ NT ; PT )jP0 = p] (3.8)
and
Gk(y; p) = g(y; p) +
k(y; p): (3.9)
24 Chapter 3: Inventory Models with Randomly Fluctuating Prices
Note that k(y; p) is the expected discounted total future pro ts for the remaining
periods. Since we allow backorders, the inventory level for the next period upon
ordering decision y is y ô€€€ NT ; which in fact can be negative. For any period k, the
risk-neutral rm aims to maximize Gk(y; p), which is the sum of the expected one-
period pro t, g(y; p); and expected discounted future pro ts,
(y; p); resulting from
the ordering decision. Additionally, since we assume that the seller serves all arriving
customers, the revenue term is independent of decision variable y. Since there is no
salvaging, for each (x; p) pair the terminal value function is
VM+1(x; p) = ô€€€pxô€€€
where xô€€€ = (ô€€€x)+.
Optimal Ordering Policy
Now we present the structural properties of Gk(y; p) and the form of the optimal
policy. In the following discussion, 4f(x) = f(x + 1) ô€€€ f(x) represents the forward
di erence of a discrete function f:
Theorem 3.1 For 0 k M, Vk (x; p) is concave in x and Gk(y; p) is concave in
y for every p and a price-dependent-base-stock policy is optimal, i.e., there exists a
base-stock level Sk(p) for each period k such that if the inventory level is less than
the base-stock level, it is optimal to raise the inventory up to Sk(p); otherwise, it is
optimal to order nothing. Moreover, optimal base-stock level for period k is given by
Sk(p) = inf
y 0 : P fNT y j P0 = pg
ô€€€p + b (p) +
4 k(y; p)
b (p) + h (p)
: (3.10)
Proof. We proceed by induction. First note that the terminal value function
VM+1(x; p) is concave in x for each p. Now assume that Vk+1(x; p) is concave in
x for some k M: Then, k(y; p) given in (3:8) is concave in y by the linear-
ity of expectation. Note also that for each p; one-period expected pro t g(y; p)
in (3:6) is concave in y since ô€€€py is linear and b (p) 0, h (p) 0 ensure that
ô€€€E [b (p) (NT ô€€€ y)+ + h (p) (y ô€€€ NT )+ j P0 = p] is concave. This makes Gk(y; p) given
Chapter 3: Inventory Models with Randomly Fluctuating Prices 25
in (3:9) and consequently Vk(x; p) concave functions of y and x, respectively. By induc-
tion argument, Vk (x; p) and Gk(y; p) are concave for each period k. Clearly, concavity
of Gk(y; p) ensures that the optimality of a base-stock policy with a base-stock level of
Sk (p) which is the maximizer of Gk(y; p). To calculate the base-stock level for period
k, we can apply the rst-order optimality condition on Gk(y; p): More speci cally,
Sk(p) = inf fy 0 : 4Gk(y; p) 0g
= inf fy 0 : 4g(y; p) +
4 k(y; p) 0g
= inf fy 0 : ô€€€p + b (p) P fNT y + 1j P0 = pg
ô€€€h (p) P fNT yj P0 = pg +
4 k(y; p) 0g
= inf
y 0 : P fNT yj P0 = pg
ô€€€p + b (p) +
4 k(y; p)
b (p) + h (p)
which is (3:10) :
We have established that a price-dependent-base-stock-type policy is optimal.
This is consistent with similar models with discrete dynamics such as Chen et al.
(2007). Next, we present the more explicit single-period solution. First, we de ne
expected discounted price process as
zt(p) = E
eô€€€rtPtjP0 = p
:
Corollary 3.1 The optimal base-stock level at period M is given by
SM(p) = inf
y 0 : E
(b (p) + h (p) +
PT ) 1fNT yg j P0 = p
ô€€€p + b (p) + zT (p)
:
(3.11)
Proof. Note that because of the terminal value function VM+1(x; p) and (3:8) ;
4 M(y; p) = ô€€€
4 E
PT (NT ô€€€ y)+ j P0 = p
=
E
PT (NT ô€€€ y)+ j P0 = p
ô€€€
E
PT (NT ô€€€ y ô€€€ 1)+ j P0 = p
=
E
PT
(NT ô€€€ y)+ ô€€€ (NT ô€€€ y ô€€€ 1)+
j P0 = p
=
E
PT 1fNT y+1g j P0 = p
=
E
PT
ô€€€
1 ô€€€ 1fNT yg
j P0 = p
= zT (p) ô€€€
E
PT 1fNT yg j P0 = p
: (3.12)
26 Chapter 3: Inventory Models with Randomly Fluctuating Prices
Substituting (3:12) in (3:10) for k = M yields (3:11) :
Note that if ô€€€p + b (p) + zT (p) 0, the optimal base-stock level will be SM(p) =
0: Although it does not a ect the concavity of the expected pro t function, it is
reasonable to assume that ô€€€p+b (p)+zT (p) 0. ô€€€p+b (p)+zT (p) can economically
be interpreted as the expected cost of ordering one less unit. If it is negative, it is
optimal to order zero.
We have an explicit formula for the base-stock level of the last period. Therefore,
we can analyze the behavior of the optimal base-stock level SM(p) as a function of
initial price p. It is clear that the stochastic behavior of the market prices conditional
on the initial price and the behavior of the deterministic rate function (:) play a
key role. We make the following three assumptions in which the rst two are very
plausible for a real-life inventory system and the third can be justi ed in the context
of the speci c model setup.
Assumption 3.1 Pt stochastically increases in the initial price P0 = p:
Assumption 3.2 (:) is a decreasing function.
Assumption 3.3 ô€€€p + b (p) + zT (p)is decreasing in p:
Theorem 3.2 If assumptions 3.1, 3.2 and 3.3 hold, SM (p) is decreasing in initial
price p:
Proof. Consider the characterization of SM(p) in (3:11). Note that increasing the
initial price stochastically increases the prices Pt between [0; T] by Assumption 3.1.
This consequently decreases the number of sales NT stochastically by Assumption
3.2. This in turn implies that the left-hand side of the in mum, increases in p. By
Assumption 3.3, ô€€€p + b + zT (p) is decreasing which makes SM(p) decreasing in p:
Assumption 3.1 requires that the future prices are stochastically higher if the
initial price is higher, which is highly intuitive and satis ed by the most practical
stochastic price processes. For instance, let us assume that price follows a geometric
Brownian motion process:
Pt = P0evt+ Wt
Chapter 3: Inventory Models with Randomly Fluctuating Prices 27
with drift v and volatility where Wt is a Wiener process with E [Wt] = 0 and
V ar (Wt) = t: Then, Assumption 3.1 trivially holds.
Assumption 3.2, on the other hand, requires that the deterministic rate function
(:) is a decreasing function of price. Although there may be cases that violate this
assumption in volatile markets as explained before, it is a very common assumption
in the literature that the customer demand decreases as the price increases. We only
need this assumption to show the monotonicity of SM (p). Price-dependent base-stock
policy is an optimal ordering policy regardless of the structure of (:) :
Assumption 3.3 requires that sum of the expected discounted price increase until
time T and the unit backorder cost is decreasing in the initial price. Note that we can
interpret both b (p) and zT (p) ô€€€p as the loss from ordering one less unit. The latter
is due to the di erence between two successive ordering prices (discounted) while the
former is by the de nition of backorder cost. Therefore, Assumption 3.3 essentially
implies that total loss from ordering one less unit should be lower for higher initial
market prices. If ô€€€p + b(p) + zT (p) does not decrease in initial price p; one can nd
cases where optimal base-stock level does not decrease as initial price increases.
The result in Theorem 3.2 can also be proved by showing that 4GM(y; p) is
decreasing in p; i.e., GM(y; p) is submodular under Assumptions 3.1, 3.2 and 3.3.
However, in either approach, a conclusion can not be drawn for intermediate periods
k < M: This is also consistent with the ndings of Kalymon (1971).
3.3 Model with Lost-Sales
In this section, we explore the lost sales case where we assume that any arriving
customer that can not nd an available item is lost. This case is more challenging
than the backorder case because the expected revenue now depends on the ordering
policy. To our knowledge, few results on the structure of the optimal policy exist for
the lost sales case with price-dependent demand even for simpler models.
