Regression analysis

Model specification

To analyze the impact of experimental conditions on performance and convergence property of rules, we specify the following regression model for each rule:

” •_ K™› œ 4 •b∙ ž !_2Ÿ 5™4 •{∙ ž !_P Ÿ ™4 •¡∙ ž !_¢p£H™4 •¤∙ ž !_¥ p™

4 •¦∙ ž !_23™4 •§∙ ž !_¢p ™4 •¨∙ ž !_G ™4 •©∙ ž !_\RIª™ 12"

4 •«∙ ž !_¥£u ™4 •b ∙ ž !_2pp™4 •bb∙ ¬ 4 - ∙ ^¬ ∙ ®™_ 4 K™›

” •_ K™˜ : Deviation from maximum profit for scenario l and replication d in planning cycle m;

ž !_2Ÿ 5 : Heterogeneity of elasticities across products (0: equal, 1: unequal); ž !_P Ÿ : Heterogeneity of sales level across products (0: equal, 1: unequal);

ž !_¢p£H : Heterogeneity of growth parameters across products (0: equal, 1: unequal); ž !_¥ p : Heterogeneity of carryover coefficient across products (0: equal, 1: unequal); ž !_23 : Heterogeneity of elapsed time since launch across products (0: equal, 1:

unequal);

ž !_¢p : Shape of growth function (0: asymmetric, 1: symmetric);

ž !_G : Shape of market response function (0: multiplicative, 1: mod. exponential); ž !_\RIª : Initial budget allocation (0: equal, 1: percentage-of-sales);

ž !_¥£u : Competitive situation (0: No competition, 1: Nash competition); ž !_2pp : Estimation error (0: non-included, 1: included);

®™ : Vector of all simulation factors for scenario l;

œ, •, - : (Unobserved) parameters;

e : Error term;

z = 1, 2, …, 10 (number of planning cycles); l = 1, 2, …, 1024 (number of scenarios); and d = 1, …, Dl (number of replications).

All scenarios of our simulation experiment in which we do not incorporate an estimation error are independent, i.e. _K™›~] 0, ¯{", with the variance σ2. This allows us to apply OLS for estimating equation (12) for the models of the naïve solution and the percentage-of-sales rule. But in all scenarios in which we incorporate an estimation error we apply the technique of common random numbers. As we use the same error terms for each scenario and each planning cycle we have to account for correlation among regression errors (Kleijnen 1988). The error terms for each replication across scenarios as well as across planning cycles within a scenario are correlated, while the error terms across replications within a scenario are uncorrelated, i.e. _K™›~] 0, ¯_™›{", with the variance ¯_™›{, and Cov(elz,elz’)=σlz,lz’ for Ÿ¬ ° Ÿ¬±.

Therefore, we estimate equation (12) for the models of the numerical optimization and the attractiveness heuristic by using two step GLS which allows us to account for the serial correlation and correlation across scenarios (Greene 2006).

Equation (12) assumes a linear convergence process. We also tested as a log-linear process, i.e. z is replaced by Log(z). Estimation results were very similar, so that we do not discuss them here in detail.

5.2. Results

The results of our models are shown in Table 5a and 5b. The constant can be interpreted as the average in our basic scenario, i.e. if all dummies in equation (12) equal zero: a homogeneous parameter set, an asymmetric growth function, a multiplicative market response

function, no estimation error, and (if the factor of initial budget is included) an equal initial allocation. The coefficient values of the main effects show how the performance of the allocation rule would change on average in terms of deviation from maximum profit if the corresponding simulation factor changes to the specific experimental condition. A positive value means a worse, a negative value a better performance of the allocation rules. For example, a coefficient value of 0.1 for the simulation factor unequal elasticities means that the rule would have a higher deviation from the optimal solution of 10 % on average if we have a scenario characterized by heterogeneous elasticity across the portfolio instead of a homogeneous elasticity set. The interaction effects with the time variable show the impact of the factors over time, i.e. their influence on the convergence properties. A negative coefficient indicates a faster convergence process under the specific experimental condition, while a positive coefficient indicates a slower process.

By comparing the constant across all allocation rules in Table 5a and 5b we see that the attractiveness heuristic has the lowest value which confirms that it outperforms all other allocation approaches in our basic scenario on average.5

The effect of heterogeneity across the product portfolio is reflected by the coefficients of the product characteristics. The three simpler rules, i.e. the naïve solution, percentage-of-sales, and the attractiveness heuristic, are all strongly and negatively affected in their performance by heterogeneity in elasticities and elapsed time since launch. But while the interaction effect with time for the naïve solution is positive, i.e. the performance is getting even worse over time, the interaction effects for percentage-of-sales and the attractiveness heuristic are negative, i.e. the negative effect on the performance diminishes if the rule is applied subsequently. This is a reasonable finding as marketing responsiveness as well as life cycle effects are reflected in the sales outcome which is incorporated into both rules. The attractiveness heuristic even directly includes the elasticity and the growth multiplier (see equation (6)). As expected, heterogeneity in the sales level has a negative effect on the performance of the naïve solution, while percentage-of-sales and the attractiveness heuristic are able to capture the heterogeneity. The effect on performance becomes even positive for these two rules. Heterogeneity in the carryover coefficients affects the performance of all three rules negatively, but this effect decreases over time. As expected, numerical optimization generally performs better if the portfolio is characterized by heterogeneity as it has negative coefficients for the elasticity, the sales base, the growth parameters, and the

5

Note that the constant of the numerical optimization method is not directly comparable as it only includes scenarios with an estimation error.

elapsed time since launch. Only heterogeneity in the carryover coefficient affects the performance of numerical optimization negatively.

The type of the growth model specifications is meaningful for all budgeting methods apart from numerical optimization. As expected, better allocation solutions are provided in the scenarios characterized by a symmetric growth function.

Percentage-of-sales provide superior results in scenarios characterized by a multiplicative response function, while the type of the response function has no impact on the performance of the naïve solution and the attractiveness heuristic. Contrary, numerical optimization works much better in case of the modified exponential function which confirms that the sales outcome based on this model specification is less affected by noisy demand parameters. Nash competition has a negative effect on the performance of the naïve solution, the attractiveness heuristic, and numerical optimization. Contrary to our expectations, it has a negative effect on the percentage-of-sales rule, which is probably the result of an asymmetric portfolio structure of the two competitors, i.e. the larger products of company A compete with the smaller products of company B and vice versa. Based on this rule the budget is more heavily allocated to the products which generate larger sales so that an asymmetric portfolio structure across competitors avoids negative substitution effects and therefore is closer to the optimal solution.

In line with our expectations, percentage-of-sales as initial budget allocation leads to a superior performance of the attractiveness heuristic. But numerical optimization provides better results if the initial budget allocation is determined by equal distribution. This finding is contrary to our first expectations but may be explained by the replacement of the initial marketing stocks over time. Numerical optimization and the true optimal solution both suffer equally from a suboptimal initial solution but start replacing these stocks subsequently by new stocks based on their calculations. In case of a better initial allocation the suboptimal share of the marketing stocks is faster build down so that they can be replaced by new stocks. But while this is the true optimal stock for the profit maximum solution, the numerical optimization approach builds up another suboptimal stock in case of noisy demand parameters. Therefore, the deviation to the profit maximum is larger in case of superior initial marketing stocks.