METAL LPs

Two very simple Linear Programming problems from the METAL Series (Mathematics for Economics: enhancing Teaching and Learning).
http://www.metalproject.co.uk/METAL/Resources/Films/

The Chocolatier Problem¶

In [1]:

using JuMP

In [2]:

m1=Model()

Out[2]:

$$ \begin{alignat*}{1}\min\quad & 0\\ \text{Subject to} \quad\end{alignat*} $$

In [3]:

@defVar(m1,x[1:2])

Out[3]:

$$ x_{i} free \quad\forall i \in \{1,2\} $$

In [4]:

@addConstraint(m1,4x[1]+18x[2]<=1296)
@addConstraint(m1,12x[1]+6x[2]<=1824)
@addConstraint(m1,x[1]>=0)
@addConstraint(m1,x[2]>=0)

Out[4]:

$$ x_{2} \geq 0 $$

In [5]:

@setObjective(m1,Max,55x[1]+89x[2])

Out[5]:

$$ 55 x_{1} + 89 x_{2} $$

In [6]:

print(m1)

Max 55 x[1] + 89 x[2]
Subject to
 4 x[1] + 18 x[2] ≤ 1296
 12 x[1] + 6 x[2] ≤ 1824
 x[1] ≥ 0
 x[2] ≥ 0
 x[i] free ∀ i ∈ {1,2}

In [7]:

solve(m1)

Out[7]:

:Optimal

In [8]:

print(getValue(x))

[130.5,43.0]

In [9]:

println("Objective is: ",m1.objVal)

Objective is: 11004.5

Tomatoes vs Lettuces Problem¶

In [10]:

m2=Model()

Out[10]:

$$ \begin{alignat*}{1}\min\quad & 0\\ \text{Subject to} \quad\end{alignat*} $$

In [11]:

@defVar(m2,y[1:2])

Out[11]:

$$ y_{i} free \quad\forall i \in \{1,2\} $$

In [12]:

@addConstraint(m2,10y[1]+15y[2]<=420)
@addConstraint(m2,y[1]+2y[2]<=50)
@addConstraint(m2,y[1]<=10)
@addConstraint(m2,y[2]>=12)
@addConstraint(m2,y[1]>=0)

Out[12]:

$$ y_{1} \geq 0 $$

In [13]:

@setObjective(m2,Max,6.25y[1]+20y[2])

Out[13]:

$$ 6.25 y_{1} + 20 y_{2} $$

In [14]:

print(m2)

Max 6.25 y[1] + 20 y[2]
Subject to
 10 y[1] + 15 y[2] ≤ 420
 y[1] + 2 y[2] ≤ 50
 y[1] ≤ 10
 y[2] ≥ 12
 y[1] ≥ 0
 y[i] free ∀ i ∈ {1,2}

In [15]:

solve(m2)

Out[15]:

:Optimal

In [16]:

print(getValue(y))

[-0.0,25.0]

In [17]:

println("Objective is: ",m2.objVal)

Objective is: 500.0