Prepared by Viktor Plamenov
In the realm of sensory and consumer science, the consistent profiling of food and beverage products is essential. Central to this consistency is the art and science of blending different batches or barrels of foundational ingredients. Take, for instance, the wine sector: vintners routinely blend diverse batches of wine to assure a steady flavor profile under specific brand names. In a similar vein, chocolatiers meld cocoa beans from varied sources to guarantee a standardized taste. This whitepaper examines the methodologies to derive optimal blending ratios for base products, ensuring alignment with a predetermined target profile, thus paving the way for streamlined product customization. To illustrate our blending approach, winemaking serves as our primary example. That being said, the techniques discussed in the report extend to a multitude of other mixing processes in the broader food and beverage sector.
1. Introduction
The art of wine blending is a delicate balance between tradition, expertise, and sensory analysis. Historically, the creation of a harmonious blend has largely relied on the palate and experience of master blenders. Their task, often handed down through generations, is to mix different wines to achieve a consistent and desirable flavor profile. However, as the wine industry evolves and scales, there's an increasing demand for systematic and replicable methods to achieve optimal blends, especially when trying to match a particular target profile. In this context, a manual approach, while valuable, may face limitations when applied to large-scale operations. Leveraging mathematical models and optimization techniques not only streamlines the blending process but also ensures a more precise alignment with the desired profile. Through this approach, we aspire to bridge the gap between traditional artistry and modern-day analytical precision in the realm of wine blending.
2. Formulation
The optimization program consists of two main parts - an objective function and a set of constraints (e.g. business/chemistry related). The objective function could be interpreted as the business goal we try to achieve, whereas the constraints are all the business related limitations that need to be taken into account. Problem without any constraints belong to the class of unconstrained optimization problems, which tend to be considerably easier to solve compared to constrained optimization. In order to work with less ambiguity we need to introduce a bit of mathematical notation.
2.1 Objective function
The class of objective functions to be examined in this particular use case could be more broadly categorized into two buckets depending on whether we optimize absolute errors or absolute percentage errors. This might become an important distinction whenever the sensory attributes are on completely different scales.
2.1.1 Scale-Sensitive
(1) where i = 1, 2, ..n and n is the number of sensory attributes and j = 1, 2, 3, ...m, where m denotes the number of wines to be used in the blending process. τi denotes the sensory attribute i of the target wine. The objective here is to minimize the sum of absolute errors from each sensory target attribute τi and the linear combination of attributes across all blended wines, ie Xj . An alternative formulation would be to square the errors rather than take their absolute values:
(2) tends to be slightly easier from a mathematical perspective, but the business objective of the procedure remains intact.
2.1.2 Scale-Invariant
When absolute proximity to the target profile is the goal the above formulation might be preferred. Until now we examined objectives where the scale differ- ence of the sensory attributes was ignored and we aimed to be as close as possible to the target profile in terms of absolute error. When there is a substan- tial scale difference, the optimizer will favor sensory attributes with a larger magnitude. One possible way to circumvent this limitation is to rescale the error and convert it in percentage form. This will force the optimizer to treat a 5 percent improvement across a small and large-scale sensory attribute the same.
2.2 Constraints
2.2.1 Proportionality Constraint
The blending problem is defined over a set of decision variables representing the wine proportions. It is imperative to impose specific constraints to ensure the feasibility and practicality of the solution. Primarily, a proportionality constraint is imposed, ensuring that summation of all wine proportions equates to one. This can be mathematically represented as:
Furthermore, to maintain the physical relevance of the proportions, each decision variable must be non- negative, denoted as:
Various additional constraints may be necessary, such as limiting the number of wines in the blend or prescribing a specific range for the proportions. As operational requirements evolve, these constraints can be methodically integrated into the optimization framework.
2.2.2 Indicator Constraints
To ensure that the number of wines in the blend does not exceed a certain threshold, we introduce binary indicator variables. These variables are defined such that they assume a value of 1 if a specific wine is utilized in the blend (regardless of its proportion) and 0 otherwise. Mathematically, the constraint can be expressed as:
Here, M symbolizes the maximum permissible num- ber of wines in the blend. The indicator function, I(x), is distinctly defined for the decision variables as:
By implementing this framework, we can regulate the number of wines in the blend, ensuring it aligns with operational or quality criteria.
