LIBRARY
OF THE
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
mmsm
A Model
RETAIL OUTLET LOCATION:
of the Distribution Netv;ork Aggregate
Performance
642-73
Philippe A. Naert*, and Alain
January, 1973
V.
Bultez**
MASS.
tN'ST. -\fr.\C[
MAR 121973
DtWEY LIBRARY
A Model
RETAIL OUTLET LOCATION:
of the Distribution Netv;ork Aggregate
Performance
642-73
Philippe A. Naert*, and Alain
V.
Bultez**
January, 1973
*
Assistant Professor of Management Science, Sloan School of Manageirent,
M.I.T.
**
Charge de cours suppleant, Faculte Uni versi taire Catholique de Mons
(F.U.CA.M.)
IllC
UUUiiUlO HiJii
the data base.
UU
uitUiiiS
iiwiv
»J«-*i
V.
\j
*
t_MiMi^iti
r»..w
[--tx^V »'-'-«-.
w..^
They are also indebted to the National Bureau of Economic
Research (N.B.E.R.) and the Sloan School of Management for computer funds
allocated.
i)^-
RECFIVFD
JUN
'
M.
I.
T.
19
1973
LIBRARIES
1
Locatjon of sales outlets Is of major concnrn to
such
orp,
rTnl>!ations
oil
as
companies, banks, etc. The problem
ten approached in two steps.
p,
The total
r.^irkct
is
divided into
expand or to contract. Thus for example in period
(i)
1
add
'.•;ill
2 t
outlets in area
n.
of-
r<.>-
stape amounts to decidinp in which repions to
and the first
ions,
Ir,
area 2,
in
n^
1,
the company
t,
etc.
We will
i
1
refer to the first step as the ap pre pate location prohlern. The second stape consists in chooLlnp specific sites for these new cut-
these decisions are
lets.
In practice,
dent,
especially because they are
orpani ;^a t i on
.
macle
rep,
arded as rather in da pen-
at
different levels in the
The number of new outlets is
a
corporate decision,
whereas specific sites are selected at the repional level, subject
however to approval by the corporate headnucrtsrs. Many companies
feel
that,
at
least for the time beinp,
aggregate and detailed
this paper our sole crncern will be with aggi'cgs-
location. Our procedure will be closely related
loped by Hartung and Fisher
ness,
of
neither worth the effort ncr the
.
In
te
linFing of the
1
^
cost
is
problerris
hiierarchical
and suffered from
the model parameters.
a
f/'].
Hov-jcver,
t'j
a
model deve-
their work lacked robust-
variety of deficiencies :n the estimation
In
section
2
^.'e
will review the Hartunp-
Fisher [hereafter M-F] model and the various weaknesses associated
v^/ith
it.
section
In
1
vie
section
3
the estimation problems wjll be examined.
will propose various changes to the r'^dsl whicti
637560
v.'
In
ill
I
maKc it robust,
a
ond we will
use data from
a
major oil corpany in
European country to est i ma to the parr.mGtcrr. and validate cur ap-
proach
.
2
^
•
THE HARTDNH-FISHrR rODFL
The market is
to
reductrd
quasi duopoly,
a
we
consider our brand.l, versus competitive brands taken
c.
Buyinr behavior is described as
X.
=
probability that
in period
t-1,
person
a
will usual
top,
ether,
first order Markov chain.
a
transition prnbabilit, ins are defined as follows
The
that is,
i-;ho
7-
u
:
usual I y buys brand
buy brand
i
i
in period
t.
Adding the word "usually" broadens the definition used by H-F because it alio v; 3 for incidental purchases of
-This
important for
usually buys brand
v;ho
on
is
a
a
a
competitive brand.
product such as gasoline. Take
1.
On
a
given day he is running
thruway and tanks at the next s'ervice area.
not i,
If
a
person
cLit
of gas
the brand is
we should not conclude yet that brand switching has occurred
The other transition probabilities are similarly
defined
:
probability of switching from
probability of remaining
a
to
buyer of
pro ability of switching from
[j
c
i
to
i
a
c.
competitive bran
r
,
Market share of bronri
t
Is
m=x.*m_+o.-m
i,t-1
i.t
c,t-1
1)
1
In
in period
1
1
steady state,
rn
i
m
(2)
i,ei
=
a
/
.
