Price-demand Modeling
A Tool to Support Inventory and Production Decisions for Competing Products
Peter Schuur
1
, Asli Sencer
2
and Bertan Badur
2
1
Department of Industrial Engineering and Business Information Systems, University of Twente, Enschede, the Netherlands
2
Department of Management Information Systems, Bogazici University, Istanbul, Turkey
Keywords: Price-demand Models, Inventory and Production Management, Market Share Attraction Models.
Abstract: Aim of this positioning paper is to explore the existing price-demand models that have been applied in
inventory and production management so far and to identify new potential structures that may have been
applied in marketing research but have not been touched yet in inventory and production research. Our
focus will be on dynamic pricing structures for competing products, since our exploration so far revealed
that they have not been studied exhaustively in price-demand inventory literature. Specifically, we propose
a price-demand-inventory model framework for optimizing joint pricing, production, inventory and
transportation decisions in a supply chain with multiple products, multiple factories, and multiple markets
operating in multiple periods. The factories operate in a cooperative environment where these decisions are
given centrally so as to maximize the total profit. Competition between the various product types is
achieved centrally by setting market specific prices for each product type in each market in each period. To
support this price setting we introduce several promising price-demand structures for competing products.
1 INTRODUCTION AND
LITERATURE SURVEY
In the realm of dynamic pricing, studies that
consider the effect of pricing on the demand of a
product have been made for decades, in parallel with
the research on inventory and production
management. Marketing researchers identify the
four determinants of the demand by price, product,
place, and promotion, referred to as the classical
four Ps of marketing. Among these, price has always
been considered as the most significant factor in
affecting the demand of a product. Extensive
research has been done into formulating the
relationship between the demand and the marketing
mix variables (Lilien et al., 1992).
In the inventory and production management
literature, models have been developed with
exogenous demand assumptions where the demands
are either planned orders or estimated by forecasting
models. Ample attention has been given in literature
to stochastic models where the demands are
considered to be random variables with known
probability distributions. However, most of these
stochastic models tend to ignore the underlying
determinants of demand.
The past decades, research into models for joint
decision making for price and inventory
management has received increasing attention. The
earliest study that incorporates price as a decision
variable into inventory theory is by Whitin (1955)
where the total profit is maximized in an EOQ
setting. After the 1990’s the joint research in these
areas has gained a considerable acceleration due to
the increasing pressure to attain flexibility and
responsiveness in the supply chains operating under
fierce global competition. This motivation grew by
the advances obtained in information and
communication technology that provided facilitators
for the applicability of dynamic pricing strategies,
namely: (i) increased availability of demand data
that led to better customer segmentation; (ii)
introduction of faster and cheaper labelling in retail
stores and e-procurement environments; and (iii)
development of better decision support tools with
higher speed and accuracy. Thus, more and more
analytical models were needed for developing
optimal management strategies (Elmaghraby and
Keskinocak, 2003); (Chen and Simchi-Levi, 2012).
The broad research area of price-demand and
inventory management models is comprehensively
reviewed with different perspectives in (Chen and
317
Schuur P., Sencer A. and Badur B..
Price-demand Modeling - A Tool to Support Inventory and Production Decisions for Competing Products.
DOI: 10.5220/0004924403170321
In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 317-321
ISBN: 978-989-758-017-8
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
Simchi-Levi, 2012); (Yano and Gilbert, 2005);
(Elmaghraby and Keskinocak, 2003); (Petruzzi and
Dada, 1999). In general, the basic choices in joint
price-demand and inventory models are identified as
the following set of structural attributes. Primarily,
the structure of the chosen price-demand model
plays a significant role in the setting of the problem
and the optimal policy obtained. Other attributes are:
(i) incorporating capacity restrictions on the quantity
ordered; (ii) using cost structures like fixed ordering
costs or price adjustment costs; (iii) incorporating
pricing policies like static pricing where the price is
fixed for a few periods once it is set; (iv) the
structure of the inventory replenishment policy,
namely the availability of multiple replenishments in
the planning period or a single replenishment like in
revenue management; (v) the number of planning
periods being single or multiple; (vi) the number of
products being single or multiple; and (vii) the
structure of the supply chain being competitive,
coordinated or cooperative. As for the latter, the
perspective of the model is essential: does it support
the buyer, the seller, or the system as a whole.
In this study our aim is to explore the existing
price-demand models that have been applied in
inventory and production management so far and to
identify new potential structures that may have been
applied in marketing research but have not been
touched yet in inventory and production research. In
particular, our focus will be on dynamic pricing
structures for competing products, since our
exploration so far revealed that they have not been
studied exhaustively in price-demand inventory
literature.
