1) Product

A product encompasses anything offered to a market to satisfy a want or need. This includes physical goods, services, experiences, events, persons, places, organizations, ideas, or properties [8] . The key components of a product are:

Theories of Product

Three major theories provide insights into product dynamics:

a) Product Life Cycle (PLC) Theory: Suggests that products evolve through stages—introduction, growth, maturity, and decline [8] .

b) Feature-Based Segmentation Theory: Proposes that products are designed to meet specific customer needs [5] .

c) Brand Equity Theory: Highlights the importance of strong branding in fostering customer loyalty and differentiation [12] .

Empirical Evidence

Empirical evidence underscores the role of product-related factors in consumer behavior:

2) Place

Place or distribution refers to the processes involved in making a product or service accessible to the target market through various channels, intermediaries, and logistics systems [15] . The key components of distribution include:

Theories of Distribution

Several theories provide a framework for understanding distribution dynamics:

a) Channel Management Theory: Posits that effective distribution channels enhance product availability [15] .

b) Logistics Management Theory: Suggests that efficient logistics lead to improved customer satisfaction and streamlined operations [16] .

Marketing Coverage Theory: Highlights that distribution strategies are shaped by the size and complexity of the target market [8] .

Empirical Evidence

3) Price

Price represents the amount of money customers must pay to acquire a product or service [8] . It plays a critical role in the marketing mix and encompasses several key features:

Theories of Price

Three prominent theories guide the understanding of pricing:

a) Price Elasticity Theory: Explains that changes in price significantly impact demand [19] .

b) Value-Based Pricing Theory: Suggests that prices should reflect the product’s perceived value to customers [20] .

c) Pricing Strategy Theory: Highlights how pricing strategies affect customer behavior and revenue generation [8] .

Empirical Evidence

4) Promotion

Promotion encompasses the activities and strategies used to communicate the value and benefits of a product, service, or idea to potential customers, aiming to influence their purchasing decisions and behaviors [23] . The components of promotion include:

5) Theories of Promotion

a) Promotion Mix Theory: Combines advertising, sales promotion, public relations, and personal selling as key elements of the promotion strategy [8] .

b) Advertising Effectiveness Theory: Highlights how advertising shapes customer attitudes and behaviors [24] .

c) Diffusion of Innovations Theory: Explains how innovations spread through promotion and effective communication [25] .

6) Empirical Evidence

a) Studies in Nigeria

b) Studies in Other African Countries

c) Studies Using Geometric Programming

Recent studies have applied geometric programming to optimize the marketing mix:

Common Methodologies Across Studies

The studies cited above share several methodological features, including:

Similarities and Differences

Contribution of This Study

This study offers a unique contribution by integrating multiple regression and geometric programming to provide a comprehensive understanding of the relationships between marketing mix elements and performance in the Nigerian brewing industry.

This research uses secondary data collected over a 10-year period from 2013 to 2022, which is considered a long-term period for business analysis [29] . The data was primarily extracted from the annual reports of Nigerian Breweries plc, with the exception of price data, which was directly obtained from the company. In this context, “price” refers to the average price of all drinks produced by NBL per carton/crate. All other variables such as Product, Promotion and Place are represented as Production, Promotion and Distribution Cost respectively and are quantified in billions of naira.

To effectively describe the data, both graphical methods and correlation analysis will be employed. The following techniques will be utilized:

Following the descriptive analysis, as indicated above, inferential statistics will be applied using the Ordinary Least Squares (OLS) regression method. This method estimates the relationships among variables by minimizing the sum of squared differences between observed and predicted values. OLS regression is particularly effective in identifying how changes in independent variables affect a dependent variable.

The Model

$Y_t={\beta }_o+{\beta }_1{X}_1{}_t+{\beta }_2{X}_2{}_t+{\beta }_3{X}_3{}_t+{\beta }_4{X}_4{}_t+{\epsilon }_t$ (1)

$t=1,2,3,\cdots ,10$.

where:

$Y_t$ = Revenue

$X_1$ = Product

$X_2$ = Promotion

$X_3$ = Place

$X_4$ = Price

${\beta }_o$ = Intercept

${\beta }_i;i=1,2,3,4$ = slope coefficients of $X_1,X_2,X_3,X_4$ respectively.

$\epsilon$ = error associated with $Y$

$t$ = time (in years).

Assumptions:

i. ${\epsilon }_t$ follows  $N\left(0,{\sigma }^2\right)$

ii. $E\left({\epsilon }_t{\epsilon }_{t^{\prime }}\right)=0;t\ne t^{\prime }$

iii. $E\left({\epsilon }_t|X_i\right)=0$

iv. No multicollinearity.

