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An extended algorithm for optimal product positioning
An extended algorithm for optimal product positioning Similar applications arise in the political sciences. Political parties are confrontated with the problem to present the ideal candidate for the office of prime minister, president or chancellor. The candidates have to be positioned into the space of personal attributes so as to attract the majority of voters. Formally, the product positioning problem requires the following data: J: a set of mutually independent attributes which are considered as potentially relevant to constitute brand preferences; the attribute space is RF, where f" gives the number of elements (cardinality) in J, eta/: an estimate of the kth (k E K) customer's perception of the existing brands i E I, expressed by the coordinate value of the jth ( / E J) coordinate, ck/: an estimate of the kth (k E h0 customer's perception of an ideal product as expressed by the coordinate value of the/th ( / E Jr) coordinate, s~: a salience, measuring the relative importance of each attribute / E J to the customer k ~. K, where K: a set of customers, I : a set of existing brands. The input data can be provided by the application of various multidimensional sealing techniques (MDS) to customer product evaluations [8]. If we have achieved a valid mapping of the ~rceptions in the attribute space by such techniques, we need some rule by which customers effect product choice. One rule is given by Zeleny [ 12]. The alternative closest to the ideal product perception is preferred to all other alternatives in the market. This normatic result from axioms of rational behaviour has been observed in a number of empirical investigations. A considerable problem, however, is how to measure closeness. We choose, to solve it by use of a weighted Minkowski-metric
SOnke A L B E R S
Institut 15" Betriebswirtschaflslehreder Chrisffan-AlbrechtsUniyer~'tatKiel, Olshausenstrane40-60,19-2300 Kiel, FederalRepublic of Germany Received February 1978 Revised July 1978 Optimal product p~itioning in an attribute space has been formulated according to the axiom of choice as a mixed integer nonlinear programming problem. To solve it, Albers and Brockhoff have designed the special purpose algorithm PROPOSAS. It works under simplified c.ssumptions: Euclidean metric, equally weighted dimensions of the attribute space and equal sales per customer. The following article shows that the basic ideas of PROI~3SAS are flexible enough to be expanded to cover a weighted Minkowski-metrie as well as different revenues from the customers. Furthermore, the calculation of a new upper bound is deserilx d which reduces CPU-time considerably. 1. The problem Optimal product positioning means to determine the levels of the attributes of a new product to be introduced into a market such as to meet best certain objectives as specified by a firm. The attributes could in principle be considered as objective or as subjective characteristics of certain products [ 10, p. 923]. The product positioning problem does not really occur if all customers evaluate all product attributes by the rationale: the more (less) the better 0.e. the juicier, the better). In the first ease, the direction of product development efforts is obvious: the firm shoukl develop new products that provide more of at least one of the attributes without providing the ct, stomers with less of the other attributes. Rather, the product positioning problem occurs where customers recognize ideal attribute levels [9] (i.e. for the sweetness of wine one would not like to say: the more (less) the better). Positive and negative deviations from the ideal levels are evaluated less favourably. For such problems one would like to evaluate product co:~cepts such that latent niches in a mzrket could be discovered. These provide the potential position of a new product.
di~=l[~skjlctq-eta/lm]llml,
iELkEK.
(1)
It provides a wide range of geometric measures of closeness. Note that m = 1 gives the city-bloc-metric and m = 2 gives the Euclidean metric. The explanatory
© North-Holland Publishing Company European Journal of Operational Research 3 (1979) 222-231. 222
,%Albers / An extended algorithm for optimal product positioning
power of both metrices has been demonstrated in various papers [11 ]. The usefulness of attribute saliences has been advocated by Shocker and Srinivasan [10] and Zeleny [12]. Coming back to the axiom of choice [ 12], the kth (k E K) customer is assumed to buy the product with minimal distance to his ideal product as mea-, sured by the weighted Minkowski-metric. Be y/: ~ e coordinate value for the location of the new product on / E J, the customer chooses the new product, if I[~ stjlca¢-Y/lm]llml
(2)
kEK,
/EJ
where dtc =min{dik: i E 1 } ,
(3)
kEK.
is the critical distance. By introducing the following binary variables: 1
if (1) holds,
0
if (1) does not hold,
x~ =
k~K
,
(4)
the product positioning problem may be formulated as the following optimization problem [7]: r k
kEK
•
X k =~ max!
2. Weighted Minkowski-metrices
sk/(Ick/- Y/I)mlVml -
kEK
dk < (1 - Xk)" M ,
(6)
,
xtcE{0,1},
with ctq q •J) as its center and dk as its radius if it is to be bought. Maximizing the objective function (5) is equivalent to search for a point that lies within the common intersection of a maximum number of spheres. As the property of pairwise nonempty intersections within a set of spheres (which can be checked easily) represents a necessary condition for a common point within that set of spheres, the problem is decomposable into: - the implicit enumeration procedure ENUSOS, that generates explicitely all sets of spheres • which fulfill the necessary condition for a common point, • and which would probably lead to a solution better than the best one found during the preciding computations - the procedure INTSEA, that determines a common point (if one exists) for a certain set of spheres. The algorithm PROPOSAS has been criticized for the fact that it was based on the restricting assumption (a) through (c). In the present p a ~ r we show how the basic ideas of PROPOSAS can be extended to overcome the critique.
(5)
subject to: I[ ~ icy
223
kEK,
(7)
where M is a finite upper bound on the left-hand-side of(6). This mixed integer nonlinear programming problem has real variables y / ( / E J), and binary variables x k (k E K). It can be solved for the special case (a) st# = 1 (equally weighted dimensions ] E J of the attribute space for each customer (k E K)), (b) m = 2 (Euclidean metric), (c) rt~ " 1 (equally weighted revenues from each customer k E K), by the algorithm PROPOSAS (see [21). Basic to the algorithm is a geometric interpretation of the problem. According to (1), a customer k E K chooses the new product if the distance of the position of the new product to his ideal product perception is less than the critical distance dk. Thus, the position of the new product must be within a sphere
Let us call Pk the set of all positions of the new product that will cause a customer k E K to buy the new product. Using the unweighted Euclidean metric (ski = 1 (k E K, ] E J), m = 2) Pk geometrically is a sphere (in the plane a circle, see Fig. I(A)). If we allow for different weights of the attributes, we have to introduce the weighted Euclidean metric. Then the set Pk becomes a solid that looks like an ellipsoid (see Fig. I (B)). Furthermore, if we introduce different exponents m ~ 2, so that we arrive at the weighted Minkowskimetric, the shape of the set Pk alters with respect to m. Using the following exponents, we get the following solids in the plane: Exponent
Solid reflects a:
m=l m=2 m-4 m .., oo
rhombus (see Fig. I(C)) ellipsoid (see Fig. I(B)) television screen (see Fig. I(D)) rectangle
224
$. Albert / An extouted alSorithm /or optimal ts,oduet ~ltiontng
CB)
8 t.
