A Resource Allocation Method using a Non-parametric Approach

A Resource Allocation Method
using a Non-parametric Approach
ICIBM2018
Tomohiro Noguchic,
Nobuyuki Tachibanaa,
Susumu Kadoyab,
Takashi Namatamed
a,b,c BrainPad Inc., Analytics Service Division
d Department of Indastrial and Systems Engineering, Chuo University
2018/06/14

Analytics Innovation Company
©BrainPad Inc.
Strictly Confidential
1
Outline
1. Background & Objectives
2. Budget Allocation
3. Non Parametric Models
4. Proposed Method (iFDH)
5. Empirical Test
6. Conclusion

©BrainPad Inc.
2
Background & Objectives
• Since the difficulty of specifying advertising contribution structure, we studied non
parametric method, Data Envelopment Analysis (DEA) , to apply the problem.
• Then, we propose a new MMM methodology that can be applied to a non-convex
shape frontier.
• Efficient budget plans are increasingly becoming important in mature developed
countries, and the market mix model (MMM) plays an important role.
• When products are advertised, in many cases, multiple advertising media are used.
However, the sales contributions cannot be decomposed into individual advertising
media effect since we do not directly know the relationship between the product
sales and the investment costs in each media.
• Despite these limitations, to allocate the budget based on the media contribution,
we must assume the relationship between the individual media investments and
the product sales.
• Conveniently, we use parametric formula to estimate the relationships. (such like
regression model) However, cross effects exist among a plurality of mediums
make difficult to estimate the accurate parameters (Even when there is no cross
effect it is difficult to estimate parametrically though).
BackgroundObjectives

©BrainPad Inc.
3
Budget Allocation
Sales cannot be decomposed into individual advertising media based on their contributions.
Therefore, it is difficult for decision-makers to decide their budget.
Decision Makers
Our Total Budget is 15M$
TV
WEB
Media
Call Center
WEB Page
EC
Sales
Medius Budget
TV 10M?
WEB 5M?
TV
Sales
Sales
Cost of TV Cost of WEB Total Cost
Media A
Unobservable
Medea B
Unobservable
A + B
Observable
① ② ③WEB
Sales Basically using a parametric
formula to estimate the model
parameters like regressions.

©BrainPad Inc.
4
It is common that sales response to advertisement costs eventually saturates.
Considering that BCC and FDH are available for budget allocation
Non-Parametric Approach | DEA (Data Envelopment Analysis)
• DEA is a multi-criteria evaluation method that can select the most favorable alternatives
from a large set when there is no parametric assumption among variables. It uses a
mathematical programming technique where the most favorable alternatives form an
effective frontier
• Outline the model properties for DEA(CCR and BCC) and FDH(Free Disposal Hull), one-
input one-output case
CCR
The initial DEA is a CCR model, which
assumes constant RTS
Evaluate all DMUs based on the most
effective DMU.
But this methodology is not applicable to
the case of decreasing returns to scale
(DRS)
Sales
Media Cost
BCC
BCC is more suitable in capturing the
frontier line in the case of a concave
type.(We assume that the sales
response function to advertisement
costs can be a sigmoid or concave
shape based on numerous marketing
theories)
FDH
Free Disposal Hull
model assumes the free
disposability relaxing the convexity
assumption in defining
the production possibility set from
the observations. FDH is more
suitable in capturing the frontier
line in the case of a sigmoid
Media CostMedia Cost
Sales Sales
DMU : Decision Making Unit

©BrainPad Inc.
5
Problem of BCC and FDH
【BCC】
•BCC can not correspond to Non Convex. When Sales response is Sigmoid, there
is a Gap
【FDH】
•The frontier is represented as a step-like form. This means that there is a region
where the output does not increase even if the input does. The step-like frontier
shape is inadequate except in the case where sales are saturated
Input 1
Input2
Media Cost
Sales
Gap
Input2 > Input1. But got same output
Media Cost
Sales
In low cost area,There is a GAP
between estimation frontier
andrreal sales response curve