Normality Assumption

The normality assumption is crucial for mathematical convenience [33] model validity [34] and prediction accuracy [35] . Violating the normality assumption can lead to several consequences, including:

To test the normality assumption, [37] will be employed, using the following formula:

$W=\frac{{{\displaystyle \sum }}_{i=1}^n{\left(x_i-\bar{x}\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(x_i-\bar{x}\right)}^2+{{\displaystyle \sum }}_{i=1}^n{\left(x_i-x_i^{\ast }\right)}^2}$ (2)

where

$x_i$ = individual data points

$\bar{x}$ = sample mean

$x_i^{\ast }$ = expected normal order statistics (from a standard normal distribution).

If the hypothesis of normality is tested at a 5% level of significance, and the p-value is greater than 0.05, we accept the hypothesis that the data is normally distributed; otherwise, we reject it.

Homoscedasticity and Autocorrelation Assumptions

The homoscedasticity assumption implies that all error terms, ${\epsilon }_t$ have the same variance. The presence of heteroskedasticity does not destroy the unbiasedness and consistency properties of the Ordinary Least Squares (OLS) estimators; however, it results in these estimators being inefficient. Consequently, the t-tests and F-tests based on OLS estimators can yield misleading results, leading to erroneous conclusions [38] . In this research, homoscedasticity will be addressed using White’s Robust Standard Errors. In STATA software used for the regression analysis, the model is adjusted to account for heteroskedasticity by use of option, robust (r), in the regress command [39] . Autocorrelation refers to the situation where the disturbance term related to any observation is influenced by the disturbance term of another observation. While autocorrelation does not affect the unbiasedness, consistency, and asymptotic normality of OLS estimators, it does render them inefficient. This inefficiency makes the application of t-tests, F-tests, and Chi-square tests inappropriate [38] . In this work, after the regression with r, autocorrelation will be tested using Durbin-Watson (DW) d Statistic given in [38] as:

$d=\frac{{{\displaystyle \sum }}_{t=2}^n{\left(e_t-e_{t-1}\right)}^2}{{{\displaystyle \sum }}_{t=1}^n{e}_t^2}$ (3a)

where

d = DW statistic

$e$ = estimated residuals

$t$= time

$0<d<4$

If d = 2, then, there is no autocorrelation; less than 2 indicates positive autocorrelation; while greater than 2 indicates negative autocorrelation. To ensure robust statistical inference in the presence of both heteroskedasticity and autocorrelation, the application of Heteroskedasticity and Autocorrelation Consistent (HAC) or Newey-West Standard Errors is essential [38] . These standard errors provide a reliable adjustment to conventional estimates when dealing with time series data that may exhibit serial correlation and non-constant variance.

The Newey-West formula for estimating HAC standard errors is outlined in [33] as:

$S_{\ast }=S_o+\frac{1}{T}\underset{j=1}{\overset{L}{{\displaystyle \sum }}}\underset{t=j+1}{\overset{T}{{\displaystyle \sum }}}\left(1-\frac{j}{L+1}\right)e_t{e}_{t-j}\left[x_t{x^{\prime }}_{t-j}+x_{t-j}{x^{\prime }}_t\right]$ (3b)

where: $S_{\ast }$ = HAC standard errors

$S_o=\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{e}_i^2{x}_i{x^{\prime }}_i}$ (4)

T = Number of observations in  a time series data

L = maximum lag, which must be determined in advance to be large enough

that autocorrelations at lag longer than L are small enough to ignore

$n=T$

Multicollinearity Assumption

Multicollinearity arises when there is an exact or near-exact linear relationship among the independent variables. In a business context, it reflects how many economic series tend to move together over time due to shared influences, such as the overall trade cycle or prevailing economic conditions [40] . In this study, multicollinearity is expected to be present, given that data collection spans a period marked by significant political shifts and associated economic policies in Nigeria. Although OLS estimators are BLUE (Best Linear Unbiased Estimators), multicollinearity can increase their variances and covariances, complicating precise estimation. As a result, one or more regressors may have statistically insignificant t-values.