In analogy with the backorder model, let us write the expected total revenue
28 Chapter 3: Inventory Models with Randomly Fluctuating Prices
during a period as a function of initial price p and order-up-to decision y as
r(y; p) = E
"
NXT ^y
n=1
eô€€€rTnf (PTn) j P0 = p
#
=
Xy
n=1
E
eô€€€rTnf (PTn) 1fTn Tg j P0 = p
(3.13)
where a^b = min(a; b): Note that only the revenue term is di erent than the previous
model by which we now collect revenues until the rm runs out of inventory, i.e., until
the arrival of (NT ^ y)th customer. The total expected discounted one-period pro t
can be written similarly as
g(y; p) = ô€€€py + r(y; p) ô€€€ c (y; p) : (3.14)
We write the dynamic programming equation for period k as in (3:7) where Gk(y; p)
is given in (3:9) and with a slight change in the future expected pro t which is given
as
k(y; p) = E
Vk+1((y ô€€€ NT )+ ; PT )jP0 = p
:
Since there is no backordering, the inventory level can not be negative in the next
period. It should be zero if the demand turns out to be more than the total inventory
in the current period.
As in the backorder case, we assume that the salvage price is zero: Therefore, the
terminal value function for the lost-sale model is
VM+1(x; p) = 0: (3.15)
Optimal Ordering Policy
In this section, we present the structural properties of Gk(y; p) and the form of the
optimal policy. Note that we can use the transformations
(y ô€€€ NT )+ = y ô€€€
Xy
n=1
1fTn Tg (3.16)
Chapter 3: Inventory Models with Randomly Fluctuating Prices 29
and
(NT ô€€€ y)+ = NT ô€€€ y + (y ô€€€ NT )+ = NT ô€€€
Xy
n=1
1fTn Tg: (3.17)
Moreover, trivially,
y =
Xy
n=1
1:
Then, by using (3.16) and (3.17), (3.14) becomes
g(y; p) =
Xy
n=1
E
1fTn Tg
ô€€€
eô€€€rTn PTn + b + h
ô€€€ p ô€€€ h j P0 = p
ô€€€ b (p)E [NT j P0 = p] : (3.18)
One-period expected discounted pro t function g(y; p) consists of a nite sum where
the upper limit of the summation is the decision variable y: and a constant. Clearly,
the behavior of this function is directly determined by the behavior of the inner terms.
In the following discussion, the terms decreasing and increasing refer to weak
monotonicity.
Assumption 3.4 E
1fTn Tg
ô€€€
eô€€€rTnf (PTn) + b (p) + h (p)
j P0 = p
is decreasing in
n:
Theorem 3.3 Under Assumption (3:4) ;Gk(y; p) is concave in y and Vk(x; p) is con-
cave in x for every p and a base-stock policy is optimal, i.e., there exists a base-stock
level Sk(p) for each period k such that if the inventory level is less than the base-stock
level, it is optimal to raise the inventory up to Sk(p); otherwise, it is optimal to order
nothing. Moreover, optimal base-stock level for period k is given by
Sk(p) = inf fy 0 : P fNT y j P0 = pg
ô€€€p + b (p) + E
1fTy+1 Tgeô€€€rTy+1f
ô€€€
PTy+1
j P0 = p
+
4 k(y; p)
b (p) + h (p)
)
:
(3.19)
30 Chapter 3: Inventory Models with Randomly Fluctuating Prices
Proof. We prove the result by induction. First note that terminal value function
VM+1 (x; p) is trivially concave. Now assume that for any k M; Vk+1(x; p) is concave.
Note that forward di erences of the one-period pro t function in (3.18) is
4g (y; p) = E
1fTy+1 Tg
ô€€€
eô€€€rTy+1f
ô€€€
PTy+1
+ b (p) + h (p)
ô€€€ p ô€€€ h (p) j P0 = p
which is also decreasing in y under Assumption (3:4) : This makes g(y; p) concave
in y since bE [NT jP0 = p] is a constant. Since Vk+1(x; p) is concave by induction,
k(y; p) is concave which makes Gk (y; p) concave. This in turn implies that Vk(x; p) =
max
y x
Gk(y; p) + px is concave in x: By induction, it is true that Vk(x; p) and Gk(y; p)
are concave for all periods k and initial price p which suggests the existence of an
optimal price-dependent base-stock type policy for this inventory model. Similar
to the backorder model, optimal base-stock level for any period k can be found by
analyzing the forward di erence of Gk(y; p): More speci cally,
Sk(p) = inf fy 0 : 4Gk (y; p) 0g
= inf
y 0 : E
1fTy+1 Tg
ô€€€
eô€€€rTY +1f
ô€€€
PTy+1
+ b (p) + h (p)
j P0 = p
ô€€€p ô€€€ h (p) +
4 k(y; p) 0g
= inf
y 0 : E
1fTy+1 Tg j P0 = p
p + h (p) ô€€€
4 k(y; p) ô€€€ E
1fTy+1 Tgeô€€€rTy+1f
ô€€€
PTy+1
j P0 = p
b (p) + h (p)
)
= inf fy 0 : P fNT y + 1 j P0 = pg
p + h (p) ô€€€
4 k(y; p) ô€€€ E
1fTy+1 Tgeô€€€rTy+1f
ô€€€
PTy+1
j P0 = p
b (p) + h (p)
)
= inf fy 0 : P fNT y j P0 = pg
ô€€€p + b (p) + E
1fTy+1 Tgeô€€€rTy+1f
ô€€€
PTy+1
j P0 = p
+
4 k(y; p)
b (p) + h (p)
)
:
For the single-period problem, the optimal order quantity is
SM(p) = inf fy 0 : P fNT y j P0 = pg
Chapter 3: Inventory Models with Randomly Fluctuating Prices 31
ô€€€p + b (p) + E
1fTy+1 Tgeô€€€rTy+1f
ô€€€
PTy+1
j P0 = p
b (p) + h (p)
)
: (3.20)
Note that Assumption 3.4 is the necessary condition for g (y; p) to be concave. A
su cient condition, on the other hand, is the case of expected discounted price zt(p)
being decreasing in time: We give the motivation in the following result.
Proposition 3.1 If the expected discounted price zt(p) given initial price p is de-
creasing in t; then E
1fTn Tg
ô€€€
eô€€€rTnf (PTn) + b (p) + h (p)
j P0 = p
is decreasing in
n.
Proof. Note that (b (p) + h (p))E
1fTn Tg j P0 = p
is decreasing in n as arrival
times Tn's form an increasing sequence which makes 1fTn Tg decreasing. Now de ne
' (t; p) = E
eô€€€rtf (Pt) 1ft Tg j P0 = p
Note that if zt(p) is decreasing in t; ' (t; p) is decreasing in t as 1ft Tg is decreasing
in t. Now, we can write
E
eô€€€rTnf (PTn) 1fTn Tg j P0 = p
= E [' (Tn; p)]
which is decreasing in n since Tn is increasing in n:
Proposition 3.1 is very easy to verify for most price processes. For instance, for
the geometric Brownian motion process given earlier, the expected discounted price
at time t is zt(p) = pe( +1
2 2ô€€€r)t. Observe that if + 1
2 2 ô€€€ r 0; then expected
discounted price is nonincreasing in time and Assumption 3.4 is satis ed.
Unfortunately, the situation may be more complicated when Assumption 3.4 does
not hold and its violation may lead to non-base-stock situations even in very simple
cases. For instance, consider the following case where f(p) = 2p, b = h = 0 and
(t) = 40; i.e., customer arrivals are Poisson, independent of the prices. Also assume
that the price process is deterministic but is a function of time, such that
Pt =
8<
:
50 ô€€€ 80t 0 t < 0:5
80t ô€€€ 30 0:5 t 1
:
32 Chapter 3: Inventory Models with Randomly Fluctuating Prices
Figure 3.2: A price process that leads to a non-base-stock system.
This simple price process yields a non-base-stock system as observed in Figure 3.2
which plots the expected total pro ts as a function of order quantity. We observe two
critical points, which are local maxima y(1) = 12 and y(2) = 37: The optimal policy
in this case is to order Q = 12 ô€€€ x units when 0 x < 12; to order nothing when
12 x 20, to order Q = 37ô€€€x units when 21 x < 37 and to order nothing when
x 37: It turns out that the revenue function may lose properties such as concavity
(and even quasiconcavity) if the prices exhibit changing or upward patterns.
Note that we investigated optimal ordering policies for several rather general mod-
els. In the next part, we assume a speci c form for the price and demand processes
Chapter 3: Inventory Models with Randomly Fluctuating Prices 33
and derive some managerial analysis.
