Assessment and Selection of Mathematical Trends to Increase the Effectiveness of Product Sales Strategy

Abstract

This paper explores the application of a mathematical trend model to analyze product sales performance. A logistic trend model was utilized to analyze product sales performance, employing monthly sales data collected over three years. The model assessed impacts across various phases of the product life cycle. Significant sales trends were identified and modeled from historical data, demonstrating how sales dynamics mirror broader economic phenomena and consumer behaviors. In addition to logistic trends, linear and quadratic trends were also evaluated. To assess the significance of the sales trends for three products, the Mann–Kendall test was applied. The results indicate a statistically significant positive trend in the sales of product A. For evaluating the quality of data fit in model comparison, the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) were deemed appropriate. The analysis revealed that the logistic model effectively delineates different sales phases—from introduction to maturity—and highlights opportunities for optimizing strategic sales planning and customer satisfaction in alignment with market demands. The study’s findings are crucial for businesses seeking to enhance product lifecycle management and boost sales forecasting precision.

1. Introduction

The rapidly evolving advanced markets and unpredictable changes in customer behavior continuously reshape customer needs. Customer information plays a crucial role, as it contributes to the development of commercial products and services. The overall organizational growth of a company and its responses to insights regarding the needs of both current and potential customers must focus on anticipating future customer demands as well as enhancing the added value of products and services. Aligning strategic behavior with target market selection is a fundamental component of any organization.

In the face of rapid advancements and the unpredictability of modern markets, the introduction of predictive models like the Gompertz curve has become imperative. Unforeseen economic scenarios, such as changes in VAT, various global and local crises, and evolving geopolitical landscapes, significantly influence consumer behavior. These shifts necessitate robust analytical tools to navigate and predict market dynamics effectively. Concurrently, continuous innovations within product technologies further compel the adaptation of advanced predictive models. The Gompertz model, with its ability to forecast growth and saturation trends, provides essential insights that help businesses stay resilient and proactive in a rapidly changing economic environment.

The Gompertz model is a widely used framework for describing growth phenomena characterized by an initial period of exponential growth that gradually slows as it approaches a maximum saturation level. This model is distinctive for its asymmetric, sigmoidal shape, which captures the often observed slowing of growth as a population approaches its carrying capacity or other limiting factors [1]. First introduced by Benjamin Gompertz in 1825, the model has since been applied across a variety of disciplines, from biological processes to technological advancements and market trends [2].

The Gompertz model is vital for aligning sales strategies with market demands, forecasting demand for new products, and managing life cycles of existing ones. By predicting stages of product adoption and saturation, it helps businesses optimize strategies for maximum impact. Implementations such as quality support, rapid problem resolution, and personalized services like targeted recommendations and customizable options, enhance customer satisfaction and loyalty. Informed by the Gompertz model, these strategies exceed customer expectations, boosting brand loyalty and market share.

One of the key advantages of the Gompertz model is its flexibility in modeling complex growth dynamics with relatively few parameters. As [3] explains, the model “provides robust long-term predictions,” even when only limited early-stage data are available. This makes it particularly useful in forecasting trends in various industries, where early data points may be scarce, but reliable projections are necessary for decision-making. The model’s ability to predict future growth based on historical data are further enhanced by its capacity to adjust as more data accumulates [4].

Another significant advantage is the model’s performance in comparison to simpler symmetric models. While models like the logistic model assume symmetric growth, the Gompertz model’s asymmetric nature allows it to better represent real-world dynamics, particularly those with slower rates of growth as the system matures [1]. Furthermore, recent studies have proposed generalizations of the Gompertz model, incorporating additional parameters to capture a more diverse range of growth behaviors, increasing its overall applicability and accuracy [5].

In the face of rapid advancements and the unpredictability of modern markets, the introduction of predictive models like the Gompertz curve has become imperative. Unforeseen economic scenarios, such as changes in VAT, various global and local crises, and evolving geopolitical landscapes, significantly influence consumer behavior. These shifts necessitate robust analytical tools to navigate and predict market dynamics effectively. Concurrently, continuous innovations within product technologies further compel the adaptation of advanced predictive models. Additionally, the evolution of market conditions and the introduction of new technologies demand adaptive forecasting models, as demonstrated in newer approaches to understanding technology lifecycles [6].

The Gompertz model serves as a crucial tool in aligning sales strategies with market demands by accurately forecasting the demand for new products and effectively managing the life cycles of existing products. By predicting the various stages of product adoption and saturation, the model enables businesses to tailor their strategies for maximum impact. The model’s application in diverse sectors, such as broadband for predicting service adoption rates [7], illustrates its versatility and effectiveness in strategic planning. Furthermore, innovations in predictive modeling techniques are enhancing the accuracy and applicability of models like Gompertz in dynamic markets [8,9].

The Mann–Kendall (MK) test is a widely used non-parametric statistical tool designed to detect monotonic trends in time series data. Its popularity stems from its robustness and minimal assumptions about data distribution. As [10] highlights, the “Mann-Kendell test is one of the most popular non-parametric trend test based on ranking of observations”. The MK test does not require data to follow a normal distribution and is resilient against missing values and irregular time points, making it highly applicable across diverse domains [10].

The application of the Gompertz model provides top management with predictive insights that significantly influence strategic decision-making. By accurately forecasting sales growth and saturation points, the model informs decisions on expanding into new markets, where understanding the potential customer base and demand lifecycle is crucial. The model’s insights into market dynamics and consumer behaviors enable strategic collaborations and multi-channel strategies, crucial for reaching broader markets [11]. Additionally, the continuous refinement of predictive models, adapting to modern market dynamics, underscores the evolving nature of strategic tools like the Gompertz model [12].

The Gompertz model equips top management with predictive insights for strategic decision-making, forecasting sales growth and saturation points crucial for market expansion and understanding customer demand cycles. It supports collaborative strategies by offering insights into market dynamics and facilitates multi-channel sales strategies to reach a broader customer base. Guided by these insights, companies can optimize market positioning and enhance their competitive advantage.

