A comparison of semiparametric and heterogeneous store sales models for optimal category pricing

Abstract

Category management requires sales response models helping to simultaneously estimate marketing mix effects for all brands of a product category. We, therefore, develop a general heterogeneity seemingly unrelated regression (SUR) model accommodating correlations between sales across brands. This model contains a latent class SUR model, the well-known hierarchical Bayesian SUR model and the homogeneous SUR model as special cases. We further propose a hierarchical Bayesian semiparametric SUR model based on Bayesian P-splines which comprises a homogeneous semiparametric SUR model as nested version. The results of an empirical application with store-level scanner data indicate that the flexible SUR approaches of modeling price response clearly outperform the various parametric (homogeneous and heterogeneous) SUR approaches with respect to not only predictive validity but also total expected category profits. In particular, functional flexibility turns out to be the primary driver for improving the predictive performance of a store sales model as heterogeneity pays off only once functional flexibility has been accounted for. Furthermore, since both flexible SUR models perform nearly equally well with respect to expected category profits, a uniform pricing strategy which is much less complex to implement than micromarketing can be recommended for our data.

Appendices

Appendix 1: General heterogeneity SUR model

1.1 The marginal model

We use the partly marginalized Gibbs sampler suggested by Frühwirth-Schnatter et al. (2004) where the random effects are integrated out when sampling S and $\gamma =(\alpha ,\beta ^{G}_{1},\ldots ,\beta ^{G}_{K})$. If segment membership is known the random effects $\beta _{i}$ can be rewritten as

$$\begin{aligned} \beta _{i}= & {} \beta ^{G}_{k}+b_{i},\quad b_{i}\sim N(0,Q^{G}_{k}). \end{aligned}$$

(22)

Under the assumption that $b_{i}$ and $\epsilon _{i}$ are independent (Frühwirth-Schnatter 2006) the marginal model is obtained by substituting (22) into model (4):

$$\begin{aligned} y_{i}=X_{i}\alpha + W_{i}\beta ^{G}_{k} + \tilde{\epsilon _{i}}, \quad \tilde{\epsilon _{i}}\sim N(0,W_{i}Q^{G}_{k}(W_{i})^{\prime }+\varSigma \otimes I_{T_{i}}). \end{aligned}$$

(23)

Introducing indicator variables

$$\begin{aligned} H^{(k)}_{i}= & {} {\left\{ \begin{array}{ll} 1, &{} \text {if } S_{i}=k,\\ 0, &{} \text {otherwise}, \end{array}\right. }\quad k=1,\ldots ,K, \end{aligned}$$

(24)

leads to the following representation of the marginal model:

$$\begin{aligned} y_{i} = Z_{i}\gamma +\epsilon ^{*}_{i},\quad \epsilon ^{*}_{i} \sim N(0,V_{i}) \end{aligned}$$

(25)

with the design matrix

$$\begin{aligned} Z_{i}= & {} \left( X_{i}\ \ W_{i}H^{(1)}_{i}\ \ \cdots \ \ W_{i}H^{(K)}_{i}\right) \end{aligned}$$

(26)

and the covariance matrix

$$\begin{aligned} V_{i} = W_{i}H^{(1)}_{i}Q^{G}_{1}\left( W_{i}\right) ^{\prime }+ \cdots + W_{i}H^{(K)}_{i}Q^{G}_{K}\left( W_{i}\right) ^{\prime }+\varSigma \otimes I_{T_{i}}. \end{aligned}$$

(27)

1.2 Prior specifications

The parameter vectors $\alpha$ and $\beta ^{G}_{k}$ ($k=1,\ldots ,K$) are assumed to be a priori independent leading to a joint prior $N(\mu _{0\gamma },\varSigma _{0\gamma })$ for $\gamma =(\alpha ,\beta ^{G}_{1},\ldots ,\beta ^{G}_{K})$ with $\mu _{0\gamma }=(\mu _{0\alpha },\mu _{0\beta ^{G}_{1}},\ldots ,\mu _{0\beta ^{G}_{K}})^{\prime }$. $\varSigma _{0\gamma }$ is a block diagonal matrix with diagonal elements $(\varSigma _{0\alpha },\varSigma _{0\beta ^{G}_{1}},\ldots ,\varSigma _{0\beta ^{G}_{K}})$. We determined the prior means $\mu _{0\alpha }$ for the fixed effects $\alpha$ and $\mu _{0\beta ^{G}_{k}}$ for the group-specific effects $\beta ^{G}_{k}$ ($k=1,\ldots ,K$) by estimating a homogeneous SUR model via Gibbs sampling implemented in the R function rsurGibbs() contained in the R package bayesm (see Rossi et al. 2005 for details).