Multicollinearity is less problematic if R² is high and the regression coefficients remain individually significant, indicated by high t-values [38] . In marketing research, multicollinearity can reveal equally plausible media-buying options [41] . Various methods are available to assess multicollinearity, including Tolerance (TOL), Variance Inflation Factor (VIF), and Artificial Neural Networks (ANN) [42] . This study will apply the VIF, as detailed by [38] :

$\mathrm{var}\left({\hat{\beta }}_j\right)=\frac{{\sigma }^2}{{{\displaystyle \sum }}_{j=1}^k{\left(X_j-\overline{X_j}\right)}^2}VI{F}_j$ (5)

where

${\hat{\beta }}_j$ = estimated partial regression coefficient of the regressor $X_j$

$VI{F}_j=\frac{1}{1-R_j^2}$

$R_j^2$ = $R^2$ in  the regression of $X_j$ on the remaining $\left(k-1\right)$ regressors

$k$ = number of regressors.

The VIF for a variable indicates the degree to which its variance is inflated due to non-orthogonality with other variables in the model [33] . A VIF greater than 10 suggests potential issues with multicollinearity (Torres-Reyna, n.d.). While no definitive solutions exist for eliminating multicollinearity, several approaches are commonly used, including: (i) using prior information, (ii) combining cross-sectional and time-series data, (iii) omitting highly collinear variables, (iv) transforming the data, and (v) obtaining additional data [38] .

Omni-Directional Variance Test (OVTEST)

The third assumption, $E\left({\epsilon }_t|X_i\right)=0$, indicates that the error term and each independent variable in the model are uncorrelated. This assumption relates to testing for omitted variable bias, which can be evaluated using the Ramsey RESET test [39] . The test procedure is as follows:

Regression Equation:

$Y={\beta }_o+{\beta }_1{X}_1+{\beta }_2{X}_2+{\beta }_3{X}_3+{\beta }_4{X}_4+\epsilon$ (6)

Auxiliary Equation: 

$Y={\beta }_o+{\beta }_1{X}_1+{\beta }_2{X}_2+{\beta }_3{X}_3+{\beta }_4{X}_4+{\beta }_5{\hat{Y}}^2+{\beta }_6{\hat{Y}}^3+\epsilon$ (7)

OV Statistic = $n{R}^2$ (from auxiliary equation).

The test statistic follows an approximate F distribution or an asymptotic Chi-squared distribution, with degrees of freedom equal to the number of auxiliary terms added (e.g., 2 in (7). The null hypothesis (Ho) is accepted if the p-value is greater than the specified level of significance, indicating that the model is likely correctly specified [43] .

Link Test

Similar to the OVTEST, the Link Test is a diagnostic tool used to detect model misspeci-fication [44] . The procedure is as follows:

Regression Equation:

$Y={\beta }_o+{\beta }_1{X}_1+{\beta }_2{X}_2+{\beta }_3{X}_3+{\beta }_4{X}_4+\epsilon$

Auxiliary Equation:

$Y={\beta }_o+{\beta }_1{X}_1+{\beta }_2{X}_2+{\beta }_3{X}_3+{\beta }_4{X}_4+{\beta }_5\hat{Y}+{\beta }_6{\hat{Y}}^2+\epsilon$ (8)

Link Statistic = $n{R}^2$ (from auxiliary equation). The Link Test statistic follows an approximate F distribution or an asymptotic Chi-squared distribution, with degrees of freedom equal to the number of auxiliary terms added. We accept Ho if the p-value is greater than the specified level of significance, suggesting that the model is correctly specified.

The primary distinctions between the Ramsey RESET test and the Link Test are as follows:

a) The Ramsey RESET test is more general, addressing a broader range of misspecification types, while the Link Test specifically addresses link function misspecification.

b) The Ramsey RESET test includes cubed fitted values among the auxiliary terms, whereas the Link Test does not.

These assumptions will be explored and investigated in section 4.0

Optimization model in marketing mix problem takes the form $f\left(x\right)=S\left(x\right)-C\left(x\right)$, see [45] , and [46] where $f\left(x\right)$ is the profit, $S\left(t\right)$ is the sales and $C\left(t\right)$ is the cost. The function is called signomial function [47] because it allows the cost coefficient of the function to have negative values, unlike posynomial function that allows only the positive cost coefficient; when the function is optimized, we have signomial programming, which is an extension of geometric (posynomial) programming, see [48] and [49] . In this study, we do not intend to continue with signomial programming because it does not attain global optimal solution, rather, we intend to convert to geometric programming, with the assurance of obtaining a global optimal solution. The attainment of global optimal solution is achieved by the constraint equation being bounded above by unity in the case of the primal solution to the problem. In the case of the dual solution and by the fundamental duality theory, the maximization of the dual problem program is the same as the minimizing of the primal problem [50] and the dual program is a concave function constrained by linear constraints, which is easier to handle. Therefore, instead of minimizing the primal objective function [51] , which is a convex function and more difficult to work on, we can maximize the dual objective function subject to orthogonality and normality conditions, which are combined to form a non-homogenous system of linear. Therefore, we maximize the profit function to have:

