Media Mix Optimization

Applying a Quadratic Knapsack Model

Ulrich Pferschy

, Joachim Schauer

and Gerhild Maier

Department of Statistics and Operations Research, University of Graz, Universitaetsstrasse 15, 8010 Graz, Austria

UPPER Network GmbH, Seering 7/2, 8141 Unterpremst¨atten, Austria

Keywords:

Advertising Media Optimization, Quadratic Knapsack Problem, Genetic Algorithm.

Abstract:

In this contribution we present an optimization model for deciding on the best selection of advertising media

to be used in a promotional campaign. The effect of each single medium and each pair of media is estimated

from the evaluation data of past campaigns taking into account a similarity measure between the attributes and

goals of campaigns. The resulting discrete optimization model is a Quadratic Knapsack Problem which we

solve by a genetic algorithm. Then campaign budget is assigned to each selected advertising medium based

on a statistical estimation from previous campaigns. Our optimization tool is integrated in the marketing

management software solution MARMIND.

1 INTRODUCTION

Marketing is a crucial aspect for every company to

sell its products, whatever industry or market it is

concerned with. However, as a famous quotation

(sometimes attributed to Henry Ford) states: “Half the

money I spend on advertising is wasted, the trouble is,

I don’t know which half“. Indeed, it is a central ques-

tion of marketing management how to use the budget

of a promotional campaign. In particular, the avail-

able options have increased considerably in the last

decade with new possibilities such as targeted social

media advertising and context sensitive web banners.

Thus, the suitable selection of advertising media for

a promotional campaign, i.e. deciding on the media

mix, has become an increasingly complex task with

only limited information on the actual impact of a

medium on the goals of the campaign.

Contributions to ﬁnding the best media mix were

given for particular industry sectors, e.g. in (F¨are

et al., 2004) and (Reynar et al., 2010), and from an

optimization point of view in several papers going

back to (Balachandran and Gensch, 1974) and more

recently e.g. by (Sorato and Viscolani, 2011), (Nobi-

bon et al., 2011) and (S¨onke, 2012).

The software platform MARMIND produced and

offered by UPPER Network

provides a wide range

of tools to support the daily tasks of a marketing de-

www.uppernetwork.com

partment from planning to realization. In collabora-

tion with the University of Graz, Austria, an opti-

mization tool was developedand added to the solution

which computes a suggestion for the media mix of a

planned promotional campaign. This tool is now an

integral part of MARMIND and starts being used by

marketing managers.

A central question of marketing planning concerns

the effect and efﬁciency of advertising media (see e.g.

the survey paper (Vakratsas and Ambler, 1999) and

(Pergelova et al., 2010) on internet advertisements).

While many statistical methods have been employed

to ﬁnd partial answers to this questions, these require

survey data or other means of market research, which

is usually not available for the full range of marketing

options available to the decision maker in a typical

planning scenario. Therefore, we aim to gain infor-

mation from past campaigns.

The main outline of the optimization tool works

as follows. MARMIND keeps a data base of all past

promotional campaigns with ratings of their overall

success and an evaluation of the different goals of

the campaign. Based on these observations of past

campaigns, we estimate the effect of every advertis-

ing medium for the currently planned campaign. To

this end we take the “similarity” between planned

and past campaigns into account. Moreover, we de-

rive estimations for the pairwise effect of advertis-

ing media, since many media inﬂuence each other or

363

Pferschy U., Schauer J. and Maier G..

Media Mix Optimization - Applying a Quadratic Knapsack Model.

DOI: 10.5220/0004825803630370

In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 363-370

ISBN: 978-989-758-017-8

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

are dependent on each other and thus cannot be sepa-

rated into unconnected decisions. Based on these ef-

fect estimations we draw up an optimization model

which turns out to be a Quadratic Knapsack Problem

(QKP). After solving this model by an improved ge-

netic algorithm we assign the available budget to the

selected advertising media by considering the propor-

tional budget allocation of past campaigns.