As summarized in (Chen and Simchi-Levi,
2012), the most common price-demand structure
used in joint price-demand-inventory models is the
linear demand
,∈
0,
⁄
,
0,b0 where demand
is a linearly
decreasing function of price . In spite of its
computational simplicity, this structure is questioned
in terms of its validity due to the linearity
assumption and possibility of obtaining negative
demands at high price levels. At this point, the
exponential demand structure
exp
,0,0 is introduced as an applicable
alternative. The third common structure is the iso-
elastic demand
,0,1 where
the demand elasticity is the same for all price
levels. On the other hand, the logit demand structure
exp
/1exp
is an
alternative to all of the above structures by allowing
a fixed potential demand which is multiplied by
the probability of buying a product at a price . It
should be noted that all these models can be
extended to reflect the effect of complement or
substitute products’ prices on the demand of a
certain product in a competitive environment.
In a stochastic setting, random demand
,
is
defined as a function of the price and a random
noise . Standard approaches include the additive
model
,
, with
0, and the
multiplicative model
,
,with
0and)1. Yet, there are hybrid structures
of the additive and multiplicative models (Chen and
Simchi-Levi, 2012).
Moreover, we have the price-demand models
with intertemporal effect, i.e., the models that
incorporate the effect of prices in the previous
periods on the current demand of an item. Ahn, et al.
(2007) and Gümüş and Kaminsky (2010) provide
models with substitute products and multiple periods
where the total demand in a period is the sum of the
linear demand function of price, the intertemporal
effect, and the substitution effect.
In addition to the price-demand models
mentioned above, marketing theory includes several
other approaches. Among these, market share
attraction (MSA) models are used to calculate the
demand of a product in competitive environments
moderated by 1 substitute products. Assuming
that there is a fixed potential demand
, the demand
of substitute product ,
, for
1,2,…, is obtained by multiplying by the market
share
of product . Here, the market share
is
defined as the ratio of the so-called attraction
of
product with the total attractions of all products,
i.e.,
∑
. Depending on how the attraction
caused by the price of an item is formulated, MSA
models can have different forms (Lilien et al., 1992).
The above MSA approach is a good example of a
technique that has proven its value in marketing
theory, but is still of limited importance when it
comes to price-demand inventory and production
models.
Aim of this positioning paper is to unfold a line
of research to change this. Specifically, we propose
a price-demand-inventory model framework for
optimizing joint pricing, production, inventory and
transportation decisions in a supply chain with
multiple products, multiple factories, and multiple
markets operating in multiple periods. The factories
operate in a cooperative environment where these
decisions are given centrally so as to maximize the
total profit. Competition between the various
product types is achieved centrally by setting market
specific prices for each product type in each market
ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems
318
in each period. To support this price setting we
propose several price-demand structures for
competing products, many of which are based on
MSA.
The remainder of the paper is organized as
follows. In Section 2 we recall from literature an
illustrative modeling framework for price-demand-
inventory models and discuss its merits as well as
possible extensions. In Section 3 we propose our
price-demand-inventory framework for competing
products and present our research agenda. We
conclude in Section 4 with a discussion of our
approach.
In the next section, let us discuss an interesting
modeling framework for price-demand-inventory
models.
2 THE GENERALIZED
STRUCTURE OF
PRICE-DEMAND-INVENTORY
MODELS
In their comprehensive review, Chen and Simchi-
Levi (2012) provide a general modeling framework
for a single product in a periodic review
deterministic setting. Specifically, a firm is
considered that makes pricing and replenishment
decisions of a single product over a finite planning
horizon with periods. Letting the price vector
,
,...,
where
denotes the price in
period , 1,2,…, and the demand vector
,
,…,
where
is the demand in period as a function of the selling
price
in period , two mathematical models are
introduced that operate recursively.
Decision variables
: Order quantity in period
: Price in period
Auxiliary variables
: Demand as a function of the selling price
in period
: Inventory level in period
: Binary variable showing whether an order is
placed in period
Parameters
:Fixed ordering cost in period
: Variable ordering cost in period
: Unit holding cost in period
: Upper bound on the order quantity in period
,
: Lower and upper bounds on the price in
period
For given optimal total costs for ordering,
production, and inventory holding
, with
,
,…,
, Model 1 is
used to optimize the prices in each period as shown
below:
Maximize Profit =
∑
such that
∈
,
,1,2,…,.
Model 2 is used to find the optimal total costs for
ordering, production, and inventory holding,
as shown below:
=Minimize
∑
such that
, 1,2,…,
(1)
, 1,2,…,
(2)
0,
∈
0,1
,
,
0
(3)
Although the above framework is inspiring, it lacks
a number of features that are frequently encountered
in practice:
Capacity considerations to fulfill the demand
and/or options to take care of backlog are
missing. Instead, demand is always satisfied
completely per each period and may not be met
partially or postponed
The model is single product so that competition
between various product types is not accounted
for
The model is single market
The model does not distinguish between
producers or production sites
Transportation is not included
Since according to our findings the above
framework is representative for the state of the art in
deterministic price-demand-inventory modelling we
judge it worthwhile to endeavor a research journey
in order to discover how the above shortcomings can
be mended.