$\text{Maximize}f\left(y\right)={\displaystyle {\prod }_{k=0}^m{\displaystyle {\prod }_{j=1}^{n_k}{\left(\frac{C_{kj}}{y_{kj}}{\displaystyle {\sum }_{j=1}^{n_k}{y}_{kj}}\right)}^{y_{kj}}}}$ (9)

Subject to:

${\displaystyle {\sum }_{j=1}^{n_0}{y}_0{}_j}=1$ (10)

${\displaystyle {\sum }_{k=0}^m{\displaystyle {\sum }_{j=1}^{n_k}{a}_{kij}{y}_k{}_j}}=0$; $i=1,2,\cdots ,n$ (11)

where $y_0{}_j$ = the dual decision variable from the objective function, $y_k{}_j$ = dual decision variables from the constraint equation, m = number of variables, $n_k$ = number of terms in the constraint equation, n₀ = number of terms in the objective function. The necessary condition for optimality is that the constraint equation must be bounded above by unity and the sufficient condition for optimality are the normality and orthogonality conditions. Equations (10) and (11) are combined to form equation (12);

$Ay=B$ (12)

where A is an ( $m\times n$) coefficient matrix, y is a vector of dual decision variables of order ( $n\times 1$) and B is a vector of constants of order ( $m\times 1$), see [52] . At stationary point the optimal solution becomes:

$f\left(x^{*}\right)=f\left(y^{*}\right)={\displaystyle {\prod }_{k=0}^m{\displaystyle {\prod }_{j=1}^{n_k}{\left(\frac{C_{kj}}{y_{kj}^{*}}{\displaystyle {\sum }_{i=1}^{n_k}{y}_{kj}^{*}}\right)}^{y_{kj}^{*}}}}$ (13)

Determination of Optimal Weights of the Dual Decision Variables

We determine the optimal weight of the dual decision variable from equation (12) as follows:

$y^{*}=A^{-}B$ (14)

$y^{*}=A^{-1}B$ (15)

Equation (14) is a case where the exponent matrix is rectangular while equation (15) is a case where the exponent matrix is a square. Using R-program, we obtain $y^{*}$ as follows:

$A<\text{-matrix}\left(c\left(a_{11},a_{12},\cdots ,a_{1n},a_{21},a_{22},\cdots ,a_{2n},a_{m1},a_{m2},\cdots ,a_{mn}\right),\text{nrow}=m,\text{ncol}=n\right)$ (16)

$R<\text{-}t\left(A\right)$ (17)

$L<\text{-}R\%\ast \%A$ (18)

$solve.default\left(L\right)\text{-}>P$ (19)

$T<\text{-}P\%\ast \%R$ (20)

$B<\text{-matrix}\left(c\left(0,0,0,0,0,1\right),\text{nrow}=m,\text{ncol}=1\right)$ (21)

$y<\text{-}\left(T\right)\%\ast \%B$ (22)

$y^{*}=\left[\begin{array}{c}{y}_1^{*}\\ ⋮\\ y_n^{*}\end{array}\right]$ (23)

where y* is the optimal dual decision variables, is the Moore-Penrose generalized inverse, see [52] .

Determination of Optimal Objective Function f (y*)

From equation (13), we determine the optimal value of the objective function using R-program as follows:

$f\left(y^{*}\right)={\left(\frac{C_1}{y_1^{*}}\right)}^{y_1^{*}}\times {\left(\frac{C_1}{y_2^{*}}\right)}^{y_2^{*}}\times \cdots \times {\left(y_3^{*}+y_4^{*}\right)}^{y_3^{*}+y_4^{*}}\times \cdots \times {\left(y_{n-1}^{*}+y_n^{*}\right)}^{y_{n-1}^{*}+y_n^{*}}$ (24)

where $f\left(y^{*}\right)$ is the optimal value of the objective function which correspond to the optimal profit for the Nigerian brewery PLC.