©BrainPad Inc.
6
Figures of procedure | Proposed Method
x2
y
x1
x2
y
x1
x2
y
x1
x2
y
x1
DMU A DMU B DMU C DMU D
Input 1 80 30 90 50
Input 2 20 70 10 50
Input 1 0.8 0.3 0.9 0.5
Input 2 0.2 0.7 0.1 0.5
Input 1 160 60 180 100
Input 2 40 140 20 100
x2
y
x1
Step0：Observe DMUs Step1: Calculate FDH Step2:Delaunay triangulation.(Smoothing)
Step3: The individual media investment set
in monetary amounts
Step4: assigned to the regions Step5: The achievable maximum
sales are calculated
y
xView from the side
✖✖ M$

©BrainPad Inc.
7
Propose a method to compensate for the disadvantages(step like frontier) of FDH
which can express non-convexity
Our proposed procedure is as follows:
1. Calculate the FDH frontier using the observed DMU set
2. Select only the efficient DMUs by FDH. Then, applying Delaunay triangulation to their
inputs as a vertex set, divide the frontier area into triangles (higher-dimensional simplex
such as tetrahedron). Here, we do not consider output values
3. Optimal allocation candidates are the observed DMU allocations. An advertising budget
is given, the candidate allocation can be transferred to the individual media investment
set in monetary amounts
4. The allocation candidates are assigned to the regions where they belong, which is the
divided frontier by the triangulation
5. In terms of the assigned region in Step 4, estimate the hyperplane, which includes the
vertex of the triangulation. In this hyperplane estimation process, not only the inputs, but
also the output values are considered. Consequently, the FDH frontier is smoothed by
this hyperplane.
The achievable maximum sales are calculated on the hyperplane for the allocation
candidates in their assigned areas
Our Proposed Method

©BrainPad Inc.
8
Empirical Tests 1/2
Test purpose :
• Compare the four estimation errors of the model(Parametric, BCC, FDH,
Proposed method).
• Through this test, To show that nonparametric is appropriate when the structure
is unknown and that the proposed method gives a more accurate estimate
Procedure of test :
① Define the true relationships between the advertising costs and sales by a set of
equations that are simplified versions of the ADBUDG* formula (call true model)
② Generate scenarios based on this true model(make a test datasets)
③ Outputs can be calculated based on following Tested models .”Parametric Model”, “DEA
model”, “FDH model” ,and ”Our proposed model(Interpolated FDH | iFDH)”
④ Comparing the model accuracy

©BrainPad Inc.
9
Empirical Tests 2/2
Test overview :
• There are two advertising media
• Assume following cases:
• The interaction between the two media and without any interaction
• The individual response function of sales to the advertisement costs is considered to be a sigmoid
shape or concave
Media A Media B
True
Media A Media B
False
Media BMedia A
sigmoid & concave w/ DRS
Media BMedia A
sigmoid & sigmoid
Considering the six cases (three function combinations & interaction)
Have interaction or Not :
x1 x2 x1 x2
Media BMedia A
concave w/ DRS & concave w/ DRS
x1 x2