Special Case: Demand is Independent of Prices
For the lost-sale model, we consider a single-period special case to derive some closed
form characterizations. Assume that customer arrival process is independent of
stochastic price movements and let N be a Poisson process with rate . Also as-
sume that P is a geometric Brownian motion process with a mean price process given
by
E [Pt j P0] = P0e t; for t 0
where 2 R: For the sake of simplicity, we also assume that inventory holding and
backorder costs as well as the interest rate is zero, i.e., h(p) = b(p) = r = 0: Assume
also that selling price function is proportional to the prevailing prices and f (p) = p
where 1 and < . Then, we have the following result.
Corollary 3.2 Optimal base-stock level is given by
y (p) = inf
8><
>:
n 0 : P
NT n
1 ô€€€
1
ô€€€
n+1
9>=
>;
where NT Poisson (( ô€€€ ) T).
Proof. Note that by (3:20) ; optimal base-stock level is given by
y (p) = inf
n 0 : E
1fTn+1 Tg PTn+1jP0 = p
ô€€€ p 0
= inf
n 0 : E
1fTn+1 Tg pe Tn+1jP0 = p
ô€€€ p 0
: (3.21)
Assume that Tn Erlang(n; ô€€€ ) and NT Poisson (( ô€€€ ) T). Since Tn+1
Erlang(n + 1; ); (3.21) can be written as
y (p) = inf
8><
>:
n 0 : p
Z
[0;T ]
n+1tneô€€€ t
n!
e tdt ô€€€ p 0
9>=
>;
34 Chapter 3: Inventory Models with Randomly Fluctuating Prices
= inf
8><
>:
n 0 : p
ô€€€
n+1 Z
[0;T ]
( ô€€€ )n+1tneô€€€( ô€€€ )t
n!
dt ô€€€ p 0
9>=
>;
= inf
(
n 0 : p
ô€€€
n+1
P
Tn+1 T
ô€€€ p 0
)
= inf
(
n 0 : p
ô€€€
n+1
P
NT n + 1
ô€€€ p 0
)
= inf
8><
>:
n 0 : P
NT n
1 ô€€€
1
ô€€€
n+1
9>=
>;
: (3.22)
Note that if = 0; i.e., the price process is a martingale, then (3:22) reduces to
y (p) = inf
n 0 : P fNT ng
ô€€€ 1
where NT Poisson( T) :
We remark here that if 0; i.e., expected discounted price process is nonin-
creasing, Assumption 3.4 is satis ed by Proposition 3.1. Observe that if < 0, then
both left-hand and right-hand side of (3:22) will be increasing functions of n which
does not guarantee the unimodality of the objective function.
Some Managerial Insights
Note that as retail markup increases, optimal base-stock level increases. For
the special case where customer arrivals are independent of selling prices, it is
straightforward that increasing the potential revenue from a sale increases the
optimal order quantity. Note that this result may not be valid in the general
model where customer arrival rate depends on the prevailing random prices.
For instance, in the case of decreasing rate function , the behavior of p ( p)
becomes important. We perform an in-depth analysis in Section 4 where markup
is also a decision variable.
Assume that < 0: Then, it is evident from (3:21) that as decreases, optimal
base-stock level decreases, which is intuitive in the sense that as selling prices
Chapter 3: Inventory Models with Randomly Fluctuating Prices 35
are decreasing and customers keep arriving at the same rate, it is optimal to
reduce the ordering amounts.
3.4 Partially Backorder Setting
Backorder and lost-sale models that we examined in Section 3.2 and 3.3, respectively,
can be considered as two extreme cases for the rm's operations. The reason is that
in the backorder case we analyzed, all unsatis ed customers are assumed to accept
backordering with probability 1 and to pay the prevailing selling price. In the lost-sale
case, on the other hand, each unsatis ed customer is assumed to be lost completely.
Since we explicitly use the arrival times and selling prices in the revenue calculation,
we can easily extend these two extreme cases to other models where there are partial
backorders and to another case where backordered customer pays the price at the
time of next replenishment. These extensions are straightforward combinations of
these models. In the former case, for example, we can write the total revenue as a
summation similar to (3:3), but each summation term is multiplied with a probability
of backorder. In the latter case where backordered customers agrees to pay the price
at the end of the sales period, we can write the expected total discounted revenue as
r(y; p) = E
"
NXT ^y
n=1
eô€€€rTnf (PTn) + (y ô€€€ NT )+ eô€€€rT f (PT ) j P0 = p
#
which can be treated similar to the lost-sale model.
In the next section, we relax the unit demand assumption and analyze a compound
Poisson demand case.
3.5 Compound-Poisson Demand Case
So far, we assumed that each arriving customer demands a unit of the product. In this
section, we extend this model to a case where each arriving customer requires ran-
dom amounts of the product independent of the arrival process. Stochastic amounts
of each demand forms an independent and identically distributed random sequence
36 Chapter 3: Inventory Models with Randomly Fluctuating Prices
fXn; n 1g which are drawn from a continuous distribution having a cumulative dis-
tribution function F: Customer arrival process is the same as in the models previously
analyzed in which it is a doubly stochastic Poisson process modulated by the market
price process P. We again assume that the decision maker sets the order-up-to levels
at the beginning of each period and as customers arrive, the selling price is determined
according to price process P.
In compound-Poisson demand model, more interesting case is the lost sales case.
Since the revenue terms will be independent of the ordering decision, the backorder
case will be very similar to the previous backorder model with unit demands. There-
fore, we start our analysis with the lost sale case. In particular, we assume that at
any period, if the last arriving customer's demand exceeds on-hand inventory, the
customer is partially satis ed. The remaining part of this sale along with future sales
in that period are assumed to be lost forever. Finally, we assume that the customers
will always require a positive amount, i.e., there is no possibility that they will require
nothing. To this end, we de ne the following new notation. We let
Dn =
Xn
k=1
Xk
to denote the cumulative demand including the nth customer. We additionally de ne
N(y) = inf fn 1;Dn yg
to denote the order of the last customer who makes a purchase (full or partial) for y
units to be depleted. Observe that this quantity is independent of the period length
T: It basically indicates how many customers should arrive for the current inventory
to be sold. Note that in case of unit demand (i.e., Xn = 1; 8n 1), N(y) = y.
With the introduction of new notations, the expected total discounted pro t in
kth period given initial price p is given in (3:9) where the one-period expected pro t
now becomes
g (y; p) = ô€€€py + r(y; p) ô€€€ c (y; p) (3.23)
Chapter 3: Inventory Models with Randomly Fluctuating Prices 37
where the expected total discounted revenue is
r(y; p) = E
2
4
NX(y)ô€€€1
n=1
eô€€€rTnXnf (PTn) 1fTn Tg
+eô€€€rTN(y)
ô€€€
y ô€€€ DN(y)ô€€€1
f
PTN(y)
1fTN(y) TgjP0 = p
i
(3.24)
and expected total inventory-related costs (lost-sale and holding) is
c (y; p) = E
b (p) (DNT ô€€€ y)+ + h (p) (y ô€€€ DNT )+ j P0 = p
: (3.25)
The rst summation inside the expectation in (3:24) is the total discounted revenue
collected from fully satis ed customers during the period. The subsequent term,
on the other hand, is the revenue collected from the possibly-last customer who is
partially satis ed. Note also that, the only distinction between inventory-related
expected costs between unit demand (3.5) and compound Poisson demand case (3.25)
is that instead of NT , we now write DNT to denote total amount of demand in a
period. Here we also remark that in case of unit demand, (3.24) reduces to (3.13)
since N(y) = DN(y) = y.
As in the previous models, we denote future pro ts as
k(y; p) = E
Vk+1((y ô€€€ DNT )+ ; PT )jP0 = p
and the value function for period k and the boundary condition as (3:7) and (3:15) ;
respectively.