The optimal primal decision variables are obtained from the relationship in equation (25), see [53] ;

$\frac{y^{*}f\left(x^{*}\right)}{C_{kj}}={\displaystyle {\prod }_{i=1}^m{x}^{a_{ij}}}$ (25)

Determination of Optimal Weights of the Primal Decision Variables

From equation (25), we have;

$C_j{\displaystyle {\prod }_{i=1}^m{\left(x_i\right)}^{a_{ij}}}=y_j^{*}f\left(x^{*}\right)$ (26)

Taking the Ln of both sides of equation (27), we have

$\frac{y_j^{*}f\left(x^{*}\right)}{C_j}={\left(x_1\right)}^{a_{1j}}\cdot {\left(x\right)}^{a_{2j}}\cdots {\left(x\right)}^{a_{mj}}$ (27)

Taking the Ln of both sides of equation (27), we have

$\ln P=a_{1j}\ln x_1+a_{2j}\ln x_2+\cdots +a_{mj}\ln x_n$; $i=1,\cdots ,m$ (28)

Let

$w_i=\ln x_i$ (29)

we have

$\ln P=a_{1j}{w}_1+a_{2j}{w}_2+\cdots +a_{mj}{w}_n$ (30)

where is the vector of constant, B. We obtain the optimal weights of the primal decision variables by taking the exponential of equation (29) to have;

$x_i^{\ast }={\text{e}}^{w^{\ast }}$ (31)

where the elements of the column vector w* are subset of real number R, [51] [53] .

By employing these methodologies, this research aims to provide a comprehensive analysis of financial metrics related to Nigerian Breweries plc, contributing valuable insights into its operational performance over the specified period.

All statistical analyses for this study will be conducted using STATA 15.0 and R software.

Table 1 presents NBL’s ten-year financial statement on revenue, product, promotion, place, and price. The data was collected for 10 years (2013-2022). The data has been adequately described in Section 3.1. The graph of the data as well as the correlation matrix will further help describe the data.

Figure 1 presents the graph matrix of revenue and marketing mix variables. All covariates exhibit a strong linear relationship with revenue. Particularly, the graph of price versus place has the same shape with price versus revenue which is a bit concave downward suggesting diminishing return. As price increases, the marginal effect on place (distribution) and revenue decreases. Also, the graph of place versus promotion shows initial concave downwards slope and later straightened out to a straight line. This implies that the concave downward slope at the beginning indicates that, at low levels of promotion, increases in place (distribution) have diminishing returns. As promotion increases, the relationship becomes more linear, suggesting consistent returns. The linearity between place and product suggests consistent returns; increases in product are directly proportional to increases in place (distribution). These phenomena are more explained using the correlation coefficient.

Table 2 presents multiple correlation of data. Below each coefficient is the p-value. The result shows that all the covariates are highly significantly linearly correlated with revenue even at 1% level of significance. The covariates are also significantly linearly correlated among themselves. In particular, the strong positive correlation (86%) between price and place and between price and revenue indicate a significant relationship between them even though there appears some quadratic effect as discussed by the graph matrix above. The same can be said about place versus promotion (r = 96%) and product versus place (r = 98%). This high linear relationship among the covariates may be problematic in the regression analysis that follows.

Table 3 presents multiple regression of revenue and marketing mix variables. With an R² value of 99.28%, one expects all the covariates to be significant but promotion did not explain revenue (p = 0.860) even when its correlation coefficient with revenue is 93.78%. This calls for further examination of the result, first in terms of multicollinearity, which the correlation matrix indicated above and which result is presented.

Table 4 presents VIF of marketing mix variables. Clearly, the result shows a very high degree of multicollinearity. As was said earlier, a VIF value of more than 10.0 is not acceptable. So, all the variables failed the VIF test. We therefore take some steps, as explained in section 3, to remove the multicollinearity effect from the data.

After applying the various procedures for transforming a multicollinearity set of independent variables, and noting that “place” has the highest VIF value, and noting also the preceding discussions about its relationship with other covariates, the ratio transformation of the covariates coupled with the log transformation of the response gave the most meaningful result and it is presented on Table 5 .

Table 5 presents multiple linear regression of Log(revenue) and ratios of covariates, where:

Log revenue = log of revenue i.e. Log (revenue)

prodR2 = product/place

promoR2 = promotion/place

priceR2 = price/place

Normalizing the other marketing mix variables with “place” allows comparison of their relative effects. From the above results, 97% of the total variation in log revenue is explained by the covariates. The covariates are significant at 1% level of significance except “promoR2” which is significant at 10% level of significance. The price coefficient also has the required negative sign suggesting that the higher the price, the lesser the quantity of the products that are purchased. The coefficients of the covariates in the table represent elasticities, measuring percentage changes. The coefficient of the constant term (19.8902) shows the expected log (revenue) when all the covariates are zero. The prodR2 coefficient of 0.3484 means that a 1% increase in production relative to place leads to a 0.3484% increase in log (revenue). The promoR2 coefficient of 0.4407 means that a 1% increase in promotion relative to place leads to a 0.4407% increase in log (revenue). The priceR2 coefficient of −13565.41 means that a 1% increase in price relative to place leads to a 13565.41% decrease in log (revenue). The implication of the findings above are that (1) increasing production or promotion efforts relative to distribution (place) may lead to moderate revenue growth and (2) increasing prices relative to distribution may significantly harm revenue. Checking if there is multicollinearity effect in the result above, the following VIF table is produced in Table 6 .

