In the sales environment for expensive machinery and technological lines for the food industry, precision in matching a Customer–Salesperson pair (so-called smart routing) is becoming a key factor of competitive advantage. This study analyzes the fundamental problem of instability in the classification of salespeople in systems based on artificial intelligence.
We compare two dominant mathematical paradigms: the classical, geometric K-means algorithm and the advanced, statistical Gaussian Mixture Model (GMM). The analysis demonstrates that, in the case of hybrid profiles—which account for a significant part of the salesperson population—the deterministic approach (K-means) generates categorization errors resulting from rigid decision boundaries, which leads to fluctuations in the sales process. The probabilistic approach (GMM), offering so-called soft clustering, provides the operational stability required to implement a predictable decision-support system.
Relationship Engineering in the Machinery Industry
The sector producing advanced machinery for food processing is characterized by specific sales dynamics. Unlike retail sales, where transactions are quick and impulsive, the process of purchasing a packaging line or dosing system lasts for weeks and involves many decision-makers. The value of a single contract is often measured in hundreds of thousands or millions of Polish zlotys. In such an ecosystem, the role of the call center and sales department extends far beyond simple order handling. It becomes a strategic node where engineering competence must be perfectly synchronized with negotiating skills.
The overriding objective of data analytics in this area is to maximize the probability of conversion (closing a sale) by intelligently routing incoming inquiries to the most appropriate salespeople. This is an optimization process that requires the creation of a mathematical model describing reality—the profiling of customers and salespeople.
The fundamental challenge faced by Data Science departments is not acquiring the data itself—we use Whisper-1 transcriptions, sentiment analysis, and lexical models—but interpreting it over time. Stability is a key parameter for evaluating the quality of every model deployed in a production environment. A system that classifies an employee as a “Technical Expert” one week and as an “Aggressive Negotiator” the next, solely on the basis of a marginal change in behavior, is useless from the perspective of process management. Such fluctuations make it impossible to build reliable sales-probability matrices and, consequently, to forecast revenue.
This article advances the thesis that the source of this instability is not the quality of the input data, but the incorrect selection of the mathematical method used to group (cluster) objects. We will analyze why the market standard (K-means) fails when confronted with human nature and why probabilistic methods (GMM/LCA) represent a necessary evolution in enterprise-class systems.
The Nature of the Data: The Problem of a Continuous Behavioral Spectrum
To understand the essence of the problem, we must examine the structure of the data on which we operate. Psychometric models such as the Big Five (OCEAN) or HEXACO, as well as behavioral analyses of influence techniques (according to Cialdini), provide us with continuous rather than discrete variables.
In materials engineering, we rarely encounter pure elements. Likewise, in sales psychology, we rarely encounter “pure archetypes.” A salesperson is not a binary variable. They are not either an “Advisor” (1) or a “Seller” (0). In reality, every sales-department employee is a unique mixture of different styles and competencies. We can imagine a multidimensional space in which each dimension represents a different characteristic—for example, assertiveness, empathy, or technical knowledge.
Most of the population is not located at the extremes of this space (in its corners), but in its center. These are so-called hybrid profiles. Such an employee may display analytical behaviors to a degree of 30%, relational behaviors to a degree of 30%, and directive behaviors to a degree of 40%. This is a natural phenomenon: people adapt to situations, and their style is the resultant of personality and the current context.
The analytical problem arises when the business demands categorization. The Sales Director needs a report in which employees are assigned to specific groups in order to manage teams. The Data Scientist therefore faces the following task: how can this complex, multidimensional, continuous reality be reduced to a finite number of groups (clusters) without losing key information? This is precisely where two mathematical worlds collide: Euclidean geometry and probabilistic statistics.
The Deterministic Approach: K-means and the Illusion of Hard Boundaries
The K-means method, often supported by PCA (Principal Component Analysis) dimensionality reduction, is the algorithm most frequently used in business. Its popularity results from its ease of implementation and speed. However, it is a deterministic method based on hard geometric logic.