We now analyze the structural properties of Gk (y; p) to nd the structure of the
optimal ordering policy. However, it is di cult to perform a probabilistic analysis as
in the unit demand case since there are additional random variables such as N(y) and
Xn: We proceed with a sample path analysis on Gk (y; p) : For now, consider N(y); Tn
and Xn as the realizations of these random variables. Observe that the rm will only
be able to sell an additional in nitesimal amount dy when TN(y) T; since otherwise,
the last customer will arrive after this period (although we might have a positive
amount of inventory). This is due to the de nition of N(y): If TN(y) T, the rm
38 Chapter 3: Inventory Models with Randomly Fluctuating Prices
sells dy units with a total revenue of dyeô€€€rTN(y)f
PTN(y)
: Therefore, we can write
the marginal expected revenue as
r0 (y; p) = lim
dy#0
r (y + dy; p) ô€€€ r(y; p)
dy
= E
h
eô€€€rTN(y)f
PTN(y)
1fTN(y) TgjP0 = p
i
: (3.26)
Note that in this analysis, the possibility that DN(y) is exactly y is ruled out. However,
this is not an issue since P
DN(y) = y
= 0 as Xn's are assumed to be continuous
random variables and, by de nition of N(y); the last customer is always partially
satis ed. We remark that in the unit demand lost-sale case, pro t-to-go function for
any period is concave under Assumption (3:4). For the compound Poisson demand
case, we also need a condition to ensure concavity.
Assumption 3.5 The expected discounted price zt(p) is decreasing in t.
Theorem 3.4 Gk (y; p) is concave in y and Vk(x; p) is concave in x and a base-stock
policy is optimal, i.e., there exists a base-stock level Sk(p) for period k such that if the
inventory level is less than the base-stock level, it is optimal to raise the inventory up
to Sk(p); otherwise, it is optimal to order nothing. Moreover, optimal base-stock level
for period k is
Sk (p) = inf fy 0 : P fDNT y j P0 = pg
ô€€€p + b (p) + E
h
1fTN(y) Tgf
PTN(y)
eô€€€rTN(y) jP0 = p
i
+
0
k(y; p)
b (p) + h (p)
9=
;:
(3.27)
Proof. First note that the terminal value function VM+1 (x; p) = 0 is trivially concave.
Now assume that for some k < M; Vk+1 (x; p) is concave. Then,
k(y; p) is concave.
Note also that N(y) is increasing in y and the same reasoning as in Proposition
(3:1) applies here; that is, (3:26) is decreasing if the expected discounted price is a
decreasing function of time, i.e., if zt(p) is decreasing in t: Therefore, r(y; p) given in
(3:24) is concave. Moreover, since both h(p) and b(p) are positive parameters, it is
Chapter 3: Inventory Models with Randomly Fluctuating Prices 39
clear that ô€€€E
b(p)(DNT ô€€€ y)+ + h(p) (y ô€€€ DNT )+ jP0 = p
is also a concave function
and the one-period pro t function g (y; p) given in (3:23) is concave. Since both
g (y; p) and
k(y; p) are concave, so is Gk(y; p). This in turn makes Vk(x; p) given
in (3:7) concave. By induction, Vk(x; p) is concave for all k: Then, it is clear that
Gk(y; p) is concave for all k: To characterize the optimal base-stock levels, consider
the rst-order optimality condition for Gk (y; p),
G0
k (y; p) = ô€€€p + r0(y; p) + b (p) P fDNT > y j P0 = pg
ô€€€ h (p) fDNT y j P0 = pg +
0
k(y; p)
= ô€€€p + b (p) +
0
k(y; p) + E
h
eô€€€rTN(y) PTN(y)1fTN(y) TgjP0 = p
i
ô€€€ (h (p) + b (p)) P fDNT y j P0 = pg
= 0
which can also be written as
P fDNT y j P0 = pg =
ô€€€p + b (p) + E
h
eô€€€rTn PTN(y)1fTN(y) Tg
i
+
0
k(y; p)
b (p) + h (p)
:
(3.28)
However, since the distribution of DNT has a mass at y = 0; (3:28) should be corrected
as (3:27). Note that if Xk = 1 for all k; then (3:27) reduces to (3:19) since N (y) = y
and DNT = NT :
In the case of complete backordering, the extension to compound Poisson demand
is much simpler. As before, the revenue terms do not depend on the order-up-to
decision y. Therefore, the analysis for this extension will be exactly the same as in
Section 3:2 when we replace NT with DNT and take expectations accordingly.
3.6 Fixed Ordering Cost Case
In previous sections, only variable unit purchase costs were incorporated in the pro t
function. However, it is well-known that independent of the order size, a prevalent
xed cost may be incurred for each order. This may be a xed cost of using a vehicle
of transportation for procurement, etc. Previous analysis can be extended to the case
40 Chapter 3: Inventory Models with Randomly Fluctuating Prices
where there is a xed order cost of K > 0 for each positive order amount. In this
case, the value function for period k becomes
Vk(x; p) = G
k(x; p) + px (3.29)
where
G
k(x; p) = max
Gk(x; p); max
y x
Gk(y; p) ô€€€ K
:
The rst function inside maximum operator refers to not ordering. In the last period,
we assume that there is no xed cost and the terminal value function is again given
by VM(x; p) = ô€€€K ô€€€ pxô€€€ for the backorder case and VM(x; p) = 0 for the lost-sale
case. Existence of a xed order cost fundamentally changes the structure of the
problem since we do not necessarily have concave pro t functions as in the linear
order cost case. Therefore, a base-stock policy is not usually suboptimal for this case.
For this problem, the pro t-to-go function that is being maximized at each period is
Kô€€€concave.
Theorem 3.5 Gk(y; p) is K-concave for any initial price p and a price-dependent
(s; S) policy is optimal, i.e., there exists sk (p) Sk (p) such that whenever the inven-
tory level x is below sk(p); it is optimal to order up to Sk(p); otherwise it is optimal not
to order. The optimal order-up-to level is given by (3:10) and (3:19) for the backorder
and lost-sale cases, respectively and the reorder level is given by
sk(p) = inf fx 0 : Gk(x; p) Gk(Sk(p); p) ô€€€ Kg :
Proof. Note that the proof is valid for both backorder and lost-sale cases under
Assumption 3.4 for the latter. Let us proceed with the backorder case. Note that
VM(x; p) = ô€€€K ô€€€ pxô€€€ is K-concave. Now assume that Vk+1(x; p) is K-concave in x
for some k Mô€€€1: Then k(y; p) given in (3.8) is K-concave which makes
k(y; p)
Kô€€€concave which in turn makes Gk(y; p) in (3.9) K-concave since g (y; p) is concave.
Then G
k(x; p) is also Kô€€€concave in x and this leads to Vk(x; p) being Kô€€€concave
(Porteus (2002)). This clearly implies that a price-dependent (s; S) policy is optimal.
Chapter 3: Inventory Models with Randomly Fluctuating Prices 41
3.7 Some Relevant Price Processes
In this section, we review some of the important nancial price processes that are
used to model the movements of nancial instruments, commodities, exchange rates,
etc. One of the most important nancial price models is the geometric Brownian
motion. In this model, the stock price at time t is given by the following stochastic
di erential equation
dSt = dt + dWt
where W is a Wiener process, and are the drift and volatility terms. This model
is the basis for Black-Scholes option pricing formulas and due to its simplicity, the
calculations with this process are relatively easy and lead to closed-form solutions
(see Baxter and Rennie (1996)).
Another well-known price model is the Ornstein-Uhlenbeck process where the
prices follow
dSt = ô€€€ ( ô€€€ St) dt + dWt:
In this model, the prices tend to revert to their long-term mean with a degree
of volatility and a reversion rate parameter : This model is particularly useful
when one models commodity price processes as they are known to exhibit some
mean-reversion (Baxter and Rennie (1996)). A more specialized model developed
by Schwartz and Smith (2000), on the other hand, uses both of the above models
to represent the commodity price movements by taking into account both long and
short-term behaviors. In the short-term, the commodity prices show mean-reversion
properties whereas in the long term they revert to an equilibrium. In particular, it is
assumed that market prices follow
Pt = e t+ t (3.30)
where t is an Ornstein-Uhlenbeck process
d t = ô€€€ tdt + dW( )
t
42 Chapter 3: Inventory Models with Randomly Fluctuating Prices
which models the short-term deviations by reverting towards zero. On the other hand,
long-term equilibrium level t is a Brownian motion process
d t = dt + dW( )
t :
Moreover, W( )
t andW( )
t are correlated Wiener processes with a correlation coe cient
of ; i.e.,
dW( )
t dW( )
t = dt
(see Schwartz and Smith (2000)).