About the remaining parameters we stay nearly noninformative choosing $\varSigma _{0\alpha }=\varSigma _{0\beta ^{G}_{1}}=\cdots =\varSigma _{0\beta ^{G}_{K}}=50I$, $a_{0\varSigma }=1$ and $B_{0\varSigma }=0.005I$ and $e_{0k}=1$ for $k=1,\ldots ,K$. The covariance matrices $Q^{G}_{k}$ $(k=1,\ldots ,K)$ are defined as diagonal matrices with diagonal elements $(\tau ^{2}_{k1},\ldots ,\tau ^{2}_{kr})$ (Gelfand et al. 1995, p. 483) where $r=\sum ^{M}_{m=1}r_{m}$ corresponds to the dimension of $\beta _{i}$. Thus, inverse gamma priors ${IG}(a^{k}_{0h},b^{k}_{0h})$ are placed on the variance parameters $\tau ^{2}_{kh}$ $(h=1,\ldots ,r)$ with hyperparameters $a^{k}_{0h}=b^{k}_{0h}=0.001$ ($h=1,\ldots ,r$, $k=1,\ldots ,K$). For our data used in the empirical application, we found that applying diagonal covariance matrices $Q^{G}_{k}$ $(k=1,\ldots ,K)$ considerably improved the model fit of the hierarchical Bayesian SUR model and the general heterogeneity SUR model (measured by the model likelihood). In the context of a single (HB) regression model, Andrews et al. (2008) have shown that the predictive performance of the model can be improved (and besides model complexity will be reduced) by not estimating covariances between store-specific effects. The respective assumption that, for example, the price sensitivity of a store i is independent of the display activity in another store $i^{\prime }$ is reasonable in the context of sales data belonging to one retail chain since customers usually shop in the store which is closest to their home (cf. Andrews et al. 2008, p. 25). If the covariance matrices $Q^{G}_{k}$ ($k=1,\ldots ,K$) were non-diagonal matrices we could choose $a_{0Q^{G}_{K}}=r+1$ and $B_{0Q^{G}_{K}}=rI$ for $k=1,\ldots ,K$ as hyperparameters for the inverse Wishart prior.

Parameter estimates are based on the last 2000 iterations of the MCMC algorithm. Due to high computing times the burn-in period was limited to 1000 iterations for the one-segment models and the unidentified versions of the two-segment models, and to 8000 iterations when the latent class SUR model had to be identified via constrained permutation sampling. Trace plots of sampled coefficients indicated that these numbers of burn-in iterations are sufficient to ensure convergence of the MCMC algorithm.

1.3 Bayesian inference

The joint posterior of the latent random effects $\beta ^{I}=(\beta _{1},\ldots ,\beta _{N})$, the latent segment indicator $S=(S_{1},\ldots ,S_{N})$ and all unknown parameters $\phi =(\alpha ,\beta ^{G}_{1},\ldots ,\beta ^{G}_{K},$ $Q^{G}_{1},\ldots ,Q^{G}_{K},\eta _{1},\ldots ,\eta _{K},\varSigma )$ given the data $y=(y_{1},\ldots ,y_{N})$ is proportional to

$$\begin{aligned} \pi (\beta ^{I},S,\phi | y)\propto & {} f(y| \beta ^{I},S,\phi )\pi (\beta ^{I}| S,\phi )\pi (S|\phi )\pi (\phi ). \end{aligned}$$

(28)

For a fixed number K of segments the following sampling scheme results (compare Frühwirth-Schnatter et al. 2004, 2005, and the derivation of steps 4 (a2) and (b) in Weber (2015)):

• 1. Sample $S_{i}$, $i=1,\ldots ,N$, from

  $$\begin{aligned} P(S_{i}=k|\cdot )\propto & {} \eta _{k}\cdot f(y_{i}| \alpha ,\beta ^{G}_{k},Q^{G}_{k},\varSigma ),\quad k=1,\ldots ,K, \end{aligned}$$

  (29)

  where the likelihood $f(y_{i}| \alpha ,\beta ^{G}_{k},Q^{G}_{k},\varSigma )$ is represented by the density of the normal distribution

  $$\begin{aligned} N(X_{i}\alpha + W_{i}\beta ^{G}_{k},W_{i}Q^{G}_{k}(W_{i})^{\prime }+\varSigma \otimes I_{T_{i}}). \end{aligned}$$

• 2. Sample $\eta$ from the Dirichlet distribution $D(e_{0,1}+N_{1},\ldots ,e_{0,K}+N_{K})$ with $N_{k}=\#\{S_{i}=k\}$.