0 ,,,,,e
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N
.Ill
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.U
.fill
.60
.70
.S.
.S
i
i
i
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I
I
I
I
I
I.S. l
O .o e
e
i
f
.o
*4
te
J
.,=
g
(c)
(D)
|
Fig. I ( A ) - ( D ) .
The consequences of using different metrices to the product positioning problem are important [6] as can be realized from Figs. I(B)-(D). Further psychological research seems necessary to support the proper choice of the exponent m. In ~ e algorithmic treatment, the changes of the geometric properties of Pk lead to modifications of the optimization procedure concerning the computation of painvise iatersections and the determination
of a common point for a set of solids. (a) Computing the matrix of pairwise intersections. There does not exist a sufficient criterion that would show pairwise intersections of solids as it is the case when considering spheres. Therefore, we first have to apply necessary criteria for indicating pairwise intersections definitely. These are given
225
.~. Albers / An extended algorithm for optimal product positioning
by the following properties: (¢~) Two solids k and I intersect eachother if the distance between their centers using the maximum salienees in each dimension ( / E J) is less than the sum of their critical distances de and dr, i.e. if
I[~
max{stq,
st/}"
(IcA7-
with respect to the computation of the weighted center of gravity. The modified procedure INTSEA II is given by:
Step 1: Initialize weights. Q is the set of solids for which a common point is to be determined. Let
ctil)ml llml < d e + dl ,
1
k, I E K .
(8)
~ ) Two solids k and I do not intersect eachother if the distance between their centers using the minimum saliences in each dimension ] E J is greater than the sum of their critical distances de and dr, i.e. if
(9)
It may occur that neither (8) nor (9) are full'died as we have only considered necessary conditions. If this cases occurs we have to apply the procedure INTSEA to search for a common point. Such a point would indicate that the two solids must intersect eachother. If a common point has not been found we can conelude that the two solids cannot intersect. Thus, the matrix A = (ae I e2) used in ENUSOS !I is defined as follows: 1 if solids k l and k 2 fulf'di (8) or if INTSEA provides a common point for solids k I and k2, kl, k 2 E K , aele2 =
--"
"--ma~
[ 1 F i ~
0 if solids k, and k2 fulf'dl (9) or if INT~E"/~ does not provide a common point for solids kt and k2, kl, k2 E K.
This matrix A serves as a starting point for the enumeration procedure ENUSOS II, which is otherwise equivalent to the one described in Albers and Brockhoff [21.
kEQ
It
•
Step 2: Compute the weighted center of gravity with coordinates Y/by applying the saliences s;,i and the weights we:
-
~e~ocej_" (se/) '/m_'wk
Y/=
I [ ~ min{s~7, st/} • (Ictq - Cti!)m]l/rnl > de + dr, k, 1E K .
Wk de
~k,EQ (SRj) l/m" ~/¢
'
j EJ.
Step 3: Find the set of solids T C Q such that its elements have the point y/(from Step 2) in common: Dk=l[~Sei(lcei-yil)m]l/ml,
]EJ
kEQ,
T = { k E Q : Dk < d k } . Let/" be the cardinality of T. Step 4: Test whether Step 3 improves the current solution: (S'is the maximum number of solids with a common point found so far). If/'>~
storeZ=T,'~='f, y j = ~ i ,
jEJ.
Step 5: Test of termination (if met, go back to Step 8 of ENUSOS il, otherwise go to Step 6) if T = Q, the common point isYl (/E J). Then terminate. If no convergence occurs (see Albers and Brockhoff [2]) or convergence conflicts with control parameters (i.e., weak convergence, precision), terminate. Step 6: Compute improved weights: De wk • = We • de ,
kEQ.
(b) Determining the common point in INTSEA.
Continue with Step 2.
The common point of a set of solids can be found by determining a weighted center of gravity of the centers of the solids. Then, the problem is to find the correct weights. The weights will be adjusted in an iterative process according to the fact whether the present center of gravity lies without or within the respective solids. As compared with the original procedure the search procedure has to be modified only
It is found empirically that only very few iterations of this type are required in most problems•
3. Improved criterion for separation The implicit enumeration scheme for sets of solids (spheres) ENUSOS described in Albers and
$. Albers ~An extended algorithm for optimal product positioning
226
Brockhoff [21 uses upper bounds/3k on the number of solids that may have a common point within solid k. These bounds serve as a criterion for separating partial solutions (see Step 6 in ENUSOS). The process of separation can be improved by computing stronger bounds. Lct ~ be the lower bound, i.e. the maximum number of solids with a point in common found so far, and let bgt be the number of solids (k and I included) that intersect with both solids k and L Then ~I, has been computed in ENUSOS by the number of elements bla with bla > ~ (1 ~ K). The underlying idea of the new upper bound/3k is the property that we must have a number of elements bla with bti i>/3k which often cugs deeper. The following example may illustrate the new upper bounding technique: Let us consider 12 solids and for k = 5 the number of solids bst (k = 5 and I included) that intersect with both solids k = 5 and I. They are given in the following table: l
1 2 3 4 5 6 7 8 9 10 11
bst
- 5 7 -9
-7
-8
7
4
12
7
Let ~ -" 3, then, using the old upper bounding technique we compute the number of solids with bst > 3 and get/3k = 8. Applying the new technique we first have to compute the values X~ giving the number of elements b~t with bkt >- i. By that we arrive at the following vector: i
456789
~,si
8
7
6
6
2
1
Now we can conclude that the upper bound can only be ~k = 6 because the value 6 gives the maximum number of solids l E K for which the number of solids intersecting with both solids k and I are greater or equal than #k = 6. Using the improved bounds/~k we separate the n,ost promising solids. By that we will fmd good solutions faster, which in turn, cuts computational time. Applying this new upper bounding technique we only have to modif) Step 4 of ENUSOS by the follovdng operations: Modified Step 4: (Test for termination.) Let K = S be the set of ~ solids (customers), k = IKI the number of elements in K , and G(k) the set of integer constants from 1 to k.
Compute the upper bounds ~k (k C K) on the number of solids that intersect with soli6 k by the following operation: Aia = I{bja t> i: 1 E K}[,
k E K, i E G(k),
/3k = max{ i E G(k): Aia I> i},
s* = K k E K : & ~ +
k E K.
1}1,
If s* g~', then go to Step 10,go to Step 5 otherwise.