Operating Mechanism: Voronoi and the Division of Space
The K-means algorithm treats the classification problem as the task of dividing space into disjoint regions. It operates iteratively, seeking to determine central points (centroids) that minimize the sum of squared distances of all data points from their nearest center.
This can be compared to sorting physical objects—for example, stones of different shades—into several buckets. Each stone must go into one and only one bucket. The algorithm takes a stone, measures its “distance” (similarity) to the reference in each bucket, and puts it wherever it fits best. A stone cannot be in two buckets at the same time.
As a result, the feature space is divided into so-called Voronoi cells. These are polyhedral regions separated by sharp mathematical boundaries. Within a cell, the algorithm treats all points as “the same.” A point located at the very center of a cluster and a point located at its absolute edge receive the same label.
The Problem of Boundary Regions
The main weakness of K-means in psychometric applications is how it treats boundary cases. Let us return to our hybrid salesperson, whose profile consists of 33% type A characteristics, 33% type B characteristics, and 34% type C characteristics. From a geometric point of view, this data point is almost equally distant from the three centroids, but it is minimally closer to centroid C. Operating in “Hard Voting” mode, the K-means algorithm makes a binary decision: it assigns the salesperson 100% to cluster C. The membership indicator takes the form of a vector.
At this point, there is a drastic loss of information. The system “forgets” that this employee possesses 66% of the characteristics of other types. Worse still, the system begins to treat this person in exactly the same way as an employee who is 95% type C. This is a generalization error known in engineering as a quantization error.
Consequences: System Instability
The most serious consequence of the hard approach is sensitivity to noise. Human behavior is stochastic (random) by nature. A salesperson may be slightly more energetic one week and slightly more withdrawn another week. These natural fluctuations cause the point representing the salesperson in feature space to vibrate continually.
For a hybrid salesperson (33/33/34), a minimal shift in parameters—for example, resulting from one poorer conversation—may change the proportions to 34/33/33. For a person, this is an imperceptible change (1%). For the K-means algorithm, however, it means crossing the mathematical boundary of a Voronoi cell. As a result, the salesperson’s label changes from type C to type A from one week to the next. From the routing system’s perspective, this is an earthquake. The control algorithm, which previously routed customers matching profile C to this salesperson, suddenly stops doing so and starts routing profile A customers instead.
This leads to the phenomenon of “cluster flickering.” The sales-probability matrix, which should form the foundation of the strategy, becomes a chaotic image. Technical directors see changing performance bars that result not from market changes but from computational artifacts of the K-means method.
The Probabilistic Approach: GMM and Mixed Reality
The answer to the limitations of Euclidean geometry is Bayesian statistics and density modeling. This approach is represented by Gaussian Mixture Models (GMM) and, in the case of categorical variables, Latent Class Analysis (LCA).
Operating Mechanism: Distributions and Soft Clustering
GMM does not divide space into rigid pieces. Instead, it assumes that the observed data result from several hidden generative processes, each of which has the character of a normal distribution (Gaussian curve). Using an industrial analogy: rather than sorting stones into buckets, GMM operates like a spectrometer analyzing the composition of a metal alloy. If we examine a steel sample, the device does not tell us, “this is iron” or “this is carbon.” It tells us: “this sample consists of 98% iron, 1.5% carbon, and 0.5% admixtures.”
Using the Expectation-Maximization (EM) method, the GMM algorithm fits multidimensional shapes (ellipsoids) representing probability density to the cloud of data. The final result is the key difference. For each data point, GMM does not return a label, but a vector of probabilities of membership in each cluster. This is known as soft clustering.
Absorption of Variance
Let us return to our problematic hybrid salesperson (33% A, 33% B, 34% C). The GMM algorithm will “understand” this person’s nature. The model’s output will be a vector. What happens the following week, when that one-percentage-point behavioral fluctuation occurs and the proportions change to 34/33/33? GMM will update the output vector.
The difference is fundamental. In the K-means approach, there was a 100% change of category (an abrupt change of label). In the GMM approach, there was only a marginal adjustment of the probability weights. The system recorded the change but retained stability. The salesperson is still perceived as the same complex individual, not as a completely different person.