In our numerical setup, we use a risk-neutral probability measure that makes
the price process given in (3.30) a martingale to test the e ect of price volatilities
on the optimal expected pro ts and optimal controls. This is particularly interesting
since changing the volatility related parameters of a martingale price process does not
change its expected values in time, which we desire in order to capture the sole e ect
of volatility. To nd a risk-neutral version of (3.30), we rst de ne two independent
Brownian motions W1;W2 and equivalently write
W( )
t = W1
t
and
W( )
t = W1
t +
p
1 ô€€€ 2W2
t :
Applying Ito's formula to Pt one can nd that
dPt =
ô€€€ t + + + +
2
2
+
2
2
Ptdt
+ ( + ) PtdW1
t +
p
1 ô€€€ 2PtdW2
t :
Cameron-Martin-Girsanov Theorem for n-factor models state that there exists a prob-
ability measure Q such that
dPt = 1PtdW(1)
t + 2PtdW(2)
t (3.31)
is a martingale where
1 = ( + )
Chapter 3: Inventory Models with Randomly Fluctuating Prices 43
and
2 =
p
1 ô€€€ 2
and W(1)
t ;W(2)
t are two independent Brownian motions with respect to Q (see, for
example, Baxter and Rennie (1996)). Note that analytical Ito's solution for (3:31) is
Pt = P0eô€€€1=2( 2
1+ 2
2)t+ 1W(1)
t + 2W(2)
t (3.32)
In our model, we will use (3:32) as our market price process.
The next section gives a numerical example and a sensitivity analysis on the price
parameters for the lost-sale model that is analyzed in Section 3.3.
3.8 Numerical Analysis
In this section, we conduct a numerical analysis on the lost-sale model and aim to
investigate the e ect of some parameters (especially price related parameters) on the
value function. For the demand process, we use three di erent rate functions, namely
exponential, normal and piecewise linear rate functions. The exponential rate function
is assumed to have the form
E (p) = Eeô€€€ p (3.33)
where is a sensitivity parameter for the arriving customers. Note that this sort
of a rate function applies to the cases where individual customer arrivals form an
independent Poisson process with rate E and arriving customers have i.i.d. reserva-
tion prices which are exponentially distributed random variables with parameter :
Similarly, the normal rate function is assumed to have the form
N (p) = N
1 ô€€€
p ô€€€ N
N
(3.34)
where is the cumulative distribution function of the standard normal random vari-
able and N and N are the mean and standard deviation, respectively. Finally, the
piecewise linear rate function is of the form
L (p) = (A ô€€€ B p)+ (3.35)
44 Chapter 3: Inventory Models with Randomly Fluctuating Prices
where A represents a potential arrival rate and B represents the customer sensitivity.
We use these functions to test the e ect of di erent rate functions to price changes
on the optimal expected pro ts.
For the price process P; we use the risk-neutral model in (3:31) and employ a sim-
ulation approach to estimate the one-period expected pro ts since a direct analytical
approach is challenging in the lost sale model for this price process. Steps of this
Monte-Carlo simulation is as follows:
Simulation of Price and Arrival Paths
To simulate the price process given in (3:31) ; we use n = 100 equally-spaced dis-
cretization of each unit of time, i.e., the interval [0; 1]. In addition, we use N = 2000
as the replication number which is statistically signi cant and does not lead to large
computing times. Here are the steps to simulate price and arrival processes.
Using the incremental independence and Gaussian properties of Wiener pro-
cesses, we rst generate N Normal(0; 1=n) random variables for both W(1)
t and
W(2)
t : This is due to the fact that for k = 1=n
Wtk ô€€€Wt(kô€€€1) Normal(0; 1=n):
Cumulative sum of these incremental realizations gives random paths for Wiener
processes W(1)
t and W(2)
t :We then use these Wiener realizations in the analytical
Ito's solution given in (3:32) to generate the desired price process.
Remember that conditional on the price path, doubly stochastic process of cus-
tomer arrivals reduces to an ordinary nonhomogeneous Poisson process. There-
fore, for each sample path of market prices P, we generate a nonhomogeneous
Poisson arrival stream. More speci cally, we utilize the thinning algorithm
of Lewis and Shedler (1979). According to thinning algorithm, for each price
path, we nd the maximum realized intensity max and generate a Poisson pro-
cess with this maximum rate. At each arrival time, we additionally generate an
Chapter 3: Inventory Models with Randomly Fluctuating Prices 45
independent uniform random variable
U Uniform(0; 1)
and accept the arrival if U < U= max where U is the realized intensity at time
U; i.e., U = (PU) for the particular rate function. This way, the stream of
accepted arrivals form a nonhomogeneous Poisson arrival vector.
Note that using these price and arrival time realizations, expected revenue and
pro t functions can easily be computed. To solve the dynamic programs outlined in
earlier sections, we also use a simulation approach along with a state space reduction
approximation which are explained next.
Dynamic Programming Approximation
Since we use a two-factor geometric Brownian process to model the market price
process, unbounded state space, i.e., R+ is very problematic for solving the dynamic
programming equations to optimality. To overcome this problem, we use another
discretization for the price state to compute value functions. In particular, at any
period, we equally discretize possible price realizations to nd price states. Moreover,
we increase the number of possible discretized states as we proceed in periods. The
steps of this approximation are as follows:
For the rst period k = 1; we nd maximum and minimum values in realizations
of PT and divide the corresponding interval into l = 100 equal intervals whose
middle points are assumed to be price states. Then, the probability distribution
is also evident since as each price realization as a result of simulation falls into
a particular interval.
For any intermediate period k we use the same logic. However, since the gap
between minimum and maximum price realizations increase, we now divide the
corresponding interval into kl equal intervals. In this approach, one can think
of state space for random prices as an horizontal and increasing cone.
46 Chapter 3: Inventory Models with Randomly Fluctuating Prices
Numerical Setup and Sensitivity Results
Throughout this numerical analysis, we take initial price P0 = 100, selling price
function f(p) = p with = 4, xed holding cost h = 5; xed lost-sale cost b = 20 and
the interest rate r = 0: For the demand rate functions, we use A = 380; B = 0:8 for the
linear case, E = 160; = 0:0025 for the exponential case and nally N = 120 and
mean and standard deviation of the normal distribution as N = 400 and N = 100,
respectively. Note that each rate function gives the same result for p = P0 = 100:
However, each of them have di erent robustness to changes in price which will be
important in the following sensitivity analysis.
For the lost-sale model, we rst analyzed how the optimal expected pro ts change
with respect to the magnitude of price volatility. Note that, since the price process
given in (3:32) is a martingale, i.e., constant in expectation, altering the values of
and increases only the volatility of within-period prices. We take M = 4, i.e,
we solve a 4-period dynamic programming recursion and with = 0:3; = 0:05,
we change the value of from 0 to 0:2: As observed in 3.3, for each rate function
in (3:33), (3:34) and (3:35) we observe that the optimal expected pro ts decrease as
the price volatility increases. This suggests that price volatilities are undesirable for
the rm. There are also di erences in the magnitude of the e ect of volatility for
these rate functions. Clearly, this is due to the robustness of these functions to price
changes. However, for each of the three rate functions, we observed the negative e ect
of increased volatility on the optimal expected pro ts. This observation holds for the
vast majority of the cases with plausible parameter values. Only in some extreme
cases where, for instance, a more volatile price process leads to much higher arrival
rates, the optimal expected pro ts increase. In terms of optimal base-stock levels,
on the other hand, we do not particularly observe any monotonicity with respect to
price volatility.
A similar sensitivity analysis can also be conducted to observe the e ect of corre-
lation parameter : We observe that as the value of increases, the value of optimal
expected pro ts decrease. This is again due to the fact that a higher means a more
Chapter 3: Inventory Models with Randomly Fluctuating Prices 47
Figure 3.3: E ect of price volatility on optimal expected pro ts.
volatile price process. More speci cally, the variance of Pt given in (3:32) is
V ar (Pt) = P2
0
e 2
+2 + 2
ô€€€ 1
which is an increasing function of :
In another numerical setup, we compare our proposed model with a deterministic
approximation model to test the e ectiveness of modeling price
uctuations explicitly.
In particular, we take the model in Kalymon (1971) as benchmark, where prices are
constant within sales periods, however they are still random with the same probability
distribution at the end (and beginning) of each period. We again use the price process
given in (3:31) with = 0:3 and = 0:05. Note that, since we use a martingale
price process, the expected prices do not di er from the initial price in time. For a
given volatility level ; we nd the optimal base-stock levels for both models and use
them in the proposed model that considers within-period price
uctuations to make
a consistent comparison between resulting expected pro ts. We use the piecewise
linear rate function given in (3:35) as the rate process and assume that B = 0:8:
48 Chapter 3: Inventory Models with Randomly Fluctuating Prices
Figure 3.4: Deviation from optimal results when approximate model is used.