4. Different revenues from customers Two arguments, namely: a customer would accept different price-categodes, - a customer would buy different quantities of the new product should be recognized in the positioning problem. Thus, the decision-maker is concerned with different revenues from the customers. Consequently we do not search for a maximum number of solids with a common intersection but for a common point within a set of solids the elements of which lead to a maximum revenue. To cover this extension we have to modify the implicit enumeration scheme ENUSOS II with respect to the calculation of bounds. The computation of upper bounds on the maximum revenue when customer k buys the new product is based on the calculation of bounds on the maximum number of solids that may have a point in common within solid k as described in Section 3. If the latter bound /3~ is determined one can calculate for every integer 1 ~ i ~ ~ the upper bounds 7 ~ which are the sum of the i maximal revenues from the set of customers 1E K with bta >- i. Then the maximum of 7ta, i
$. AIbers / An extended algorithm for optimal product positioning
1
I 1
2
3
4
5
6
7
8
9
10
11
12
-
5
7
-
9
-
7
-
8
7
4
7
0.9
2.2
0.8
2.4
3.3
0.6
2.2
1.1
1.0
1.9
2.1
0.7
,,
bst rt
I
Using the upper bounding technique from Section 3 we calculate the upper bound at = 6. In a second step we can determine the values 3/~i which give the sum of the i maximal revenues rt (! E K) from the set of customers with bla/> i. i
14
5
6
10.6
9.9
I ............
I
Tsi] 9.8
ber of solids which may have a common point within the solids kl and/62. Step 3 (Corrections of the matrix A): Compute the upper bounds #g (k E K) on the number of solids that intersect with solid k by the following operations (see Section 4.): ~,~/= I{btt I> i: l E K}I,
Then, the maximum value 7~a- (1 g i < / / t ) gives the upper bound u~ on the revenue the firm would achieve if it positioned the new product within solid k = 5. In our example the upper bound u~ takes the value 10.6. As the improvement of the current solution cannot be tested any longer via the number of solids for which a common point has been found, Step 4 in INTSEA should be deleted. Now, the test of improvement is implemented directly in ENUSOS. Thus, the new procedure ENUSOS Ill can be described in detail by the following steps.
k ~ K, i ~ G(k),
/3k = max{/E G(k): kta i> i}, where k = IKI: the number of elements in K and G(k): the set of integer constants from 1 to k. Compute the upper bounds Uklk2e (k v k2 E K) on the revenue that can be achieved from the customers if the new product would be placed within the solids kl audk2. IfKt is defined as a subset of K with exactly I elements (IKtl =/) that fulfill the condition #k I> l (k E/(t), then we get
Step 1 (Relaxation: A set of solids that intersect eachother pairwise is considered as a relaxation for a set of solids with a nonempty intersection): Determine A = (atd,2) from
kit2
227
max{ ~ ~ aklk3 "ak2ka'Sk3 } k3~K !
e
if subsets/ft exist, k 1, k2, 1 ~ K,
_~_
if the solids kt and k2 fulfill (8) or if INTSEA provides a common point for solids kl and k2, kt, k2 E K,
~klk21
if the solids kl and k2 fulfill (9) or if INTSEA does not provide a common point for solids kl and k2, kt, k2 E K.
The elements 7klk21e give the maximum sum of ! different revenues Ska from a set of customers k3 ~ K whose corresponding solids intersect with solids k z and k2 and with at least 1 - 2 other solids.
The matrix is symmetric around the diagonal, and a~lt 2 = 1 for kt = k2. Its elements denote a nonempty intersection of solids kl and k2 ifat, lk 2 = 1. Step 2 (Supporting calculation of bounds): Compute B = (btd,2) from: l ~ r a t d " at2t
ifatqk2 = 1 , k 1, k 2 E K ,
0
ffatqt2 = 0 ,
btlk2 =
kvk2EK.
The elements of B give an upper bound on the num-
0
if no subsets/~t exist or ifaklk 2 = 0, k v k2, I E K . '
e =m e Utqk2 ax{Ttlk2 t.. I E K , / ~>min{/~kt, ~k2}},
kv k2 E K . Compute the upper bounds u~t (kl ~ K) on the revenue that can be achieved from the customers if the new product would be Jlaced within: the solid k l. If the elements "/klk2t aJe in decreasing order, then 3'lit should give the value c f the lth element in this sequence. uklr =max{T~:lt'lEK, l<~:~ I},
klEK.
22~
$. a~,en ~an e x t ~
d s o a t ~ for opamat~
Correct the matrix A by:
~ao,~
Step 7 (Fathoming):
attt2 = 0
if uetlt2 ~ I z ,
~J, k2 ~ K ,
atlts=atstl=O
ff ur~1 ~ 1 z ,
kt, ks E K ,
k~Q
where I z is the maximal revenue Oower bound) that has been found in a preceding computation (naturally, / : = 0 at the start). This step excludes every nonempty intersection of solids kt and k2 with a common point that could only be within a set of solids the elements of which give a sum of revenues less than or equal to the lower bound. Go back to Step 2, if any such corrections are performed, otherwise go to Step 4. Step 4 (Test for termination): Compute the upper bound s* on the number of solids that may have a point in common: w
hi = I{/Ik t> i: k E K}I,
i E G(k),
$* = max{/E G(k): hi ~>i}. Compute ff~.eupper bound us on the maximal revenue: If the elements 7~d are in decreasing order then the elements ~ should give the value of the lth element in this sequence.
u s = max{~/: I E K , l ~ s * } .
with
V=S\(QuR). Let ft be the set of solids that may have a point in common:
f~=Quv
w i t h ~=lfZl.
Now the upper bound $* can be computed similar to Step 4.
hi = I{~k i> i: k ~ n}l,
i ~ G0,0,
$* = max{/E G(w): hi t> i}. Compute the upper bound uS on the maximum revenue: If the elements 7~d are in decreasing order for kl E K, then the elements ~ should give the value of the lth element in this sequence. u s' = max{'ffl: I E K , I ~ s * } . If u s ~ I z, then go to Step 9, otherwise continue. If V ~ {~}, then go to Step 6, otherwise go to Step 8. Step 8: Apply INTSEA to determine a common point for the solids q E Q, if such a point exists. Call ~ the set of solids for which a common point has been found.
Ifu s ~ Is, then $o to Step 10, otherwise go to Step 5. Step 5 (Initia~ation): Q = ~: set of soEds that intersect eachother pairwise and have been separated in a preceding computation, R - ~: set of solids the elements of which do not intersect with at least one solid in Q, V = K: set of solids that have not yet been inspected for inclusion in Q or R, Z = ~: set of solids for which a common point has been found that has been a maximum sum of revenues. Step 6 (Separation). Find a solid ~which is the element of V with the minimal index, and which has a maximum value for its upper bound u~.