Implications for Routing and Prediction
The use of a probabilistic approach changes how the decision matrix is built. Instead of a simple “If Cluster A, then Customer X” table, we build an advanced weighting function.
The probability of sales success for a given salesperson is calculated as the sum of the products of that person’s cluster memberships and the effectiveness of those clusters. If a salesperson is a “mixture,” their predicted effectiveness will be the weighted average of the effectiveness of the component archetypes. This makes the routing system resistant to noise. Minor fluctuations in the salesperson’s mood do not cause a revolution in customer assignments. We obtain a system that evolves smoothly rather than abruptly. This is a critical characteristic of industrial- and business-process control systems.
Comparative Analysis: Parameter Overview
The following analysis identifies the key technical and business differences between the methods under discussion.
Decision Boundaries
- K-means creates linear (when the Euclidean metric is used) and sharp boundaries. An object lying one micrometer from the boundary is treated in the same way as an object at the center, and completely differently from an object lying one micrometer beyond the other side of the boundary.
- GMM creates fluid boundaries. The probability of membership changes continuously (as a gradient). The model understands the concept of assignment “uncertainty.”
Cluster Shape
- K-means assumes that clusters are spherical (circles in 2D, spheres in 3D). This means that the variance of features in all directions must be similar. This is rare in sales data: one characteristic, such as “technical orientation,” often has a much wider spread than another, such as “politeness.” K-means handles this poorly, often artificially splitting natural, elongated groups.
- GMM, thanks to the covariance matrix, allows clusters to take the form of ellipsoids with any orientation and degree of flattening. The model can detect that a group of “Technical Salespeople” is very diverse in terms of “Extraversion” but very consistent in terms of “Conscientiousness.”
Data Requirements and Computational Cost
- K-means is an extremely fast and convergent algorithm. It requires less data to produce a result, although that result may be affected by simplification bias.
- GMM is a more computationally complex algorithm. Estimating parameters—particularly the full covariance matrix—requires a larger number of data samples to avoid singularity or overfitting. It requires more advanced engineering expertise for calibration, such as the use of AIC/BIC criteria to select the number of components.
Asymmetry of Data Volume: The Impact of Extreme Disproportion (500+ vs. 1)
In operational practice, we face a fundamental asymmetry that calls into question the validity of using any classical clustering method. We are dealing with two populations with extremely different densities of information:
- salespeople—a stable population in which every object is described by 500+ conversations. The law of large numbers works in our favor here; the variance of the mean is low;
- customers—a highly dynamic population in which 90% of objects have only one conversation, while the remainder generally do not exceed three.
This disproportion gives rise to a justified doubt: is psychological profiling on the basis of one conversation methodologically correct at all? Is an attempt to assign a customer to a cluster after fifteen minutes of contact not simply reading tea leaves?
The Flaw in K-means: Hallucination of Certainty
The K-means algorithm is “blind” to sample size. It treats a point determined from one conversation just as seriously as a point determined from 500 conversations. If the customer was upset about a failure during that one conversation, K-means will unhesitatingly assign the customer to the “Demanding Customer” cluster (100% certainty). This is false certainty. The system builds a profile from noise rather than signal. This leads to incorrect routing decisions that may be worse than random assignment.
The Role of GMM: Detecting a Lack of Knowledge (Safety Brake)
GMM likewise cannot conjure up information that does not exist. With a sample of N=1, we cannot calculate variance for a specific customer. GMM does, however, allow us to model this situation by means of priors (a priori distributions).
For a customer with one conversation, the probabilistic model will return a result with maximum entropy (uncertainty). Instead of “Type A customer,” we will obtain: 25% type A, 25% type B, 25% type C, and 25% type D. This is the mathematical equivalent of saying: “I have no idea who this person is.”
Critical conclusion. Under conditions in which 90% of customers are in a “Cold Start,” GMM acts as a safety mechanism. It is not used for precise targeting, because there are no data, but to prevent the system from making foolish decisions based on the illusions produced by K-means. This suggests that routing for this 90% group of customers should be based on universal (safe) rules, while advanced clustering should be activated only after a threshold of, for example, three or four conversations has been exceeded.
