Networks and Spatial Economics
https://doi.org/10.1007/s11067-019-09481-6
The Follower Competitive Location
Problem with Comparison-Shopping
Vladimir Marianov 1
& H.
A. Eiselt 2 & Armin Lüer-Villagra 3
# Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract
In competitive settings, firms locate their stores to take advantage of consumers’
behavior to maximize their market share. A common behavior is comparison-shopping:
in this behavioral pattern, consumers visit multiple stores that sell non-identical products, which are mutual substitutes, before making their purchase decision. This behavior has never been included in location-prescribing models for competitive firms. Given
existing branches of one firm, we address the location problem of a follower firm that
locates its own branches. We present insights on the instance used by ReVelle in his
maximum capture formulation, provide computational experience with one thousand
100-node instances, and consider a realistic case using a 353-node network of Santiago,
Chile. The results are compared in terms of the demand captured by each firm and the
locational patterns that result from different consumer behaviors.
Keywords Competitive location . Comparison-shopping . Follower problem
1 Introduction
Facility location researchers and practitioners deal with finding the best possible
locations for all kinds of infrastructure. The book by Laporte et al. (2015) offers a
* Vladimir Marianov
marianov@ing.puc.cl
H. A. Eiselt
haeiselt@unb.ca
Armin Lüer-Villagra
armin.luer@unab.cl
1
Department of Electrical Engineering, Pontificia Universidad Católica de Chile, Av. Vicuña
Mackenna, 4860 Santiago, Chile
2
Faculty of Business Administration, University of New Brunswick, Fredericton, NB, Canada
3
Engineering Science Department, Universidad Andres Bello, Santiago, Chile
Marianov V., Eiselt H. A., Lüer-Villagra A.
comprehensive view of the field while the contributions in Eiselt and Marianov (2015)
discuss selected examples of applications. An important area in the field of location
theory deals with competitive scenarios. In his seminal paper, Hotelling (1929) studied
a market in the shape of a line segment (a so-called “linear market”) with uniformly
distributed demand, on which two competing facilities locate and sell the same product.
Both firms locate a single facility each, and their decision variables are the location of
their facility as well as the mill prices they charge. Customers are assumed to patronize
the facility at which they will pay the lowest full price, i.e., the mill price plus the
transportation costs, which are assumed to be linear in the distance. The main thrust of
Hotelling’s work dealt with the question whether or not there is an equilibrium in the
situation he described. A (Nash) equilibrium is defined as a pair of decisions of the
competitors, which is stable, i.e., neither competitor can benefit by unilaterally changing his decision. Hotelling’s conclusion was that such equilibrium exists with both firm
locating their facilities at the center of the market, a result that was dubbed the principle
of minimum differentiation.
In the early days following Hotelling’s paper, a number of his followers considered
the principle of minimum differentiation the ultimate explanation for the frequently
observed clustering or agglomeration of competitive facilities in practice. However, the
decades following Hotelling’s paper showed many limitations of the analysis. First, the
result was rather fickle similar to the equilibrium a ball is in when located on a plain
surface: even a minor change of the slope of the surface will destroy the equilibrium (at
least in the absence of friction). Secondly, it turned out that small changes in the
assumptions could result in dispersed locations (see, e.g., Chamberlin 1933; Lerner and
Singer 1937; Eaton and Lipsey 1975; Okabe and Suzuki 1987; Brown 1989). Thirdly,
D’Aspremont et al. (1979) in their paper demonstrated that Hotelling’s conclusion with
his own assumptions did not hold and that the original problem did not possess an
equilibrium.
Another strand of analysis dates back to the work of the economist von Stackelberg
(1943). The competitive situations in his work comprise two (classes of) competitors:
the one(s) to act first are the leaders, while those that act after the leader has made a
decision and this decision has become public knowledge, are the followers. Once
decisions have been made, they are irreversible. Note that the scenario is asymmetric:
while the leader has to consider the potential decisions of the follower, the follower
does not need such foresight; he only takes the situation as given and optimizes his own
objective. It is noteworthy that the follower’s problem is a conditional location problem
(“optimize your own objective, given that the leader has already located at known
sites”), while the leader must include the follower’s reaction in each of his decisions,
resulting in a bi-level optimization problem (see, e.g., Aras and Küçükaydın 2017). The
first to apply such a leader – follower model to competitive location problems (albeit
with fixed and equal prices) are Prescott and Visscher (1977). While these authors still
worked on Hotelling’s linear market, Drezner (1982) considered competitive location
problems in the plane, while Hakimi (1983) discussed similar problems on networks.
Hakimi also coined the expression centroid for the leader’s problem and medianoid for
the follower’s problem. Later, ReVelle (1986) solved the follower problem (dubbed the
maximum capture or MAXCAP problem) on a network using an integer programming
formulation. These contributions started a large body of research dedicated to the
subject of competitive location by operations researchers, most of it solving one of
The Follower Competitive Location Problem with Comparison-Shopping
the von Stackelberg problems, sometimes considering location and pricing, as in Kress
and Pesch (2016). Berglund and Kwon (2014) present a different asymmetric approach,
in which a von Stackelberg firm competes with Cournot-Nash firms. For reviews, see,
e.g., Eiselt et al. (2015), Kress and Pesch (2012) for problems on networks; and
Drezner (2014) for problems in the plane. Eiselt (2011) as well as Marianov and
Eiselt (2016) analyze competitive location and agglomeration results from the point
of view of location researchers.
One of the key features in any competitive location problem concerns customer
behavior. While Hotelling assumed that customers would purchase the good from the
source with the lowest full price, most of his successors, who did not include price
competition in their models, reduced this assumption to its simplified version, in which
customers purchase from the source closest to them. A number of authors have made
different assumptions, which essentially fall into two categories. First, there are those
authors who have dropped the “individual trip” assumption made by almost all
researchers in location analysis. In other words, consumers are not necessarily assumed
to make special trips for each product they attempt to purchase. This bundling is
typically referred to as multipurpose shopping or, as the case may be, multi-stop
shopping. Dellaert et al. (1998) provide some insight into multipurpose shopping
behavior from a conceptual point of view, while Hodgson (1990) introduced the
concept of flow capturing (or flow interception). In it, customers no longer choose
the facility closest to their respective home locations, but a facility closest to a trip they
are on anyway, e.g., the daily trip to work. In other words, the relevant customer facility distances are no longer point-to-point distances, but path-to-point distances.
This concept applies when customers handle drop-offs (as in the case of childcare
facilities) or pickups (such as gas fill-ups) along the way between home and work.
Lately, there has been a revival of this idea when locating alternative fuel stations, as in
Miralinaghi et al. (2017). This idea can be seen as a limited version of location-routing
problems, see, e.g., Nagy and Salhi (2007). Marianov et al. (2018) investigate the
effects of multipurpose shopping on store location.
Another aspect of customer behavior concerns information gathering. In the case of
retail location models, this would include internet searches, flyers, ads in media, and
information collection by visiting the stores. Most authors in this subfield consider in
price search. Contributions such as those by Guo and Lai (2014) include not only
internet search, but also internet purchases in their model. The present paper belongs
into this category, as we deal with customers comparing products in stores before
making any purchase, i.e., comparison-shopping. More specifically, we assume a
customer behavior that includes two levels of comparisons: in the first pre-trip planning
stage, customers compare observable primary features of the products in which they are
interested. This will include price as well as specifications that are quantifiable and
typically published, such as the size of an item, its weight, its primary product features,
etc. Given the results of the research, the consumer will plan a trip. Then, in the second
during-trip stage, customers will examine the secondary product features (such as color,
flavor, and specific fit in the case of clothing) in detail and make their decision to
purchase or not to purchase accordingly. This behavior will be described in detail in the
second section of this paper.