Case studies detailing the successful application of the Gompertz model in the retail industry are presented. These examples illustrate the model’s effectiveness in real-world settings, showing how businesses have applied the Gompertz curve to optimize marketing strategies and improve product lifecycle management.

Sales strategies play a critical role in determining the success of both online and brick-and-mortar retailers. One significant challenge facing traditional retailers is “showrooming, a phenomenon in which customers use brick-and-mortar stores to assess products and then purchase them from online retailers for lower prices” [13]. This consumer behavior has led to declining sales in physical stores, prompting businesses to adopt competitive strategies such as price matching. By offering customers the same lower prices found online, price matching aims to “prevent customers from practicing showrooming”. However, despite its ability to retain some customers, it also “enables customers to pay online retailers’ lower prices, thereby harming brick-and-mortar stores’ profit margins” [14].

To counter these challenges, some online retailers implement product differentiation, offering unique products unavailable in traditional stores. Research suggests that “product differentiation can be a win-win strategy for both traditional and online retailers under certain conditions” [14]. This strategy reduces direct competition and minimizes the impact of price matching by limiting identical product offerings across different retail channels. Furthermore, “the coexistence of showrooming and price matching makes the benefits of implementing product differentiation for online retailers inconclusive”, indicating the complexity of finding an optimal approach to sales strategy [14].

Overall, managing sales in a competitive market requires a balance between pricing strategies and product differentiation. Businesses must evaluate consumer behavior, market conditions, and the competitive landscape to develop effective sales strategies that sustain profitability and customer loyalty.

Matching strategy refers to the process of allocating resources, individuals, or entities based on predefined preferences, optimizing outcomes for all participants involved. It plays a critical role in various fields, including market operations, labor economics, virtual power plants, and decision-making behavior [15]. According to Wang [16], matching strategies can be broadly categorized into the shotgun approach, where individuals or entities engage in a broad, less selective matching process, and the rifle approach, which focuses on precision and targeting specific matches.

Similarly, in case–control studies, flexible matching algorithms help avoid bias by ensuring representative control groups [17]. Despite its advantages, matching strategies face challenges in human decision-making. Research shows that individuals often default to suboptimal matching due to immediate reward biases. However, structured guidance, such as informative cues, can shift behavior towards optimal decision-making [18,19].

Recent innovations and research related to the Gompertz model are discussed, highlighting adaptations and extensions designed to better align with current market dynamics. This discussion provides insights into the continuous evolution and increasing relevance of the Gompertz model in scholarly and applied research, reflecting its adaptability and enduring significance in the field.

Matching strategies are therefore vital across multiple disciplines, optimizing resource allocation and improving decision-making efficiency. Their applications continue to expand, driven by advancements in technology and algorithmic precision [20].

Modeling logistics trends refine matching strategies by aligning supply chain and inventory management with forecasted market demands, optimizing logistics for efficient resource allocation, cost reduction, and improved service delivery. This strategic integration enables businesses to adapt quickly and maintain competitiveness in dynamic markets.

2. Problem Formulation

In today’s highly competitive market, enterprises must establish themselves as reliable partners by fostering strong customer relationships through specialized services and support. Achieving long-term success requires businesses to focus on customer retention, sustained profitability, and brand identity, as these factors not only ensure business stability but also attract new customers. The effectiveness of product sales strategies is influenced by the synergy of four key areas: the product life cycle, mathematical/logistics trend modeling, matching strategy, and competitiveness of product sales (Figure 1).

The product life cycle provides a structured approach to understanding the different stages a product undergoes, from introduction to decline, allowing businesses to anticipate market dynamics and adjust their strategies accordingly. Mathematical and logistic trend models, such as the Gompertz curve, offer a valuable tool for predicting sales behavior, helping companies optimize production, pricing, and resource allocation. The matching strategy ensures alignment with customer expectations and industry trends, reinforcing brand credibility and enhancing customer satisfaction. Lastly, competitiveness in product sales is achieved through continuous innovation, efficient market positioning, and strategic pricing decisions.

In this paper, we analyze and evaluate the impact of these four key elements on sales performance, focusing on the effectiveness of the Gompertz curve as a predictive tool for product sales trends. The study investigates how different sales strategies, including product life cycle management and the application of matching strategies, influence market share and profitability.

The interaction of these four key elements plays a crucial role in shaping the success of product sales. By integrating product life cycle analysis with trend modeling, matching strategies, and competitive positioning, enterprises can enhance profitability and market share, ensuring long-term sustainability and growth.

3. Background

Overall, managing sales in a competitive market requires a balance between pricing strategies and product differentiation. Businesses must evaluate consumer behavior, market conditions, and the competitive landscape to develop effective sales strategies that sustain profitability and customer loyalty.

3.1. Product Life Cycle Model

The Product Life Cycle (hereinafter referred to as PLC) curve represents the sales trajectory of a product over time while it remains on the market. It consists of four fundamental phases, which can be characterized as follows (Figure 2).

Start: This phase marks the product’s entry into the market, characterized by low sales volumes. As a new product is introduced, a high degree of variability in its parameters can be expected. From a time perspective, it is desirable to keep this phase as short as possible [21].

Growth (Improvement): In this stage, the product gains popularity among customers, gradually becoming well-known and sought after. On the one hand, sales experience the highest growth rate; on the other hand, the variability of product parameters decreases as standardization takes place [21].

Maturity: At this point, product sales reach their peak, and the product becomes a standard offering in the market. High competition is expected. Ideally, a product should remain in this phase for as long as possible. Maintaining a competitive edge requires continuous product innovation [21].

Decline (Regress): Sales begin to decline due to innovations introduced by competitors. The product gradually exits the market, leading to production discontinuation. However, from a customer service perspective, it is essential to ensure after-sales support, including warranty services and spare parts availability [21].