2.2.3 Hard vs. Soft Constraints
In optimization, constraints are typically categorized as either hard or soft. Hard constraints are non-negotiable and any solution violating them is labeled infeasible. These are essential for maintaining operational or business standards. On the other hand, soft constraints, while desirable, are not strictly binding. Solutions can violate these constraints but at the expense of incurred penalties, usually augmenting the objective function’s cost. In essence, while hard constraints are imperative, soft constraints are more akin to desirable guidelines.
2.3 Optimal Blend
This section offers an insight into the optimization procedure used to determine the optimal blends. While we won’t delve extensively into mathematical optimization’s detailed theory, we aim to elucidate the underpinnings of select methods. Various mathematical strategies can achieve these results; here, we illustrate the Lagrange Multipliers method for nonlinear constrained optimization. Notably, this approach faces challenges with larger problems, prompting a preference for numerical methods. The entire procedure can be succinctly expressed using the Laplacian operator, ie using the first-order partial derivatives, which entails formulating a system of nonlinear equations and setting them to zero to pinpoint the extreme points (minima/maxima/saddle point).
Recall the values of τi and αij directly from the sensory data. The primary decision variables in our analysis are xj , which denote the blending proportions of the wines. While λ is present, it is primarily for mathematical considerations and can be set aside. Differentiating the objective function and settings its Jacobian to zero:
which translates into the following system of equations:
The last step is to solve this system of equations in order to obtain the optimal values of the blending proportions. In order to construct the blended profile, we need to use the optimal blending proportions and multiply them with the matrix of sensory coefficients for all wines used in the blending.
where S is the sensory attributes matrix of the wines used in the blending, and B is the matrix containing the sensory profile of the optimal blend.
3. Numerical Example
A numerical example is prepared to demonstrate the mechanics behind the method.
Consider a set of three distinct wines to blend with four features, i.e., n = 4, m = 3. The chosen features are Acidity, Intensity, Oak, Tannin. For this illustration, let our target sensory profile be referred to as Wine A. The wines permitted for blending are labeled as Wine B (x1), Wine C (x2), and Wine D (x3). The target profile is τ = [1.45, 11.05, 6.0, 5.95], and the sensory matrix of the three wines S is provided as follows:
Here formulation (2) is used for the objective function. Writing this explicitly yields the following expression:
Then, define the optimization problem as:
Figure 3.1: Objective function values across different blending proportions. Due to the proportionality constraint x1+x2 +x3 = 1, we could infer the value of x3 by knowing the values of x1 and x2. The red point represents the value where the error is minimized.
To address the equality constraint, we can apply the method of Lagrange multipliers. The associated Lagrangian is given by:
Here, λ is the Lagrange multiplier associated with the equality constraint.
Setting the partial derivatives in (5) to zero and solving the system of equations yields the following optimal blending proportions:
Recall the target profile was given by τ = [1.45, 11.05, 6.0, 5.95]. The sum of squared errors for each wine plus the blend are given below:
In the present analysis, Wine B exhibited a sensory profile that closely aligned with the reference, wine A. In contrast, Wines C and D demonstrated marked deviations from the desired profile. Notably, the blend resulted in a further 21 percent reduction in the sum of squared errors in comparison to the closest matching existing profile.
4. Conclusion
The evolution of the wine industry calls for an integration of time-honored artistry with advanced analytical methods. While the expertise of master blenders remains essential, the incorporation of mathematical programming allows for a more objective examination of blending relationships. This approach ensures efficiency and scalability in real-world applications and provides a platform for swift experimentation. It empowers blenders to pose "what-if" scenarios in a formalized manner. By merging the best of both worlds, we can uphold the traditions of wine blending while capitalizing on the precision and adaptability that modern techniques offer.
5. References
H. Paul Williams, Model Building in Mathematical Programming, Fifth Edition, UK, 2013
Nkosephayo T. Manyatsi, W. K. Solomon and J.S. Shelembe, Optimization of blending ratios of wheat-maize-sprouted mungbean (Vigna radiata L.) composite flour bread using D-optimal mixture design, Cogent Food & Agriculture, 2020
M. Meyer et al, Innovative Decision Support in a Petrochemical Production Environment, Interfaces, 41(1): 79–92, Jan.–Feb. 2011.