The economic literature has addressed multipurpose and comparison-shopping, as in
Eaton and Lipsey (1975, 1979, 1982), McLafferty and Ghosh (1987), Ghosh and
Marianov V., Eiselt H. A., Lüer-Villagra A.
McLafferty (1984), Arentze et al. (2005), Mulligan (1987), O’Kelly (1981, 1983), Thill
(1982), and Wolinsky (1983). However, to the best of our knowledge comparisonshopping has never been dealt with in prescriptive facility location models see, e.g.,
Santos-Peñate et al. (2019) and Pelegrín et al. (2018).
Our contribution to the literature consists in including, for the first time, comparisonshopping in a competitive facility location problem. We propose a model that solves the
follower problem for a market-share-maximizing firm locating one or more stores in the
presence of existing competitor’s stores. The follower problem has been addressed in
the literature, and it has an importance by itself: it requires being solved by a new
entrant in a market in which one or more incumbents are already present. It is also a
sub-problem of the leader problem. Since comparison-shopping is a consumer behavior
that has never been considered before in this context, it seems reasonable to solve this
simpler problem first, to observe the effects of comparison-shopping without interference by other issues.
We use a duopoly for the sake of simplicity and clean analysis. If more chains are
involved, there could be effects related to e.g. the market size and border effects, which
become undistinguishable from the effects of comparison-shopping alone.
Our analysis compares the locations and demand captures that result from a model
with comparison-shopping to those with single- purpose shopping trips. Using the
simplest possible setting, we show that comparison-shopping results in a larger market,
more frequent co-location, and stronger agglomeration of competing stores.
There are similarities between the multipurpose shopping problem (Marianov et al.
2018) and the comparison-shopping problem investigated here. The first similarity is
that both are in essence bi-level problems that we formulate at once as one-level
optimization problems. Secondly, in both cases the stores of different firms tend to
be located close to each other. However, there are also significant differences, which are
shown in Table 1.
In summary, the multipurpose and comparison shopping models have common
properties, but there are significant differences in the concepts behind them. The main
Table 1 Differences between Marianov et al. (2018) and the present paper
Marianov et al. (2018)
Present paper
Cooperative model
Competitive model
Products are not mutual substitutes
Products are mutual substitutes
Customers can purchase both
products on one trip
Customers purchase at most one product on their trip
Monopolistic markets
Shared markets
One decision stage: visit one store or Two decision stages: visit one store or two, then, if two stores were
both
visited, what product to purchase.
No customer choice between
products of the same type
Customers have the choice between products of the same type
Certainty a priori about the utilities
Uncertainty about utilities: utility depends on the product choice and, if
an on-site inspection of the products is not satisfactory, customers
may not make any purchase at all, thus resulting in a negative utility
The Follower Competitive Location Problem with Comparison-Shopping
contributions of both papers are different. In both cases, though, the inclusion of more
refined assumptions about customer behavior result in better representations of reality.
The remainder of the paper is organized as follows. In Section 2, we describe the
problem. Section 3 develops the model. Section 4 contains the computational experience, and Section 5 presents conclusions and future extensions.
2 The Problem
A firm or chain L (the “leader”) has several stores located in a market, at known
locations, all of them selling the same product. As long as no confusion can arise, we
will refer to the chain and its product as L. A second chain F (the “follower”) wants to
enter the market by locating one or more stores and selling product F. Products L and F
are heterogeneous mutual substitutes, i.e., they differ only in secondary features. If a
consumer wants to buy a unit of product, he will purchase L or F, but not both. This
paper solves the follower problem, i.e., that of the newcomer firm F wanting to
maximize its market, given that the branches of firm L have already been located.
The products are non-essential, meaning that consumers may choose not to make
any purchase. In addition, consumers are not willing to spend more than a maximum
amount of resources (the reservation price) to acquire a unit of the product. Given that
both firms apply mill pricing, the reservation price includes the price consumers pay for
the product at the branch of the firm they patronize, and the generalized cost for the
round trip, which includes the cost of travel, parking, the customer’s value of time, and
other features. Rather than single consumers, we assume consumer clusters with similar
characteristics.
In the first stage of the customers’ decision-making process, customers decide
irrevocably to either forego the purchase, make a trip to a single store (single-stop trip
– SST), or engage in comparison-shopping by visiting two stores in the two possible
directions (two-stop trip – TST). All of this is decided based on perceived expected
utility comparisons. Suppose now that a customer has decided to patronize a single
store. After visiting the store and obtaining details of the product in question, he will
decide, with some probability α, to make the purchase. Alternatively, he may decide
not to make the purchase and return home. We assume that customers do not change
their mind along the way in the sense that they, after deciding on a single stop trip,
obtain the new information at the store and now decide to engage in comparisonshopping. Wolinsky (1983) assumes the same type of shopping behavior, and Drezner
et al. (1996) define a choice rule as consistent, if a decision to patronize a store does not
change along the way.
Suppose now that by way of the utility function, a customer decides to engage in
comparison-shopping. This means that the customer will visit the first store—in the
order decided upon in the utility function—followed by a visit to the second store. At
this point, the customer has full information about the products. The customer now
makes the decision to purchase the product with a given probability β ≥ α, in one of the
visited stores. In the second stage, the consumer decides what product to purchase.
Note that β ≥ α because the probability of finding a suitable product (and buying it) in
visiting branches of both stores is higher than finding an acceptable product in a single
store (Huff 1963; Stahl 1982).
Marianov V., Eiselt H. A., Lüer-Villagra A.
In order to describe the second stage of comparison, it is necessary to
analyze product features and the way customers compare them. There are two
main classes of features: those that are observable from anywhere and those
that are not. Clearly, price is a feature that can easily be observed from
anywhere, meaning that customers can sit at home and compare products as
far as their respective prices are concerned. Some of the product features are
more difficult to ascertain and compare: not all flyers, newspapers, or websites
that describe the different products list the same features, and intangibles, such
as the “feel” for a product, the fit of a piece of clothing, or the ease to use a
product, can only be established at a store. In addition, we consider that
consumers have taste uncertainty (their taste can change from one purchase to
the next), and they might have different preferences regarding these features or
completely dislike some products. These are the reasons why they compare.
The second stage of comparison involves features, not maximum utility. If the
maximum expected utility rule were used, there would be no reason to visit two
stores instead of one, as a single trip would always have a higher utility than a trip
visiting two stores. There would be no room for comparison. Furthermore, at the
time of the first decision stage, there is uncertainty about what product will be
chosen by the consumer. Thus, we require including the uncertainty and lack of
information in the expression of the utility, by assuming that, on starting the trip,
the consumers do not know what product they will finally choose. This is a
fundamental difference with the contribution by Marianov et al. (2018). In the
second stage, the consumer does not know what product he will purchase, because
they have different desirable or secondary features, about which he does not have
information. He needs to estimate the utility by assigning a percentage of patronage, say 50%, to each competitor. We use this figure, corrected by the cost of each
alternative (purchasing at the second visited store or coming back to the first
visited store and making the purchase there), as we will see in the next Section.
