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A Resource Allocation Method using a Non-parametric Approach

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Efficient budget plans are increasingly becoming important in mature developed countries, and the market mix model (MMM) plays an important role. We propose a new MMM methodology using a non-parametric approach that can be applied to a non-convex shape frontier. The Free Disposal Hull (FDH) is known to be applicable to a non-convex frontier, but the FDH frontier surface is stair-like stepwise. Thus, when we assume a monotonically increasing output with respect to increases in the input, the stepwise surface is inadequate for a sales response function for advertisement. The smoothing technique of the FDH frontier that we propose improves the performance in terms of budget optimization.

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A Resource Allocation Method using a Non-parametric Approach

  1. 1. 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
  2. 2. 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
  3. 3. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 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
  4. 4. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 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.
  5. 5. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 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
  6. 6. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 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
  7. 7. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 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 DMU A DMU B DMU C DMU D Input 1 0.8 0.3 0.9 0.5 Input 2 0.2 0.7 0.1 0.5 DMU A DMU B DMU C DMU D 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$
  8. 8. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 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
  9. 9. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 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
  10. 10. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 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
  11. 11. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 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 a=1 Concave with DRS & Concave with DRS 1 1.484 0.017 0.064 0.020 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 Concave with DRS & Sigmoid 1 1.675 0.239 0.469 0.133 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
  12. 12. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 11 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 a=1 Concave with DRS & Concave with DRS 1 1.484 0.017 0.064 0.020 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 Concave with DRS & Sigmoid 1 1.675 0.239 0.469 0.133 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 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
  13. 13. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 12 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 a=1 Concave with DRS & Concave with DRS 1 1.484 0.017 0.064 0.020 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 Concave with DRS & Sigmoid 1 1.675 0.239 0.469 0.133 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 2 2 2. The DEA model is more accurate than the original FDH. But some case of sigmoid type combination, FDH is more accurate.
  14. 14. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 13 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 a=1 Concave with DRS & Concave with DRS 1 1.484 0.017 0.064 0.020 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 Concave with DRS & Sigmoid 1 1.675 0.239 0.469 0.133 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 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.
  15. 15. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 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 a=1 Concave with DRS & Concave with DRS 1 0.02 0.06 -0.05 0.00 0.05 -0.05 0.00 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 Concave with DRS & Sigmoid 1 0.24 0.47 -0.23 0.11 0.21 -0.10 0.13 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
  16. 16. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 15 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 a=1 Concave with DRS & Concave with DRS 1 0.02 0.06 -0.05 0.00 0.05 -0.05 0.00 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 Concave with DRS & Sigmoid 1 0.24 0.47 -0.23 0.11 0.21 -0.10 0.13 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.
  17. 17. Analytics Innovation Company ©BrainPad Inc. Strictly Confidential 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.

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