The specification of the product life cycle phase is important for the following reasons:

• Choosing an appropriate marketing strategy and tools of the marketing mix (4P: Product, Place, Price, Promotion): Understanding the phase in which a product resides helps determine the most effective marketing tactics, including product development, distribution channels, pricing strategies, and promotional activities.
• Determining the product price: The product’s life cycle phase directly influences pricing strategies. In the introduction phase, the price may be set high for early adopters or low to encourage adoption, while in maturity, competitive pricing becomes more critical.
• Forecasting future sales: Knowledge of the product’s life cycle stage aids in predicting future sales patterns. As the product progresses through its phases, businesses can better estimate demand, plan for inventory, and make informed decisions on production and distribution.

Additionally, understanding the product life cycle through trend analysis shapes competitiveness decisions, allowing companies to navigate the various stages from introduction to decline. This knowledge informs decisions on product innovation, marketing, and pricing, helping businesses extend product profitability, meet consumer demands more effectively, and strengthen market positioning.

The increasing complexity of the environment in which organizations must operate and thrive, along with evolving customer demands, necessitates an understanding of key variables that influence both strategic development and business efficiency. Forecasting serves as a critical tool in reducing uncertainty about future developments. The accuracy and reliability of decisions are significantly influenced by information derived from forecasting methods, as forecasted data directly impact the formulation of an organization’s strategic direction.

In economics, time series analysis is one of the most fundamental methods for examining economic data. The primary objective of time series analysis is to develop an appropriate model that enables management to make informed predictions for future periods based on historical data. A well-constructed time series model allows for the simulation of time-dependent variables in a manner that minimizes discrepancies between observed values—such as those in financial markets—and values generated by the model.

Economic indicators are frequently monitored on a daily, monthly, or quarterly basis. Time series data at different time intervals are derived from each other; for instance, monthly series are created from daily observations, and quarterly series are constructed from monthly data. When selecting an appropriate time series analysis method, factors such as the objective of the analysis, its economic relevance, and the type of time series must be considered, as there is no universally applicable model. Time series emerge as a result of both significant and minor influencing factors. These data can be analyzed using both descriptive and inductive approaches, providing valuable insights for decision-making and strategic planning.

The classical time series model is based on the decomposition of data into four key components:

• Trend (development).
• Seasonal component.
• Cyclical component.
• Random (residual) component.

Among these, the trend is the most fundamental characteristic of time series development. It represents the long-term trajectory of a variable, reflecting its underlying directional movement over time. The simplest conceptualization of a trend is a linear increase or decrease, represented by a straight line fitted to actual data points. However, more complex mathematical curves can be used for smoothing time series data, providing a more accurate representation of real-world patterns.

The most commonly used trend models include [22]

• Linear trend—represents a constant rate of increase or decrease over time:

$${Tr}_t=a_0+a_1·t$$

(1)

2.

Quadratic Trend—captures acceleration or deceleration in the trend, allowing for curvature in the trajectory:

$${Tr}_t=a_0+a_1·t+a_2·t^2$$

(2)

3.

Hyperbolic Trend—models trends that initially change rapidly but gradually level off over time:

$${Tr}_t=a_0+{\displaystyle \frac{a_1}{t}}$$

(3)

4.

Exponential Trend—suitable for processes exhibiting exponential growth or decay:

$${Tr}_t=a·b^t$$

(4)

5.

Modified Exponential Trend—a variation in the exponential trend that incorporates an additional constant to adjust for baseline effects:

$${Tr}_t=k+a·b^t$$

(5)

6.

Logistics Trend—commonly used for modeling constrained growth, such as product adoption or population saturation:

$${Tr}_t={\displaystyle \frac{k}{1+a·b^t}}$$

(6)

7.

Gompertz trend—a sigmoid-shaped trend model that is asymmetrical, often applied in forecasting sales, population growth, and biological processes:

$${Tr}_t=k·{\alpha }_0^{{\alpha }_1^t}$$

(7)

The most common approach to describing the product life cycle is through trends with asymptotic bounds, specifically S-curves or sigmoidal curves. The S-curve function serves as a universal mathematical model for various growth processes characterized by an initial slow phase, followed by rapid growth, and finally, a stabilization phase.

S-curves are trend functions commonly used to model the sales trajectory of a new product. Initially, sales grow slowly due to limited consumer awareness. Over time, as a result of advertising and market penetration, sales accelerate before eventually stabilizing at a constant level, reflecting market saturation.

This pattern is best represented by a modified exponential trend, which can be applied to different transformations of the time series. When applied to the logarithmic transformation of sales data $ln{y}_t$, it results in the Gompertz S-curve. Alternatively, when applied to the reciprocal transformation ${\displaystyle \frac{1}{y_t}}$ it yields the logistic trend or Pearl–Reed trend, both of which effectively capture the dynamics of product adoption and market saturation.

Non-parametric tests are commonly employed for trend analysis, including the Mann–Kendall test, Mann–Whitney–Wilcoxon test, Pettitt test, Standard Normal Homogeneity Test (SNHT), Cumulative Sums (CUSUM) test, and the Two-Sample Kolmogorov–Smirnov test, among others. These tests are particularly valuable when they not only assess the statistical significance of a change point but also identify the precise timing of the change. The non-parametric Mann–Kendall test is employed to detect monotonic trends in time series data. The null hypothesis posits that the observations are independent and identically distributed, indicating no trend. In contrast, the alternative hypothesis suggests the presence of a monotonic (either increasing or decreasing) trend in the data.

The Mann–Kendall test is a non-parametric method used to identify significant trends in time series data. It is specifically designed to detect monotonic trends—either consistently increasing or decreasing—over the observed period. The core principle of the test involves ranking the observations and evaluating the relative ordering of data points to determine the direction and significance of the trend.

$$x_i=f\left(t\right)+{\epsilon }_i$$

(8)

where

f (t)—is a continuously monotonically increasing or decreasing function of time,

ε_i,—the residuals are assumed to originate from the same distribution with a mean of zero.