This is the expected utility of the two-store trip.
It is important to mention that performing this estimation using other rules, e.g.,
discrete choice and gravity models (see, e.g., Hodgson 1978; Drezner and Drezner
1996; Fernández et al. 2007; Marianov et al. 2008) is trivial. The results should not
change significantly.
Consumers are located at discrete points of a plane or at nodes of a network
representing the region. Each point concentrates a known amount of homogeneous
demand although, if there are different demand segments at each point, the point can be
replaced by as many copies of itself as demand segments, each copy housing homogeneous demand. Throughout this contribution, we assume that all customers at point i
have the same reservation price which indicates the value they assign to, i.e., the
maximal price they are prepared to pay for a single unit of the good in question. We
use reservation prices associated with the customer location, based on the general
income level of the area, as reservation prices indicate the ability and willingness to
pay for a product. Also, we use all customer locations as candidate locations for stores
of both chains, which can be co-located. Figure 1 shows the problem setting and the
representation we use in this paper of the customers and store locations.
As an example of the customers’ decision-making process, consider a customer,
who is interested in a pair of dress shoes. The shoes would nicely complement a given
The Follower Competitive Location Problem with Comparison-Shopping
Fig. 1 The problem setting with customers, leader and follower locations. A bold square denotes a leader
location, a thin square, customers captured by the leader in single trips. A bold triangle denotes a follower
location, a thin triangle, customers captured by the follower in a single trip. Diamonds denote co-location of
leader and follower. Circles are customer locations. A dashed circle border denotes they engage in two-stop
trips and the demand is shared between both chains
outfit, but their purchase is not necessary to complete it, making them a non-essential
good. Based on his location, the customer has a specific reservation price, is aware of
the general price level of the two chains and plans accordingly.
3 The Follower Location Model with Comparison-Shopping
For ease of reading, we first define our notation.
Sets
N
S
I⊆N
J⊆
N
K⊆
N
The set of nodes
{p, q}, where p and q take one of the values L or F, i.e., leader store location or
product, or follower store location or product.
The set of demands, indexed by i. Without loss of generality, we assume I = N
The set of candidate locations of the follower’s F stores, indexed by j. Without
loss of generality, we assume J = N
The set of known locations of the leader’s L stores, indexed by k.
Parameters
ri
πp
gij
α
β
The reservation price of the consumer i ∈ I.
The price of product p.
The generalized cost of a trip from i to j, with i, j ∈ N
Probability of finding a suitable product and purchasing it in an SST.
Probability of finding a suitable product and purchasing it by visiting two stores.
Marianov V., Eiselt H. A., Lüer-Villagra A.
ε
upij
upq
ijk
The largest utility difference that is considered as a computational zero.
Utility for consumer i of making an SST to a store p located at site j.
Utility for consumer i of making an TST to a store p located at site j, and then to
a competing store q located at site k. We assume that p ≠ q.
The probability of making
of product q at the second visited store k
a purchase
ppq
ijk
and going back home. 1−ppq
ijk is the probability of returning to the first visited
store j and purchasing product p there.
The number of customers (a proxy for demand) at node i.
The number of stores located by the follower, F.
ai
nF
Additional sets
Ki
n
k t ∈KjuLik t ≥ uLik ; ∀k∈K
o
set of locations of leader’s stores that provide the
highest utility to customers at i among all the leader’s stores. Let kt be a
representative location of this set.
n
o
j∈ J juijF ≥ uLik t þ ε , set of all candidate locations j for an F store, such that
N iF
the utility for customers at i for purchasing at j in an SST, if a store were
located there, is higher than purchasing product L at any k t ∈K i, also in an SST
trip.
n
o
j∈ J juijF ≤ uLik t −ε , set of locations j whose utility for consumers at i is strictly
F
Ni
F
N L¼
i
lower that the utility of purchasing at k t ∈K i , both in SST trips.
n
h
io
j∈ J juijF ∈ uLik t −ε; uLik t þ ε , set of all candidate locations j for an F store,
such that customer i, purchasing at j in an SST, has the same utility as
purchasing at k t ∈K i .
F
F
N iF ∪N i ∪N L¼
¼J
i
Decision variables
xj
yi
vi
zi
yqp
ijk
One if an F store is located at j, and zero otherwise. The only location variable.
One if for the customers at i the highest utility choice is making a purchase at an
F store in an SST, and zero otherwise.
One if for the customers at i the highest utility choice is making a purchase at an L
store in an SST trip and zero otherwise. Defined only if ∃uLik t > 0.
One if the utility for customers at i making an SST purchase at k t ∈K i or at an F
store j is the same, and zero otherwise. Defined only if ∃uLik t > 0.
One if the highest utility choice for customer i is visiting first a q store at j and
then a p store at k in a comparison-shopping or TST, and zero otherwise. Defined
only for uqp
ijk > 0.
The Follower Competitive Location Problem with Comparison-Shopping
We now formalize the utilities introduced above. We remark that, as it is customary
(see, e.g., Huff 1963), we compute utilities perceived by a consumer for visiting a store
and, later, these utilities are used to compute the capture of a store relative to other
alternatives.
3.1 Single Stop Trip Expected Utility
The expected utility of a customer at i of making an SST to a store p located at site j, is:
h
i
h
i
upij ¼ α ri −π p − g ij þ g ji þ ð1−αÞ − g ij þ g ji :
ð1Þ
The first term is the utility of purchasing a product p in an SST to site j, weighted by the
probability α of making the purchase. The second term is the utility of not making the
purchase, having made the trip. If desired, the price could be made dependent on the
store.
Note that we assume that consumers do not have an a priori preference among
products, which leaves the prices and trip costs as the only drivers of consumers’
decisions when choosing between products. However, the probability α can be made to
be dependent on the chain. When consumers make only SSTs, we assume they choose
their highest expected utility store. As locations of the leader’s stores are known, to
capture consumers at i, the follower F chooses a site j and a price πF that make uijF
> uLik for all sites k where leader’s L stores are located.
3.2 Two-Stop Trip Expected Utility
If the expected utility of visiting two stores is positive, the customers could visit a
leader’s L store followed by a follower’s F store, or the other way around. Once at the
second visited store, consumers have complete information on both products, and they
choose whether to purchase, and at which store. With a certain probability ppq
ijk the
purchase is made at the second visited store q and the customer returns home. With a
probability (1 –ppq
ijk ), the customer goes back to the first store p, makes the purchase and
returns home from there.
The total expected utility that a consumer starting the search at i will perceive, from
making the entire trip to a first store p located at j and a second store q located at k, is:
h n
oi
pq
pq
q
p
¼
β
r
−
g
þ
g
þ
p
ð
g
þ
π
Þ
þ
1−p
þ
g
þ
π
upq
g
i
ij
jk
kj
ji
ijk
ijk ki
ijk
h n
oi
þ ð1−β Þ − gij þ gjk þ gki
ð2Þ
As before, the first term is the utility of making the purchase at some store, while the
second term is the utility of visiting both stores and not making any purchase. Once at
the second store, the consumer has full information on the quality at both stores. If it
happened that (gki + πq) = (gkj + gji + πp), i.e., the total costs plus prices of the two
alternative actions were the same, ppq
ijk would be ½, since we assumed that
preferences are evenly distributed. However, when the equality does not hold, we
Marianov V., Eiselt H. A., Lüer-Villagra A.
assume that the customers distribute between the alternatives in an inverse proportion
to their total cost, i.e.,
ppq
ijk
gkj þ g ji þ πp
¼
ðgki þ πq Þ þ gkj þ gji þ πp
ð3Þ
In terms of purchases at each store, the proportion
of the demand originating at i that
pq
pq
purchases product q at k is pijk , while 1−pijk is the proportion of the demand
purchasing product p at store j.