• 3. $\alpha ,\beta ^{G}_{1},\ldots ,\beta ^{G}_{K}$ and $\beta ^{I}$ are conditionally independent and can be sampled within two blocks.

  • (a) Sample $\gamma$ from the normal distribution $N(\mu _{\gamma },\varSigma _{\gamma })$ with

    $$\begin{aligned} \varSigma _{\gamma }= & {} \left( \sum ^{N}_{i=1}\left( Z_{i}\right) ^{\prime }\left( V_{i}\right) ^{-1}Z_{i}+\left( \varSigma _{0\gamma }\right) ^{-1}\right) ^{-1}, \\ \mu _{\gamma }= & {} \varSigma _{\gamma }\left( \sum ^{N}_{i=1}\left( Z_{i}\right) ^{\prime }\left( V_{i}\right) ^{-1}y_{i}+\left( \varSigma _{0\gamma }\right) ^{-1}\mu _{0\gamma }\right) . \end{aligned}$$

  • (b) Sample $\beta _{i}$, $i=1,\ldots ,N$, from the normal distribution $N\left( \mu _{\beta _{i}},\varSigma _{\beta _{i}}\right)$ with

    $$\begin{aligned} \varSigma _{\beta _{i}}= & {} \left( \left( W_{i}\right) ^{\prime }\left( \varSigma \otimes I_{T_{i}}\right) ^{-1}W_{i} + \left( Q^{G}_{S_{i}}\right) ^{-1}\right) ^{-1}, \\ \mu _{\beta _{i}}= & {} \varSigma _{\beta _{i}}\left( \left( W_{i}\right) ^{\prime }\left( \varSigma \otimes I_{T_{i}}\right) ^{-1}\left( y_{i}-X_{i}\alpha \right) + \left( Q^{G}_{S_{i}}\right) ^{-1}\beta ^{G}_{S_{i}} \right) . \end{aligned}$$

• 4. The covariance matrices $Q^{G}_{1},\ldots ,Q^{G}_{K}$ and $\varSigma$ are conditionally independent so that sampling can be done within two blocks.

  • (a1) If the covariance matrices $Q^{G}_{k}$, $k=1,\ldots ,K$, are non-diagonal matrices they can be sampled from the inverse Wishart distribution ${IW}(a_{Q_{k}},B_{Q_{k}})$ with

    $$\begin{aligned} a_{Q_{k}} = a_{0Q^{G}_{k}}+N_{k}/2, \quad B_{Q_{k}} = B_{0Q^{G}_{k}}+\frac{1}{2}\sum ^{N}_{i=1}H^{(k)}_{i} (\beta _{i}-\beta ^{G}_{k})^{\prime }(\beta _{i}-\beta ^{G}_{k}). \end{aligned}$$

  • (a2) If the covariance matrices $Q^{G}_{k}$, $k=1,\ldots ,K$, are diagonal matrices of the form $Q^{G}_{k} = \mathrm{diag}(\tau ^{2}_{k1},\ldots ,\tau ^{2}_{kr})$ $(k=1,\ldots ,K)$ the variance parameters $\tau ^{2}_{kh}$, $k=1,\ldots ,K$ and $h=1,\ldots ,r$, can be sampled from the inverse gamma distribution ${IG}(a_{\tau _{kh}},b_{\tau _{kh}})$ with

    $$\begin{aligned} a_{\tau _{kh}} = a^{k}_{0h}+N_{k}/2, \quad b_{\tau _{kh}} = b^{k}_{0h}+\frac{1}{2}\sum ^{N}_{i=1}H^{(k)}_{i} (\beta _{ih}-\beta ^{G}_{kh})^{2}. \end{aligned}$$