Therefore, the variance of the distribution is assumed to be constant in time. The Mann–Kendall test statistic s is calculated using the given equation:

$$S={\sum }_{i=1}^{n-1}{\sum }_{j=i+1}^{n}sgn\left(x_j-x_i\right)$$

(9)

where

n—total number of data points in the time series,

m—cluster of data points with the same data value,

t_i—total observation of range i at a specific time.

When the total sample size n exceeds 10, the standardized test statistic Zs is calculated using the following formula:

$$Z_s=\left\{\begin{array}{cc}{\displaystyle \frac{s-1}{\sqrt{Var \left(s\right)}}}& \mathrm{I}\mathrm{f} \mathrm{S}>0\\ 0& \mathrm{I}\mathrm{f} \mathrm{S}=0\\ {\displaystyle \frac{s+1}{\sqrt{Var \left(s\right)}}}& \mathrm{I}\mathrm{f} \mathrm{S}<0\end{array}\right.$$

(10)

A positive value of Z_s indicates an increasing (positive) trend in the time series, whereas a negative Z_s value signifies a decreasing (negative) trend.

3.2. Logistics Trend

The logistic trend is widely used in demand modeling for durable goods, as well as in analyzing the development, production, and sales of specific product categories. It belongs to the family of S-curve trend functions, which effectively capture the fundamental developmental phases of a cycle. A cycle refers to the time period spanning from the emergence of new driving forces, such as technologies or products, to their eventual decline.

The logistic trend can be divided into the following five phases [23,24]:

• Phase 1—The period during which new progressive forces begin to emerge.
• Phase 2—The stage where these progressive forces fully assert themselves and start to decisively influence further development.
• Phase 3—The period when progressive forces have completely dominated market development; however, opposing forces also begin to arise, mitigating their impact.
• Phase 4—The stage in which emerging opposing forces gradually gain permanent dominance over the existing forces, significantly slowing down developmental tendencies.
• Phase 5—The point at which development halts until new progressive forces emerge to drive another cycle.

This structured progression illustrates the evolutionary nature of market dynamics and provides a valuable tool for forecasting product life cycles, technological advancements, and economic trends.

The logistic trend is commonly used for modeling the development, production, and sales of certain products, particularly for modeling demand for long-term consumption goods. The logistic trend model is not uniquely defined, and several versions are used in practice. The most commonly used model is of the form [23,24]:

$${Tr}_t={\displaystyle \frac{k}{1+{\alpha }_0·{\alpha }_1^t}}+{\epsilon }_t \hspace{1em}\hspace{1em}\mathrm{for} t=1,2,\dots ,n$$

(11)

where

${Tr}_t$ is the value of the dependent variable at time t,

${\alpha }_0, {\alpha }_1$ are unknown parameters (α₀ a constant determining the initial value and α₁ a constant determining the growth rate),

k is the unknown boundary parameter (maximum possible value, ${Tr}_t$ it is the asymptote of the function and expresses the so-called saturation level,

ε_t is a random error, $t=1,2,\dots ,n$ is a time variable.

There are several ways to calculate estimates of the parameters ${\alpha }_0$, ${\alpha }_1$ and k of the unknown parameters ${\alpha }_0$, α₁ and k. They can be obtained, for example, by linearizing the relation [23,24]:

$${Tr}_t={\displaystyle \frac{k}{1+{\alpha }_0·{\alpha }_1^t}}⟹{\displaystyle \frac{1}{{Tr}_t}}={\displaystyle \frac{1+{\alpha }_0·{\alpha }_1^t}{k}}={\displaystyle \frac{1}{k}}+{\displaystyle \frac{{\alpha }_0}{k}}·{\alpha }_1^t$$

(12)

From there, after substituting ${Tr}_t^{\prime }={\displaystyle \frac{1}{{Tr}_t}}$, $k^{\prime }={\displaystyle \frac{1}{k}}$, ${\alpha }_0^{\prime }={\displaystyle \frac{{\alpha }_0}{k}}$ it is possible to obtain a linearized trend function of a modified exponential trend:

$${Tr}_t^{\prime }=k^{\prime }+{\alpha }_0^{\prime }·{\alpha }_1^t$$

(13)

The estimates of the parameters $k^{\prime }$, ${\alpha }_0^{\prime }$ and ${\alpha }_1$ can be calculated using the partial sum method (we divide the individual members of the time series into 3 equally large groups). For its use, it is necessary to fulfill the condition:

$$S_2-S_1>0 \wedge S_3-S_2>0\vee S_2-S_1<0 \wedge S_3-S_2<0$$

It is advisable to divide the individual members of the series into three equally large groups with a range of m. If the number of members is not divisible by three without a remainder, then, when creating the groups, we omit the first or first two values. Then, we calculate the sums of these three groups:

$$S_1={\sum }_{t=n-3m+1}^{n-2m}{y}_t\hspace{1em}\hspace{1em}{S}_2={\sum }_{t=n-2m+1}^{n-m}{y}_t\hspace{1em}{S}_3={\sum }_{t=n-m+1}^n{Y}_t$$

(14)

For unknown parameters a₀, a₁, k the following applies [14,16]:

$$a_1=\sqrt[m]{{\displaystyle \frac{S_3-S_2}{S_2-S_1}}}$$

$$a_0={\displaystyle \frac{\left(S_3-S_2\right)·\left(a_1-1\right)}{{\left(a_1^m-1\right)}^2·a_1^{n-2m+1}}}\hspace{1em}\hspace{1em}k={\displaystyle \frac{S_2-a_0·a_1^{n-2m}·\frac{a_1^m-1}{a_1-1}}{m}}$$

(15)

Estimates of the parameters a₀, a₁ and k can also be calculated using the selected points method.