This utility is computed for both possible directions, i.e., visiting first a store q
followed by a p store, and the other way around. It may happen that utilities for
different trips are the same. Some tie-breaking rules of behavior are required. We
assume that, if the utility of an SST is equal to the utility of a TST, the customer will
engage in a TST. In addition, if there are trips i-L-F-i and i-F-L-i with equal utilities, the
customer will prefer visiting F first. Preliminary tests have shown that this last rule does
not have significant effect on the market capture of both competitors, while allowing a
simpler model.
3.3 The Model
The formulation of the Follower Location model with Comparison-Shopping is as
follows:
FLCS : Max Z F ¼ ∑ αai yi þ 1
i
þ
∑
i∈I ; j∈ J ;
k∈K
s:t: Z L ¼ ∑ αai vi þ 1
i
z
2 i
þ
yi ≤
zi ≤
z
2 i
þ
∑
i∈I; j∈ J ;
k∈K
LF
βai pLF
ikj yikj
FL
βai 1−pFL
ijk yijk
∑
i∈I ; j∈ J ;
k∈K
LF
βai 1−pLF
ikj yikj þ
∑ xj
ð4Þ
∑
i∈I; j∈ J ;
k∈K
FL
βai pFL
ijk yijk ð5Þ
∀i∈I
ð6Þ
∀i∈I
ð7Þ
j∈N iF ∧
uijF > 0
∑ xj
F
j∈N L¼
i
The Follower Competitive Location Problem with Comparison-Shopping
∀i∈I uLik t ≤ 0
ð8Þ
yFL
ijk ≤ x j
∀i∈I; j∈ J ; k∈K
ð9Þ
yLF
ikj ≤ x j
∀i∈I; k∈K; j∈ J
ð10Þ
vi ≤ 0
yi ≥ x j −
xr −
∑
r∈N iF j
uirF > uijF
vi ≥ 1−
xr −
∑
r∈N iF j
uirF > uLik ℓ
zi ≥ x j −
∑
−
∑
xr −
−
∑
xr −
∑
∑
r∈ J ; k∈Kj
LF
uLF
itr > uikj
yFL
irt
k t ∈K i j
uLik t > uLF
ikj
xr −
∀i∈I; k ℓ ∈K i uLik ℓ > 0
F F
∀i∈I; j∈N L¼
uij > 0
i
xr
∑
1−
∑
xr
∀i∈I; ∀ j∈N iF uijF > 0
r∈ J ; t∈Kj
FL
uFL
irt > uijk
r∈ J ; t∈Kj
FL
uFL
irt ¼ uijk ;
tþr < kþ j
r∈N iF ∪N iL¼ F j
uirF > uLF
ikj
∑
r∈ J ; t∈Kj
F
uLF
itr ≥ uij
1−
∑
k t ∈K i j
uLik t > uFL
ijk
r∈ J ; t∈Kj
FL
uLF
itr ≥ uijk
yLF
ikj ≥ x j −
xr −
∑
r∈ J ; t∈Kj
F
uFL
irt ≥ uij
xr −
∑
r∈N iF ∪N iL¼ F j
uirF > uFL
ijk
xr −zi
∑
r∈ J ; t∈Kj
L
uLF
itr ≥ uik ℓ
r∈ J ; t∈Kj
L
uFL
irt ≥ uik ℓ
xr −
xr
∑
r∈ J ; t∈Kj
F
uLF
itr ≥ uij
xr −
∑
r∈N iF j
uirF > uijF
yFL
ijk ≥ x j −
xr −
∑
r∈ J ; t∈Kj
F
uFL
irt ≥ uij
∑
r∈ J ; t∈Kj
LF
uLF
itr ¼ uikj ;
tþr< kþ j
ð11Þ
ð12Þ
ð13Þ
ð14Þ
∀i∈I; ∀ j∈ J ; k∈KjuFL
ijk > 0
xr
∑
r∈ J ; t∈Kj
LF
uFL
irt > uikj
yLF
itr
∀i∈I; ∀ j∈ J ; k∈KjuLF
ikj > 0
ð15Þ
Marianov V., Eiselt H. A., Lüer-Villagra A.
yi þ vi þ z i þ
∑
j∈ J ;k∈KjuLF
>0
ikj
yLF
ikj þ
∑
j∈ J ;k∈KjuFL
>0
ijk
∑ xj ≤nF
yFL
ijk ≤ 1
∀i∈I
ð16Þ
ð17Þ
j
LF
x j ; yi ; vi ; zi ; yFL
ijk ; yikj ∈f0; 1g ∀i∈I; j∈ J ; k∈K:All y and v variables defined
for corresponding positive utilities
ð18Þ
Objective (4) maximizes the market share of chain F. The first term is the demand of
consumers that make SSTs. The second term includes customers that make TSTs
starting by visiting one of the L stores. The third term are the consumers making TSTs
that start at an F store. There are respectively proportions α and β of consumers that
make a purchase in SST and TST trips. Note that this is not a multiobjective problem
and (5) is not an objective, but a constraint, that counts the market share captured by
chain L using similar terms to those in the objective. Constraints (6) state that capturing
SST consumers at i by F is possible only if there are F stores with a utility that is greater
than the highest utility L stores. Constraints (7) allow capturing SST consumers from i
by F stores with the same utility as the highest utility L stores. Later, variable zi is
weighted by ½ in the objective, to represent the capture of only half of the sales.
Constraint (8) allows capture of a consumer at i by an L store only if is one of the
highest utility L stores. Constraints (9) and (10) ensure that capture by TSTs is possible
if there are TSTs with positive utility. Constraints (11) to (15) are “highest utility
assignment constraints” (Marianov et al. 2018). These constraints correspond to the
problem solved by the customer and characterize customers’ behavior, i.e., the maximization of utility in a sequential search, and there is one set of these constraints for
each possible choice of the customers at i, as explained next. Constraints (11) force the
variable yi to take the value one if an SST to an F store is the highest utility choice for i.
Note that this only can happen if an F store is located at a site j∈N iF . If firm F locates
such a store (i.e., xj = 1 in the first term on the right-hand side), and if the utility of the
trip is positive, the first term on the right-hand side is one. If the remaining terms are
zero, the constraint forces yi = 1. If any of the remaining terms on the right-hand side is
a non-zero, it will mean that there is a higher utility choice for the customer, and yi is
not forced to take the value one. These higher utility choices are, in the same order they
appear in the constraints:
i) One or more F stores located at points r∈N iF provide a higher utility on an SST.
ii) One or more TSTs i-r-t-i, visiting an F store located at r and then an L store at t ∈ K,
has at least the same utility as the SST to j or higher.
iii) One or more TSTs i-t-r-i, visiting an L store located at t ∈ K and then an F store at r
has at least the same utility as the SST to j or higher.
The Follower Competitive Location Problem with Comparison-Shopping
Constraints (12) force the variable vi to take the value one if an SST to an L store is the
highest utility choice for i. The first term of these constraints is similar to that in
constraints (11), except that the location of all L stores is known. The following three
terms are the same as in constraints (11), and the last term is one if there are F stores
with the same utility as the highest utility L stores, when customers make SSTs.