Other parameters are assumed as P0 = 100; = 4; b = 20; h = 5; T = 1: For three
potential customer arrival rates (A = 340, A = 360, A = 380), Figure 3.4 shows the
percentage deviation from optimal expected pro ts for di erent volatility ( ) levels.
Figure 3.4 shows that as prices get more volatile, then the bene t of using the
proposed model that explicitly considers within-period price
uctuations increases.
We also note that the bene t of using the proposed model greatly increases when the
potential arrival rate A decreases since a lower A implies an arrival rate process which
is more prone to price increases. In other words, if A is low, then there will be more
occurrences with zero arrival rate if price are more volatile.
Although it is intuitive, we also remark here that as period length T increases,
then the gap between deterministic approximation and the proposed model increases.
This can be observed in Figure 3.5 which plots the percentage deviation from optimal
expected pro t with respect to changes in within-period length when second approx-
imate model is used. This is for the single-period model and the period length is
increased from T = 0:6 to T = 3 for three di erent volatility levels.
Chapter 3: Inventory Models with Randomly Fluctuating Prices 49
Figure 3.5: E ect of period length on the gap with the approximate model.
We also observe that as the number of periods increase, the percentage devia-
tion from optimal expected pro ts decreases. This is also intuitive as the decision
maker has additional opportunities to react to price changes as the number of or-
dering periods are higher. The e ect of the number of periods can be observed in
Figure 3.6 which plots the percentage deviation from optimal expected pro t when
the benchmark model is used as an approximation with respect to number of periods.
3.9 Summary
In this chapter, we analyzed an inventory management problem where purchase and
selling prices are described by a continuous-time stochastic price process which also
in
uences the customer demand. In contrast with most of the existing literature,
within each period demand arrives continuously and is in
uenced by the continuous
price process. In this setting, sales revenues depend on individual arrival times of
demands and not only on total accumulated demand. We show that for the backorder
case, price-dependent base-stock policies are optimal under standard assumptions.
50 Chapter 3: Inventory Models with Randomly Fluctuating Prices
Figure 3.6: E ect of number of periods on the gap with the approximate model.
This implies that in any ordering period, the rm's order-up-to decision is only a ected
from observed market price, which stochastically describes the evolution of upcoming
prices and customer arrivals. Moreover, this also extends to the more challenging lost
sales and compound Poisson cases under additional plausible conditions. A violation
of these may lead to non-base-stock environment even for very simple price cases. In a
numerical setting where market prices are modeled as a two-factor price process, it is
observed that price volatility has a signi cant e ect on expected pro ts and optimal
inventory policies. In particular, if expected future prices remain the same, increased
volatility leads to smaller expected pro ts. In addition, these examples illustrate
that modeling within-period price
uctuations in contrast with available models in
existing literature is more advantageous as the level of price volatility increases. This
is observed by comparing the proposed model and several approximate models that
ignore price
uctuations. It is also observed that as period lengths increase or number
of ordering opportunities decrease, then the advantage of explicit modeling increases.
Chapter 4
MARKUP PRICING IN THE PRESENCE OF PRICE
FLUCTUATIONS
Besides a successful inventory management policy, pricing is one of the most e ec-
tive tools that a rm has in order to increase its revenues. The impact of a successful
pricing strategy lies in its e ect on sales as price is one of most critical determi-
nants of customer demand. By e ectively controlling the demand, rms also have
the potential to create a more e cient supply chain due to decreased variability and
better management of the mismatch between supply and demand. Even more value
can be created by integrating pricing decisions with inventory, production and dis-
tribution decisions. As much as it is essential for manufacturing rms to coordinate
these decisions rather than employing a decentralized approach, integration of pric-
ing and inventory decisions has the potential of thoroughly increasing supply chain
e ectiveness (Chan et al. (2004)).
In classical inventory-pricing models, rms are assumed to control the sales prices
freely. However, for some rms, the nature of the inventory item may prevent them
from controlling the selling prices fully. An example is for rms selling and/or pro-
ducing commodity-based products whose underlying prices are market-determined
and prevailing input price volatilities pass to customers to an extent. If one sells a
commodity-based item, for instance, material, labor and overhead costs as well as
prevailing commodity prices need to be considered in determining the selling price
to stay pro table. Although it may be argued that pro t margins are only based
on marginal costs (more speci cally, wholesale price at the time of inventory replen-
ishment), the decision maker can not overlook prevailing market prices at the time
of customer demand due to high competition. On the other hand, accounting for
52 Chapter 4: Markup Pricing in the Presence of Price Fluctuations
volatilities in pricing decisions is rather overlooked in the literature.
In this chapter, we further delve into backorder model analyzed in Section 3.2 and
investigate how altering the selling prices a ects rm's pro tability. In particular,
we now assume that the rm sells a rather specialized product to its customers and
has the power to in
uence the sales prices. With this model, we contribute to the
literature by incorporating the e ect of
uctuating commodity prices into inventory
and pricing decisions. Unlike traditional models that use a demand function that
has an error term and a deterministic part which changes with respect to pricing
decision, we assume that the rm sets a proportional markup for constantly changing
market prices. Again, customer arrivals are modeled as a process that is modulated
by the stochastic price process as well as the markup decision. In this section, we also
investigate how the level of
uctuations in market prices a ect optimal performance
measures.
4.1 The Model
As in Chapter 3, we again assume that stochastic evolution of the market prices is
given by the process P = fPt ; t 0g with state space R+ = (0;1): For this section,
we do not necessarily assume that P is Markovian and time-stationary. However,
we assume that P has continuous sample paths. We use a particular multiplicative
form for the selling price function f and assume that the selling price for the item
at any time t is merely Pt where > 1 is the rm's markup. This implies that the
selling prices are both determined by the stochastic price process and markup. The
rm sets the proportional markup and ordering decision y at the beginning of sales
horizon t = 0. Note that unlike traditional pricing models in the literature, in this
model the rm cannot fully determine the selling price. To a degree, the rm has
to pass price
uctuations to the customer, but has the freedom to in
uence it via a
proportional constant. This is an appropriate setting if the rm sells products where
inherent price
uctuations somehow pass to the customer which is typical in jewelry,
or gold retailers. In addition, this setting applies to the non-price-taker rms which
Chapter 4: Markup Pricing in the Presence of Price Fluctuations 53
sell exclusive products so that they have more freedom to control markups.
For this model, to account for random prices and their a ects on customer demand,
we again assume that individual customers arrive according to a doubly-stochastic
Poisson process with a stochastic intensity measure = f t
= ( Pt) ; t 0g where
(:) is a nonnegative deterministic function of random selling price. Note that we
changed the notation for stochastic intensity process to to denote its connection
to markup control. For the operational setting, we use the compound Poisson model
analyzed in Section 3.5 where at each arrival, customers demand a random amount
of product independent of the price process. Remember that we let X = fXn; n 1g
denote the stochastic individual demands where we assume that each Xn is positive,
independent and identically distributed with common expectation :
We also denote the customer arrival process as N = fN
t ; t 0g where super-
script denotes its connection to our control variable . We additionally de ne Dn
as the cumulative demand by nth customers so that
Dn =
Xn
k=1
Xk:
With this notation, total random demand during sales season is DN
T
:
We assume a backorder setting and assume that if there is not enough on-hand
inventory, newly arrived customers are charged at the prevailing selling price and
satis ed at time t = T: This case is applicable to situations where the rm sells
exclusive products such that arriving customers may be unable to nd elsewhere.
Jewelry stores, for instance, usually take orders for diamond rings etc. to be supplied
later, yet their selling prices are determined considering the current market prices of
diamond and gold at the time of customer order, not the market prices at the time of
delivery. To keep the model simpler, we do not assume any physical holding cost or
salvage revenue in our analysis, i.e., h = 0, yet by discounting all future cash
ows, we
are capturing the opportunity costs associated with the rm's capital investment. We
assume that the rm incurs a penalty cost of b for each unit of backordered demand.
The rm needs to raise the inventory level up to zero by purchasing at the market
price PT at time T in case of backorders during the sales season. .
54 Chapter 4: Markup Pricing in the Presence of Price Fluctuations
We assume that b + E[PT ] ô€€€ P0 > 0 where P0 is the initial price at time 0, which
states that underage cost, i.e., cost of ordering one less unit is positive. If this is
not satis ed, then the rm does not order at all and simply backorders each arriving
customer. The objective of the rm is to maximize the expected total discounted
pro t by simultaneously setting order-up-to level and proportional markup.