This logistic trend model is frequently used to forecast the life cycle of products and technologies. It is applicable not only to demand and sales but also to production, storage, and distribution. Its symmetric nature and well-defined parameters allow analysts and managers to gain a deeper understanding of market behavior and develop more effective inventory management and marketing strategies [24].

Individual logistic curves that express the course of sales of various products differ in the course of individual phases as well as the length of these phases. The Gompertz curve is also among the S-curve forecasting models. Unlike the logistic function, it is asymmetric; the center of gravity of values in time is behind its inflection point.

3.3. Specification of the Gompertz Curve

The Gompertz curve is described by an equation that depends on two parameters, a and b, which influence its shape. Parameter a represents the maximum value that sales tend to reach, while parameter b characterizes the rate of growth or saturation. One of the key properties of the Gompertz curve is its ability to predict both the initial and final behavior of the system. It is often used to forecast sales growth and to analyze market dynamics as well as the adoption of new technologies [23,24].

The Gompertz curve depends on two parameters, α₀ and α₁, which influence its shape. Parameter α₀ represents the maximum sales level that the trend is expected to reach, acting as the upper asymptote, while parameter α₁ characterizes the rate of growth or the speed at which the process, such as sales or market penetration, progresses over time. These parameters define the asymmetrical S-shape of the Gompertz curve, making it particularly useful for modeling growth processes where initial adoption is slow, followed by a phase of rapid expansion, and ultimately stabilizing as market saturation is reached [23,24].

The Gompertz trend belongs to the group of S-curves and has the following form:

$${Tr}_t=k·{\alpha }_0^{{\alpha }_1^t}+{\epsilon }_t$$

(16)

where

α₀, α₁, k are unknown parameters,

k—the unknown boundary parameter k gives, for example, the upper limit of sales of goods at its saturation, t = 1, 2, …, n is a time variable,

ε_t is a random error.

To perform parameter estimation, we first linearize the function [23,24]:

$$ln{Tr}_t=\ln k+{\alpha }_1^t·ln{\alpha }_0$$

(17)

where

$${Tr}_t^{\prime }=ln{Tr}_t, k^{\prime }=\mathit{ln}k, {\alpha }_0^{\prime }=ln{\alpha }_0$$

(18)

The parameters of the modified exponential trend can be calculated using the previously mentioned partial sum method. Unlike the logistic trend, the function’s progression is asymmetrical, with most of the values occurring after the inflection point (Figure 3). The curve has an upper positive asymptote at parameter k.

When assessing the suitability of a model, priority should be given to substantive considerations supplemented by statistical criteria. We usually examine the size of the differences $y_t={Tr}_t$ (the actual values of the variable ${Tr}_t$ at time t) and ${\widehat{y}}_t$ (the adjusted values of the variable ${Tr}_t$ at time t), i.e., the residuals $y_t-{\widehat{y}}_t$ at time t = 1, 2, …, n. The accuracy of the model can be evaluated using basic measures of accuracy (absolute and relative measures), for example, using the Mean Square Error. The Mean Square Error (MSE) expresses the average squared difference between the estimated values and the actual values. It is calculated using the following formula [23]:

$$MSE={\displaystyle \frac{1}{n}}·\sum {\left(Tr-{\widehat{Tr}}_t\right)}^2$$

(19)

where

Tr is the actual value,

${\widehat{Tr}}_t$ is the estimated value.

n is the total number of observations.

It can be stated that the Mean Square Error (MSE) estimates the amount of error in a statistical model; for example, how accurately the regression curve fits the observed (actual) values. For example, a company has a model that predicts sales revenue for a one-year period. At the end of the year, the company has actual sales figures. Subsequently, it is possible to calculate the MSE, which ultimately determines how well the model predicts (describes) the outcome. This evaluation helps in assessing the accuracy and reliability of the forecasting model, and the lower the MSE, the better the model’s predictions align with actual results.

3.4. Matching Strategy

The purpose of the matching strategy is to understand customer needs and ensure satisfaction not only for customers but also for competitors, while adapting to market trends. By fulfilling these conditions, companies can make decisions regarding pricing, promotions, and overall strategies. The goal for businesses is to align product sales effectively, which leads to increased opportunities and maximized profits. The implementation of the matching strategy is suitable for both small and large companies. Creating competitive prices and strengthening customer trust contributes to improving price adjustments and overall product sales [25,26].

Advantages of the matching strategy are [25,26]:

• Higher Accuracy: Focused on information about product sales. Sales data must be accurate, up-to-date, and relevant. The results reveal the trend in product sales development.
• Improved Competitiveness: Enables companies to study and evaluate competitors and specify market trends. The collected data and information are used to make decisions about product pricing and promotions, ultimately leading to maintaining a competitive advantage.
• Increased Sales Efficiency: Relying on manual or online product promotion. Online promotion frees up time and resources for other important tasks.
• Specification of Pricing Strategies: By examining and comparing product sales, sellers can make decisions about pricing strategies, which ultimately ensures the company’s competitiveness in the market.
• Improved Product Development: Sellers understand what competitors offer. Competition contributes to decisions about pricing strategies, and which features to include in their own products, ensuring competitiveness and attractiveness for customers.
• Adequate and Reliable Customer Information: By tracking and analyzing sales data, sellers can gain better insights into customer behavior, including buying habits and preferences. This information is important for product development and marketing.
• It is essential to note that clearly defined prices increase consumer confidence. This implies that customers have certainty in obtaining the best offer. The response to acquired trust contributes to higher customer loyalty as well as repeat business. Appropriately set prices are more appealing to price-sensitive customers, resulting in a higher conversion rate and increased sales. Consumers are more likely to make a purchase only when they are certain they are receiving the best price.
• The matching strategy, though conceptually outlined, aligns quantitatively with the demand tracking strategy applied over the study period of three years. The developed models, by integrating time-series analysis and regression techniques, demonstrate the connection between this demand strategy and the selection of appropriate models. These models effectively map demand fluctuations to sales outcomes, illustrating how strategic adjustments in product positioning can impact market performance. It is noted that seasonal components are observable within the models; however, a comprehensive analysis and evaluation of these is needed.