Constraint (13) makes variable zi = 1 if SSTs to some F store have the same utility as
SSTs to the highest utility L stores. Constraints (14) force yFL
ijk ¼ 1 if the highest utility
choice for customers at i is a TST i - F store - L store - i, and their explanation is similar
to that of the preceding two sets of constraints. Note that, if there are two different trips,
say i-j-k-i and i-r-t-i, with the same utility, picking any of them is indifferent, and the
last term of the constraint will choose the one with t + r < k + j to break the tie.
Constraints (15) are equivalent to (14), for TSTs i - L store - F store - i. Note that there
could be a tie between TST utilities from i-j-k-i and i-k-j-i trips, which would make the
model choose a suboptimal solution, due to constraint (16). We avoid that by making
the consumers to choose visiting an L store first, if such situation occurs. This is
enforced by the fifth terms in constraints (14) and (15).
Constraints (16) force the capture of consumer i by exactly one facility in an SST, or
a pair of facilities and the order of their visit in a TST. Constraint (17) sets the number of
stores to be open by the chain F, and constraints (18) defines the domain of the decision
variables.
4 Computational Experience
This section reports computational results first on a small, 30-node instance, and then
from one thousand 100-node random instances. The number of variables required for
the model is O(4|N| + 2|N|2|K|), the number of constraints is O(4|N| + 4|N|2|K| + |N|2 +
2|N||K| + 1), and the problem is NP-hard, as we show next.
Proposition 1 The problem is NP-hard.
Proof: Let K, the set of leader sites, be the empty set. The problem reduces to the
Maximum Covering Location Problem, which is NP-hard (Megiddo et al. 1983) ■.
Although the problem is NP-hard, we run the integer-programming model using
AMPL and CPLEX 12.8, allowing it to use only one thread in each run. The computer
was an HPE Proliant DL360 G9 server with two Intel® Xeon® CPU E5–2630 v4 @
2.20GHz, 160 GB of RAM, and Debian 9 Operating System.
We first comment on the observability of the parameters, or how these parameters
can be estimated in practice. Utilities depend on the product mill price, the reservation
price, cost of travel and, in the case of a trip visiting two stores, the probability ppq
ijk . The
price of the products is observable. There is literature available on how to estimate
consumers’ reservation price (see Jagpal 2008, Jedidi and Jagpal 2009, Breidert 2006
and references therein). The cost of travel and value of time has also been modeled and
empirically measured (see Small 2012; Oregon Department of Transportation 2014;
Victoria Transport Policy Institute 2016, and many references therein.) Finally, in this
paper we estimate the value of ppq
ijk by assuming that if both product are perceived equal,
a 50% of the consumers will consider adequate the product in store L, and 50%, that in
Marianov V., Eiselt H. A., Lüer-Villagra A.
store F. Furthermore, once the consumer compares both products and he is at the
second visited store, his decision also depends on the difference between the full costs
of making the purchase in both stores. We remark that this rule can be trivially replaced
by any other rule that allocates customers to products (e.g., Huff rule), including
imbalances in the preferences. The rule can also be estimated using the data available
to retailers from their previous experience as well as market studies.
The probabilities α and β are the probabilities of the customers finding and
purchasing a suitable product when engaging in single and multiple-stop trips, respectively. The rationale for applying these probabilities is the empirical fact that the
number of items of the kind the consumer desires, that are likely to be available during
the trip, have an influence on the utility of a trip (Huff 1963). Thus, a TST, with more
available items, will have a higher utility β. The probability α was set to 0.5, assuming
that 50% of the consumers will find acceptable the product they find at the store in a
single-stop trip. The value of β covered the range 0.5 to 0.9, reflecting the fact that as
consumers are more selective, they value more engaging in comparison-shopping and
are less satisfied with visiting a single store. Market studies reflect how selective the
customers are. This is in accordance with Huff (1963), although later, he uses the
shopping center’s square footage of selling space as a proxy for the number of items.
Stahl (1982) uses a similar approach to compare markets with different numbers of
competing stores.
4.1 Small Instance
We first made runs on a small 30-node instance to show some of the location patterns
that result from the assumptions we made. Recall that firm L, the leader, locates without
foresight, i.e., not considering that there could be another firm entering the market later
on. If only the leader is present in the market, consumers’ reservation prices, together
with the price of the product define how far they are willing to travel to make the
purchase at the single provider. The Maximum Capture Location Model by Church and
ReVelle (1974) can thus be used to solve the leader’s problem with no foresight on his
part. The entering firm F, the follower, wants to enter the market and maximize its
market share. If there is no comparison-shopping, the maximum capture (MAXCAP)
model by ReVelle (1986) solves the problem for the optimal locations of the stores of
the entering follower chain, which is what the usual competitive models solve. By
setting α = 1 and β = 0 in our model, we reproduce this situation. For a review of
competitive models, see Eiselt et al. (2015). In addition, our model includes the context
in which there is comparison-shopping, which we model using values of β larger than
0.5.
The 30-node instance is the same as in ReVelle (1986). Without loss of generality, all
nodes are both demand points and location candidates. We used α = 0.5, and values of
β ranging from 0.5 to 0.9. The run time of each instance never exceeded 0.094 s, and
the average time was 0.022 s. In Fig. 2, the reservation price is 300, and the prices at
both chains are 150. The Figure shows the situation in which both chains locate three
stores each, for β = 0.5, 0.7 and 0.9, in that order.
The Figure shows how, for equal prices, as the probability β of finding a suitable
product by visiting two stores increases, the follower tends to co-locate with the leader,
as this strategy facilitates the comparison and attracts more customers. Figure 2a shows
The Follower Competitive Location Problem with Comparison-Shopping
the case in which there is no additional gain for visiting two stores over visiting just
one, as α = β. There is no incentive for the follower to co-locate with the leader, as this
would mean sharing the market in equal proportions. Rather, the follower locates to
obtain a regionally monopolistic market, in which consumers never visit more than one
store. As the probability β of the consumers making a purchase increases, enabling
comparisons by moving closer to the competitor increases the expected sales, so that
the follower tends to co-locate with the leader. Both total and each competitor’s market
(the number of consumers making a purchase) increase as follows: leader 59, follower
68 (β = 0.5, Fig. 2a), leader 78, follower 87 (β = 0.7, Fig. 2b), and leader 169, follower
169 (β = 0.9, Fig. 2c). Equal demands on both competitors reflect the fact that they are
co-located and offer their products at the same price. Note how the number of nodes
whose demand does make purchases, increases when moving to the right in Fig. 2.
This behavior is similar for different numbers of stores, as Fig. 3 shows. Note that
this case could be representative of the follower desiring to locate stores over a time
horizon, in which case the intermediate locations would be robust, i.e., are a subset of
the final set of locations.
Table 2 shows the demand for different numbers of stores (#) located by the leader
and the follower (L and F). The third and following columns show the leader and
follower demand captured in SSTs and TSTs, as well as the total demand captured by
each competitor. For β = 0.5, all captured demand consists of single-stop shoppers,
while for β = 0.9, the captured demand not only significantly increases, but it includes
mainly two-stop shoppers.