,t
(1
i
-
X
11
+
.
o
.
and thus
,
.t-1
=
)
c^.
1
/
fo
ic
The transition probabilities
the decision variables of company i.
.examp
1
a
v-/ill
)
be
functions of
such as advertising, cxpendi-
e
.
=
i
s.
(a.,
1
p.,
I'^ic
d.,
i
a
,
d
cc
,
p
H-F consider only the number of sales outlets as determinants of
the
(3)
transition probabilities. The following functions
^i
=
1^1
•
d^
/
(d^
*
d
/
[d
+
.
1
c
^i^
d
.
1
)
v;ere
post u
la-
5.
For
given value of
a
d
both
,
and show decreasing returns.
and a.
X
The
increasing with ri,
are
limits of
X.
11
are
a.
and
k
respectively. However,
K
in
d
c
the
and
K
and hence
,
d
<»
->•
and
X.
restricted to he betv^een zero and one. For example, H-F
are not
larger than one.
and
K
for
and of o.
i
Since
is
X
probability, its dependence on
a
should be constrained in such
interval.
[0,1]
"Unless
The H-F model
aware of that.
H-F are well
k
=
1.0,
=
k
values of d./(d
1
+
c
d,].
a
d.
way that its value will
lie
therefore loc^s robustness.
They state that,
X.
and o,
Ho v; ever,
are not probabilities for all
equations
and
'3)
can be
(4}
i
ted.
More general functions can be substituted for equations
(4)
(3)
and
without invalidating later results. The authors found that
3
for their problem equations
Section
4
v-jill
be
(4]
were sufficiently accurate".
devoted to examining ways through
v-.
ch
the model
can be mads robust.
Let us now relate market share to the parameters
k
and
k. Substituting
{3}
and
(4]
for
X
and
o
in
['^)
gives
6.
(5)
i.e
Lpt q
-
C1
sales of
and
i,
Note that H-F replac::'
industry sales. Market share
=
Q
m^
in
I.e
m.
q./
=
We should observr; that
by-^1
a./O.
(51
this implies the assumption that observed market share is equal to
steady state market share.
will return to this issue in ssctiori
V-'e
terms, H-F obtain.
q^/d^
(6]
From (65
=
k^
Q
•
(d^
/
follows that for
it
piven value of
a
d
'-
,
c
and
average sal as per outlet will increase with ri, and
ty when
p
=
(d^
-2.a0,
+
anc)
y«d.) goes
therefore,
per outlet when d./d
^
1
=
c
to
^ero. With
=
k
f.
v/ithi
p
<
D,
oes to infini-
4.44 and
=
k
0.64,
the model>woulri predict infinite sales
For values
1/2.8.
d
.
1
/d
>
c
1/2.8,
the mo-
The function pattern is depicted
del would predict negative sales.
inFigurol.
Insert Fioure
cbD'jt here
1
On the other hand
pf
1
-^'itlet
l.ambin
will decrease when
applied
u'-"
i
tlic
H-F model on
a
f
p
>
nufitaer
,
of
the model
average sal^'
outlets incrftastrG.
brand of p;asoline in
a
European
•
7.
cruntrv
[10,
chanter 7^, and founci values of
and
0.9 5
7.
y
=
However, even with
difficulties. With
d.
(for
larp,c
given
a
k
one may still
0,
>
p
d
},
run into
approaches
q.
k*
Q/u
sales exceed industry sales.
The next
rameters
k
issue deals with the estimation of the pa-
and k„. Equation
(B)
is
nonlinear In
K
k,
and
could not be estimated by linear regression. H-F applied
of
trans for rr.ations to
[6)
and ended up with the folio in
i-j
and
a
series
rr
function
(7)
H-F use
linear regression to estimate
a
and k_ are then uniquely determined from
and
a
B.
and
The paramete-s
B.
probleins absociated witii the Gj.LimdLion pioceduie
These will be examined in section
in an
1^
There
usyd
arf
ij
y
H-F.
3.
aggregate retail outlet location model in
to maximize discounted return of firm (i).
i^jhich
k
various
the ob,
.
Let
J
=
the number of regions
(J=1,...,J1
T
=
the tine horizon
(t-0....,T]
T
=
the disco nt factor
r;:
=
return per unit sold in
ri'T
LI
q.
,
d.
=
...