©BrainPad Inc.
10
Empirical Tests | Result
Parametric DEA FDH IFDH
a=0 Concave with DRS & Concave with DRS 1 1.528 0.011 0.048 0.009
2 1.040 0.005 0.038 0.006
3 0.571 0.002 0.022 0.001
4 0.694 0.001 0.021 0.002
5 0.633 0.002 0.026 0.003
Concave with DRS & Sigmoid 1 0.989 0.212 0.420 0.115
2 1.138 0.278 0.929 0.192
3 1.062 0.035 0.152 0.035
4 1.372 0.004 0.044 0.008
5 1.639 0.002 0.030 0.003
Sigmoid & Sigmoid 1 0.851 0.985 1.162 0.589
2 0.894 0.921 0.792 0.348
3 1.068 0.190 0.248 0.057
4 0.959 0.032 0.217 0.019
5 0.739 0.009 0.101 0.009
2 1.051 0.007 0.056 0.008
3 0.819 0.002 0.041 0.002
4 1.041 0.003 0.022 0.003
5 0.950 0.003 0.033 0.005
2 2.341 0.224 0.881 0.132
3 2.996 0.018 0.129 0.022
4 3.813 0.011 0.105 0.016
5 4.573 0.003 0.048 0.003
Sigmoid & Sigmoid 1 1.600 1.261 1.162 0.585
2 1.337 1.594 0.705 0.322
3 2.647 0.299 0.470 0.069
4 3.188 0.055 0.326 0.041
5 2.204 0.021 0.199 0.021
Average 1.563 0.215 0.299 0.093
Model Type
BudgetResponse Function Type CombinationInteraction

©BrainPad Inc.
11
2 1.040 0.005 0.038 0.006
3 0.571 0.002 0.022 0.001
4 0.694 0.001 0.021 0.002
5 0.633 0.002 0.026 0.003
2 1.138 0.278 0.929 0.192
3 1.062 0.035 0.152 0.035
4 1.372 0.004 0.044 0.008
5 1.639 0.002 0.030 0.003
Sigmoid & Sigmoid 1 0.851 0.985 1.162 0.589
2 0.894 0.921 0.792 0.348
3 1.068 0.190 0.248 0.057
4 0.959 0.032 0.217 0.019
5 0.739 0.009 0.101 0.009
2 1.051 0.007 0.056 0.008
3 0.819 0.002 0.041 0.002
4 1.041 0.003 0.022 0.003
5 0.950 0.003 0.033 0.005
2 2.341 0.224 0.881 0.132
3 2.996 0.018 0.129 0.022
4 3.813 0.011 0.105 0.016
5 4.573 0.003 0.048 0.003
Sigmoid & Sigmoid 1 1.600 1.261 1.162 0.585
2 1.337 1.594 0.705 0.322
3 2.647 0.299 0.470 0.069
4 3.188 0.055 0.326 0.041
5 2.204 0.021 0.199 0.021
Average 1.563 0.215 0.299 0.093
Model Type
1
1. The parametric models’
accuracies are lower than those
of the non-parametric model.
Parametric models is
remarkable in the case where
interaction exists between the
two media. This means that the
structure complexity causes
difficulties in calibration of
parameters in the parametric
model

©BrainPad Inc.
12
2 1.040 0.005 0.038 0.006
3 0.571 0.002 0.022 0.001
4 0.694 0.001 0.021 0.002
5 0.633 0.002 0.026 0.003
2 1.138 0.278 0.929 0.192
3 1.062 0.035 0.152 0.035
4 1.372 0.004 0.044 0.008
5 1.639 0.002 0.030 0.003
Sigmoid & Sigmoid 1 0.851 0.985 1.162 0.589
2 0.894 0.921 0.792 0.348
3 1.068 0.190 0.248 0.057
4 0.959 0.032 0.217 0.019
5 0.739 0.009 0.101 0.009
2 1.051 0.007 0.056 0.008
3 0.819 0.002 0.041 0.002
4 1.041 0.003 0.022 0.003
5 0.950 0.003 0.033 0.005
2 2.341 0.224 0.881 0.132
3 2.996 0.018 0.129 0.022
4 3.813 0.011 0.105 0.016
5 4.573 0.003 0.048 0.003
Sigmoid & Sigmoid 1 1.600 1.261 1.162 0.585
2 1.337 1.594 0.705 0.322
3 2.647 0.299 0.470 0.069
4 3.188 0.055 0.326 0.041
5 2.204 0.021 0.199 0.021
Average 1.563 0.215 0.299 0.093
Model Type
2
2
2. The DEA model is more accurate than the original FDH. But some case of
sigmoid type combination, FDH is more accurate.

