The goal of the Diet Problem is to select foods that satisfy daily nutritional requirements at minimum cost. This problem can be formulated as a linear program, for which constraints limit the number of calories and the amount of vitamins, minerals, fats, sodium, and cholesterol in the diet. Danzig (1990) notes that the diet problem was motivated by the US Army's desire to minimize the cost of feeding GIs in the field while still providing a healthy diet.
The Diet Problem can be formulated mathematically as a linear programming problem using the following model.
$F$ = set of foods
$N$ = set of nutrients
$c_i$ = cost per serving of food $i$, $\forall i \in F$
$a_{ij}$ = amount of nutrient $j$ in food $i$, $\forall i \in F, \forall j \in N$
$Nmin_j$ = minimum level of nutrient $j$, $\forall j \in N$
$Nmax_j$ = maximum level of nutrient $j$, $\forall j \in N$
$V_i$ = the volume per serving of food $i$, $\forall i \in F$
$Vmax$ = maximum volume of food consumed
$x_i$ = number of servings of food $i$ to consume
Minimize the total cost of the food
$\min \sum_{i \in F} c_i x_i$
Limit nutrient consumption for each nutrient $j \in N$.
$Nmin_j \leq \sum_{i \in F} a_{ij} x_i \leq Nmax_j$, $\forall j \in N$
Limit the volume of food consumed
$\sum_{i \in F} V_i x_i \leq Vmax$
Consumption lower bound
$x_i \geq 0$, $\forall i \in F$
We begin by importing the Pyomo package and creating a model object:
In [1]:
from pyomo.environ import *
infinity = float('inf')
model = AbstractModel()
The sets $F$ and $N$ are declared abstractly using the Set component:
In [2]:
# Foods
model.F = Set()
# Nutrients
model.N = Set()
Similarly, the model parameters are defined abstractly using the Param component:
In [3]:
# Cost of each food
model.c = Param(model.F, within=PositiveReals)
# Amount of nutrient in each food
model.a = Param(model.F, model.N, within=NonNegativeReals)
# Lower and upper bound on each nutrient
model.Nmin = Param(model.N, within=NonNegativeReals, default=0.0)
model.Nmax = Param(model.N, within=NonNegativeReals, default=infinity)
# Volume per serving of food
model.V = Param(model.F, within=PositiveReals)
# Maximum volume of food consumed
model.Vmax = Param(within=PositiveReals)
The within option is used in these parameter declarations to define expected properties of the parameters. This information is used to perform error checks on the data that is used to initialize the parameter components.
The Var component is used to define the decision variables:
In [4]:
# Number of servings consumed of each food
model.x = Var(model.F, within=NonNegativeIntegers)
The within option is used to restrict the domain of the decision variables to the non-negative reals. This eliminates the need for explicit bound constraints for variables.
The Objective component is used to define the cost objective. This component uses a rule function to construct the objective expression:
In [5]:
# Minimize the cost of food that is consumed
def cost_rule(model):
return sum(model.c[i]*model.x[i] for i in model.F)
model.cost = Objective(rule=cost_rule)
Similarly, rule functions are used to define constraint expressions in the Constraint component:
In [6]:
# Limit nutrient consumption for each nutrient
def nutrient_rule(model, j):
value = sum(model.a[i,j]*model.x[i] for i in model.F)
return model.Nmin[j] <= value <= model.Nmax[j]
model.nutrient_limit = Constraint(model.N, rule=nutrient_rule)
# Limit the volume of food consumed
def volume_rule(model):
return sum(model.V[i]*model.x[i] for i in model.F) <= model.Vmax
model.volume = Constraint(rule=volume_rule)
Putting these declarations all together gives the following model:
In [7]:
!cat diet.py
In [8]:
!cat diet.dat
Set data is defined with the set command, and parameter data is defined with the param command.
This data set considers the problem of designing a daily diet with only food from a fast food chain.
Pyomo includes a pyomo command that automates the construction and optimization of models. The GLPK solver can be used in this simple example:
In [9]:
!pyomo solve --solver=glpk diet.py diet.dat
By default, the optimization results are stored in the file results.yml:
In [10]:
!cat results.yml
This solution shows that for about $15 per day, a person can get by with 4 cheeseburgers, 5 fries, 1 fish sandwich and 4 milks.