,
d.,
as
existing
r.
rep,
ion J,
period
but superscripted for regions nnd time
q.
upher of outlets at time
D
Jt
and in period
t
Jt
.Jt
The return maximization model is then
Max
Z
I
J
t
.f
^i'
.f
subject to
jt
I
it
c~
J
'
qf
=
K^
df
=
dJ°
^
jt
-
i
Jt
'i
n^
<
^
"
b
'
qJ^
.
.
I
/
[dJ^
nJ^
t
.
..dfl
Vj.t
Vj.t
Vj.t
9.
The readRr wil]
increasing
'•it
function of
d
specified as
costs,
i.e.
notice that with
.
<
p
and therefore unle^js
D,
it
L
Z
j
t
n^
c.
\i^
*
^J
the objective func-
are deduced,
,
i.
1
constraints.
It
p,
i\'en
by
also important to note th^t even when the
is
discounted coiistruction costs ore deduced,
may still be monotone increoslnp,
3.
carefully
is
r;:
an
is
net return and unless the discounted construction
a
tion is monotcjne increasing and the solution is rrerely
the
n-j
the objective function
for d^
,
|d
<
/
\i\
•
ISSUES IN ESTIMATING THE H-F MfinEL
timated values of
rif;;ht
a
B
in equation
(7)
hand side of the equation while in fact the number of retail
outletsis expected
left
and
to
have
hand side of equation
a
causal effect on market share and not
(7)
is
q./'J
=
1
tion
(7)
can be rewritten
From
have
ri./d
1
on
c
a
causality point of
+
d
c
/d.}.
Or.
equ
1
on
the
viev^/
I'^ft
of
we would
like to
the equation.
Scne
1
simple manipulations of equation
equation,
1
as
the ripht and m.
"^
(q,/Q)(1
(6)
result in the followinp, lineai
10,
1/m.=Y*<5'(cl/ri.)
c
(9)
1
where
y
(1
=
the market
In
equation
1
and
P)/nc,
-
6
share function,
(5)
m,
(i-/:th
ned fron equations
(7),
Or in ternG
1/ci.
=
K,
and
k
K=
1/
can estimate it directly as nbtainer;
wt.'
m.
r-eplacinp,
and
(9],
of
}
hv nonlinear
er.
tinatjnn
(5K The estimations were pcrforme
5
on
the TROLL systpn
on r^arquardt's
paramet ers [12
The estimation proceriurG in TROLL
,
is
br-seri
alporithni for least-squares estimation of nonlinear
1.
Wg estimated eauations
brand of gasoline in
a
(7),
(91,
and
Furopcan coun'try- There arc
for
(5)
35
a
major
qu^Tterly
observations, from the first quarter of 1962 to the third quarter
of
1970.
on
its
The statistical results for the nonlinear model
linearized form around the ontimu'n, i.e.
are based
the minimum of the
6
residual sum of squares
.
Let
Pt
The general model is written as.
=
(0,
...
C
...
0„)
^
vnctcr of
Let
e
bG the final
InrDst-squnr'RS
er-
tirpatG of
Q
close to
Vlith
,
3
9f (y, ,ei
(in)
E(\'
f(y^,o]
=
}
\—-~-A
T
p
3f
with
f
-
p
1
I
-
^^
e=G
^
^
(X^.O)
pt
f^
f
=
(x^ .0)
-
f
j:
t
t
,
pt
p
p
can be v.Titten as
(10)
jqufltion
t
wherB,
z
pt
P
f.
-
V
f
•
following
matrix
X
P)
for the nonlinear model are
The statistics
a
(T
linear function of the parametBrs
t
similar to those of
-
a
lineai-
Thus,
rep;ross'ion.
if we
definr;
the
:
P1
12
/here
T
is
the number of observations
^ariance-covariancc n>atrix of
is
??
tbe residual
mean sqLiare
is
V
[
available,
0)
=
s
-n
2-1
•
i
f-,
(F'F)
,
the
e s 1
3
in v-zhich
ra
t
7
:
e
r
1?.