4. The Implementation of Trends and Results

The company under analysis is involved in the production and sale of low-voltage plastic cable distribution cabinets, with a focus on the electrical engineering sector. The company’s primary activities are centered on the manufacturing, assembly, and delivery of cable distribution boxes. Additionally, the company provides technological equipment and protective components that are part of the main products.

Within its strategic group, the company operates in a competitive environment, where both domestic and international competitors are present. The market demands substitute products with similar functions but varying types of materials and designs. The objective of this analysis is to examine the sales of three specific products: product A—fuse boxes, product B—power distribution cabinets, and product C—enclosed terminal blocks over the period of 2020–2024. The dataset utilized in this study comprises sales data for three products, observed over a five-year period from 2020 to 2024. The focus is on evaluating initial sales performance and identifying existing trends. Based on the observed sales dynamics, it is essential to formulate and implement an appropriate strategic approach tailored to each product’s market trajectory.

Product sales are analyzed through mathematical trends and the Gompertz curve. It is necessary to determine the appropriateness of the relevant trends. Specifying an adequate trend will contribute to the selection of an appropriate product sales strategy. The first trend applied is a linear trend.

From Figure 4, it can be seen that a linear trend is not suitable for the sales of product A. The identified trend is described by the equation:

$${Tr}_t=31.2401+0.3950·x_t$$

(20)

Subsequently, a nonlinear trend is applied, which is shown in Figure 5.

The figure presents the application of a nonlinear trend to the sales of product A. The specified nonlinear trend inappropriately describes the sales of product A. The specific trend is expressed by the equation:

$${Tr}_t=23.3101+1.1624·x_t-0.0126·x_t^2$$

(21)

It follows from Figure 4 that a linear and a nonlinear trend are not suitable. Prediction interval 95% for product A for linear trend (18.811; 67.351). Prediction interval 95% for product A for nonlinear trend (8.001; 57.060).

For the specified time series, an exponential trend, specifically the Gompertz curve, was implemented. Sales of Product A show a slow start followed by a steady sales increase. There are visible deviations in certain periods that may reflect fluctuations in demand for product A. An outlier is observed in October 2021 with a sales quantity of 5074 units, which is just below the upper bound of the prediction interval. Over time, sales continued to fluctuate within the predicted interval. Figure 6 illustrates the time series, which shows both increases and decreases in the number of products sold. The dashed lines illustrate the 95% prediction intervals. These intervals are notably wide, reflecting the inherent variability in the dataset and the confidence level of the predictions made by the models. The prediction interval is 95% for product A for Gompertz trend (38.609; 53.892).

The exponential trend presented by the Gompertz curve:

$${Tr}_t=51.3191·{0.4584}^{{0.9389}^t}$$

(22)

The linear and quadratic models used in the analysis are observed to yield almost identical functions, as evidenced by the parameter a₂ in the quadratic trend approaching zero, thereby rendering it effectively linear. To facilitate a clear comparison between models, we present the statistical significance and estimated coefficients for the linear, quadratic, and Gompertz models, as shown in Equations (21)–(23). The statistical significance (p-value) has been determined for the models presented. For the linear model, a significance of 0.0000 is observed; for the nonlinear model, the significance is noted at 0.0000; for Gompertz model, the significance is noted at 0.0000.

After applying the Gompertz function to the sales data, the model parameters were estimated through an iterative process, ensuring that the best possible fit was achieved. The model fit metrics were determined (

Table 1. Model summary—product A.

Model	Sum of Squared Error (SSE)	Mean Squared Error (MSE)	Significance	AIC/BIC
Linear	1342.8433	22.3807	0.0000	360.73/364.92
Nonlinear	660.0689	11.0022	0.0000	320.12/326.41
Gompertz	584.0173	9.7336	0.0000	312.78/319.06

), leading to the following summary:

The Akaike Information Criterion (AIC) and the Bayesian Schwarz Information Criterion (BIC) are indicators that can be used to select a suitable model for the data under study. The lower value of the criterion, the more suitable the model.

To verify the sales trend of Product A, the Mann–Kendall test (

Table 2. Z_S values at 10%, 05%, and 1% significance level.

Zs		Critical Value	Significance Level
3.8242	>	1.6449	at 10% significance level
3.8242	>	1.9600	at 10% significance level
3.8242	>	2.5758	at 10% significance level

) was applied. When performed on the current dataset, the test yielded a variance S = 226 and a standardized test statistic Z_s = 3.8242. The corresponding p-value of the test was p = 0.0001, which indicates that the trend is statistically significant at all three significance levels of 10%, 5%, and 1%.

Table 2 presents the trend values at the 10%, 5%, and 1% significance levels. It can be observed that, at the 10%, 5%, and 1% significance levels, the Z_s value falls within the acceptable range, indicating that the sales records exhibit a statistically significant trend. Given that the value of S (variance) is 226, this further supports the presence of a positive sales trend for Product A.

Sales of product A—fuse boxes are presented in Figure 6, which confirms a more appropriate trajectory within its industry. The mean square error (MSE) is 9.7336, the lowest for the Gompertz trend, indicating the degree of deviation between the model’s predicted values and the actual measured data.

Figure 6 illustrates the sales trend of product A over time:

• In the growth phase (January 2020) 1 < t < 22 (October 2021) an exponential increase in sales of product A is observed, which requires an expansion of production capacity to meet the growing demand.
• At the saturation point t = 22 (October 2021) sales of product A stabilize, which indicates that the peak in sales of product A has been reached.
• in the phase t > 23 (November 2021) there is a decrease in sales of product A, which signals a decrease in demand.
• Although sales of product A are within the prediction interval, Figure 6 shows that sales of the product oscillate around the calculated exponential curve.

The observed pattern highlights the need for product innovation, either through design improvements, enhanced features, or targeted marketing strategies, to sustain competitiveness and prevent a long-term decline in demand.