Now, let R
T denote the total random revenue until time T; which is generated by
summing individual revenues from sales. More speci cally,
R
T =
N
XT
n=1
eô€€€rTn PTnXn
where Tn denotes the arrival time of nth customer and, as stated before, N
T is the
total number of individual customers arrived by time T when markup is : Also,
PTnXn is the random revenue obtained from the nth customer. We also discount all
individual revenues to time 0 by multiplying them with eô€€€rTn where r is the interest
rate per unit time. The risk-neutral rm is concerned with the expected revenue until
time T as a function of retail markup , which we denote as
rT ( ) = E [R
T ] :
In a compact form, expected total sales revenues until time T can be written as
rT ( ) =
ZT
0
eô€€€rtE [ Pt ( Pt)] dt (4.1)
where = E [Xn] : Note that this is a special case of expected total revenue given in
(3:3) with f (p) = p and E [Xn] = whose derivation is given in Appendix. We will
also frequently use the probability distribution of N
T although it does not appear in
(4:1) : But rst let us de ne P = fPt; t 2 [0; T]g as the random prices in the sales
horizon [0; T]. With this notation, N
T , total number of individual customers who
arrived during [0; T] is Poisson with conditional random mean
M
T = E [N
T j P] =
ZT
0
( Pt)dt: (4.2)
Chapter 4: Markup Pricing in the Presence of Price Fluctuations 55
Expected total sales, on the other hand, is
dT ( ) = E
DN
T
= E
2
4
N
XT
n=1
Xn
3
5 = E [M
T ]
=
ZT
0
E [ ( Pt)] dt: (4.3)
Assuming that there is no initial inventory, we write the expected total pro t as
a function of markup and order-up-to level y as
g(y; ) = ô€€€P0y + rT ( ) ô€€€ E
(b + PT ) (DN
T
ô€€€ y)+
(4.4)
where the rst term denotes the total purchase cost, the second term denotes the
expected total discounted revenue and the last term denotes the backorder and re-
purchase costs. The objective of the decision maker is to solve
max
>0;y 0
g(y; )
by choosing a proportional markup 2 (0;1) and an order-up-to level y 2 [0;1)
Next section characterizes the form of the optimal inventory-markup pricing policy.
4.2 Optimal Inventory Control & Markup Pricing
In this section, we analyze the behavior of the expected pro t function g (y; ) with
respect to y and and corresponding optimal inventory and markup pricing strategies.
We begin by analyzing the optimal inventory policy for a xed markup decision :
Note that for the backorder case, assuming a general sales-price function and a unit-
demand setting, we already characterized the form of optimal ordering policies in
Section 3.2. On the other hand, we only presented results and derivations for the
lost-sale compound Poisson case in Section 3.5. Below, we give the result for the
backorder and compound-Poisson demand case.
56 Chapter 4: Markup Pricing in the Presence of Price Fluctuations
Optimal Inventory Policy for a Given Markup
In the following sections, we will use (y; ) = P
DN
T
= y
and (y; ) = P
DN
T
< y
to denote the probability density and cumulative distribution functions of DN
T
eval-
uated at y; respectively.
Theorem 4.1 Given markup ; g (y; ) is concave in order-up-to level y and a base-
stock policy is optimal, i.e., it is optimal to order up to the optimal base-stock level
y ( ) if initial inventory is less than y ( ); otherwise, it is optimal to order nothing.
The optimal base-stock level is given by
y ( ) = inf
y 0 : E
(b + PT ) 1n
DN
T
y
o
b + E [PT ] ô€€€ P0
: (4.5)
Proof. The rst and second-order derivatives of g(y; a) with respect to y is given by
gy(y; ) = ô€€€P0 + E
(b + PT ) 1n
DN
T
y
o
= ô€€€P0 + b + E [PT ] ô€€€ E
(b + PT ) 1n
DN
T
<y
o
(4.6)
and
gyy(y; ) = ô€€€E
(b + PT )
@
@y
E
1n
DN
T
<y
o j P
(4.7)
= ô€€€E
(b + PT )
@
@y
(y; jP)
(4.8)
= ô€€€E [(b + PT ) (y; jP)] : (4.9)
Observe that (4:7) is negative for all y and : Since expected pro t is concave, a
base-stock policy is optimal. For each markup level ; the optimal base-stock level is
the maximizer of g(y; ) which is found by,
y ( ) = inf
y : ô€€€P0 + b + E [PT ] ô€€€ E
(b + PT ) 1n
DN
T
<y
o
0
= inf
y : E
(b + PT ) 1n
DN
T
y
o
b + E [PT ] ô€€€ P0
:
Chapter 4: Markup Pricing in the Presence of Price Fluctuations 57
Concavity of the objective function ensures that a base-stock inventory policy is
optimal given markup and optimal base-stock level is given by (4:5). In the next
section, we analyze the behavior of expected total pro t function with respect to the
markup for xed inventory level.
Optimal Markup for a Given Inventory Level
In this section, we make the following reasonable assumptions and prove a series of
lemmas that will lead to our main characterization. Analogues of these assumptions
in models without price volatilities are quite common in pricing literature, Ziya et al.
(2004).
Assumption 4.1 (x) is convex decreasing.
Assumption 4.2 x (x) is concave.
The next two lemmas establish that the expected total revenue is concave and
expected total sales is convex in markup level.
Lemma 4.1 The expected total discounted revenue rT ( ) is concave in markup :
Proof. Note that since x (x) is concave, Pt ( Pt) is concave in for each Pt
which makes eô€€€rtE [ Pt ( Pt)] ; hence rT ( ) given in (4:1) is concave in :
Lemma 4.2 The expected total demand (sales) dT ( ) is convex decreasing in markup
:
Proof. Since (:) is convex decreasing, ( Pt) is convex decreasing for each Pt which
makes E [ ( Pt)] ; hence dT ( ) given in (4:3) convex decreasing in :
From now on, we also use the notation
4kE
(y ô€€€ Dk)+
= E
(y ô€€€ Dk+1)+ ô€€€ (y ô€€€ Dk)+
and
42
kE
(y ô€€€ Dk)+
= E
(y ô€€€ Dk+2)+ ô€€€ 2 (y ô€€€ Dk+1)+ + (y ô€€€ Dk)+
58 Chapter 4: Markup Pricing in the Presence of Price Fluctuations
to denote the rst and second-order forward di erences. We will also make use of the
following lemma in forthcoming analysis.
Lemma 4.3 E
(y ô€€€ Dk)+
and E
(Dk ô€€€ y)+
are integer convex in k:
Proof. For any discrete function to be integer convex, the second-order forward
di erences should be positive. Note that,
4kE
(y ô€€€ Dk)+
= E
(y ô€€€ Dk+1)+ ô€€€ (y ô€€€ Dk)+
= E
(y ô€€€ Dk ô€€€ Xk+1)+ ô€€€ (y ô€€€ Dk)+
:
Using
(a ô€€€ b)+ = a ô€€€ min fa; bg
for any a; b 2 R; we can write
4kE
(y ô€€€ Dk)+
= ô€€€E
min
Xk+1; (y ô€€€ Dk)+
: (4.10)
As k increases, ô€€€E
min
Xk+1; (y ô€€€ Dk)+
increases so that the second-order dif-
ference with respect to k is nonnegative, i.e., 42
kE
(y ô€€€ Dk)+
0: Similarly,
E
(Dk ô€€€ y)+
= E
Dk ô€€€ y + (y ô€€€ Dk)+
is also integer convex in k:
In the following characterizations, (M
T )0 = @
@ M
T ; (M
T )" = @2
@ 2M
T ; r0T
( ) =
@
@ r
T and r00
T ( ) = @2
@ 2 r
T denote rst and second order derivatives of M
T and r
T
given in (4.2) and (4.1) respectively.