Note that
a^
is
n
no
ir,
Icn^Gr
an
unhiased
c.i
timet
of
n
thR disturbance variance and that even when the en- or tf;rm
,
normally diGtributnd,
result,
valid in
they
the
u'.
nal
t-,
therefore he
The
share of brand
:
rf;
to
Worse even,
k
is
nep,
a
statistics are not
these statistics will be reported here
rep,
arded as mere
suits for tirand
less
riod of observations,
f:
normally dlntrihut. ed. As
Innp, er
and Durbin -Watson-
F-,
Howpver,
psner.-^l.
shoLilri
no
is
than
risen valur^s.
are presented
per cent.
5
its outlet
otive.
i
com. pa
Yet,
in
Taiile
I.
ovpr the
i%'hole
pe-
share was hip her than
8
per cent.
The values of the '"'urbin-VJatson sta-
tistic indicates autncon- elation of the residuals, and hence problems with the model
specification. There are several possifile
explanations. One is that
im,
bles may have been left out.
port ant additional explanatory variaA
previous study of these data shows
8
that this is not the case
.
More
liRely reasons are the misspecifir
tion of the transition probability functions,
and the assumption
that observed market shares are enuilibrium values.
In
other cases one may have better luck. For exam9
pie,
VIP.
applied the H-F model on another brand (a)
,
and found
sults similar to those in the H-F paper as illustrated in Table
The Durbin- V/atson statistic,
10
autocorrelation
ho v; over,
ap,
ain indicates
reII
significant
Insert Tables
In
the next
which thR model can
formulations
4
,
to
\r
I
?.nd
aiiojt
il
s c c t i o n
'
here
we w
j
1 1
rxp
1
n rb
made robust. Wo will apply
c
t
u'
both
w
o
ri
vs
in
these rridrl
the data for brand i.
ALT E;_P h!_AT TVF FO RniJt -VrrO^N S
In
this section we will
redefine the transition
oxponential functions
prohahility functions. First, as
relative number nf outlets in section 4.1.,
funct
i
ons
j
section 4.2.
n
How to
rn
and next
as
of
the
lopistic
ake use of these models will he
further examined in section 4,3.
4.1.
Exponential model
Define
A
as
.
i
d./d
approaches infinity,
A.
approaches one. Fnuation (1?) thus
relates the relative number of outlets D.,
bility
X
.
in
,
a
Similarly,
(13}
=
\
1
c
0.
-
the transition protia-
to
robust way.
EXP(-a
•
c
D
defined as
is
X
]
c
And therefore.
o
=
c
EXPC-a. 'D.)
1
1
,
a
_,
i
=
EXPf-a
c
14
TarKet share at
at
tirrie
related to market share
ic
t
.
timp t-1.
(14)
rn
=
i.t
.
.
[1
E
-
XP
(
a
-
n
•
.
1
EXP(-
+
J
D
•
c
i.t
.
,
E
-
]
XP
[
-
a
•
P
c
,
c.t
1
)
m
•
,
,
i,t-1
.
^
c.t
Assume now for
a
moment, as H-F did, that ohsorvcd
markot share values pre Gquilibriurt values, for piven values oT
D.
.
I.t
(15)
and
D
^
c.t
.
1.
^
i.t.e
That is,
EXP{-a
=
•
c
Applying
a
linear mod e
(1G)
1
D
^]
c.t
/
fEXP(-a,
•
D.
,)
a
logjt transformation to
-
m.
EXP(-a c
+
I.t
i
D
•
^}1
c.t
results in
(15)
.
l0R[m.
^
'^'x.t.G
/
(1
)]
i.t.e-*
.
a ,•
=
i
D.
,
I.t
a
n
•
c
.
c.t
The results of the estimation of
ted in Table III.
The estimated a.
is
negative which would seum to
indicate that the assumption on enuilibrium
It
is
not
at
should be c]Gar that many assumptions
equilitjriurn form
For ex a n^ pie,
(16).
-
the ccrsumption patterns arc adequately stable.
b)
-
the unit-tirnj period is ouffiniently
of
a
not
tf-
e
provided that
a)
-J
-
have to be made in order to accept
satisfied in this case
-
tive effect
all Wr-irrcjntec
marketing campaign
long so that the disrup-
launched in period
[t-1,t]
-
on
the
B
teady- st nt
<;
narknt shares can resorh In
new equilibrJum achieved within thq same period
a
c)
-
ft-l.tl,
the unit-tirnc period is short ennuph so that no compntitive
reaction can interfere
witfi
retain equation (15).
we may
The
this new ccuilihrium,
restrictive and somewhat contra-
dictory chnr-acter of this no n -exhaustive ret of assumptions cxplciin
why vm should turn to dynamic forms.