3 A PRICE – DEMAND –
INVENTORY FRAMEWORK
FOR COMPETING PRODUCTS
In this section we introduce a price-demand-
inventory model framework for optimizing joint
pricing, production, inventory and transportation
Price-demandModeling-ATooltoSupportInventoryandProductionDecisionsforCompetingProducts
319
decisions in a supply chain with multiple products,
multiple factories, and multiple markets operating in
multiple periods. The factories operate in a
cooperative environment where these decisions are
given centrally so as to maximize the total profit.
Competition is between the various product types
that in every period aim for generating demand in
each of the multiple markets. Competition is
achieved centrally by setting market specific prices
for each product type in each market in each period.
The formulation of the model is as follows:
Indices
i: index of factory (i=1,2,…,I)
j: index of market (j=1,2,…,J)
k: index of product type (k=1,2,…,K)
t: index of time period (t=1,2,…,T)
Parameters
: Production costs per unit for product type k
produced at factory i in period t
: Inventory holding costs per unit for product
type k stored at factory i in period t
: Transportation costs per unit for product type
k sent from factory i to market j in period t
: Stockout costs per unit for product type k to
be sold in market j in period t
: Production capacity of factory i in period t
Variables
: Amount of product type k sent from factory i
to market j in period t
: Amount of product type k produced at factory
i in period t
: Inventory of product type k stored at factory i
in period t
: Price per unit of product type k in market j in
period t
Auxiliary function
,
,…,
:Quantity of product type
k demanded by market j in period t given all prices
,
,…,
Problem formulation:
Maximize
,,
,,
,,
,,,
,
,…,
,,
such that
,∀,
(4)
,
,…,
,∀
,,
(5)
,∀,,
(6)
,
,
,
≥ 0, ∀,
,,
(7)
Here, the objective maximizes profit of all factories
together. Restriction (4) gives capacity constraints
per factory per period. Restriction (5) specifies the
demand per market per product type per period
given all product type prices in the market and
period under consideration. Restriction (6) gives the
inventory balance constraints on product type level
for all factories.
Crucial in the above model framework are the
price-demand functions
,
,…,
specifying the amount of product type k demanded
by market j in period t given all product type prices
in that same market and period.
Choices for
form the cornerstone for the
research announced in this positioning paper. So, let
us give them due attention. For brevity of writing we
suppress the indices j and t. Thus, we write
instead of
and
instead of
. Contrary to
the price-demand functions discussed in Section 1,
we are interested in those functions that model
competition between products.
A computationally simple choice that we will
examine is:
(1) the linear competitive price-demand model :
(8)
where
and
are nonnegative constants. This
choice results in a non-linear programming problem
with linear constraints and a quadratic (first term of
the) objective. This demand model may fit for a
price sensitive market in which total demand of all
product types together is fluctuating. However, in
some markets in practice, total product demand is
fixed. For instance in a health insurance market,
where all citizens have to be insured, the total
demand is fixed. Hence the demand
depends
ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems
320
solely on the market share. For those markets we
will examine the following choice:
(2) the market share attraction (MSA) competitive
price-demand model with fixed total demand:
with denoting total market demand
and with the market share of product type k being
given by
∑
where the attraction is a
function of all product type prices:
,
,…,
. Of special interest is linear
attraction given by:
1
(9)
where the
are nonnegative constants. This
choice results in a non-linear programming problem
which is computationally harder than the previous
choice. Clearly, MSA is also an interesting approach
when total market demand is not fixed. Therefore,
we will also examine a third choice for
:
(3) the market share attraction (MSA) competitive
price-demand model with fluctuating total demand:
with
as in (2) and with the total
market demand
depending on the average price:
1
(10)
where is a (small) positive constant and denotes
total priceless market demand. The idea behind this
choice is that total market demand
is affected
unfavorably by the average price level.
4 OUR RESEARCH APPROACH
For each of the above competitive price-demand
models we start by examining the computational
tractability of the corresponding non-linear
programming problem. We experiment with
heuristic search methods such as multi-start local
search. Gauging our approach on small problems we
scale up to larger ones. Next, we interpret the results
in order to discover simple approximate heuristic
rules. These results may indicate in what market
situations our models are valuable tools. We
conclude our research by facing the challenge of
parameter calibration.
Clearly, the above competitive price-demand
models are also interesting in other settings. For
instance, they can serve as a decision support tool in
electronic reverse auctions (ERAs), where
competing product supplier agents may place price
bids which are evaluated by the various markets. In
their bid evaluations, the market players may base
their product demands on one of the above price-
demand models. These bid evaluations may be
fortified by using recently developed machine
learning techniques (Den Boer, 2013). It is our
intention to include settings like these in our
research.
ACKNOWLEDGEMENTS
This research has been supported by Scientific
Research Foundation (Project #: 6949) of Bogazici
University, Istanbul, Turkey.
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