4.2 Computational Experience with Larger Instances
We run the model for 1000 instances with 100 demand nodes each. The region is a
100 × 100 square over which the demand is randomly located according to a uniform
distribution on each axis, from an integer uniform distribution from 0 to 100. We chose
to generate 1000 random instances because these cover many of the possible configurations of demand distribution. The distances are Euclidean, but network or any other
practical norms or distances work the same. The competitors could locate one, two or
three stores. The reservation price is 200. The probability α is 0.5. The following
Tables synthesize the results.
Fig. 2 Leader and follower locate three facilities each. In all cases, πL = πF = 150, α = 0.5. a β = 0.5, b β =
0.7, c β = 0.9. Circles: customer locations. Dashed border indicates customers visited in TSTs. Squares: bold
for leader location, thin for customers captured by the leader. Triangles: bold for follower location, thin for
customers captured by the leader. Diamonds: bold if leader and follower co-locate, thin if both leader and
follower capture customers
Marianov V., Eiselt H. A., Lüer-Villagra A.
Fig. 3 The leader locates three facilities. In all cases, πL = πF = 150, α = 0.5, β = 0.9. a follower locates 1
store, b follower locates 2 stores, c follower locates 3 stores. Same notation as before
Table 3 shows the capture for three leader stores and one, two and three follower
stores, when both firms have equal prices. We keep the notation used in Table 2 and the
figures correspond to averages over the 1000 instances. The average number of colocated stores is denoted Co-loc. The last column shows the average run time required
for solving one instance.
For any number of follower stores, Table 3 shows that as β increases, so does the
capture. There is a significant follower advantage in terms of capture for low values of
β. As β increases, the follower stores co-locate with the leader’s stores. When both
have the same number of stores, the capture of both firms becomes the same. Furthermore, with only two stores and for low β, the follower still obtains an amount of
demand that is larger than the demand captured by the leader with three stores. As β
increases, the two follower stores tend to co-locate with two of the leader’s facilities
and the third leader’s facility gives him an advantage. For only one follower store, its
location follows the same behavior: for low β it acts as a regional monopoly but, as β
increases, it eventually co-locates with one of the leader’s stores.
Table 2 Demand for different numbers of stores and values of β
β = 0.5
β = 0.9
L
#
F
#
L
SST
F
SST
L
TST
F
TST
L
Tot
F
Tot
L
SST
1
1
22
28
0
0
22
28
1
2
22
54
0
0
22
54
1
3
22
76
0
0
22
2
1
42
26
0
0
42
2
2
42
49
0
0
2
3
42
68
0
3
1
59
26
0
3
2
59
49
3
3
59
3
4
59
πL = πF = 150
F
SST
L
TST
F
TST
L
Tot
F
Tot
0
0
86
86
86
86
0
28
86
86
86
113
76
0
45
86
86
86
131
26
20
0
86
86
106
86
42
49
0
0
135
135
135
135
0
42
68
0
18
135
135
135
153
0
59
26
38
0
86
86
123
86
0
0
59
49
18
0
135
135
153
135
68
0
0
59
68
0
0
169
169
169
169
85
0
0
59
85
0
18
169
169
169
186
The Follower Competitive Location Problem with Comparison-Shopping
Table 4 displays the captured demand for three stores per chain, at equal prices of
120, 150 and 180 and a range of values of β. For β = 0.5, both competitors’ stores
receive always only SST purchases, and the follower has a strong advantage in captured
demand (almost double), because the leader located without foresight and the information on leader locations is available for the follower when locating his own stores.
As β increases, the total capture of both chains increasingly includes TST demand,
the total capture increases for both competitors, and their markets approach each other
in size, as co-location increases. When β = 0.9, the total capture is entirely due to TST
shoppers and both chain markets are practically the same, indicating very strong
predominance of co-location. Note that there is a column denoted as “Av min dist”,
which is the Average Minimum Distance between closest competitors’ stores. For each
solution, the closest B store is found for each A store and these distances are averaged.
The mean value over the 1000 instances is reported. As β increases, this distance, a
proxy for agglomeration, decreases because the follower’s stores approach the leader
stores. As the price increases, the demand captured by both competitors decreases
because the utility of all kinds of trips also decreases. Especially, for very high prices,
e.g., 180, agglomeration and TSTs demand decrease. It is worth noting that although
captured demand decreases with price, the profit does not necessarily follow the same
trend, and the customers’ total utility decreases. When the prices set by both chains are
different, the location patterns change, but general trends remain. Co-location is not
Table 3 Average captured demand for 1000 runs on random instances
β
L
SST
F
SST
L
TST
F
TST
L
Tot
F
Tot
Co-loc
Time(s)
One Follower store
0.5
406
265
0
0
406
265
0
0.03
0.6
404
261
4
4
408
265
0
0.04
0.7
345
173
106
106
451
279
0
0.04
0.8
270
18
345
345
615
362
1
0.05
0.9
265
0
505
505
771
505
1
0.05
Two Follower stores
0.5
378
491
0
0
378
491
0
0.04
0.6
373
484
7
7
380
491
0
0.04
0.7
272
331
185
185
457
515
1
0.05
0.8
138
54
606
606
743
660
2
0.07
0.9
121
0
925
925
1046
925
2
0.09
Three Follower stores
0.5
354
694
0
0
354
694
0
0.03
0.6
344
682
12
12
356
694
0
0.04
0.7
223
494
232
232
454
726
1
0.05
0.8
47
133
773
773
821
906
2
0.06
0.9
1
3
1263
1263
1264
1267
3
0.08
Leader: three stores. Follower: one, two and three stores. α = 0.5, πL = πF = 150
Marianov V., Eiselt H. A., Lüer-Villagra A.
Table 4 Average captured demand for 1000 runs on random instances
β
L
SST
F
SST
L
TST
F
TST
L
Tot
F
Tot
Co-loc
Av min dist
Time (s)
πL = πF = 120
0.5
637
1117
0
0
637
1117
0
24
0.07
0.6
583
1055
67
67
650
1122
0
23
0.09
0.7
250
531
698
698
948
1228
2
13
0.17
0.8
24
69
1501
1501
1525
1570
3
2
0.26
0.9
1
3
1997
1997
1998
2001
3
1
0.41
0.03
πL = πF = 150
0.5
354
694
0
0
354
694
0
20
0.6
344
682
12
12
356
694
0
20
0.03
0.7
223
494
232
232
454
726
1
15
0.04
0.8
47
133
773
773
821
906
2
4
0.06
0.9
1
3
1263
1263
1264
1267
3
0
0.07
πL = πF = 180
0.5
167
306
0
0
167
306
0
23
0.01
0.6
158
297
10
10
168
307
0
22
0.01
0.7
122
248
69
69
191
317
1
19
0.01
0.8
66
149
208
208
274
357
2
12
0.01
0.9
20
49
400
400
420
449
2
4
0.01
Three stores each chain. α = 0.5
always the best strategy, but as β increases the tendency towards agglomeration
remains. Table 5 shows this situation.
To analyze l the location patterns in more detail, we made runs varying the value of
β in steps of 0.1. Figures 4, 5 and 6 show the changes in capture, both in SSTs and
TSTs, as well as the co-location and average minimum distances, as β changes.