This does not
imply that we
have to disrecnrd the steady-state aspects w/hen we are about to
make decisions on where to add new outlets. The results in Tahle
III merely
indicate the market dynamics should be taken into ac-
count in estimatinp, the parameters.
Insert Table
Equation (14)
I
I
I
about her3
is
.estimated in two different ways.
we applied the Senuentia]
1
(SUMT)
intrinsically
nonlinear, and was
First, we used TF^DLL.
Secondly,
Unconstrained MinJmizotinn Technique
2
.
rjonlincar programming can bn used for nonlinear estima-
tion in the following way. Let the model be.
r-i
is
i n i
mi
z
:
achieved by solving the nonlinear programming problem belcw
(17)
min
i:J^![
cj(e)
t =
1
,
-
;rt
TV
Tnhl'.'
>quation
(Ih]
usinj;
Table IV and
stiov-.'S
Fi
twe re5u]t;; of the
TROLL and SU^iT.
Figure
2
n;; 1 1
illu?.
ma t
i
tratcs
on of
and
\.
X
c
1
VJith
A
tht:
currRnt number of outlets,
.
ThGSG valuR'5 are very
dieted equilibrium market share
in
d./Cdj
+
ri
c
i
1
],
and assuminr
in share would be
share
hip, h,
is
ri
c
'
one per cent
a
needed to incr-easc outlet
one percent.
5
look
at
the problem
regional basis. For this particular product, we have informa-
tion on four different regions.
from
unit
worth v-jile depends on industry sales volume,
More interesting of course is to
on
increase
remains constant, equilibrium mar-
and the number of new outlets
tjy
A
.
which is to be Rxpectcd
per cent. Whether such an increase
Ket share would increase by
profit,
D^
=
a
high of about
per cent
in
ID
sho'.'vn
Adding outlets in region
It
differ
r ^ c ros s
per cent in one region to
another region.
per outlet added are
in
a
about
low of
Incremental year'ly regional sales
Table
V
for each of the four regions.
contributes thehighest marginal return.
quite possible that the response parameters
is
regions.
gulshed from met rope
The company's outlet share varies
]
i
For example,
ru
r-
a
1
areas might be
d
i
:;
t
In our particular application
t an areas.
i
n
in-
17.
formation by region wos
..
vailable only on an annual basis over
period of six years. This was insufficient for the purpose of
a
es-
timating responsR coefficients by re ion.
f,
Insert Table V about h^re
4.2.
L^O£ist_i c__modRj
Market share as
a
function of relative number of
outlets is often thouc^t of as having an
oil
in
companies,
for example,
S
shaped form. Various
have been able to observe such
curvsHS
S
plots of sales or market share as functions of the number or
1
3
the relative number of outlets
The theoretical arguments in favor of such
ped relationship at the market share
the form of equation
bility level.
Thus,
[15)
at
-
level
-
a
S
sha-
already introduced in
may also hold at the transition proba-
one extreme,
if
the oil
company has too few
filling-stations consumers will notice them too infrequently and
will often be obliged to tank up at other companies stations; es
a
consequence, their loyalty will be very low. As the number of stations increases consumers will be able to tank up at the company
petrol pumps
locatod in various geof.r aphi
c a
1
areas and their loyalt
will be enhanced accordingly. At the other extreme,
continues to extend its distribution network,
tructed wil]
of
i,::"r
to
if
the company
each new station conr
attract consumers from the remaining hard cn:-
competitors customers.
Estimatinc thg pararr.ctors of equation (20)
If
thR
hir.
toricel data come from
shares and markrt
v/ill
raise on
stable marknt,
a
i.e.
isr-ue.
p.
our cutlet
relatively little variatjility.
F^hares'-. hovv
will be severe nulticnllinee'rity probl^uos. As mnritioned in
3,
a
the independent
vr.
first orrinr Taylor
ted at
the
thei-e
sp':;tion
riahles in the linf;ar equation derived
t>/:pansiori
current solution.
first
are
Let
order derivatives evalua-
current solution be
the
frorr.
o
a.
=
1
3m
[
a
.
=
tween
^.t-1
b
^^i.t^^i=^l
.
=
b
1
/3a.
and 3m
An
/
i.t
.