In-house tests indicate that the media mix se-

lected by the optimization tool gets highly positive

appraisals from experts in the ﬁeld. The various possi-

bilities of parametrization allow a ﬂexible adaptation

for every domain.

2 FORMAL PROBLEM

FORMULATION

In our setting a promotional campaign is described by

a number of attributes, some of them represented by

nominal values such as target groups, product classes

and general strategic goals, others expressed by nu-

merical values such as desired market share, increase

in revenue, etc.

Formally, a promotional campaign t is deﬁned by

a k-dimensional vector of parameters t(1),. .. ,t(k),

where for some ﬁxed k

′

with 0 ≤ k

′

≤ k there are nom-

inal values t(1),. .. ,t(k

′

) and positive cardinal values

t(k

′

+ 1),. . .,t(k). A campaign may also consist of

only a subset of these parameters and leave the re-

maining entries of the vector empty.

To express and measure the goals of promotional

campaigns there is set of operative goals g

,. . ., g

ℓ

de-

ﬁned such as number of new customers, awareness

level, number of repeat customers, etc. Each promo-

tional campaign t is assigned a subset G

of these op-

erative goals with ℓ

:= |G

|. For convenience we im-

pose an upper bound ℓ

≤ L on the number of selected

goals, which is of moderate size in practice (think of

single digit numbers), i.e. L ≪ ℓ. Furthermore, the

chosen goals in G

are ranked in a total ordering to

indicate their relative importance. This preference re-

lation between goals is represented by a rank number

) for each goal g

∈ G

, where r

= ℓ

signiﬁes the

most important, i.e. highest ranked, goal and r

= 1

the least important. Clearly, each number in 1,. .. , ℓ

is assigned to exactly one goal as a rank r

Finally, there is a total budget B

given for the pro-

motional campaign t.

After completion of the promotional campaign t

the responsible manager should be able to state the

degree of achievement of each operative goal g

∈ G

of the campaign by assigning a numerical value repre-

senting the achieved percentage of the goal. For sim-

plicity we will assume that this value is scaled into an

achievement level a

) ∈ [0,1] with a

) = 1 in-

dicating perfect achievement of goal g

. In addition,

the marketing manager will be asked to evaluate the

overall success of a completed promotional campaign

by assigning a discrete value s

∈ {1,.. ., S}, where S

indicates the best outcome and 1 the worst. Usually,

S is a single digit number.

Of course, it would be desirable to extract more

information on the impact of the applied advertising

media. However, one should keep in mind that an

overly complicated feedback system will often be ig-

nored or ﬁlled with data of low quality. Practical ex-

perience suggests to keep the evaluation system as

simple as possible.

To reach the goals of a promotional campaign

there are n different advertising media m

,. . ., m

available (n ≈ 200), e.g. TV spots for different sta-

tions, newspaper ads in various publications, ﬂyers,

catalogs, social media ads, promotional events, etc.,

each with different characteristics.

After choosing the parameters and operative goals

of a promotional campaign the central task of the mar-

keting manager as a decision maker consists of the

selection of a subset of advertising media and the al-

location of a budget b

to each selected medium m

such that the deﬁned goals are met to a high degree

while the available budget B

is not exceeded. The

decision on this so-called media mix is crucial for the

success of any campaign.

Unfortunately, the effect of each advertising

medium on the deﬁned goals in connection with the

selected parameters of the promotional campaign are

mostly impossible to be quantiﬁed. Moreover, the ef-

fects of different media can not be separated but are

highly interdependent, e.g., a promotional event with

a celebrity will hardly have any effect without appro-

priate news coverage, and an evening TV spot will be

better remembered if its tune is repeated by a morning

radio spot. Under these circumstances, only educated

guesses and general rules of thumb gained from expe-

rience can be used by the decision maker to allocate

the promotional budget.