Figure 7 shows the nonlinear sales trend of product B—power distribution cabinets. Based on the sales data for product B, the quadratic trend appears to be a more appropriate model to capture the sales dynamics. Statistical analysis of the trends shows that the quadratic trend explains 62.71% of the variability in the data, indicating a higher level of accuracy compared to the linear model (35.56%). Furthermore, all model coefficients are statistically significant, with p-values of 0.0000, confirming the robustness of the estimates. The standard deviation of the residuals is 18.823, reflecting the degree of variation in the model predictions. Although the quadratic model is more consistent with the observed sales patterns of product B, its predictive ability remains limited, especially when predicting long-term fluctuations in demand.

$${Tr}_t=151.2000-1.9663·x_t+0.0092·x_t^2$$

(23)

Prediction interval 95% for B (18.567; 206.930). From a product life cycle and nonlinear trend perspective, sales of product B initially show a gradual increase during the first year, reaching a peak in December 2020. Two data points (t = 1 (January 2020) and t = 12 (December 2020) lie outside the prediction interval, and one point (t = 32 (August 2022) lies at the lower boundary of the prediction interval, indicating deviations from the expected trend.

Following a period of fluctuation, the sales of product B have experienced a significant decline, indicating a decline in product attractiveness. Given this downward trend, product B either requires innovation to regain market interest or should be discontinued to optimize resource allocation and market competitiveness.

The sales of product C are also described by three types of trends, namely linear, nonlinear and Gompertz trend.

The linear sales trend of product C (Figure 8) is described by the equation:

$${Tr}_t=37.5310+0.9264·x_t$$

(24)

Figure 9 presents the quadratic trend of product C sales. Statistical analysis of the trends shows that the quadratic trend is suitable at the level of 96.86% and the linear trend at the level of only 70.41%. For the quadratic trend, the model coefficients are statistically significant, with p-values of 0.0000. The standard deviation of the residuals for the quadratic trend is 3.402, which reflects the degree of variation in the model predictions.

$${Tr}_t=14.6695+3.1388·x_t-0.0363·x_t^2$$

(25)

The quadratic trend is more consistent with the observed values of product C sales. The peak of product C sales was reached at time t = 29 (May 2022), specifically the number of products sold was 8125 pieces. Subsequently, a slight decrease in product C sales can be seen.

The exponential trend presented by the Gompertz curve:

$${Tr}_t=78.8736·{0.1083}^{{0.8695}^t}$$

(26)

Based on the values of the Akaike Information Criterion (AIC) and Bayesian Schwarz Information Criterion (BIC) indicators, it is possible to select a suitable trend for the data under study. The values in

Table 3. Model summary—product C.

Model	Sum of Squared Error (SSE)	Mean Squared Errror (MSE)	Significance	AIC/BIC
Linear	6334.4830	105.5747	0.0000	453.81/457.99
Nonlinear	659.7190	10.9953	0.0000	320.09/326.37
Gompertz	534.4357	9.0573	0.0000	308.46/314.74

show that the lowest AIC/BIC values are for the Gompertz trend.

The Mann–Kendall analysis (

Table 4. Z_S values at 10%, 05% and 1% significance level—product C.

Zs		Critical Value	Significance Level
2.0906	>	1.6449	at 10% significance level
2.0906	>	1.9600	at 10% significance level
2.0906	<	2.5758	at 10% not significance level

), when performed for the current dataset, found that, for S (variance) = 124, Z_S = 2.0906. The resulting p-value of the test is p = 0.0366, which means that the trend is significant at the 5% and 10% significance levels.

However, at the 1% significance level, the data do not show a significant trend. Therefore, with 90% and 95% confidence, it can be concluded that the current sales of product C demonstrate a significant trend. Given that the value of S (variance) is 124, this further supports the presence of a sales trend for product C.

Figure 10 presents the sales of product C—enclosed terminal blocks. The lowest value of the mean square error (MSE) is 9.0573 precisely at the Gompertz trend, i.e., the deviation between the predicted values and the actual values of sales of product C. Sales of product C are described as follows:

• In the first, i.e., growth phase, it is possible to see increasing sales of product C, which is related to increasing demand. There is only one value outside the prediction interval, namely t = 1 (January 2020), when 2.835 pieces of products were sold.
• In the saturation phase t = 29 (May 2022), sales of product C reached a peak. At the peak, 8 125 pieces were sold.
• At time 30 < t < 60, a slight decrease can be seen, but the values oscillate around the predicted sales curve of product C.

From Figure 10, it is possible to see stable sales of product C until December 2024.

In the case of the examined product sales, a matching strategy has been applied, ensuring a steady and balanced implementation of sales for product A. From a strategic perspective, expanding sales into new markets presents a viable opportunity for growth. Additionally, strategic investments in product innovation could prove beneficial, potentially increasing demand and further boosting sales of product A. Conversely, a significantly negative sales trend is observed for product B, with a sharp decline in sales at the end of 2024. This drop is primarily attributed to low market interest in this product category. Another contributing factor may be the winter season, which presents issues in assembling power distribution cabinets, thereby reducing demand.

To effectively respond to identified market trends, companies should strategically adjust their product offerings to meet evolving consumer preferences and behaviors. This includes refining product features or introducing new variations tailored to the market’s needs. Additionally, optimizing pricing strategies for different product lifecycle stages can maximize revenue while retaining buyer interest. Focusing distribution efforts on channels and locations predicted to see increased demand ensures products are accessible to likely buyers.

For exploring growth opportunities, trend analysis can help identify untapped or growing market segments, enabling companies to strategically introduce tailored products and potentially capture significant new market shares. Forming strategic partnerships can amplify these efforts, providing synergistic benefits like expanded customer reach and enhanced product value. Moreover, investing in technology and innovation informed by long-term trends equips companies to meet future consumer demands, maintaining a competitive edge and ensuring sustainable growth.