1
state,
i.
./3b,
important issue in nonlinear estimation
din^ good initial values.
used the follov/inj' procedure.
VJe
is
In
fin-
stead
the elasticity of market share with respect to the relative
number of outlets is.
(24)
°i,t
'i.t-1
8rn.
,
1
a.
(22)
(23)
a.
A
.
1
•
m
c
,
t
,
e
t
13.
For thpsp rnasons,
a
curve relating, tran-
slu^peci
S
sition probabilitlDs to the ralative num'npr of retrtil outlets '^poms
-D.
11
(18)
11
/(a. +n.
A.
3
approaches onr, according to
Similarly
b
(19)
with
\
a
-
C
and
b
^
D
defined as
is
X
/
[o
C
CO
*
D
^
positive constant
b
shaped pattern
S
a
ciii
r,
iccc
Market share at
%.t
-'
ff"i,t
I.
i
If D
state market share
(21)
1,
^
i.t.e
a.
tirr^e
=a/(a
is
t
+D
,
°c.t
•
.
t-
.
1
+
^i
^ ^^
-
[a
c
/
(a
remains equal to
v/oulri
).
then
^c
bi
^^°^
are
b^
),
=a./(a. +D.
defined as
pmb abilities
The s^'^itchinp,
.
b.
'
'-\
+
D
c
D.
c.
.
^)
^c
°i.t^-^%
'
^c.t
^
1
and
to
D
.
steady
be,
=a(a.
ci +D.i,t^)/[a.(a
ic +D c,t^}
+
a(a.
ci +D.i.t.)
To estimate
initial values for the parameters, we assumed
a
value
value for m
in
a
=
c
0.62S and
b
?.46.
=
c
With these initio
Divergence
optimal solution.
values TROLL failcid to find an
1
ocf. urrori
initial values which we tried.
for these and for all other
The SUMT
program performed hetter.
The main reason is pro ably that the method for minimizinp, the
ti
unconstrained penalty function
second order procedure,
the Nowton-Rephson method,
is
whereas
f^larouard
1
t
'
'
only uses first order derivatives
.
Furthermore, all our SUMT
"computations were done in double precision.
ly
This may be pai-ticular-
important in view of the factthat there is
rity problem.
The
SUf-IT
tistics ere very poor,
estimates
as
have the correct sign,
all
all,
are> shov;n
in
a
and the magnitudes
real multicollinea-
Table VI. The
expected. Weverthelcss,
the estimated values are
a
estimation method
s
t
sta-
the coefficients
are reasonable.
After
not too different from our initial
subjective estimates.
-t
to
an
Table
actual market share of
sales for adding
a
V,
about
.
Vc'^le
V/II
shows the incremental
now outJet in each of the r8i:;lens.
21
sert Tabl- VI
4.3.
_no_do_l_
£P£lj_f^a.t
les
lop,
ab?ut her
i Oj2
TflblRS
tinl and
I
\1
and VIT
rpspcctively how the
shov-;
istic morials can be applied to compare in crp mental sa-
from invGstmRnt in outlets in various regions.
The ultimate^ measure of performance is the
1
profits rather
son of rofional
may differ from cne
costs,
and cost
of
rep,
tl-ian
sales
ion to another.
"""^^
•
•
^i
16*
ced
by
q
constraint
in
ri"''
•
^
q
cost structure
The
a
i:
trans [lortation
cross regions.
proposed by H-F could
(R]
=
the objertive
k^-Q-^^
'^
t
.
For example,
transition [probability functions
be adjusted for the exponential
section 4.1. We replace
nompari-
5
land will vary
purchasing,
The optimization model
0''^
e>;ponf?n-
/
(d^^
+
y
o-
function by
'
^'[^
]
is
repla-
:
c.t
c
i
,
_j
c
c.t
i,t-1
t
or if cur interest
is
only in steady state results without conceri
for the transient behavior.
J.n-
-aJ.D^
,
I'^th
in
the optimiz'tlon,
''i
the
"i.t
loeistic
model,
complications would
ai-i;:t
because the transition probability functions
77
convex for some ronrc of
are
D.
[or
D
),
c
1
and concave clEewhRrn.
NeverthG]o3S the model will remain useful. Instead
find the optima]
of trying' to
be
5.
allocation over time, the model could
applied to civaluate various outlet expansion plans.