The existing software solution MARMIND can

keep track of all tasks involved with the realization of

a promotional campaign including accounting, man-

aging orders with advertisement companies, etc. In

this contribution we describe an optimization system

developedto givethe decision maker an automatically

generated suggestion for the media mix.

There are two core features of our system: (1) an

estimation of the direct effect and the interdependen-

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364

cies between advertising media based on the evalua-

tion of past promotional campaigns by the managers,

(2) the incorporation of these values into a discrete

optimization model, which is basically a Quadratic

Knapsack Problem (QKP), possibly with additional

constraints.

3 QUADRATIC KNAPSACK

MODEL

Given the parameters and operative goals of a promo-

tional campaign t we will derive in Sections 4 and 5

an estimation of the following three values for all ad-

vertising media. For simplicity of notation we omit

the reference to the current campaign t.

1. direct effect p

on the promotional campaign

caused by selecting medium m

2. joint effect q

on the promotional campaign

caused by selecting both media m

and m

3. estimated budget b

allocated to medium m

, if it

is selected in the promotional campaign.

With these estimations we can set up the following

mathematical optimization model with binary vari-

ables x

∈ {0, 1} representing the selection of adver-

tising medium m

. The objective function consists

of a convex combination of a linear (direct effect)

and a quadratic (joint effect) term with a parameter

λ ∈ (0,1) to be chosen appropriately. As a starting

value we set λ = 0.5.

max λ

∑

i=1

+ (1 − λ)

∑

i=1

∑

j=1

(1)

s.t.

∑

i=1

≤ B

(2)

∈ {0, 1} (3)

The model (1)-(3) is the well-known Quadratic

Knapsack Problem (QKP), see e.g. (Kellerer et al.,

2004, Chapter 12) or (Pisinger, 2007).

It may seem reasonable to restrict the number of

different advertising media selected for one promo-

tional campaign by adding a cardinality constraint

∑

i=1

≤ K. (4)

However, it will turn out that the estimation of budget

allocations b

produces values of a certain proportion

w.r.t. B

which implicitly restricts the number of cho-

sen advertising media and thus makes (4) redundant.

Practical considerations also suggest that certain

advertising media (e.g. TV spots) are more costly and

require a minimum budget to make sense. Thus, we

will eliminate in a preprocessing step all advertis-

ing media whose minimum budget requirementwould

consume most of the available budget B

The ﬁnal suggestion of the media mix presented

to the user of the system follows directly from the so-

lution of (1)-(3). Exactly those advertising media m

should be used whose decision variables have value

= 1 in the solution. Allocating the ﬁnal budget

to each selected medium m

requires a bit more care

and will be treated in Section 5.2.

4 LINEAR AND QUADRATIC

EFFECT ESTIMATION

It should be pointed out that all our estimations are

based on the evaluation of past promotional cam-

paigns and are not founded on some strict stochas-

tic model. They were developed in several rounds of

interaction with practitioners and validated with real-

world case data. The fact that the convex combina-

tion of several terms allows the setting of a number of

weighting parameters should be seen as an advantage

since it permits the adaptation of the optimization sys-

tem to the special customs and practices of the partic-

ular domain the system is applied in. By no means we

can expect to deliver a “plug-and-play” system ready

for use in any domain for every type of company.

Let T(i) be the set of all past promotional cam-

paigns containing advertising medium m

. The linear

proﬁt value p

will be expressed by a convex combi-

nation of the general success attributed to medium m

in the past and the level of goal achievement reached

by similar campaigns if they included m

, i.e.

:= λ

+ (1 − λ

)pg

(5)

with λ

∈ (0,1). The ﬁrst term ps

represents the aver-

age scaled success of all past promotional campaigns

containing medium m

. The underlying argumentsays

that every medium contributed in some way to the

overallsuccess of past campaigns. Formally, we have:

|T(i)|

∑

t∈T(i)

(6)

Clearly, ps

is in [0,1].

The second term pg

considers achievement of op-

erative goals and similarity of parameters in more de-

tail and will be described in the following subsection.