An analysis of Figs. 4, 5 and 6 shows that, no matter what the price difference
between chains, for low values of β the capture is completely due to SSTs. As β
increases, the purchases due to TSTs increase and those by SSTs decrease. If the
follower’s price is low or similar to that of the leader (Figs. 4 and 5), the SSTs are
negligible when β is high. However, when the follower price is higher than the leader’s
(Fig. 6), there is always some capture by SSTs at high βs, because two-stop trips are
more expensive, and the follower can capture more demand by establishing regional
monopolies. It is interesting to note that, for low follower price (Fig. 4) and low β, the
follower frequently co-locates with the leader, as this allows sharing high demand areas
where the leader is located. As β increases, however, a point is reached at which this
strategy is not optimal (β = 0.6, in this case), and the follower changes to a monopoly
strategy. Beyond that point, again the follower approaches the leader locations as β
increases. For high follower price (Fig. 6) and when β is low, the follower stays away
from the leader as its higher price makes him an inferior choice for customers. Rather,
he remains as a regional monopoly until the benefit of comparing becomes enough for
The Follower Competitive Location Problem with Comparison-Shopping
Table 5 Average captured demand for 1000 runs on random instances
β
L
SST
F
SST
L
TST
F
TST
L
Tot
F
Tot
Co-loc
Av min dist
Time(s)
0.5
103
1267
0
0
103
1267
0.7
11
0.07
0.6
108
1251
8
10
115
1261
0.5
12
0.07
0.7
100
977
220
270
320
1247
0.2
14
0.18
0.8
18
386
856
1048
874
1434
0.5
7
0.28
0.9
1
80
1388
1695
1389
1775
0.4
6
0.32
0.5
354
694
0
0
354
694
0.0
20
0.03
0.6
344
682
12
12
356
694
0.1
20
0.03
0.7
223
494
232
232
454
726
0.9
15
0.04
0.8
47
133
773
773
821
906
2.4
4
0.06
0.9
1
3
1263
1263
1264
1267
2.9
0
0.07
0.5
422
316
0
0
422
316
0.0
23
0.01
0.6
422
316
0
0
422
316
0.0
23
0.01
0.7
262
237
132
111
394
348
0.9
16
0.01
0.8
186
168
281
236
467
405
1.3
12
0.02
0.9
117
98
491
413
608
511
1.4
8
0.02
πF = 120
πF = 150
πF = 180
Three stores each chain. α = 0.5, πL = 150. πF = 120, 150, 180
breaking this strategy (β = 0.68). From then on, the TSTs increase, as well as the
agglomeration as shown by the average minimum distance.
Fig. 4 Above: Follower and leader’s capture by SSTs, TSTs, and total. Below: Average number of co-locations
and average minimum distance between closest competitors’ stores. πL = 150, πF = 120, α = 0.5. Three stores
located by each competitor
Marianov V., Eiselt H. A., Lüer-Villagra A.
Fig. 5 Above: Follower and leader’s capture by SSTs, TSTs, and total. Below: Average number of co-locations
and average minimum distance between closest competitors’ stores. πL = 150, πF = 150, α = 0.5. Three stores
located by each competitor
Note that when prices are different, for high values of β, the co-location decreases
slightly, because the capture radius of the competitors is different, which makes the
follower seek the optimum not necessarily co-locating with the leader. Finally, Table 6
shows the capture for both competitors for different prices and values of β.
Table 6 shows that an increase in the follower price always decreases his capture.
When β is small, part of the market lost by the follower by increasing its price, goes to
the leader. As β increases however, e.g., 0.8 or above, both firms loss market when the
Fig. 6 Above: Follower and leader’s capture by SSTs, TSTs, and total. Below: Average number of co-locations
and average minimum distance between closest competitors’ stores. πL = 150, πF = 180, α = 0.5. Three stores
located by each competitor
The Follower Competitive Location Problem with Comparison-Shopping
Table 6 Average captured demand for 1000 runs on random instances. Three stores each chain. α = 0.5. πL =
150
β = 0.5
β = 0.6
β = 0.7
β = 0.8
β = 0.9
β=1
πF
F Tot
L Tot
F Tot
120
1267
103
1261
115
1247
320
1434
874
1775
1389
2151
1749
130
1048
179
1020
248
1052
323
1219
764
1545
1279
1922
1680
140
865
237
846
307
869
365
999
680
1299
1145
1673
1562
150
694
354
694
356
726
454
906
821
1267
1264
1727
1727
160
556
388
557
386
574
412
656
557
839
822
1112
1158
170
430
409
430
408
455
410
526
512
663
702
877
966
180
316
422
316
422
348
394
405
467
511
608
672
810
L Tot
F Tot
L Tot
F Tot
L Tot
F Tot
L Tot
F Tot
L Tot
follower rises the price. In usual competitive models, an increase in the price of one of
the competitors reduces only his own market.
4.3 Computational Experience with a Real Network
Santiago is the capital and the largest city in Chile, with a population of almost 7
million people. We use a 353-node network of Santiago, using its 353 census districts
(the union of several census tracks). The location of the nodes of each census district is
obtained by allocating each of them to the closest node in the 2212-node main road
network of the city, which allows using the road network distances. If two census
districts have the same closest road network node, one of them is allocated to its second
closest node of the road network.
In the resulting 353-node network, every census districts is connected to its neighbors, through arcs whose length is the shortest distance over the road network. Figure 7
shows the network, on a background representing the Metropolitan Area of Santiago.
A Floyd-Warshall algorithm was applied to the 353-node network, to obtain minimum distances between every pair of nodes. The demand in every node is obtained
from the 2002 Chilean National Census. We use a travel cost equal to its distance.
Every node is a both a demand point and a potential facility location. Tables 7, 8 and 9
show some of the results, which confirm the insights obtained with the previous runs.
We remark that the reported time is the sum of the AMPL plus the CPLEX time. In
average, the CPLEX time is between 3 and 4% of the total time.
From Table 7, we see again that as β increases, the locations change from regional
monopolies to shared markets (SST capture decreases, TST capture increases, colocation and average distance between firms both decrease.)
Table 8 shows that, for the parameter values used in these runs, the follower always
co-locates as many stores as possible with leader’s stores (for five or less stores, all the
captured demand is by TSTs, over five stores, leader and follower have the same
amount of demand captured by TSTs.)
Table 9, finally, shows that naturally, as the follower’s price increases, its captured
demand decreases. It is interesting to note that for low follower prices, the leader’s
Marianov V., Eiselt H. A., Lüer-Villagra A.
Fig. 7 The 353-node network of the Metropolitan Area of Santiago
market increases. However, as the follower increases its price over the leader’s price,
both competitors loose market, as they both become less attractive.