If ~ >I I z, then store the best solution found so far: I z ffiF;Z =Q;Yi- Y ] ; / E J, and go to Step 3, otherwise continue with Step 9. S:ep 9 (Separation): If ~ = 0, then go to Step 10. I.."~ = 1, then r* is the only element in Q, ar*k = akr* = 0 (k E K), and go to Step 3. If ~ >t 2, then recompute:
T= min{v E IF: Uvr = max{u~: k C ,']}
R :=R \{r~R: Z) a k , = ~ } u { / }
Q: = Q u{~},
v=S\(QuR)
r: = v \ {~}.
If ~ is the number of Q we have to set ~: = ~ + 1.
k¢Q
r*={kEQ: ur~=min{~ . I E Q } } , Q:=Q\{r*};
q:=~k~Q
and go to Step 6.
1
S. ADen ~An extended algorithm for optimal product positioning
Step 10 (Termination): The solids with the maximum revenue that have a point in common are in Z as stored in Step 8, the coordinates of this point are y~ (l"E J).
229
perceptions of 4 existing brands (eiF symbol O) in a two dimensional attribute space. For purpose of demonstration, the perceptions are assumed identical for all customers. We are looking at evaluations of 10 customers each of them perceiving an ideal product denoted by the symbol +. The different revenues (rT,) the firm could realize from the customers are also listed in Fig. 2. According to the different weights that the customers give to the attributes, curves of constant distance between ideal and real product
5. Numerical results To illustrate the procedure let us consider the following example (see Fig. 2). Take as givens the
+2
- t,liO -,91
-,Oe
-,?l
-.60
-,51" -,'qL'~c,3l
-.2ll
6g
i
".i'4
'*9
"6
I
"17
I )
.70
.Sil
.ge
+8
"10
g II ..=i-
Fig. 2. r I = 0.57; r2 = 0.67; r 3 = 0.78; r4 = 1.13; r 5 = 0.99; r6 - 0.97; r7 = 0.98; r8 = 1.12; r9 = 1.30; rlO = 1.31.
1.00
230
s. ~
Table 1 Computational e x ~ No.
]-
m
k
/ An ~ t ~
1
4
4 4
s(A)
4
4
5
4
6 7
4 4
8
4
9
4
10 11 12
4 4 4
No.: .w-
I: 6(A): liB:
O1"1.: SEP: IND:
De?:
20 40 60 80 100
for opt~mt ~
~ o ~
with I'ROPOSASm UB
01'1'
~
2 3
~oat~
CPU-uime
SEI'
IND
DCP
insec
16.5 11.1 14.4 8.1 8.2
4.02 5.43 8.26 8.52 . 7.94
4.02 4.57 7.95 8.17 6.57
40
6.0
4.21
4.21
3.6
40 40 40 40 40 40
20.5 49.6 68.8 80.0 90.7 99.0
8.50 15.68 20.88 25.28 34.44 37.34
8.20 15.68 19.67 24.33 34.14 37.34
12.5 19.8 122.2 135.0 151.4 9.0
1.7.
7
3
2
8.3 21.5 46.2 62.2
50 108 390 1265
11 51 40 66
9 10 40 63
4
9
75 56 1096 572 952 37
29 37 29 29 12
1 8 3
55 33 67
4
1
number of problem; number of the relevant attributes comtituth~g brand preferences; number of customers; density of the matrix A, percentage of pairwise intersectiom; upper bound on the revenae cal~lated from the matrix (A) of palrwi~ intersections; optimal value of the revenue; number of separations (Step 6); number of runs through INTSEA to indicate a pairwise intersection; number of runs through INTSEA to search for a common point.
perceptions aclueve the form of ellipsoids as in Fig. 2. The efiipsoids encompass the set of those po,,itions for a new product which will cause the corre.~ponding customer to choose the new product. In Fig. 2 one discovers two major niches for positioning a new product. They are given within the intersections of the sets of e~.lipsoids { 1, 2, 3, 6}, and {6, 7, 10}. These niches correspond to local maxima if the problem would have been formulated continuously. The principles of relaxation, separation and fathoming are combined in PROPOSAS in an efficient way so that the niche {6, 7, 10} with the highest revenue is discovered immediately. It shows the optimal position for the new product by y l = 0.283 andy2 = -0.179. Furthermore, the algorithm has been tested on various problems. To enable an evaluation of the algorithm with respect to CPU-time, a set of artifically generated problems has been solved (see Table 1). Problems (1) through (5) show that problems up to 100 customers can be solved within reasonable time if the density 6 (A) is within the range of up to 20%. Density measures the percentage of pairwise. intersections or the relative numbers of non zeroelements in the matrix A.
Problems (6) through (12) show a significant influence of 8 (A) on the CPU-time. Problems with a sparse or dense matrix A require less CPU-time than problems with a density of 40-80%. This is because the number of partial solutions that have to be separated can be approximated by a function of the type (~'~k) ) which has its maximum at ~(A) = ~k-As the procedures providing the input data are often restricted with respect to the number of customers and as a market share of more than 20% is unlikely for most applications we can conclude that PROPOSAS is applicable to the majority of practical problems. 6. Some further extensions The optimal positioning procedur,~ PROPOSAS has been extended to cover three more cases: (1) Customers may have different saliences with resl~Ct to the direction of deviation between real and ideal product perceptions, that means, the saliences are different due to the fact whether the coordinate value of the position of the new product is less or greater than the coordinate value of the ideal product position.
S. Albers ~An extended algorithmfor optimalproduct positioning
(2) In many cases the firm under consideration will operate on the market with multiple products. Then, we have to take into account that the introduction of a new product may decrease that sales of products that are already offered by the f'Lrm.The consideration of that aspects has been given in Albers and Brockhoff [3]. (3) In some cases one may be interested to introduce more than one product. As a consequence, the
optimal positions have to be deteanined simultaneously. An algorithm for this problem is given in
Albers [11. Until now, dynamic aspects and uncertainty of data as considered in Brockhoff [5], and costs of positioning as asked for in Shocker and Srinivasan [10] and Bachem and Simon [4] are not considered in the positioning procedure. The algorithm developed here, however, seems to be flexible enough to cover these aspects without loosing its computational solvability.
References [ I ] S. Albers, A mixed integer nonlinear programming procedure for simultaneously locating multiple products in an attribute space, in: R. Henn et al., eds., Methods of Operations Research, Vol. 26, (Meisenheim am Glan, 1977) 899-909.