Theorem 4.2 Assume that y is xed. Then g(y; ) is concave in and optimal
markup (y) is found by solving
r0
T ( ) ô€€€ E
(b + PT ) (M
T )0
+ E
h
(b + PT ) (M
T )0 min
nô€€€
y ô€€€ DN
T
+
;XN
T +1
oi
= 0:
(4.11)
Chapter 4: Markup Pricing in the Presence of Price Fluctuations 59
Proof. It is shown in the Appendix that the rst and second-order partial derivatives
of g(y; a) with respect to are given by
g (y; ) = r0
T ( ) ô€€€ E
(b + PT ) (M
T )0
+ E
h
(b + PT ) (M
T )0 min
nô€€€
y ô€€€ DN
T
+
;XN
T +1
oi
and
g (y; ) = r00
T ( ) ô€€€ E
(b + PT ) (M
T )00
ô€€€ E
h
(b + PT ) (M
T )00 E
h
4
ô€€€
y ô€€€ DN
T
+
j P
ii
ô€€€ E
h
(b + PT )
ô€€€
(M
T )0 2
E
h
42 ô€€€
y ô€€€ DN
T
+
j P
ii
;
respectively. By Lemma 4.1 and Lemma 4.2, r00
T ( ) < 0 and (M
T )00 > 0: Additionally,
by Lemma 4.3, the last term is also negative. Observe also that,
E
h
4
ô€€€
y ô€€€ DN
T
+
j P
i
= ô€€€E
h
min
n
XN
T +1;
ô€€€
y ô€€€ DN
T
+
o
j P
i
ô€€€E
XN
T +1
= ô€€€ :
Then the following inequality holds:
g (y; ) r00
T ( ) ô€€€ E
(b + PT ) (M
T )00
+ E
(b + PT ) (M
T )00
ô€€€ E
h
(b + PT )
ô€€€
(M
T )0 2
E
h
42 ô€€€
y ô€€€ DN
T
+
j P
ii
= r00
T ( ) ô€€€ E
h
(b + PT )
ô€€€
(M
T )0 2
E
h
42 ô€€€
y ô€€€ DN
T
+
j P
ii
0:
The last inequality is due to Lemma 4.3 and r00
T ( ) 0: Since g (y; ) 0; expected
total pro t is concave in markup for each inventory level y: We can nd the optimal
markup by setting the rst partial derivative with respect to equal to zero, i.e.,
g (y; ) = 0; which is given in (4:11) :
In Theorem 4.1 and Theorem 4.2, we characterize the optimality equations for con-
trols y and : The decision maker simultaneously solves equations (4:5) and (4:11) to
nd the optimal controls. Next proposition explains that the expected pro t function
is submodular in two control variables which leads to several monotonicity properties
for the optimal controls.
Proposition 4.1 g (y; a) is submodular. Consequently, optimal inventory level y ( )
is decreasing in and optimal markup (y) is decreasing in y:
60 Chapter 4: Markup Pricing in the Presence of Price Fluctuations
Proof. We show the submodularity of g (y; ) by showing that gy (y; ) < 0: Note
that
gy (y; ) = ô€€€E
(b + PT ) (M
T )0 ô€€€
P
DN
T
+ 1 < y j P
ô€€€ P
DN
T
< y j P
as shown in the Appendix. Note that since (M
T )0 0 and P
DN
T +1 y j P
P
DN
T
y j P
; the derivative of the expected pro t function with respect to each
variable is negative, i.e., gy (y; ) < 0: Moreover, observe that in (4:5) ; as in-
creases 1n
DN
T
<y
o increases for xed y which results in lower y ( ) : Similarly, since
gy (y; ) < 0; g (y; ) is lower for higher values of y that is g (y; ) is decreasing in
y: Then, it is clear that (y) is lower for higher values of y:
In this section, we explicitly characterize the optimality equations for the two
controls, markup and inventory level, and prove that optimal decisions are decreasing
functions of each other. In other words, if the rm has more inventory on hand, for
instance, he should charge a lower markup, which is highly intuitive. Similarly, for
higher values of markup, if given, the rm needs to hold less inventory as less number
of customers are expected to arrive during the sales season. In the next section, we
theoretically analyze the e ect of volatile market prices on the expected revenues,
sales and pro ts.
4.3 The E ect of Price Variability on Expected Pro t
In our model, we used a general stochastic price process where we assume that it has
continuous and nonnegative price paths. In this section we analyze how the optimal
expected pro ts change with respect to the variability of this price process. We use
convex ordering of random variables and stochastic processes in our analysis. The
following de nitions are from M uller and Stoyan (2002).
De nition 4.3 Let X and Y denote two generic random variables. X is said to pre-
cede Y in convex order (increasing convex order, decreasing convex order) if E [f (X)]
E [f (Y )] for all convex (increasing convex, decreasing convex) functions f; i.e.,
X
cx(icx;dcx)
Y , E [f (X)] E [f (Y )]
Chapter 4: Markup Pricing in the Presence of Price Fluctuations 61
for all convex (increasing convex, decreasing convex) functions f.
Similar to convex ordering of random variables, the de nition of convex ordering
of stochastic processes is the following.
De nition 4.4 Let X = fXt; t 0g and Y = fYt; t 0g denote two stochastic pro-
cesses. Then,
X
cx;icx;dcx
Y , E [f (Xt)] E [f (Yt)]
for all t 0 and for all convex (increasing convex, decreasing convex) functions f:
In other words, two stochastic processes are said to be convexly ordered if random
values at each time are convexly ordered. Convex orders are generally used to order
random variables in terms of their variabilities. A property of the convex orders is
the following.
Remark 4.5 If X
cx
Y , then E [X] = E [Y ] and V ar (X) V ar (Y ) : That is,
convexly ordered random variables are also ordered in magnitude of their variances
although their mean are the same.
In the rest of this chapter, we will use two market price processes, namely, P(1) = n
P(1)
t ; t 0
o
and P(2) =
n
P(2)
t ; t 0
o
to compare the expected revenues, pro ts and
sales that are previously examined. Analogously, we denote their corresponding rate
processes (intensity measures) as (1) and (2) where (i) =
n
(i)
t =
P(i)
t
; t 0
o
for i = 1; 2 and corresponding counting measures as N(i) =
n
N(i)
t ; t 0
o
: Let us
denote r(i)
T ( ) ; d(1)
T ( ) and g(i)(y; ) as the expected revenue, expected sales and
expect pro t functions under market price process P(i) for i = 1; 2:
Next two lemmas show that a more variable price process leads to lower expected
revenues and higher expected sales.
Proposition 4.2 If P(1)
cx
P(2), then r(1)
T ( ) r(2)
T ( ) for each 2 R+:
62 Chapter 4: Markup Pricing in the Presence of Price Fluctuations
Proof. Note that since Pt ( Pt) is a concave function of Pt by Assumption 4.2,
P(1)
t
cx
P(2)
t implies
E
h
P(1)
t
P(1)
t
i
E
h
P(2)
t
P(2)
t
i
:
Then it is easy to see that
r(1)
T ( ) =
ZT
0
eô€€€rtE
h
P(1)
t
P(1)
t
i
dt
ZT
0
eô€€€rtE
h
P(2)
t
P(2)
t
i
dt = r(2)
T ( ) :
Proposition 4.3 If P(1)
cx
P(2), then d(1)
T ( ) d(2)
T ( ) for each 2 R+:
Proof. Similar to the proof of Proposition 4.2, since ( Pt) is a convex function of
Pt by Assumption 4.1, P(1)
t
cx
P(2)
t implies
E
h
P(1)
t
i
E
h
P(2)
t
i
:
Then it is easy to see that
d(1)
T ( ) =
ZT
0
E
h
P(1)
t
i
dt
ZT
0
E
h
P(2)
t
i
dt = d(2)
T ( ) :
Next proposition proves that if the market price processes are convexly ordered,
so are their rate processes.
Proposition 4.4 If P(1)
cx
P(2); then (1)
icx
(2):
Proof. Let t 0 be xed and f be an increasing convex function. Then, note that
f ( ) is convex where it can be shown
f ( )0 = 0f0 ( )
and
f ( )00 = 00f0 ( ) + ( 0)2 f00 ( ) 0
Chapter 4: Markup Pricing in the Presence of Price Fluctuations 63
since is convex by Assumption 4.1 and f0 0 by the assumption. Since f ( ) is
convex,
E
h
f
(1)
t
i
= E
h
f
P(1)
t
i
E
h
f
P(2)
t
i
= E
h
f
(2)
t
i
:
Since this is true for any convex increasing f and t 0; (1)
icx
(2):
We use the following proposition from B laszczyszyn and Yogeshwaran (2009) that
links the convex ordering of intensity measure of a doubly-stochastic Poisson process
to the counting measure.
Proposition 4.5 (1)
cx;icx;dcx
(2) imp