Table 6 presents VIF of covariates ratios. We observe that the ratio transformation has completely eliminated the multicollinearity effect in the covariates. Next, we check if the data satisfies the normality assumption of the OLS regression by using the Shapiro-Wilk test, which is presented in Table 7 .

Table 7 presents Shapiro-Wilk Test for Normality. The test, with a p-value of 0.4311 shows that the data follows a normal distribution at 5% level of significance.

We next show that the model is adequate by carrying out omitted variables test (ovtest) and the link test. We first show the result of the ovtest as in Table 8 .

Table 8 presents OVTEST for omitted variables. The result shows that there are no omitted variables at 5% level of significance. The link test result is in Table 9 .

Table 9 presents Link Test for model adequacy. The link test result shows that the model is adequate since the p-value of “_ hatsq” is 0.565. It is noted that the problem of heterogeneity of variance of the model was ab initio taken care of by using robust standard errors. We lastly show if there is autocorrelation in the data as outlined in 3.3b. The result is:

Durbin-Watson d-statistic (4, 10) = 2.087351

Interpretation:

i. (4, 10) represent the number of regressors (4) and the number of observations (10)

ii. 2.087351 is the value of the DW d-statistic.

iii. since the value of DW d-statistic is approximately 2.0, then, there is no autocorrelation. We therefore adopt the model of Table 5 as the best model.

Table 10 presents proportion of contribution of the marketing mix in billions of naira. The Table present marketing mix variables as: Product (A), Promotion (B), Place (C) and Price (D); from where we optimize the profit from the interaction of these marketing mix variables over the years in question.

In developing the marketing mix model for NBL PLC, we define the following marketing mix variables as: Product (A), Promotion (B), Place (C) and Price (D). Our interest is to optimize the profit from the interaction of these marketing mix variables over the years in question, we also define Revenue (R) as the totality of income accrued from the business over the period of time. Each of the variables are randomly assigned values as follows: Let (A) = X₁, (B), = X₂, (C) = X₃, (D) = X₄, and (R) = Totality of sales, which comprise the entire marketing mix variables. Therefore, we proceed as follows:

Let $f\left(x\right)$ be the profit, then

$f\left(x\right)={\displaystyle \sum R\left(x\right)}-{\displaystyle \sum C\left(t\right)}$ (32)

$f\left(x\right)=3.4851x_1{x}_2{x}_3{x}_4-1.7154x_1-0.2887x_2-0.7234x_3-0.00007034x_4$ (33)

Hence, we can write equation (33) as a signomial programming model

$\text{Max}f\left(x\right)=3.4851x_1{x}_2{x}_3{x}_4-1.7154x_1-0.2887x_2-0.7234x_3-0.00007034x_4$ (34)

Equation (34) is an unconstrained signomial model, which does not produce a global optimal solution. Therefore, we convert the problem to geometric programming problem by reformulation as follows;

Let $u\left(x\right)=3.4851x_1{x}_2{x}_3{x}_4$ (35)

$f\left(t\right)=-\left[1.7154x_1+0.2887x_2+0.7234x_3+0.00007034x_4\right]$ (36)

Therefore, we minimize the inverse of the objective function subject to the reformulated constraint;

Minimize $x_0^{-1}$ (37)

Subject to $\frac{x_0}{u\left(x\right)}+\frac{f\left(x\right)}{u\left(x\right)}\le 1$ (38)

Equations (37) and (38) are equivalent to the geometric programming primal problem, see [32] . The geometric programming model for the problem is now written as:

Minimize $f_0\left(x\right)=x_0^{-1}$ (39)

Subject to $g_1\left(x\right)$:

$\begin{array}{l}0.2869x_0{x}_1^{-1}{x}_2^{-1}{x}_3^{-1}{x}_4^{-1}+0.4922x_2^{-1}{x}_3^{-1}{x}_4^{-1}+0.08283x_1^{-1}{x}_3^{-1}{x}_4^{-1}\\ +0.2076x_1^{-1}{x}_2^{-1}{x}_4^{-1}+0.0000202x_1^{-1}{x}_2^{-1}{x}_3^{-1}\le 1\end{array}$ (40)

Subject to equations (10) and (11) (41)

Equations (39) to (40) are the standard constrained geometric programming model for the marketing mix problem.