231
[2] S. Albers and K. Brockhoff, A procedure for new product positioning in an attn'bute space, Europ. J. Operational Res. 1 (1977) 230-238. [3] S. Albers and K. Brockhoff, A comparison of two approches to optimal positioning of new products, Working Paper 77-14, European Institute for Advanced Studies in Management, Bruxelles (1977). [4] A. Bachem and H. Simon, Optimal product positioning in an attribute space, Working Paper, University of Bonn (1977). [5] K. Brockhoff, Zur optimalen mehrperiodigen Produktpositionierung, Z. Betfiebswirtschaftliche Forschung, 30 (1978) 257-265. [6] K. Brockhoff, Alternative approaches for optimal product positioning, in: A. Shocker, ed., Proceedings of the AMA/MSI/Pitt Workshop on Analytic Approaches to Product and Market Planning (Marketing Scien~ Inst. Cambridge. M~, 1978). [7] K. Brockhoff and H. Rehder, Analytische Planung yon Produkten im Raam tier Produkteigenschaften, in: E. Topritzhofer, ed., Marketing (Wiesbaden, 1778). [81 P.E. Green and V.R. Rao, Applied Multidimensional Scaling (New York, 1972). [91 D.R. Lehmann, Television show preference: Application of a choice model, J. Marketing Res. 8 (1971) 47-55. [101 A.D. Shocker and V. Srinivasan, A consumer-based methodology for the identification of new product ideas, Management Sci. 20 (1974) 921-937. [ l l ] W.L. Wilkie and E.A. Pessemier, Issues in marketing's use of multi-attribute attitude models, J. Marketing Res. 10 (1973) 428-441. [12] M. Zeleny, The attribute-dynamic attitude model (ADAM), Management Sci. 23 (1976) 12-26.
SOnke A L B E R S
Institut 15" Betriebswirtschaflslehreder Chrisffan-AlbrechtsUniyer~'tatKiel, Olshausenstrane40-60,19-2300 Kiel, FederalRepublic of Germany Received February 1978 Revised July 1978 Optimal product p~itioning in an attribute space has been formulated according to the axiom of choice as a mixed integer nonlinear programming problem. To solve it, Albers and Brockhoff have designed the special purpose algorithm PROPOSAS. It works under simplified c.ssumptions: Euclidean metric, equally weighted dimensions of the attribute space and equal sales per customer. The following article shows that the basic ideas of PROI~3SAS are flexible enough to be expanded to cover a weighted Minkowski-metrie as well as different revenues from the customers. Furthermore, the calculation of a new upper bound is deserilx d which reduces CPU-time considerably. 1. The problem Optimal product positioning means to determine the levels of the attributes of a new product to be introduced into a market such as to meet best certain objectives as specified by a firm. The attributes could in principle be considered as objective or as subjective characteristics of certain products [ 10, p. 923]. The product positioning problem does not really occur if all customers evaluate all product attributes by the rationale: the more (less) the better 0.e. the juicier, the better). In the first ease, the direction of product development efforts is obvious: the firm shoukl develop new products that provide more of at least one of the attributes without providing the ct, stomers with less of the other attributes. Rather, the product positioning problem occurs where customers recognize ideal attribute levels [9] (i.e. for the sweetness of wine one would not like to say: the more (less) the better). Positive and negative deviations from the ideal levels are evaluated less favourably. For such problems one would like to evaluate product co:~cepts such that latent niches in a mzrket could be discovered. These provide the potential position of a new product.
di~=l[~skjlctq-eta/lm]llml,
iELkEK.
(1)
It provides a wide range of geometric measures of closeness. Note that m = 1 gives the city-bloc-metric and m = 2 gives the Euclidean metric. The explanatory
© North-Holland Publishing Company European Journal of Operational Research 3 (1979) 222-231. 222
,%Albers / An extended algorithm for optimal product positioning
power of both metrices has been demonstrated in various papers [11 ]. The usefulness of attribute saliences has been advocated by Shocker and Srinivasan [10] and Zeleny [12]. Coming back to the axiom of choice [ 12], the kth (k E K) customer is assumed to buy the product with minimal distance to his ideal product as mea-, sured by the weighted Minkowski-metric. Be y/: ~ e coordinate value for the location of the new product on / E J, the customer chooses the new product, if I[~ stjlca¢-Y/lm]llml
(2)
kEK,
/EJ
where dtc =min{dik: i E 1 } ,
(3)
kEK.
is the critical distance. By introducing the following binary variables: 1
if (1) holds,
0
if (1) does not hold,
x~ =
k~K
,
(4)
the product positioning problem may be formulated as the following optimization problem [7]: r k
kEK
•
X k =~ max!
2. Weighted Minkowski-metrices
sk/(Ick/- Y/I)mlVml -
kEK
dk < (1 - Xk)" M ,
(6)
,
xtcE{0,1},
with ctq q •J) as its center and dk as its radius if it is to be bought. Maximizing the objective function (5) is equivalent to search for a point that lies within the common intersection of a maximum number of spheres. As the property of pairwise nonempty intersections within a set of spheres (which can be checked easily) represents a necessary condition for a common point within that set of spheres, the problem is decomposable into: - the implicit enumeration procedure ENUSOS, that generates explicitely all sets of spheres • which fulfill the necessary condition for a common point, • and which would probably lead to a solution better than the best one found during the preciding computations - the procedure INTSEA, that determines a common point (if one exists) for a certain set of spheres. The algorithm PROPOSAS has been criticized for the fact that it was based on the restricting assumption (a) through (c). In the present p a ~ r we show how the basic ideas of PROPOSAS can be extended to overcome the critique.
(5)
subject to: I[ ~ icy
223
kEK,
(7)
where M is a finite upper bound on the left-hand-side of(6). This mixed integer nonlinear programming problem has real variables y / ( / E J), and binary variables x k (k E K). It can be solved for the special case (a) st# = 1 (equally weighted dimensions ] E J of the attribute space for each customer (k E K)), (b) m = 2 (Euclidean metric), (c) rt~ " 1 (equally weighted revenues from each customer k E K), by the algorithm PROPOSAS (see [21). Basic to the algorithm is a geometric interpretation of the problem. According to (1), a customer k E K chooses the new product if the distance of the position of the new product to his ideal product perception is less than the critical distance dk. Thus, the position of the new product must be within a sphere
Let us call Pk the set of all positions of the new product that will cause a customer k E K to buy the new product. Using the unweighted Euclidean metric (ski = 1 (k E K, ] E J), m = 2) Pk geometrically is a sphere (in the plane a circle, see Fig. I(A)). If we allow for different weights of the attributes, we have to introduce the weighted Euclidean metric. Then the set Pk becomes a solid that looks like an ellipsoid (see Fig. I (B)). Furthermore, if we introduce different exponents m ~ 2, so that we arrive at the weighted Minkowskimetric, the shape of the set Pk alters with respect to m. Using the following exponents, we get the following solids in the plane: Exponent
Solid reflects a:
m=l m=2 m-4 m .., oo
rhombus (see Fig. I(C)) ellipsoid (see Fig. I(B)) television screen (see Fig. I(D)) rectangle
224
$. Albert / An extouted alSorithm /or optimal ts,oduet ~ltiontng
CB)
8 t.