MediaMixOptimization-ApplyingaQuadraticKnapsackModel

365

4.1 Considering Similarity of

Campaigns

The value pg

should reﬂect the principle that it is a

good idea to repeat strategies that worked well in the

past for campaigns with similar parameters. To for-

malize this principle we will express “working well”

by the degree of goal achievement and “similar pa-

rameters” by introducing a similarity measure be-

tween campaigns.

Let

T( j) be the set of all past promotional cam-

paigns containing operative goal g

. Then the overall

goal achievement a

of a promotional campaign t will

be deﬁned as follows:

∑

j∈G

)

∑

j∈G

)· (7)









) −

T( j)|

∑

τ∈

T( j)

)













The term in the inner capital brackets computes

the difference of the goal achievement for goal g

from the average goal achievement over all promo-

tional campaigns τ containing goal g

. This number

lies in (−1,1) and is transformed to lie in (0, 1). Fi-

nally, the terms are weighted by their rank number

and scaled by the sum of rank numbers.

Now we introduce a measure to express the sim-

ilarity between two promotional campaigns t and t

′

Formally, we will deﬁne a function sim(t,t

′

) → [0, 1],

such that higher values of sim indicate closer simi-

larity of two campaigns. Measures of distance and

similarity are used in many ﬁelds of applied mathe-

matics and statistics, in particular in cluster analysis

(see e.g. (Everitt et al., 2011), (Guldemir and Sengur,

2006)). Our similarity function will deal separately

with a linear combination of nominal and cardinal pa-

rameters of campaigns expressed by sim

par and with

the similarity of the ordinally ranked operative goals

sim

goal.

sim

par(t,t

′

) :=

∑

i=1

′

∑

i=1

· sim

nom(t(i),t

′

(i))

∑

i=k

′

· sim

card(t(i),t

′

(i))

(8)

The weighting parameters c

∈ (0,1) can be used to

indicate the importance of different parameters.

Comparing nominal parameters is done simply

by an inverted Hamming distance, i.e. assigning

sim

nom(t(i),t

′

(i)) = 1 if t(i) = t

′

(i) and 0 otherwise,

for i = 1, .. . ,k

′

. Clearly, also more complicated mea-

sures such as the Jaccard index, the Sørensen coef-

ﬁcient or the Tanimoto distance might be used, see

e.g. (Tan et al., 2006).

For cardinal parameters i = k

′

+ 1, .. .,k the simi-

larity is computed from the relative deviation by

sim

card(t(i),t

′

(i)) = 1−

|t(i) − t

′

(i)|

max{t(i),t

′

(i)}

, (9)

which is clearly in [0,1]. Basically, any Minkowski

metric could be used and scaled into the correspond-

ing similarity measure.

For comparing the ordered selection of goals

between two campaigns in a similarity measure

sim

goal(t,t

′

), classical distance measures of order-

ings such as Kendall tau rank distance (similar to Ke-

meny distance) could be used (see (Sculley, 2007)

and (Kumar and Vassilvitskii, 2010) for recent con-

tributions). In our case, out of the available set of ℓ

goals each campaign is assigned only subset of goals

of small, but varying size. Hence, we use the follow-

ing rather unorthodox approach.

Deﬁne a decreasing sequence of positive bonus

points β

> β

> ... > β

and translate rank numbers

into bonus points by assigning the goal g of a promo-

tional campaign t with rank r

(g) exactly β

ℓ

−r

(g)+1

points, i.e. the best ranked goal receives β

points and

the lowest ranked goal with r

(g) = 1 gets β

ℓ

points.

The remaining points β

ℓ

,. . ., β

are not assigned at

all.