The degree of difficulty of the problem is:

$K=n-\left(m+1\right)=6-\left(5+1\right)=0$; where k = degree of difficulty, n= number of terms, m = number of variables. This problem has a zero degree of difficulty and therefore has a unique solution. But since the dual problem is a concave function constrained by linear constraints, see equation (41), and at stationary point, the Min f(x) = Max f(y); then, we apply equation (13). Forming orthogonality and normality, we apply equation (41) and since the problem has a unique solution, we determine the optimal dual decision variables from equation (15).

Forming the orthogonality and normality conditions for the dual decision variables from equation (41), we have

$-y_1+y_2+0y_3+0y_4+0y_5+0y_6=0$

$0y_1-y_2+0y_3-y_4-y_5-y_6=0$

$0y_1-y_2+0y_3+0y_4-y_5-y_6=0$

$0y_1-y_2-y_3-y_4+0y_5-y_6=0$

$0y_1-y_2-y_3-y_4-y_5+0y_6=0$

$y_1+0y_2+0y_3+0y_4+0y_5+0y_6=1$

$\left[\begin{array}{cccccc}-1& 1& 0& 0& 0& 0\\ 0& -1& 0& -1& -1& -1\\ 0& -1& 0& 0& -1& -1\\ 0& -1& -1& -1& 0& -1\\ 0& -1& -1& -1& -1& 0\\ 1& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\\ y_4\\ y_5\\ y_6\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 1\end{array}\right]$

$\begin{array}{l}A<\text{-matrix}(c(-1,0,0,0,0,1,1,-1,-1,-1,-1,0,0,0,0,-1,-1,0,0,-1,0,-1,-1,\\ 0,0,-1,-1,0,-1,0,0,-1,-1,-1,0,0),\text{nrow}=6,\text{ncol}=6)\end{array}$

$R<\text{-}t\left(A\right)$

$L<\text{-}R\%\ast \%A$

$solve.default\left(L\right)\text{-}>P$

$T<\text{-}P\%\ast \%R$

$B<\text{-matrix}\left(c\left(0,0,0,0,0,1\right),\text{nrow}=6,\text{ncol}=1\right)$

$y<\text{-}T\%\ast \%B$

$y^{\ast }=\left[\begin{array}{c}1\\ 1\\ -0.25\\ 0.0\\ -0.5\\ -0.5\end{array}\right]$

This is the optimal values of the dual decision variables. But a diagnostic check showed that $y_4$ does not contribute significantly to the objective function, rather, it leads to degenerate solution; hence, should be removed from the problem, see [54] . The new problem has a rectangular matrix and therefore result into a negative degree of difficulty problem, see [51] . Hence, we apply equation (3.14) and the solution becomes:

$\begin{array}{l}A<\text{-matrix}(c(-1,0,0,0,1,1,-1,-1,-1,0,0,0,0,-1,0,0,-1,0,-1,0,0,\\ -1,-1,-1,0,0,-1,-1,0,0),\text{nrow}=6,\text{ncol}=5)\end{array}$

$R<\text{-}t\left(A\right)$

$L<\text{-}R\%*\%A$

$solve.default\left(L\right)\text{-}>P$

$T<\text{-}P\%\ast \%R$

$B<\text{-matrix}\left(c\left(0,0,0,0,0,1\right),\text{nrow}=6,\text{ncol}=1\right)$

$y<\text{-}\left(T\right)\%\ast \%B$

$y^{\ast }=\left[\begin{array}{c}0.6452\\ -0.4839\\ 0.6774\\ -0.2258\\ 0.3548\end{array}\right]$

The signs on the optimal dual decision variables only shows the nature of interaction while the actual values measure the elasticity between the marketing decision variables, even though our interest is to optimize profit from the marketing mix variables. These signs do not affect the optimal profit [32] .

From equation (24), we compute for the optimal objective function as follows:

$\begin{array}{c}f\left(y^{*}\right)={\left(\frac{1}{0.6452}\right)}^{0.6452}\ast {\left(\frac{0.2869}{0.4839}\right)}^{0.4839}\ast {\left(\frac{0.4922}{0.6774}\right)}^{0.6774}\\ \ast {\left(\frac{0.2076}{0.2258}\right)}^{0.2258}\ast {\left(\frac{0.0000202}{0.3548}\right)}^{0.3548}\ast {\left(1.7419\right)}^{1.7419}\end{array}$ $\therefore f\left(y^{*}\right)=0.06676854=66768540\text{naira}$

The above is the optimal objective function, which is the optimal profit the business made over the period under study.