0 ,,,,,e
-~.s. -is. -:s. -:?. -is. -:5. -:u -is. "iz' "il°!"
N
.Ill
.2(I
.3#
.U
.fill
.60
.70
.S.
.S
i
i
i
I
I
I
I
I
I
I.S. l
O .o e
e
i
f
.o
*4
te
J
.,=
g
(c)
(D)
|
Fig. I ( A ) - ( D ) .
The consequences of using different metrices to the product positioning problem are important [6] as can be realized from Figs. I(B)-(D). Further psychological research seems necessary to support the proper choice of the exponent m. In ~ e algorithmic treatment, the changes of the geometric properties of Pk lead to modifications of the optimization procedure concerning the computation of painvise iatersections and the determination
of a common point for a set of solids. (a) Computing the matrix of pairwise intersections. There does not exist a sufficient criterion that would show pairwise intersections of solids as it is the case when considering spheres. Therefore, we first have to apply necessary criteria for indicating pairwise intersections definitely. These are given
225
.~. Albers / An extended algorithm for optimal product positioning
by the following properties: (¢~) Two solids k and I intersect eachother if the distance between their centers using the maximum salienees in each dimension ( / E J) is less than the sum of their critical distances de and dr, i.e. if
I[~
max{stq,
st/}"
(IcA7-
with respect to the computation of the weighted center of gravity. The modified procedure INTSEA II is given by:
Step 1: Initialize weights. Q is the set of solids for which a common point is to be determined. Let
ctil)ml llml < d e + dl ,
1
k, I E K .
(8)
~ ) Two solids k and I do not intersect eachother if the distance between their centers using the minimum saliences in each dimension ] E J is greater than the sum of their critical distances de and dr, i.e. if
(9)
It may occur that neither (8) nor (9) are full'died as we have only considered necessary conditions. If this cases occurs we have to apply the procedure INTSEA to search for a common point. Such a point would indicate that the two solids must intersect eachother. If a common point has not been found we can conelude that the two solids cannot intersect. Thus, the matrix A = (ae I e2) used in ENUSOS !I is defined as follows: 1 if solids k l and k 2 fulf'di (8) or if INTSEA provides a common point for solids k I and k2, kl, k 2 E K , aele2 =
--"
"--ma~
[ 1 F i ~
0 if solids k, and k2 fulf'dl (9) or if INT~E"/~ does not provide a common point for solids kt and k2, kl, k2 E K.
This matrix A serves as a starting point for the enumeration procedure ENUSOS II, which is otherwise equivalent to the one described in Albers and Brockhoff [21.
kEQ
It
•
Step 2: Compute the weighted center of gravity with coordinates Y/by applying the saliences s;,i and the weights we:
-
~e~ocej_" (se/) '/m_'wk
Y/=
I [ ~ min{s~7, st/} • (Ictq - Cti!)m]l/rnl > de + dr, k, 1E K .
Wk de
~k,EQ (SRj) l/m" ~/¢
'
j EJ.
Step 3: Find the set of solids T C Q such that its elements have the point y/(from Step 2) in common: Dk=l[~Sei(lcei-yil)m]l/ml,
]EJ
kEQ,
T = { k E Q : Dk < d k } . Let/" be the cardinality of T. Step 4: Test whether Step 3 improves the current solution: (S'is the maximum number of solids with a common point found so far). If/'>~
storeZ=T,'~='f, y j = ~ i ,
jEJ.
Step 5: Test of termination (if met, go back to Step 8 of ENUSOS il, otherwise go to Step 6) if T = Q, the common point isYl (/E J). Then terminate. If no convergence occurs (see Albers and Brockhoff [2]) or convergence conflicts with control parameters (i.e., weak convergence, precision), terminate. Step 6: Compute improved weights: De wk • = We • de ,
kEQ.
(b) Determining the common point in INTSEA.
Continue with Step 2.
The common point of a set of solids can be found by determining a weighted center of gravity of the centers of the solids. Then, the problem is to find the correct weights. The weights will be adjusted in an iterative process according to the fact whether the present center of gravity lies without or within the respective solids. As compared with the original procedure the search procedure has to be modified only
It is found empirically that only very few iterations of this type are required in most problems•
3. Improved criterion for separation The implicit enumeration scheme for sets of solids (spheres) ENUSOS described in Albers and
$. Albers ~An extended algorithm for optimal product positioning
226
Brockhoff [21 uses upper bounds/3k on the number of solids that may have a common point within solid k. These bounds serve as a criterion for separating partial solutions (see Step 6 in ENUSOS). The process of separation can be improved by computing stronger bounds. Lct ~ be the lower bound, i.e. the maximum number of solids with a point in common found so far, and let bgt be the number of solids (k and I included) that intersect with both solids k and L Then ~I, has been computed in ENUSOS by the number of elements bla with bla > ~ (1 ~ K). The underlying idea of the new upper bound/3k is the property that we must have a number of elements bla with bti i>/3k which often cugs deeper. The following example may illustrate the new upper bounding technique: Let us consider 12 solids and for k = 5 the number of solids bst (k = 5 and I included) that intersect with both solids k = 5 and I. They are given in the following table: l
1 2 3 4 5 6 7 8 9 10 11
bst
- 5 7 -9
-7
-8
7
4
12
7
Let ~ -" 3, then, using the old upper bounding technique we compute the number of solids with bst > 3 and get/3k = 8. Applying the new technique we first have to compute the values X~ giving the number of elements b~t with bkt >- i. By that we arrive at the following vector: i
456789
~,si
8
7
6
6
2
1
Now we can conclude that the upper bound can only be ~k = 6 because the value 6 gives the maximum number of solids l E K for which the number of solids intersecting with both solids k and I are greater or equal than #k = 6. Using the improved bounds/~k we separate the n,ost promising solids. By that we will fmd good solutions faster, which in turn, cuts computational time. Applying this new upper bounding technique we only have to modif) Step 4 of ENUSOS by the follovdng operations: Modified Step 4: (Test for termination.) Let K = S be the set of ~ solids (customers), k = IKI the number of elements in K , and G(k) the set of integer constants from 1 to k.
Compute the upper bounds ~k (k C K) on the number of solids that intersect with soli6 k by the following operation: Aia = I{bja t> i: 1 E K}[,
k E K, i E G(k),
/3k = max{ i E G(k): Aia I> i},
s* = K k E K : & ~ +
k E K.