5. Discussion

In addition to the parametric models discussed, it is important to recognize the role of non-parametric methods in trend analysis. These methods offer valuable flexibility, especially when the model form is uncertain or when traditional parametric approaches may not suffice. Non-parametric methods allow for a broader exploration of data patterns without assuming a specific model structure, which can be particularly advantageous in complex or poorly understood datasets.

In response to the observed sales fluctuations, it is important to consider not only the trend component but also the seasonal influences evident in the data. Future models could benefit from systematically incorporating and analyzing these seasonal components.

Moreover, factors such as marketing activities and shifts in consumer demand, which also appear to influence the sales trends, should be integrated into future modeling efforts. Monitoring these factors systematically could provide deeper insights into their effects on sales, enabling more accurate predictions and strategic planning. This approach would not only enhance the model’s accuracy but also its practical application in predicting and responding to market dynamics.

This study focuses on time-based modeling of sales trends using Gompertz and logistic curves, acknowledging that factors like pricing, marketing, competitor actions, and customer behavior also significantly influence market dynamics. Future studies could aim to incorporate these variables into the analysis, requiring comprehensive data collection and the use of advanced multivariate models to understand the combined effects of these factors on sales.

6. Conclusions

Predictive models, crucial for this analysis, often carry inherent assumptions that may not align with dynamic market realities. Linear models assume constant rates of change, rarely seen in volatile markets, while nonlinear models, though flexible, might not capture unseen complex interactions that impact sales. More sophisticated models, such as the Gompertz model, complement more complex models, making it a permanent and versatile tool for tracking and forecasting growth in various fields, especially for processes that exhibit slowing growth patterns. Sales of product A are characterized by a slow start, accelerating growth in the middle stage, and reaching a peak due to market saturation. The Gompertz trend allows for the impact of saturation to be taken into account, making it a reliable tool for modeling and forecasting sales throughout the product life cycle. Using this model also helps companies develop more accurate business strategies, plan cash flows, and improve their overall resilience to market fluctuations. The Mann–Kendall test was applied to verify the presence of a trend in the sales data for product A. The resulting test statistic (Z_s = 3.8242) confirms the existence of a statistically significant trend. At all three significance levels of 10%, 5%, and 1%, the test indicates that sales of product A exhibit a significant monotonic trend. Since the value of Z_s is greater than zero, it can be concluded that this trend is positive, indicating a consistent increase in sales over the observed period. The integration of trend analysis with strategic alignment enhances the understanding of sales dynamics and supports more agile responses to evolving market conditions.

In contrast, analysis of product B reveals a nonlinear trend, specifically a quadratic trend, accompanied by a significant decline in sales. This downward trend underscores the need to prioritize innovation related to product B. In the case of sales of product B, which is subject to significant competition and demand fluctuations, the quadratic model proves to be a more accurate tool for forecasting and analysis. It allows not only the prediction of future sales patterns, but also the adjustment of production and marketing strategies, thereby ensuring more efficient management of corporate resources.

From the Gompertz trend for sales of product C, it is possible to assume a gradual slowdown in growth and subsequent stabilization at a certain level. A significant strength of the Gompertz trend is the inclusion of potential sales declines for product C after the peak in demand and the possibility of more accurately predicting changes in sales dynamics, especially in a competitive environment and with evolving consumer preferences. The flexibility of the Gompertz trend allows us to take fluctuations into account and provide more accurate forecasts. To verify the existence of a trend for the sales of product C, the Mann–Kendall test was applied. The test criterion (Z_s = 2.0906) for the sales values of product C confirms the existence of a statistically significant trend at the 10%, 5% significance level. A properly applied Gompertz trend allows us to predict demand estimates and thus respond in a timely manner to market changes, optimize inventory levels and avoid overproduction, which ultimately contributes to competitiveness. The Gompertz trend appropriately captures sales fluctuations in the later stages of the product life cycle. Implementing product enhancements, along with improved promotional strategies, is expected to contribute to revitalizing sales and increasing the market appeal of this product category.

The connection between trend analysis and matching strategy contributes to the understanding of sales dynamics and the ability to flexibly respond to changing market conditions. Ensuring these models’ effectiveness on new datasets is essential for their reliability in predictive analytics. Real data show that, after the peak of demand, a decrease is observed due to external factors, such as the emergence of new competitors, changes in consumer preferences or market saturation with similar products.

The findings of this study demonstrate the importance of predictive tools in forecasting market changes and regulatory developments, enabling businesses to formulate strategic recommendations that enhance product sales and market positioning. The integration of the product life cycle curve, the Gompertz curve, and the matching strategy provides a comprehensive framework for understanding sales dynamics and market share fluctuations. By linking these elements, businesses can better anticipate demand patterns, optimize resource allocation, and refine pricing strategies, leading to more effective decision-making. The trend analysis presented in this paper directly influences the evaluation of sales strategies and their recommended adjustments. The insights gained contribute to a deeper understanding of company dynamics, facilitating adaptation and responsiveness to evolving market conditions. The identified sales trends underscore the necessity of strategic pricing mechanisms and targeted promotional efforts, both of which are essential for sustaining long-term growth.

Additionally, all models are sensitive to outliers, which can disproportionately influence predictions and skew results. This is particularly challenging when atypical data does not represent normal market behavior but arises from unusual circumstances. Properly identifying and mitigating the effects of such outliers is crucial to maintaining the models’ accuracy and relevance.

Furthermore, competitor analysis and the identification of emerging market trends are crucial in maintaining a competitive edge. Understanding the strategic actions of competitors allows businesses to position their products more effectively and strengthen their market presence. However, to sustain sales momentum, companies must prioritize continuous product innovation. The role of innovation in extending product life cycles and maintaining customer engagement cannot be overstated. By leveraging data-driven forecasting methods, companies can enhance their strategic agility, adapt to changing consumer preferences, and ultimately achieve sustainable market success.