For any pair (t,t

′

) of campaigns we determine the

intersection of selected goals and add the bonus points

accrued by every such goal in both campaigns. I.e.

if some goal g

′

is ranked on ﬁrst position in t and

on third position in t

′

, then g

′

contributes β

+ β

to the total sum, while goals appearing in only one

of the two campaigns do not contribute at all. This

sum is scaled by the maximum possible number of

points

∑

min{ℓ

,ℓ

′

}

j=1

2β

which guarantees a ﬁnal value

sim

goal(t,t

′

) in [0, 1], with the desired property that

identical orderings of goals yield a similarity of 1

while disjunctive sets of goals have similarity 0.

Finally, we put together the two similarity mea-

sures with a weighting parameter λ

sim(t,t

′

) := (1− λ

)sim

par(t,t

′

)

+ λ

· sim goal(t,t

′

) (10)

A drawback of the above deﬁnitions can be found

in the “averaging effect” which means that taking a

linear combination over many different factors may

dilute the effect of strong similarity or deviance in

some components and tends to produce moderate val-

ues for almost any pair of promotional campaigns.

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Thus, we aim at strengthening the inﬂuence

of strong or weak similarities by increasing val-

ues closer to 1 and decreasing values closer to

0. This will be done by applying the following

sigmoid function F(x) on every partial similarity

measure sim

nom(t(i),t

′

(i)), sim card(t(i),t

′

(i)) and

sim

goal(t,t

′

). F(x) is depicted in the following ﬁg-

ure. It contains a tuning parameter k which we set to

k = 10 in our implementation.

F(x) =

1+ e

(

−kx)

1+ e

· (2x − 1) (11)

constant f(x) = x

sigmoid function F(x)

0.5

It remains to put together the expressions of goal

achievement and similarity. This is done by simply

summing up achievement values of past campaigns

weighted by their similarity to the current campaign

. Formally, we have

|T(i)|

∑

t∈T(i)

sim(t,t

) · a

(12)

Again, pg

is in [0, 1].

4.2 Estimation of Media Interaction

We proceed to estimate the effectof havingtwo adver-

tising media m

and m

together in a promotionalcam-

paign. This is done by separating from the set of all

past campaigns a subset of particularly effective cam-

paigns which stood out among the remaining cam-

paigns. Then we will simply count the occurrence of

every pair of advertising media in the effective cam-

paigns relative to all its occurrences. Thereby, we aim

to detect a systematic effect of successful pairs that

happened to be chosen together in conspicuous fre-

quency among the more effective campaigns. Note

that our existing sample of campaigns is too small to

allow statistical tests on this hypothesis.

Formally, we sort the set of past promotional

campaigns in decreasing order of their goal achieve-

ment a

and determine a threshold a

such that only

a prescribed percentage of campaigns exceeds this

achievement value, e.g. 25%. Then we set:

|T(i) ∩ T( j) with a

≥ a

|T(i) ∩ T( j)|

(13)

It turned out that there are certain pairs of me-

dia that marketing managers generally want to use to-

gether and which appear in pairs in almost all cam-

paigns (if they appear at all), no matter whether the

campaigns worked well or not. This effect is not cap-

tured by (13) which was hence extended to include

the presence of pairs of media in past campaigns with

strong similarities to the current campaign t

. Let

:= {t | sim(t,t

) ≥ δ} for some similarity thresh-

old δ. Then we deﬁne the ﬁnal quadratic effect as:

′

:= λ

+ (1 − λ

) ·

|(T(i) ∩ T( j)) ∩ T

(14)

5 BUDGET ALLOCATION

5.1 Estimation of Budget Values

While it may seem quite reasonable that one can learn

from past promotional campaigns which advertising

media, resp. which combination of media, worked

well to reach certain goals for campaigns with a cer-

tain set of parameters, it is less clear how to assign

a budget value to an advertising medium after decid-

ing to use it. However, one can not separate media

selection from budget allocation since one may end

up with a collection of advertising media that can not

be realized within the given budget B

considering the

natural lower bounds on the budget for each medium.

To allow a plausible estimation of the budget val-

ues b

in the optimization model, we consider a subset

of past campaigns T

with a budget in similar range

as the current campaign t

, i.e.