1}1,
If s* g~', then go to Step 10,go to Step 5 otherwise.
4. Different revenues from customers Two arguments, namely: a customer would accept different price-categodes, - a customer would buy different quantities of the new product should be recognized in the positioning problem. Thus, the decision-maker is concerned with different revenues from the customers. Consequently we do not search for a maximum number of solids with a common intersection but for a common point within a set of solids the elements of which lead to a maximum revenue. To cover this extension we have to modify the implicit enumeration scheme ENUSOS II with respect to the calculation of bounds. The computation of upper bounds on the maximum revenue when customer k buys the new product is based on the calculation of bounds on the maximum number of solids that may have a point in common within solid k as described in Section 3. If the latter bound /3~ is determined one can calculate for every integer 1 ~ i ~ ~ the upper bounds 7 ~ which are the sum of the i maximal revenues from the set of customers 1E K with bta >- i. Then the maximum of 7ta, i
$. AIbers / An extended algorithm for optimal product positioning
1
I 1
2
3
4
5
6
7
8
9
10
11
12
-
5
7
-
9
-
7
-
8
7
4
7
0.9
2.2
0.8
2.4
3.3
0.6
2.2
1.1
1.0
1.9
2.1
0.7
,,
bst rt
I
Using the upper bounding technique from Section 3 we calculate the upper bound at = 6. In a second step we can determine the values 3/~i which give the sum of the i maximal revenues rt (! E K) from the set of customers with bla/> i. i
14
5
6
10.6
9.9
I ............
I
Tsi] 9.8
ber of solids which may have a common point within the solids kl and/62. Step 3 (Corrections of the matrix A): Compute the upper bounds #g (k E K) on the number of solids that intersect with solid k by the following operations (see Section 4.): ~,~/= I{btt I> i: l E K}I,
Then, the maximum value 7~a- (1 g i < / / t ) gives the upper bound u~ on the revenue the firm would achieve if it positioned the new product within solid k = 5. In our example the upper bound u~ takes the value 10.6. As the improvement of the current solution cannot be tested any longer via the number of solids for which a common point has been found, Step 4 in INTSEA should be deleted. Now, the test of improvement is implemented directly in ENUSOS. Thus, the new procedure ENUSOS Ill can be described in detail by the following steps.
k ~ K, i ~ G(k),
/3k = max{/E G(k): kta i> i}, where k = IKI: the number of elements in K and G(k): the set of integer constants from 1 to k. Compute the upper bounds Uklk2e (k v k2 E K) on the revenue that can be achieved from the customers if the new product would be placed within the solids kl audk2. IfKt is defined as a subset of K with exactly I elements (IKtl =/) that fulfill the condition #k I> l (k E/(t), then we get
Step 1 (Relaxation: A set of solids that intersect eachother pairwise is considered as a relaxation for a set of solids with a nonempty intersection): Determine A = (atd,2) from
kit2
227
max{ ~ ~ aklk3 "ak2ka'Sk3 } k3~K !
e
if subsets/ft exist, k 1, k2, 1 ~ K,
_~_
if the solids kt and k2 fulfill (8) or if INTSEA provides a common point for solids kl and k2, kt, k2 E K,
~klk21
if the solids kl and k2 fulfill (9) or if INTSEA does not provide a common point for solids kl and k2, kt, k2 E K.
The elements 7klk21e give the maximum sum of ! different revenues Ska from a set of customers k3 ~ K whose corresponding solids intersect with solids k z and k2 and with at least 1 - 2 other solids.
The matrix is symmetric around the diagonal, and a~lt 2 = 1 for kt = k2. Its elements denote a nonempty intersection of solids kl and k2 ifat, lk 2 = 1. Step 2 (Supporting calculation of bounds): Compute B = (btd,2) from: l ~ r a t d " at2t
ifatqk2 = 1 , k 1, k 2 E K ,
0
ffatqt2 = 0 ,
btlk2 =
kvk2EK.
The elements of B give an upper bound on the num-
0
if no subsets/~t exist or ifaklk 2 = 0, k v k2, I E K . '
e =m e Utqk2 ax{Ttlk2 t.. I E K , / ~>min{/~kt, ~k2}},
kv k2 E K . Compute the upper bounds u~t (kl ~ K) on the revenue that can be achieved from the customers if the new product would be Jlaced within: the solid k l. If the elements "/klk2t aJe in decreasing order, then 3'lit should give the value c f the lth element in this sequence. uklr =max{T~:lt'lEK, l<~:~ I},
klEK.
22~
$. a~,en ~an e x t ~
d s o a t ~ for opamat~
Correct the matrix A by:
~ao,~
Step 7 (Fathoming):
attt2 = 0
if uetlt2 ~ I z ,
~J, k2 ~ K ,
atlts=atstl=O
ff ur~1 ~ 1 z ,
kt, ks E K ,
k~Q
where I z is the maximal revenue Oower bound) that has been found in a preceding computation (naturally, / : = 0 at the start). This step excludes every nonempty intersection of solids kt and k2 with a common point that could only be within a set of solids the elements of which give a sum of revenues less than or equal to the lower bound. Go back to Step 2, if any such corrections are performed, otherwise go to Step 4. Step 4 (Test for termination): Compute the upper bound s* on the number of solids that may have a point in common: w
hi = I{/Ik t> i: k E K}I,
i E G(k),
$* = max{/E G(k): hi ~>i}. Compute ff~.eupper bound us on the maximal revenue: If the elements 7~d are in decreasing order then the elements ~ should give the value of the lth element in this sequence.
u s = max{~/: I E K , l ~ s * } .
with
V=S\(QuR). Let ft be the set of solids that may have a point in common:
f~=Quv
w i t h ~=lfZl.
Now the upper bound $* can be computed similar to Step 4.
hi = I{~k i> i: k ~ n}l,
i ~ G0,0,
$* = max{/E G(w): hi t> i}. Compute the upper bound uS on the maximum revenue: If the elements 7~d are in decreasing order for kl E K, then the elements ~ should give the value of the lth element in this sequence. u s' = max{'ffl: I E K , I ~ s * } . If u s ~ I z, then go to Step 9, otherwise continue. If V ~ {~}, then go to Step 6, otherwise go to Step 8. Step 8: Apply INTSEA to determine a common point for the solids q E Q, if such a point exists. Call ~ the set of solids for which a common point has been found.
Ifu s ~ Is, then $o to Step 10, otherwise go to Step 5. Step 5 (Initia~ation): Q = ~: set of soEds that intersect eachother pairwise and have been separated in a preceding computation, R - ~: set of solids the elements of which do not intersect with at least one solid in Q, V = K: set of solids that have not yet been inspected for inclusion in Q or R, Z = ~: set of solids for which a common point has been found that has been a maximum sum of revenues. Step 6 (Separation). Find a solid ~which is the element of V with the minimal index, and which has a maximum value for its upper bound u~.