  • (b) Sample $\varSigma$ from the inverse Wishart distribution ${IW}(a_{\varSigma },B_{\varSigma })$ with

    $$\begin{aligned} a_{\varSigma } = a_{0\varSigma }+\frac{1}{2}\sum ^{N}_{i=1}T_{i}, \quad B_{\varSigma } = B_{0\varSigma }+\frac{1}{2}\sum ^{N}_{i=1}A^{(i)} \end{aligned}$$

    with $A^{(i)} \!=\! (Y_{i} \!-\! P_{i}^{*}B_{i})^{'}(Y_{i} \!-\! P_{i}^{*}B_{i}), Y_{i} \!=\! (y_{i1},...,y_{iM}), P_{i}^{*} \!=\! (P_{i1},...,P_{iM}), P_{im} = (X_{im},W_{im}), B_{i} = diag(\gamma _{i1},...,\gamma _{iM}) \,\mathrm{and}\, \gamma _{im} = (\alpha _{m},\beta _{im})^{'}.$

The R code of the estimation procedure is available from the authors upon request.

Appendix 2: Hierarchical Bayesian semiparametric SUR model

Inference uses MCMC simulation, drawing from full conditionals of single parameters or blocks of parameters given the rest and the data. Let $y_{m}=(y_{m11},\ldots ,y_{mNT})'$ and $\eta _{m}=(\eta _{m11},\ldots ,\eta _{mNT})'$ denote the vector on the m-th response variable and the corresponding vector of predictors. Then the additive predictors in (7) can be written as

$$\begin{aligned} \eta _{m}~=~\sum _{j=1}^{M} A_{mj} X_{mj} \beta _{mj} ~+~ \sum _{l=0}^{L} \, W_{ml} \, Z \gamma _{ml}~+~V_m \delta _m, \end{aligned}$$

(30)

where $A_{mj}$ is a $n \times n$ diagonal matrix with possible entries $1+\alpha _{m1j},\dots ,1+\alpha _{mNj}$ depending on the store $i=1,\dots ,N$ a particular observation pertains to (with n being the total number of observations of brand m), $X_{mj}$ corresponds to the design matrix for the jth price effect of brand m with elements given by the B-spline basis functions evaluated at the observed prices $P_{jit}$, $W_{ml}$ is the (diagonal) design matrix for the lth parametrically and store-specifically modeled variable $D_{mit}^l$, Z is a 0/1 incidence matrix indicating if a particular observation belongs to store i, $V_{m}$ is the design vector for the homogeneous parametric effect of $E_t$, $\beta _{mj} = (\beta _{mj1},\dots ,\beta _{mjO_{mj}})'$ is the vector of regression parameters for function $f_{mj}$.

An alternative formulation in terms of the random coefficients is

$$\begin{aligned} \eta _{m}~=~\sum _{j=1}^{M} (f_{mj} + \tilde{X}_{mj} Z_{} \alpha _{mj}) ~+~ \sum _{l=0}^{L} \, W_{ml} \, Z \gamma _{ml} ~+~V_m\delta _m, \end{aligned}$$

(31)

where $\tilde{X}_{mj} = \mathrm{diag}(f_{mj}(P_{m11}),\dots ,f_{mj}(P_{mNT}))$ and $\alpha _{mj}=(\alpha _{m1j},\dots ,\alpha _{mNj})'$ the vector of random coefficients for function $f_{mj}$.

Let $\beta =(\ldots ,\beta _{mj}',\ldots )'$ and $\alpha =(\ldots ,\alpha _{mj}',\ldots )'$ denote the stacked vector of all regression parameters, $\tau ^2=(\dots ,\tau ^2_{mj},\dots )'$, $\phi ^2=(\dots ,\phi ^2_{mj},\dots )'$, $\psi ^2=(\dots ,\phi ^2_{ml},\dots )'$ the vectors of corresponding variances $\tau _{mj}^2$, $\phi _{mj}^2$, $\psi ^2_{ml}$ and $\delta =(\delta _1', \ldots ,\delta _M')'$ the stacked vector of all fixed effects parameters.