5 Conclusions and Future Work
This paper has addressed the competitive location problem, considering that consumers
engage in a behavior that is more complex than those studied so far in the location
Table 7 Average captured demand for the 353-node network of Santiago
β
L
SST
F
SST
0.50
287,348
419,394
0
0.55
287,348
419,394
0.60
287,348
419,394
0.65
227,431
335,839
0.70
84,701
175,275
277,160
277,160
361,861
0.75
33,013
93,832
413,812
413,812
446,824
0.80
0
0
611,951
611,951
611,951
0.85
0
0
695,365
695,365
695,365
0.90
0
0
842,657
842,657
842,657
842,657
L
TST
F
TST
L Tot
F Tot
Co-loc
Av min dist
Time(s)
0
287,348
419,394
0
6884
4.38
0
0
287,348
419,394
0
6884
6.14
0
0
287,348
419,394
0
6884
6.32
86,076
86,076
313,506
421,914
1
6307
6.51
452,435
3
3709
7.81
507,643
4
428
7.56
611,951
5
0
7.57
695,365
5
0
8.03
5
0
8.98
Five stores each chain. α = 0.5, r = 15,000, πF = πL = 5000
The Follower Competitive Location Problem with Comparison-Shopping
Table 8 Captured demand for the 353-node network of Santiago
L
#
F
#
L
SST
F
SST
L
TST
F
TST
L Tot
F Tot
Time(s)
5
1
255,947
0
182,284
182,284
438,230
182,284
9.32
5
2
204,259
0
361,368
361,368
565,626
361,368
8.94
5
3
133,406
0
536,768
536,768
670,174
536,768
8.92
5
4
59,593
0
708,788
708,788
768,381
708,788
8.92
5
5
0
0
842,657
842,657
842,657
842,657
8.90
5
6
0
89,377
842,657
842,657
842,657
932,034
8.89
5
7
0
170,820
842,657
842,657
842,657
1,013,477
8.86
5
8
0
250,710
842,657
842,657
842,657
1,093,367
8.89
5
9
0
319,964
842,657
842,657
842,657
1,162,620
8.88
5
10
0
386,840
842,657
842,657
842,657
1,229,496
8.89
The leader locates five stores, the follower locates between one and ten stores. α = 0.5, β = 0.9, r = 15,000, πF
= πL = 5000
literature. In particular, instead of assuming that consumers make single trips to a store
where they buy a good, we include comparison-shopping, the behavior according to
which consumers visit stores belonging to different retail chains, before deciding
whether making a purchase, and where. In our case, two chains or firms, leader and
follower, selling each one product or brand of products, locate their stores sequentially,
and consumers decide their trips once the stores are located. The products sold at the
chains are imperfect mutual substitutes, i.e., they differ in secondary characteristics as
Table 9 Captured demand for the 353-node network of Santiago
πF
L
SST
F
SST
L
TST
F
TST
L Tot
F Tot
Co-loc
Av min dist
Time(s)
1000
–
–
691,741
1,409,767
691,741
1,409,767
–
933
15.95
2000
–
–
750,965
1,195,931
750,965
1,195,931
–
844
14.03
3000
–
–
735,968
1,011,892
735,968
1,011,892
1
908
12.57
4000
–
–
759,273
874,357
759,273
874,357
2
700
11.18
5000
–
–
842,657
842,657
842,657
842,657
5
–
10.16
6000
–
–
707,790
626,845
707,790
626,845
1
704
9.27
7000
–
–
690,736
545,251
690,736
545,251
1
794
8.63
8000
–
–
681,233
477,814
681,233
477,814
1
775
8.00
9000
–
–
643,720
407,747
643,720
407,747
1
595
7.49
10,000
7494
–
630,498
365,882
637,992
365,882
1
595
7.06
11,000
59,593
50,336
535,713
289,913
595,306
340,249
2
1757
6.76
12,000
73,813
50,336
480,330
239,788
554,143
290,124
1
1915
8.22
13,000
133,406
97,362
344,167
160,143
477,573
257,505
–
4708
8.02
14,000
151,773
97,362
331,372
143,572
483,145
240,934
1
4550
7.82
The leader and the follower locate five stores each. α = 0.5, β = 0.9, r = 15,000, πL = 5000
Marianov V., Eiselt H. A., Lüer-Villagra A.
color, texture, etc. Consumers are aware of the prices charged by the chains. We assume
that consumers make a first decision at their point of origin, regarding what store or pair
of stores to visit. This decision does not change along the way. When visiting one store,
a consumer can either make the purchase or not. If visiting two stores, after comparing
the products, the consumer decides first whether purchasing the product or not, and
then, in what of the visited stores to purchase it.
The decision on what store or pair of stores to visit is made based on the utilities
perceived by the customer a priori, which depend on the reservation price, the prices at
the stores, the total distance to be traveled and the probabilities of finding a suitable
product. The trip chosen in the first stage is the one with the highest utility. In the
second stage, the decision is made based on products’ features.
We propose a model for the consumer comparison strategy and an optimization
model for the follower location problem. We run a small instance from the literature,
showing that the comparison-shopping behavior and its consideration by the follower,
increases the likelihood of agglomeration or co-location. We also run the model on one
thousand 100-node instances generated randomly, whose statistics confirm the findings
and show that the model is solvable in short run times.
The main results include
&
&
&
&
&
&
an indication that an increase of β (the probability of finding a suitable product by
visiting two stores) increases agglomeration,
an indication that an increase of β increases the demand captured by both
competitors,
other things being equal, the follower captures significantly more demand than the
leader (a result dubbed the “first entry paradox” by Ghosh and Buchanan 1988),
an increase of the price level of the follower by 50% resulted in between a 66% and
a 75% of decrease of capture,
for markets in which there is no strong drive for comparing products, an increase in
the price of the follower decreases his market and increases or keeps unchanged the
leader’s market, and
for strong comparison-shopping-oriented customers, an increase in the follower’s
price level significantly decreases the market capture of both firms.
Furthermore, if consumers are not selective (the probabilities of purchasing a suitable
product by visiting one store or two stores are close to each other), firms locate their
stores as regional monopolies. Such a store, say k, could increase the price of its
product as long as the utility of consumers in the neighborhood for purchasing at k is
higher than the utility of going to the closest competitor’s store. On the other hand, if
customers are more selective, they must compare, and as a result, stores tend to colocate in spite of the fact that they have to share the market. In this case, any price
increase comes with a proportional loss of customers. However, they do so because the
probability of obtaining a purchase is higher. This differs from the situation in which
there is multipurpose shopping (Marianov et al. 2018) in which agglomeration or
clustering is due to the fact that consumer surpluses when buying two products are
higher, and consumers have an incentive to travel farther away to get both products. In
synthesis, there are two opposing forces when locating near to the competitor: the
competition becomes stronger, as firms must share the market. However, by facilitating
The Follower Competitive Location Problem with Comparison-Shopping
comparison-shopping, the agglomerated stores attract more customers. The hope is that
this market increase counterweights the effect of sharing.
The model we present is a one-level version of a two-level problem, in which the
entering chain solves the first level (the location of stores, conditional to the location of
existing stores), and the consumers the second level (the choice of the store at which the
purchase is made. This model provides a tool to investigate the clustering behavior
observed when there is comparison-shopping, and to quantify the effects of consumers’
selectivity or preferences.
The model includes a rule on how the consumers choose the purchase once the
comparison is done. This rule can be easily be replaced by other rules (e.g., Huff), and
preferences or biases towards one or the other chain can also be included trivially.
The knowledge obtained from solving the follower problem allows us to address the
leader problem in a duopoly as future research. Note that the leader problem is a game
with three players: the leader, the follower, and the consumers. The formulation, as it is
now, is not generalizable to the case of more than two chains, as it would require more
assumptions on customer behavior. However, the information obtained from the results
on the duopoly, allows extracting the basics and later, addressing more complicated
cases. We intend to address this problem in the future. There are more potential research
directions to be explored in the context of multi-stop shopping. A first one is related to
different customers’ searching strategies, e.g., allowing changes of mind along the trip,
or taking into account the fact that the decisions have random components. Natural
extensions include more than two products and other discrete choice rules.
Acknowledgements We gratefully acknowledge the support by Grant FONDECYT 1160025 and by the
Complex Engineering Systems Institute through grant CONICYT PIA FB0816.
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