CONCLUSION
In
of
the
ag^eref,
and Fisher.
this paper we have presented
ate retail
V.'e
a
detailed study
outlet location model deve]opcd hy Hartung
found that their E^'^eral procedure was sound, but
that the specifics suffered from
the estimation of
thij
a
variety of problems
rep,
ordinp,
model parameters, and the i-obustness of the
response functions. Rem.edial action was proposed, and was applied
to
a
brand of gasoline in
a
European country.
.^^
c c
C
•rH
Q 3
1
c c
O
•H
JD
1.1
a 3
25
TABLE
III
Steady State Estimation of Exponential rodel
for brand
i
Coefficient Value
-
t
statistic
26.241
3.25
.
700
F(2 >33)
=
3i
DW
=
0.49
26,
TABLE
IV
Exponential Modnl for Brand
i
Coef f Icinnt
F{? .32)
Statistic
59. OG
2. 75
.70
68
2.76
70
69
SUMT
0. 73
7
.
63
37.3
?'3
TABLE
SUM Estimates
VI
for Logistic ModGl
CoRfficient Value
t
statisti
.019
8394
2.663
.061
.
30
FOOTNOTES
Based on discussions with various major oil companies. The
cGnciir;t
linked hierarchical models has hpen examined hy CrL.vJstcn and
of
Scott-Horton
\ 1
]
and
,
has been applied
in
the area of operatior-ril
planning and control by Green FB], New son [15], and Shwimer
A
possible application in the public sector
fias
T'^Bl.
been proposer' hy
Hausman and Naert [6].
Our notation will differ from H-F.
See
tl-F
tation
At
\
7
,
p.
R
234,
footnote ?]. The quote is given in our nn-
.
least within the
limits discussed above.
TROLL is an interactive estimation and simulation p-jckage develop 'ic
by
the Computer Research Center, National Bureau of Economic Re-
search,
For
a
Inc.
For
a
description of the estimation riethod see [3]*
further discussicn on this method,
to Draper and Smith f?,
[5.
'''
pp.
pp.
267 et sq.l
'.-.'o
may refer for
cxa'^'.pl"
and GOIDFELD and QUA^DT
49-57]
Provided that the linearized form (11) of the model is valid
6,
the final estimate of
6.
test of hypothesis conducted hy LAhLilN
^
A
^
For brand a,
^
At
observations
38
v.'
Fill.
ere available.
this stof^c an additional qualification ought to be made about
estimation procedure. The
[l-F
r.
imilarity of the results oritainsd
from regression analysis apnlied on equations
(fi)
and
as
(5)
oppo-
sed to the relative dissimilarity observed betvveon those derived
from equations
tion
[
7
)
(7)
contains
and
(5)
m.
a
,
evidenced by equation
of
and
a
6
should cause no surprise,
stochastic variable, on both sides
(fi)].
fie
from small samples
[as
nee we sfiould expect biased estimates
;
furtherrnorc;
this
in
inccnslstent since the residuals indicate
are
since equa-
a
case
a
and
6
strong autocorrela-
tion among the error terms.
All
in estimatir'g
K.
this
adds up to pointing out the bias created
and k„ via equation
(7).
Notice that no error, term was introduced in equatior
(5],
had
one
be
added the linearization of (5) would have been impos-
sible.
^
For additional
discussion on
Naert and Bultez [14
^
For
[41.
a
li.
nearizinc, such nonlinear models,
1.
theoretical exposition of SUMT, see Fiacco and Mc CormicK
The computer program is described in MylancJer et al.
[13].
set
32
'
^
Many factors making the
S
are reported by Kotler [9,
^
'*
shaped curve plausible in this industry
pp.
See Wilde and Bcightler fl7,
96-971.
pp.
22-24] for
a
discussion of the
Newton-Raphson method. The SUMT routine has various options with
regard to the technique for minimizing the penalty function. One
of
these is the Newton-Raphson method.
^^
Assuming that orofit maximization is the objective.
^^
Different types of regions, e.g. metropolitan, suburban, rural,
have dif f ei
by regions
m.igh
.
33.
REFFRENCFS
Crowston, W.O., and r.S.
[1]
S co 1 1
-
P^nr
Decision S^'stnms", £r ncer;c1inp;B of
Fall
of- TIP'S.