Posterior analysis is then based on

$$\begin{aligned}&{\pi \left( \beta ,\alpha , \tau ^2,\phi ^2, \psi ^2, \delta ,\varSigma \big | y\right) } \nonumber \\&\quad \propto f\left( y\big | \beta ,\alpha ,\delta ,\varSigma \right) \prod ^{M}_{m=1} \prod ^{M}_{j=1} \left[ \pi \left( \beta _{mj}\big |\tau ^{2}_{mj}\right) \pi \left( \alpha _{mj}\big |\phi ^{2}_{mj}\right) \pi \left( \tau ^{2}_{mj}\right) \pi \left( \phi ^{2}_{mj}\right) \right] \nonumber \\ \end{aligned}$$

(32)

$$\begin{aligned}&\qquad \times \prod ^{M}_{m=1} \prod ^{L}_{l=0} \left[ \pi \left( \gamma _{ml}\big |\psi ^{2}_{ml}\right) \pi (\psi ^2_{ml}) \right] \pi \left( \delta \right) \pi \left( \varSigma \right) \end{aligned}$$

(33)

with $f(y|\cdot )$ denoting the likelihood of the data.

Parameters are estimated via Gibbs sampling within the following blocks (Lang et al. 2003):

• 1. Sample $\beta _{mj}$, $m=1,\ldots ,M$, $j=1,\ldots ,M$. The full conditional for $\beta _{mj}$ is Gaussian, $\beta _{mj}|\cdot ~ \sim ~ N(\mu _{mj},~P_{mj}^{-1})$, with precision matrix

  $$\begin{aligned} P_{mj}= \frac{X_{mj}' A_{mj}^2 X_{mj}}{\sigma ^2_{m|-m}}~ + ~\frac{ K_{mj}}{\tau _{mj}^2} \end{aligned}$$

  and mean

  $$\begin{aligned} \mu _{mj}~ = ~P_{mj}^{-1}\left( \frac{1}{\sigma ^2_{m |-m}} X'_{mj}A_{mj} (y_{m} - o_{m})\right) . \end{aligned}$$

  Here, $\sigma ^2_{m|-m}$ is the (conditional) variance

  $$\begin{aligned} \sigma ^2_{m |-m}~=~\sigma ^2_m - \varSigma _{m,-m}\varSigma _m^{-1}\varSigma _{m,-m}', \end{aligned}$$

  derived from partitioning $\varSigma$ into

  $$\begin{aligned} \varSigma = \left( \begin{array}{cc} \sigma ^2_m &{}\quad \varSigma _{m,-m}\\ \varSigma _{m,-m}'&{}\quad \varSigma _m\\ \end{array}\right) \end{aligned}$$

  (after reordering for the mth component of the error variable). The vector $o_{m}$ is an offset vector defined in Lang et al. (2003).

• 2. Sample $\alpha _{mj}$, $m=1,\ldots ,M$, $j=1,\ldots ,M$. The full conditionals for the scaling factors are derived from (31) with $\alpha _{mj} \, | \, \cdot \sim \, N(\mu _{mj},~P_{mj}^{-1})$ and

  $$\begin{aligned} P_{mj}= \frac{Z' \tilde{X}_{mj}^2 Z}{\sigma ^2_{m|-m}}~ + ~\frac{ I}{\phi _{mj}^2} \end{aligned}$$

  and mean

  $$\begin{aligned} \mu _{mj}~ = ~P_{mj}^{-1}\left( \frac{1}{\sigma ^2_{m |-m}} Z' \tilde{X}_{mj} (y_{m} - o_{m})\right) . \end{aligned}$$

• 3. Sample $\gamma _{ml}$, $m=1,\dots ,M$, $l=0,\dots ,L$. Full conditionals are standard and omitted here.

• 4. Sample $\delta _{m}$, $m=1,\ldots ,M$. Full conditionals are standard and omitted here.

• 5. Sample $\tau ^{2}_{mj}$, $m=1,\ldots ,M$, $j=1,\ldots ,M$. Full conditionals for the variance parameters $\tau _{mj}^2$ are inverse Gamma distributions with parameters

  $$\begin{aligned} a_{mj}=0.001 + \frac{rank(K_{mj})}{2},\quad b_{mj} = 0.001 + \frac{1}{2}\beta _{mj}'K_{mj}\beta _{mj}. \end{aligned}$$

  (34)

• 6. Sample $\phi ^{2}_{mj}$, $m=1,\ldots ,M$, $j=1,\ldots ,M$. Full conditionals for the variance parameters $\phi _{mj}^2$ are inverse Gamma distributions with parameters

  $$\begin{aligned} a_{mj}=0.001+\frac{N}{2},\quad b_{mj}=0.001+ \frac{1}{2}\alpha _{mj}' \alpha _{mj}. \end{aligned}$$