From equation (27), we calculate the optimal primal decision variables as follows:

$17.5786=-\ln x_0+0\ln x_1+0\ln x_2+0\ln x_3+0\ln x_4$

$18.5395=\ln x_0-\ln x_1-\ln x_2-\ln x_3-\ln x_4$

$18.3361=0\ln x_0+0\ln x_1-\ln x_2-\ln x_3-\ln x_4$

$18.1008=0\ln x_0-\ln x_1-\ln x_2+0\ln x_3-\ln x_4$

$27.7904=0\ln x_0-\ln x_1-\ln x_2-\ln x_3-0\ln x_4$

Applying equation (3.29) in conjunction with exact solution to GP problems [53] , we have:

$\left[\begin{array}{ccccc}-1& 0& 0& 0& 0\\ 1& -1& -1& -1& -1\\ 0& 0& -1& -1& -1\\ 0& -1& -1& 0& -1\end{array}\right]\left[\begin{array}{c}{w}_0\\ w_1\\ w_2\\ w_3\\ w_4\end{array}\right]=\left[\begin{array}{c}17.5786\\ 18.5395\\ 18.3361\\ 18.1008\end{array}\right]$

Applying equation (14), we have

$A<\text{-matrix}\left(c\left(-1,1,0,0,0,-1,0-1,0,-1,-1,-1,0,-1,-1,-1\right),\text{nrow}=4,\text{ncol}=5\right)$

$R<\text{-}t\left(A\right)$

$L<\text{-}R\%\ast \%A$

$solve.default\left(L\right)\text{-}>P$

$T<\text{-}P\%\ast \%R$

$B<\text{-matrix}\left(c\left(17.5786,18.5395,18.3361,181008\right),\text{nrow}=4,\text{ncol}=1\right)$

$y<\text{-}\left(T\right)\%\ast \%B$

$w^{*}=\left[\begin{array}{c}-18.5530\\ 9.0985\\ -35.1064\\ -12.4151\\ 2.1387\end{array}\right]$

Applying equation (31), we have

$x^{*}=\left[\begin{array}{c}8.761\\ 8941.843\\ 5.669\\ 4.057\\ 8.489\end{array}\right]$

These are the optimal weights of the primal decision variables.

Analysis of NBL’s marketing mix data revealed high multicollinearity, as indicated by the VIF results ( Table 4 ). After testing various transformation methods, using “place” as the normalization variable provided the best results, allowing the coefficients to measure the impact of changes in production, promotion, and price relative to distribution. Key findings include:

(i) Product/Price Ratio: This suggests that increasing production relative to distribution capacity may modestly increase revenue (p = 0.005).

(ii) Promotion/Place Ratio: This also suggests that increasing promotional efforts relative to distribution channels may enhance revenue (p = 0.094).

(iii) Price/Place Ratio: This implies that raising prices relative to distribution may significantly harm revenue (p = 0.000).

Geometric programming analysis showed that Nigerian Breweries Plc earned a total profit of 66,768,540 naira. Among the marketing mix components, “product” had the highest contribution (8941.84) to profit, followed by “price” (8.45), “promotion” (5.67), and “place” (4.06). In other words, Product contributes 99.80% to profit, Price 0.09%, Promotion 0.06% and Place 0.05%. Relative to Place, Product contributes 220, 242.36%, Price 208.13%, and Promotion 139.66%.

The price/place relationship which shows a significant negative impact on revenue, with a large negative coefficient (−13,656.41, p = 0.000) indicates:

The analysis also shows that promotion/place and product/place ratios positively impact revenue, with promotion having a larger coefficient but less statistical significance due to initial concavity in the promotion/place graph. This concave relationship suggests diminishing returns in early promotion stages, making the effect less predictable.

The findings suggest that distribution capacity is crucial in balancing other marketing mix elements, as distribution limitations may restrict the impact of production, promotion, and pricing. Overall, to optimize marketing strategies relative to distribution, NBL and other similar industries should consider:

a) Production Alignment: Match production capacity to distribution capabilities.

b) Promotion Strategy: Tailor promotions to distribution channels, such as in-store promotions.

c) Pricing Strategy: Consider distribution costs when setting prices.