John
[31
N.R.,
nraper,
\2]
V'.h"
1
ey
an
7
Smith,
and H.
ri
t
n
"Tdc
,
11th
Dor,if',n
of
n t rrna1
1
Hi
o
rarch,
i
ca
ona
1
M pr
Repression Analyair.
.
New York
he
I
1
t i n£_
.
Sons
and P.
M,,
Eisnoi',
as
19
to
1
,
9 B G
/^'PP^J-Si'
.
Pindyck,
Generalized Approach to Estimation
"A
Implemented in the TROLL/I System", Working Paper, National
Bureau of Economic Research, Computer Research Center for Economics and Panapement Science,
[4
Fiacco,
1
ti a
1
A
.
V
.
llnconst
and G.P.
,
r
aineri
P^in
P-'c
PI
arch 1972.
CormicK, Pionlinear Programmin g
imization Techniques
.
Mew York
:
:
Soo uon
-
John Wiley
_
and Sons
[51
,
-
Ho
1 1
Plethoris
and Publishing Company,
i
n
Econom etrics
1972.
"heuristic Coupling of Aggregate and Detailed riodels
Factory Scheduling",
School
1
PJort h
:
Green. R.S.,
in
[7
93G
Goldfeld, S.P1., and R.E. Ouandt, PJonlinea r
Amsterdam
[PI
1
of r.anagement,
Unpublished Ph.
Pi.I.T.,
Hartung, P.W.. and J.L.
P.
Dissertation, Sloan
1971.
Fisher.
"Brand Switching and
r at h em
Programming in Plarket Expansion". Management Scien ce, vol.
gust 1965,
pp.
15-^9.
t t i
r.
2,
Au
j
34.
[
e]
Hausman, W.H., and
P. A.
"Optimal RuGOurce AllocaMon in
WaGrt.
Family Planning ScrvicGS DelivRry", Unpublished Working Paper,
Alfred
[9l
Kotlnr Ph.,
New York
Fin]
Lambin,
Uni ve rs
[11T
Sloan School of ManaRGment, F^.I.T., August 1972.
P.
arKe t
i
n
;;.
Decision Making
Holt. Rinchart and Wistnn,
:
J.J., r^odeles et Pro Rramns
i t
ai res
Lambin, J.J.,
of Business
[I2l
fl
,
de
"Is
vol.
Marquardt, D.,
s
rie
1971.
Harketinp.
nylander,
45,
n°
4,
sum -Version
4
11,
R.L.
:
Presses
:
n°
T
he
Jo urnal
October 1972.
:
W.C.
Paris
Gasoline Advertising Justified ?",
linear Parameters", Jou rnal of the Society
[131
,
France, 1970.
"An Algorithm for Le as t -Sq are
Mathemat ics, vol.
Bu il dinp, A pproach,
riod el
;_ A
of
Estimation of
s
In dur, trial
_5
iJon-
n d _A_p p l_i n
2, June 1963, pp. 431-4 41.
Holmes,
and G.P.
Mc Cormick.
"A
Guid^-
to
The Computer Program Implementing the Sc:qucntial
Unconstrained Minimization Technique for Nonlinear Programming",
Paper RAC-P-63, Research Analysis Corporation, October 1971.
[141
Naert.
P. A.,
and A.V.
Bultez,
"Logically Consistent r^arket Share
Models", Working Paper n° 607-77, Alfred
gement, M.I.T., June 1972
(forthcoming)
.
)
P.
also in Journa
l
Sloan School of Manao f _^
Ma rket ing Research
[15]
Newson, E.F.P.,
blished Ph.
n
[iFl
.
I
.
T
.
,
19
Shwimpr,
linp,
in
7
D.
Dissertation, Alfred
"Interaction Between
Job Shop",
P.
Wilde. D.J.,
A
p, p.
Unpublished Ph.
Sloan School of nanapomen t
ri7l
to
Finite Capacity", UnpuSloan School of ranap;er:ent
1.
J.,
a
"Lot Size Scheriulinp
and T.S.
Englcwood Cliffs, N.J.
,
f'l
.
I
.
T
.
,
rep ate and Hctaileri ScheduD.
Dissertation, Alfred
1'J72.
Beiphtler, Foundationr, of Optimization,
:
Prentice Hall,
19B7.
P.
mmm
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