  (35)

• 7. Sample $\varSigma$. The full conditional for $\varSigma$ is an inverse Wishart distribution with parameters

  $$\begin{aligned} a=1 +\frac{n}{2},\quad B=0.005I+\frac{1}{2} \sum _{i=1}^{N} \sum _{t=1}^T (y_{it}-\eta _{it})(y_{it}-\eta _{it})'. \end{aligned}$$

  (36)

  where $Y_{it} = (y_{1it},\dots ,y_{Mit})'$ and $\eta _{it} = (\eta _{1it},\dots ,\eta _{Mit})'$

The complete sampling scheme can be found in Lang et al. (2003, pp. 270–271) in combination with Lang et al. (2015). The estimation procedure of the semiparametric SUR model is implemented in the software BayesX and the code is available from the authors.

Appendix 3: Overview of model specifications

Model	Specification	Modeled effects
		Intercept	Price effects	Display and price ending effects	Holiday effect
PHomSM	Parametric, homogeneous	Store-specific	Equal across stores	Equal across stores	Equal across stores
PLCSM	Parametric, heterogeneous	Store-specific	Segment-specific	Segment-specific	Equal across stores
PHBSM	Parametric, heterogeneous	Store-specific	Store-specific	Store-specific	Equal across stores
PHetSM	Parametric, heterogeneous	Store-specific	Store-specific within segments	Store-specific within segments	Equal across stores
FHomSM	Semiparametric, homogeneous	Store-specific	Equal across stores and nonparametric	Equal across stores	Equal across stores
FHBSM	Semiparametric, heterogeneous	Store-specific	Store-specific and nonparametric	Store-specific	Equal across stores

Appendix 4: Optimization details

The optimization algorithm implemented in the R function genoud() is able to solve problems for objective functions that are nonlinear or even discontinuous in parameters and provides a high probability of finding the global optimum (Sekhon and Mebane 1998). The evolutionary algorithm starts with a population of trial solutions (in our application one trial solution corresponds to a vector of prices of the 8 brands). Then, a set of heuristic rules or operators (basically reproduction, mutation, crossover) is used to modify the trial solutions in order to increase their fitness values (i.e., the values of the function which is optimized). The selection of trial solutions for reproduction depends on their value of the objective function. The best trial solution is reproduced in each generation. The remaining trial solutions are recombined or mutated by applying the other operators (for details compare Sekhon and Mebane 1998, p. 192). That way, a new population (generation) results that “tends to be, on average, better than its predecessor” (Mebane and Sekhon 2011, p. 3). The following pseudo-code (adapted from Eiben and Smith 2003, p. 16) presents the general procedure:

Since the evolutionary algorithm is basically a genetic one where the code-strings are floating-point vectors (see, e.g., Ali and Törn 2004 for a description of genetic algorithms) its performance strongly depends on the population size (option pop.size) which must to be sufficiently large (Mebane and Sekhon 2011). In our application we chose $pop.size=3000$ which leads to relatively high computing times but increases the probability of finding the global optimum. The maximum number of generations was set to $max.generations=300$. The algorithm stops if the objective function is not improved anymore in a fixed number (wait.generations) of generations. We set $wait.generations=10$. On average 25 generations were required to solve the optimization problems. The option Domains comprises lower and upper bounds for each variable which correspond to the observed price range for each price variable in our applications. Finally, $boundary.enforcement=2$ ensures candidates that lie within the bounds specified in Domains. The R code is available from the authors upon request.

In our empirical application, optimization is done separately for each of the 89 weeks as well as separately for each store/segment in case of the heterogeneous models. The following table summarizes the size of the respective optimization problems for each model:

Model	Number of problems to solve
PHomSM, FHomSM	89 weeks $=$ 89 problems to solve
	1 problem $=$ 1 run for all brands and all stores
PLCSM (2)	89 weeks $\times$ 2 segments $=178$ problems to solve
	1 problem $=$ 1 run for all brands and all stores of 1 segment
PHBSM, FHBSM	89 weeks $\times$ 81 stores $= 7209$ problems to solve
	1 problem $=$ 1 run for all brands (in 1 store)