In a “cold start” situation, GMM should be selected, but hybrid solutions should be used.
Because of the specific nature of the “cold start” (little customer data), K-means is dangerous because it “lies” that it knows the customer’s profile. GMM is safer because it “admits” that it does not know.
A Simple Implementation Strategy
- For new customers (1–3 conversations):
- method: no advanced clustering;
- action: route to a “Universal Salesperson” (safety mechanism). Do not force a guess about the profile;
- why: one conversation is insufficient for a reliable profile.
- For returning customers (4+ conversations) and all salespeople:
- method: GMM;
- action: precise routing based on the probability of a sale;
- why: there are now enough data for statistics to work. GMM best handles ambiguous profiles (“somewhat technical, somewhat a negotiator”), which predominate in reality.
In short: reject K-means completely (it is too risky). Implement GMM as the target engine, but activate it only when you have the minimum amount of data.
This is the key conclusion: for salespeople, who have a history of hundreds of conversations, K-means is methodologically inferior and should be replaced completely by GMM. The law of large numbers works in our favor here, so the probabilistic model is stable and precise.
Case Study: Simulation of Matrix Stability
Consider a hypothetical situation in a company producing packaging machinery. We have a group of 50 salespeople.
Scenario A (K-means). In week 1, the algorithm identifies a group of “Deal Closers” comprising 10 people. Their conversion rate is 15%. In week 2, due to minor changes in conversations, three “boundary” salespeople leave this group and two others enter it. The group now has nine people, but its personnel composition has changed by 50% (when we consider the small subgroup). The group’s conversion rate suddenly falls to 12%. The Management Board concludes: “The Closers group has ceased to be effective. We must change the conversation scripts.” Error: the group’s effectiveness did not fall. It was the algorithm that changed the definition of the group, mixing effective people with ineffective ones. The management decision is wrong.
Scenario B (GMM). In week 1, the salespeople have assigned “Closer” weights. The sum of the weights is 10.0 units. In week 2, the salespeople’s weights change slightly—for example, from 0.8 to 0.75. The sum of the weights is 9.8 units. The weighted average conversion changes from 15% to 14.8%. The Management Board concludes: “The situation is stable; there are no anomalies.” Decision: the system continues operating, making slight routing adjustments.
This example clearly shows that the probabilistic method acts like a shock absorber in a car’s suspension: it smooths out irregularities in the road (the data), ensuring a smooth ride (business decisions), whereas the rigid method transfers every vibration to the bodywork.
Conclusions and Implementation Recommendations
The choice between K-means and GMM is not a purely academic choice. It is a decision about the architecture of risk in the sales system.
For the Data Scientist, the conclusion is as follows: K-means is an exploratory tool, while GMM is an operational tool. K-means is excellent for a quick overview of the data (“let us see roughly what groups we have”), but it is unsuitable for controlling an automated process in which stability is critical.
We recommend implementing a Gaussian Mixture Model (GMM) as the segmentation engine for the smart-routing system in a machinery company. Despite its higher entry threshold—more difficult implementation and greater computational requirements—this method offers:
- faithfulness of representation—it better reflects the actual, hybrid nature of sales competencies;
- temporal stability—it is resistant to noise and minor fluctuations in employee behavior;
- predictive precision—it makes it possible to build more nuanced scoring models that use complete profile information rather than merely a simplified label.
Because of the small number of customer conversations, such a GMM assignment can be applied only to salespeople.
In the world of precision machinery, we do not use a hammer where a scalpel is needed. In sales analytics, we should not use hard clustering where we are dealing with the delicate matter of human behavior and relationships. The shift to probabilistic methods is a natural evolutionary step for every organization striving for data-driven operational excellence.
Wojciech Moszczyński
Wojciech Moszczyński—a graduate of the Department of Econometrics and Statistics at Nicolaus Copernicus University in Toruń; a specialist in econometrics, finance, data science, and management accounting. He specializes in optimizing production and logistics processes. He conducts research into the development and application of artificial intelligence. For years, he has been involved in popularizing machine learning and data science in business communities.