Blending Craft Beer¶
Problem Definition¶
Duff Beer is an american brewery that sells alcoholic beverages worldwide. Duff beer has a very large market share in the US. One of the new business lines of Duff Beer is in the craft beer sector. Duff beer buys 3 types of wort from local suppliers. The availability of wort (in liters) and the price for each of them is specified in Table 1.
Table 1: daily availability and cost of wort
Wort |
Availability (liters) |
Cost (€cents/liter) |
|---|---|---|
Type 1 |
2500 |
120 |
Type 2 |
1200 |
095 |
Type 3 |
2000 |
150 |
The blend of the three types of wort must comply with the following quality requirements:
For Beer Angels, no less than 60% of Type 3 and no more of 20% of Type 2. The price of one bottle of Beer Angel is 4.10€s/liter.
For Beer Beast, no less than 15% of Type 3 and no more than 60% of Type 2. The price of one bottle of Beer Beast is 2.80€s/liter.
For Beer Cactus, no more than 50% of type 2. The selling price is 2.45€/liter.
Write a linear program to find the most profitable blend of wort to elaborate the three kinds of beers
Problem Model¶
Indices¶
We can define the following indices to express the problem in a compact form:
i: wort type \(i \in [1, 2, 3]\)
j: Beer brand \(j \in [A, B, C]\) where A notes Angels, B notes Beast, and C notes Cactus
Decision variables¶
The decision variables are:
\(x_{ij}\): Amount of wort of type i (1, 2, 3) in Beer j: A (Angels), B (Beast), C (Cactus), (ie i=[1,2,3], j=[A,B,C])
Objective Function¶
The objective function is:
\(\max z = \sum_{i=1}^{3}\sum_{j=A}^{C}(d_j - c_i)*x_{ij}\)
where \(d_j\) is the selling price of a bottle of beer j and \(c_i\) is the cost per liter of wort type i.
\(\max z =4 10(x_{1A}+x_{2A}+x_{3A})+280(x_{1B}+x_{2B}+x_{3B})+245(x_{1C}+x_{2C}+x_{3C})-120(x_{1A}+x_{1B}+x_{1C})-95(x_{2A}+x_{2B}+x_{2C})-150(x_{3A}+x_{3B}+x_{3C})\)
\(\max z = 290x_{1A}+315x_{2A}+260x_{3A}+160x_{1B}+185x_{2B}+130x_{3B}+125x_{1C}+150x_{2C}+95x_{3C}\)
Constraints¶
Subject to the following constraints:
Availability constraints The availability constraints can be expressed in a compact form as:
\(\sum_jx_{ij} \leq a_i \forall i\)
Where \(a_i\) is the availability of wort type i:
\(x_{1A}+x_{1B}+x_{1C} \leq 2500\)
\(x_{2A}+x_{2B}+x_{2C} \leq 1200\)
\(x_{3A}+x_{3B}+x_{3C} \leq 2000\)
Quality constraints The proportion of a given type of wort i’ in a type of beer j’ can be expressed as:
\(\frac{x_{i'j'}}{\sum_i(x_{ij'})} \forall i', j'\)
If we compare this to the minimum proportion \(R_{min_{i'j'}}\) we obtain:
\(\frac{x_{i'j'}}{\sum_i(x_{ij'})} \geq R_{min_{i'j'}} \forall i', j'\)
Or
\(x_{i'j'} \geq R_{min_{i'j'}}*\sum_i(x_{ij'}) \forall i', j'\)
If we take the denominator to the RHS. We can obtain a similar expression for the maximum proportion \(R_{max_{i'j'}}\):
\(x_{i'j'} \geq R_{max_{i'j'}}*\sum_i(x_{ij'}) \forall i', j'\)
From these expressions, we can derive the constraints for every type of beer:
Angels quality requirement constraints
\(x_{3A} \geq 0.6(x_{1A}+x_{2A}+x_{3A})\)
\(x_{2A} \leq 0.2(x_{1A}+x_{2A}+x_{3A})\)
Beast quality requirement constraints
\(x_{3B} \geq 0.15(x_{1B}+x_{2B}+x_{3B})\)
\(x_{2B} \leq 0.6(x_{1B}+x_{2B}+x_{3B})\)
Cactus quality requirement constraints
\(x_{2C} \leq 0.5(x_{1C}+x_{2C}+x_{3C})\)