:= {t | k

≤ B

≤ k

} (15)

with suitably chosen parameters k

< 1, k

> 1. Then

we determine for each advertising medium the rel-

ative proportion of budget allocated in the past (de-

pending on its assigned budget b

) and take the mean

over these values as an estimation of b

. Formally,

|T(i) ∩ T

∑

t∈T(i)∩T

. (16)

MediaMixOptimization-ApplyingaQuadraticKnapsackModel

367

Note that different from Section 4 we do not take

similarity of campaigns into account in this estima-

tion. Discussions with marketing managers and anal-

ysis of available data exhibit that the choice of ad-

vertising media is very much tailored to the particu-

lar goals and parameters of a campaign. But once a

medium is selected the invested budget is mostly de-

pendent on technical constraints and the “size”, i.e.

budget, of the overall campaign. But clearly, it would

be straightforward to restrict the summation in (16) to

campaigns in T

with a certain similarity to t

5.2 Actual Budget Allocation

After solving the optimization model (1) - (3) we ob-

tain a solution set S := {i | x

= 1} of all selected ad-

vertising media. Assigning the actual budget values

to all media m

∈ S could be done by simply resorting

to the estimations b

from (16).

We suggest a more reﬁned procedure taking into

account two aspects: First, the discrete solution of

optimization model will most likely leave a certain

amount of budget B

−

∑

i∈S

unused and thus miss

chances for a better utilization of the available bud-

get. Secondly, and more important, it should make

sense to consider the particular combination of me-

dia in S, which we already targeted speciﬁcally by the

quadratic coefﬁcients q

To do so, we give the relative budget proportions

in a promotional campaign t, i.e.

, more weight if

t shares more advertising media with the solution for

the current campaign t

. This is achieved by the fol-

lowing formula for every medium m

, i ∈ S:

|S| − 1

∑

j∈S, j6=i

|T(i) ∩ T( j) ∩ T

∑

t∈T(i)∩T( j)∩T

(17)

Allocating budgets according to (17) may result in

infeasible solutions or (as before) in leftover budget.

We propose the following allocation process to over-

come this issue.

The budget estimation b

in (16) can be seen as

an estimator in the strict statistical sense. Hence, we

can also compute the associated empirical standard

deviation σ

based on the sum of squared distances

from the mean and deﬁned as follows:

|T(i) ∩ T

| − 1

∑

t∈T (i)∩T



−



(18)

Now we start the budget allocation procedure by

assigning each advertising medium m

∈ S in decreas-

ing order of proﬁt values p

a conservative budget

value of b

− σ

, i.e. the estimated value reduced by

one standard deviation. Then we enter into a second

round and increase the budget to b

as long as the bud-

get B

permits, again in decreasing order of p

. Fi-

nally, if there is still budget left, we take a third round

and increase the allocated budget to b

+ σ

until B

is completely used up. Clearly, the last advertising

medium considered by this procedure may obtain a

budget allocation in between the three prescribed val-

ues by consuming all the remaining budget.

An analogous procedure is done for the more so-

phisticated budget values

(with the corresponding

empirical standard deviation

) where it can be ex-

pected to be more relevant, since there is a larger dif-

ference from the budget values used in the optimiza-

tion model. Note that in this case it may happen that

we run out of budget already in the ﬁrst round of al-

locations, since the values b

used in the weight con-

straint of the optimization model may deviate consid-

erably from

6 SOLUTION OF THE

QUADRATIC KNAPSACK

PROBLEM

The model introduced in Section 3 is a standard

Quadratic Knapsack Problem (QKP) with no addi-

tional side-constraints. This is somewhat rare, since

practical applications usually require additional con-

straints and do not ﬁt into the mould of standard mod-

els.