©BrainPad Inc.
13
2 1.040 0.005 0.038 0.006
3 0.571 0.002 0.022 0.001
4 0.694 0.001 0.021 0.002
5 0.633 0.002 0.026 0.003
2 1.138 0.278 0.929 0.192
3 1.062 0.035 0.152 0.035
4 1.372 0.004 0.044 0.008
5 1.639 0.002 0.030 0.003
Sigmoid & Sigmoid 1 0.851 0.985 1.162 0.589
2 0.894 0.921 0.792 0.348
3 1.068 0.190 0.248 0.057
4 0.959 0.032 0.217 0.019
5 0.739 0.009 0.101 0.009
2 1.051 0.007 0.056 0.008
3 0.819 0.002 0.041 0.002
4 1.041 0.003 0.022 0.003
5 0.950 0.003 0.033 0.005
2 2.341 0.224 0.881 0.132
3 2.996 0.018 0.129 0.022
4 3.813 0.011 0.105 0.016
5 4.573 0.003 0.048 0.003
Sigmoid & Sigmoid 1 1.600 1.261 1.162 0.585
2 1.337 1.594 0.705 0.322
3 2.647 0.299 0.470 0.069
4 3.188 0.055 0.326 0.041
5 2.204 0.021 0.199 0.021
Average 1.563 0.215 0.299 0.093
Model Type
3
3. The best model is the IFDH. Therefore, the inferiority of the FDH against the
DEA may be caused by the stair-like surface of the frontier.

©BrainPad Inc.
14
Additional Test ~ Effectiveness of smoothing technique
Improvement
DEA (A) FDH (B) Diff1 (A)-(B) DEA (C) FDH (D) Diff2 (C)-(D) Diff2 - Diff1
a=0 Concave with DRS & Concave with DRS 1 0.01 0.05 -0.04 0.00 0.03 -0.03 0.01
2 0.00 0.04 -0.03 0.00 0.02 -0.02 0.01
3 0.00 0.02 -0.02 0.00 0.02 -0.02 0.01
4 0.00 0.02 -0.02 0.00 0.01 -0.01 0.01
5 0.00 0.03 -0.02 0.00 0.01 -0.01 0.02
Concave with DRS & Sigmoid 1 0.21 0.42 -0.21 0.12 0.15 -0.03 0.18
2 0.28 0.93 -0.65 0.07 0.32 -0.25 0.40
3 0.03 0.15 -0.12 0.01 0.08 -0.07 0.05
4 0.00 0.04 -0.04 0.00 0.02 -0.02 0.02
5 0.00 0.03 -0.03 0.00 0.01 -0.01 0.02
Sigmoid & Sigmoid 1 0.99 1.16 -0.18 0.52 0.66 -0.14 0.04
2 0.92 0.79 0.13 0.79 0.48 0.31 0.18
3 0.19 0.25 -0.06 0.27 0.11 0.16 0.21
4 0.03 0.22 -0.19 0.02 0.06 -0.04 0.15
5 0.01 0.10 -0.09 0.01 0.04 -0.03 0.06
2 0.01 0.06 -0.05 0.00 0.02 -0.02 0.03
3 0.00 0.04 -0.04 0.00 0.02 -0.02 0.02
4 0.00 0.02 -0.02 0.00 0.01 -0.01 0.01
5 0.00 0.03 -0.03 0.00 0.01 -0.01 0.02
2 0.22 0.88 -0.66 0.06 0.36 -0.30 0.35
3 0.02 0.13 -0.11 0.01 0.08 -0.07 0.04
4 0.01 0.11 -0.09 0.01 0.06 -0.05 0.04
5 0.00 0.05 -0.05 0.00 0.02 -0.02 0.02
Sigmoid & Sigmoid 1 1.26 1.16 0.10 1.39 0.68 0.71 0.61
2 1.59 0.70 0.89 1.53 0.45 1.08 0.19
3 0.30 0.47 -0.17 0.20 0.26 -0.06 0.11
4 0.05 0.33 -0.27 0.03 0.09 -0.07 0.21
5 0.02 0.20 -0.18 0.01 0.06 -0.05 0.13
Average 0.21 0.30 -0.08 0.17 0.15 0.02 0.11
Interaction Response Function Type Combination Budget
200 DMUs 600 DMUs
We increased the number of DMUs to reduce the step level differences in the FDH
frontier, and investigate the improvement in performance before and after
increasing the DMUs