If ~ >I I z, then store the best solution found so far: I z ffiF;Z =Q;Yi- Y ] ; / E J, and go to Step 3, otherwise continue with Step 9. S:ep 9 (Separation): If ~ = 0, then go to Step 10. I.."~ = 1, then r* is the only element in Q, ar*k = akr* = 0 (k E K), and go to Step 3. If ~ >t 2, then recompute:
T= min{v E IF: Uvr = max{u~: k C ,']}
R :=R \{r~R: Z) a k , = ~ } u { / }
Q: = Q u{~},
v=S\(QuR)
r: = v \ {~}.
If ~ is the number of Q we have to set ~: = ~ + 1.
k¢Q
r*={kEQ: ur~=min{~ . I E Q } } , Q:=Q\{r*};
q:=~k~Q
and go to Step 6.
1
S. ADen ~An extended algorithm for optimal product positioning
Step 10 (Termination): The solids with the maximum revenue that have a point in common are in Z as stored in Step 8, the coordinates of this point are y~ (l"E J).
229
perceptions of 4 existing brands (eiF symbol O) in a two dimensional attribute space. For purpose of demonstration, the perceptions are assumed identical for all customers. We are looking at evaluations of 10 customers each of them perceiving an ideal product denoted by the symbol +. The different revenues (rT,) the firm could realize from the customers are also listed in Fig. 2. According to the different weights that the customers give to the attributes, curves of constant distance between ideal and real product
5. Numerical results To illustrate the procedure let us consider the following example (see Fig. 2). Take as givens the
+2
- t,liO -,91
-,Oe
-,?l
-.60
-,51" -,'qL'~c,3l
-.2ll
6g
i
".i'4
'*9
"6
I
"17
I )
.70
.Sil
.ge
+8
"10
g II ..=i-
Fig. 2. r I = 0.57; r2 = 0.67; r 3 = 0.78; r4 = 1.13; r 5 = 0.99; r6 - 0.97; r7 = 0.98; r8 = 1.12; r9 = 1.30; rlO = 1.31.
1.00
230
s. ~
Table 1 Computational e x ~ No.
]-
m
k
/ An ~ t ~
1
4
4 4
s(A)
4
4
5
4
6 7
4 4
8
4
9
4
10 11 12
4 4 4
No.: .w-
I: 6(A): liB:
O1"1.: SEP: IND:
De?:
20 40 60 80 100
for opt~mt ~
~ o ~
with I'ROPOSASm UB
01'1'
~
2 3
~oat~
CPU-uime
SEI'
IND
DCP
insec
16.5 11.1 14.4 8.1 8.2
4.02 5.43 8.26 8.52 . 7.94
4.02 4.57 7.95 8.17 6.57
40
6.0
4.21
4.21
3.6
40 40 40 40 40 40
20.5 49.6 68.8 80.0 90.7 99.0
8.50 15.68 20.88 25.28 34.44 37.34
8.20 15.68 19.67 24.33 34.14 37.34
12.5 19.8 122.2 135.0 151.4 9.0
1.7.
7
3
2
8.3 21.5 46.2 62.2
50 108 390 1265
11 51 40 66
9 10 40 63
4
9
75 56 1096 572 952 37
29 37 29 29 12
1 8 3
55 33 67
4
1
number of problem; number of the relevant attributes comtituth~g brand preferences; number of customers; density of the matrix A, percentage of pairwise intersectiom; upper bound on the revenae cal~lated from the matrix (A) of palrwi~ intersections; optimal value of the revenue; number of separations (Step 6); number of runs through INTSEA to indicate a pairwise intersection; number of runs through INTSEA to search for a common point.
perceptions aclueve the form of ellipsoids as in Fig. 2. The efiipsoids encompass the set of those po,,itions for a new product which will cause the corre.~ponding customer to choose the new product. In Fig. 2 one discovers two major niches for positioning a new product. They are given within the intersections of the sets of e~.lipsoids { 1, 2, 3, 6}, and {6, 7, 10}. These niches correspond to local maxima if the problem would have been formulated continuously. The principles of relaxation, separation and fathoming are combined in PROPOSAS in an efficient way so that the niche {6, 7, 10} with the highest revenue is discovered immediately. It shows the optimal position for the new product by y l = 0.283 andy2 = -0.179. Furthermore, the algorithm has been tested on various problems. To enable an evaluation of the algorithm with respect to CPU-time, a set of artifically generated problems has been solved (see Table 1). Problems (1) through (5) show that problems up to 100 customers can be solved within reasonable time if the density 6 (A) is within the range of up to 20%. Density measures the percentage of pairwise. intersections or the relative numbers of non zeroelements in the matrix A.
Problems (6) through (12) show a significant influence of 8 (A) on the CPU-time. Problems with a sparse or dense matrix A require less CPU-time than problems with a density of 40-80%. This is because the number of partial solutions that have to be separated can be approximated by a function of the type (~'~k) ) which has its maximum at ~(A) = ~k-As the procedures providing the input data are often restricted with respect to the number of customers and as a market share of more than 20% is unlikely for most applications we can conclude that PROPOSAS is applicable to the majority of practical problems. 6. Some further extensions The optimal positioning procedur,~ PROPOSAS has been extended to cover three more cases: (1) Customers may have different saliences with resl~Ct to the direction of deviation between real and ideal product perceptions, that means, the saliences are different due to the fact whether the coordinate value of the position of the new product is less or greater than the coordinate value of the ideal product position.
S. Albers ~An extended algorithmfor optimalproduct positioning
(2) In many cases the firm under consideration will operate on the market with multiple products. Then, we have to take into account that the introduction of a new product may decrease that sales of products that are already offered by the f'Lrm.The consideration of that aspects has been given in Albers and Brockhoff [3]. (3) In some cases one may be interested to introduce more than one product. As a consequence, the
optimal positions have to be deteanined simultaneously. An algorithm for this problem is given in
Albers [11. Until now, dynamic aspects and uncertainty of data as considered in Brockhoff [5], and costs of positioning as asked for in Shocker and Srinivasan [10] and Bachem and Simon [4] are not considered in the positioning procedure. The algorithm developed here, however, seems to be flexible enough to cover these aspects without loosing its computational solvability.
References [ I ] S. Albers, A mixed integer nonlinear programming procedure for simultaneously locating multiple products in an attribute space, in: R. Henn et al., eds., Methods of Operations Research, Vol. 26, (Meisenheim am Glan, 1977) 899-909.
231
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