Important exact solution methods for QKP were

given by (Caprara et al., 1999) and (Billionnet and

Soutif, 2004). The former approach uses Lagrangian

relaxation and is able to solve instances containing

up to 200 variables. It is especially well suited for

dense instances. (Billionnet and Soutif, 2004) uses

Lagrangian decomposition and is able to solve in-

stances of roughly the same size, however it outper-

forms the previous approach on instances of medium

and low density.

The currently best working strategy was given by

(Pisinger et al., 2007). It succeeds in reducing the

size of many instances dramatically by ﬁxing items

that will or will not occur in an optimal solution.

The reduced problem can then be solved by any al-

gorithm for QKP. Combining this approach with an

exact solution algorithm (Pisinger et al., 2007) were

able to solve instances with up to 1500 items. Unfor-

tunately, this code is not available, therefore we used

the implementation described in (Caprara et al., 1999)

for solving benchmark problems of MARMIND and

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368

managed to solve instances to optimality with up to

n = 200 advertising media in less than 10 minutes on

a simple standard PC with 2.2 GHz and 2 GB Ram.

For ensuring a good user experience UPPER Net-

work however requested that the optimized marketing

campaign of MARMIND has to be computed in less

than 3 seconds. Moreover, we recall that all data of

our QKP instances is based on estimates and does not

represent assured values. Thus, we can easily settle

for a good approximate solution.

For our optimization tool we implemented a ge-

netic algorithm and imposed a time limit of 3 seconds.

It turned out that this gave solutions for all instances

of the required size (≥ 200 items) with an average de-

viation of less than 1% from optimality.

Our algorithm is a modiﬁed version of (Julstrom,

2005) which worked well for the random test in-

stances generated according to the same method used

in (Caprara et al., 1999). (Julstrom, 2005) reports test

data for ten instances of 100 items and ten instances

of 200 items. Every instance was solved 50 times and

the algorithm was able to ﬁnd the optimal solution

value in about 90 percent of the runs, although the

running time sometimes exceeds 1 minute. Note that

our implementation was especially tuned for getting

high quality results in a very short time but often suc-

ceeded to yield results similar or better than (Julstrom,

2005).

Recently (Yang et al., 2013) published a well per-

forming metaheuristic that combined GRASP with

tabu search. On 100 randomly generated benchmark

instances that follow the same scheme as in (Caprara

et al., 1999) the metaheuristic was able to ﬁnd the op-

timal solution 99 times in less than 0.8 seconds. In

the remaining case the gap to the optimal solution

was negligibly small. Moreover, they were able to get

good solutions for instances of up to 2000 variables

(the solution quality was justiﬁed by comparison to

known upper bounds) in less than 300 seconds.

Currently, we are working on a project to system-

atically test our genetic algorithm, compare it to the

other existing methods listed above and to introduce

harder benchmark instances for QKP. The results of

this comprehensive computational study will be pub-

lished as they become available.

7 CONCLUSIONS

We developed an optimization system to offer mar-

keting managers an evidence-based suggestion for the

media mix to be used for a given promotional cam-

paign. It relies on a comparison of the current cam-

paign to past campaigns based on their parameters

and goals.

Building an optimization model with the com-

puted direct and pairwise effect estimations gives

rise to a Quadratic Knapsack Problem which can be

solved almost to optimality in all real-world scenar-

ios within a time limit of 3 seconds. The optimization

tool is currently used within the industrial software

solution MARMIND.

Future developments include a revision of some of

the effect estimations by stochastic models as soon as

a suitable set of test data derived from real world ap-

plications is available. Furthermore, the estimations

will be adjusted to include a “memory” effect, i.e.,

giving a smaller weight to campaigns in the more dis-

tant past. It may also be interesting to take trends

into accounts. Based on classical tools of statistical

analysis it should be possible to detect certain trends

of advertising media increasing or decreasing in im-

portance, or in their effect for certain goals or target

groups.

ACKNOWLEDGEMENTS

This research was supported by the Austrian Research

Promotion Agency (FFG) under project “MARMIND

media mix optimization“ and by the Austrian Science

Fund (FWF): [P 23829-N13].

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