©BrainPad Inc.
15
Additional Test ~ Effectiveness of smoothing technique
Improvement
DEA (A) FDH (B) Diff1 (A)-(B) DEA (C) FDH (D) Diff2 (C)-(D) Diff2 - Diff1
2 0.00 0.04 -0.03 0.00 0.02 -0.02 0.01
3 0.00 0.02 -0.02 0.00 0.02 -0.02 0.01
4 0.00 0.02 -0.02 0.00 0.01 -0.01 0.01
5 0.00 0.03 -0.02 0.00 0.01 -0.01 0.02
2 0.28 0.93 -0.65 0.07 0.32 -0.25 0.40
3 0.03 0.15 -0.12 0.01 0.08 -0.07 0.05
4 0.00 0.04 -0.04 0.00 0.02 -0.02 0.02
5 0.00 0.03 -0.03 0.00 0.01 -0.01 0.02
Sigmoid & Sigmoid 1 0.99 1.16 -0.18 0.52 0.66 -0.14 0.04
2 0.92 0.79 0.13 0.79 0.48 0.31 0.18
3 0.19 0.25 -0.06 0.27 0.11 0.16 0.21
4 0.03 0.22 -0.19 0.02 0.06 -0.04 0.15
5 0.01 0.10 -0.09 0.01 0.04 -0.03 0.06
2 0.01 0.06 -0.05 0.00 0.02 -0.02 0.03
3 0.00 0.04 -0.04 0.00 0.02 -0.02 0.02
4 0.00 0.02 -0.02 0.00 0.01 -0.01 0.01
5 0.00 0.03 -0.03 0.00 0.01 -0.01 0.02
2 0.22 0.88 -0.66 0.06 0.36 -0.30 0.35
3 0.02 0.13 -0.11 0.01 0.08 -0.07 0.04
4 0.01 0.11 -0.09 0.01 0.06 -0.05 0.04
5 0.00 0.05 -0.05 0.00 0.02 -0.02 0.02
Sigmoid & Sigmoid 1 1.26 1.16 0.10 1.39 0.68 0.71 0.61
2 1.59 0.70 0.89 1.53 0.45 1.08 0.19
3 0.30 0.47 -0.17 0.20 0.26 -0.06 0.11
4 0.05 0.33 -0.27 0.03 0.09 -0.07 0.21
5 0.02 0.20 -0.18 0.01 0.06 -0.05 0.13
Average 0.21 0.30 -0.08 0.17 0.15 0.02 0.11
Interaction Response Function Type Combination Budget
200 DMUs 600 DMUs
The results indicate that surface
smoothing improves the model
performance.

©BrainPad Inc.
16
Conclusion
• We propose a non-parametric model, which can be applied to a non-convex shape frontier
for MMM
• From the result of these tests, our proposed model, IFDH, shows the highest performance
from the perspective of minimizing the error
• The technique that we have introduced to smooth the FDH frontier surface seems to work
well because the DEA model performance is better than the original FDH
• To confirm this, we increased the number of DMUs to reduce the step level differences in
the FDH frontier, and investigate the improvement in performance before and after
increasing the DMUs
Some future work:
Our model has been proven to work on artificial data, it should be tested in a real-world
context. Furthermore, the robustness of the model is still subject to be confirmed.

A Resource Allocation Method using a Non-parametric Approach

Recommended

Recommended

More Related Content

Recently uploaded

Recently uploaded (20)

Featured

Featured (20)

A Resource Allocation Method using a Non-parametric Approach