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:
from pyomo.environ import *
infinity = float('inf')
model = AbstractModel()
The sets $F$ and $N$ are declared abstractly using the Set
component:
# Foods
model.F = Set()
# Nutrients
model.N = Set()
Similarly, the model parameters are defined abstractly using the Param
component:
# 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:
# 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:
# 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:
# 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 inequality(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:
!cat diet.py
from pyomo.environ import * infinity = float('inf') model = AbstractModel() # Foods model.F = Set() # Nutrients model.N = Set() # 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) # Number of servings consumed of each food model.x = Var(model.F, within=NonNegativeIntegers) # 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) # 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 inequality(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)
Since this is an abstract Pyomo model, the set and parameter values need to be provided to initialize the model. The following data command file provides a synthetic data set:
!cat diet.dat
param: F: c V := "Cheeseburger" 1.84 4.0 "Ham Sandwich" 2.19 7.5 "Hamburger" 1.84 3.5 "Fish Sandwich" 1.44 5.0 "Chicken Sandwich" 2.29 7.3 "Fries" .77 2.6 "Sausage Biscuit" 1.29 4.1 "Lowfat Milk" .60 8.0 "Orange Juice" .72 12.0 ; param Vmax := 75.0; param: N: Nmin Nmax := Cal 2000 . Carbo 350 375 Protein 55 . VitA 100 . VitC 100 . Calc 100 . Iron 100 . ; param a: Cal Carbo Protein VitA VitC Calc Iron := "Cheeseburger" 510 34 28 15 6 30 20 "Ham Sandwich" 370 35 24 15 10 20 20 "Hamburger" 500 42 25 6 2 25 20 "Fish Sandwich" 370 38 14 2 0 15 10 "Chicken Sandwich" 400 42 31 8 15 15 8 "Fries" 220 26 3 0 15 0 2 "Sausage Biscuit" 345 27 15 4 0 20 15 "Lowfat Milk" 110 12 9 10 4 30 0 "Orange Juice" 80 20 1 2 120 2 2 ;
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:
!pyomo solve --solver=glpk diet.py diet.dat
[ 0.00] Setting up Pyomo environment [ 0.00] Applying Pyomo preprocessing actions [ 0.00] Creating model [ 0.02] Applying solver [ 0.06] Processing results Number of solutions: 1 Solution Information Gap: 0.0 Status: optimal Function Value: 15.05 Solver results file: results.json [ 0.06] Applying Pyomo postprocessing actions [ 0.06] Pyomo Finished
By default, the optimization results are stored in the file results.yml
:
!cat results.yml
# ========================================================== # = Solver Results = # ========================================================== # ---------------------------------------------------------- # Problem Information # ---------------------------------------------------------- Problem: - Name: unknown Lower bound: 15.05 Upper bound: 15.05 Number of objectives: 1 Number of constraints: 10 Number of variables: 10 Number of nonzeros: 77 Sense: minimize # ---------------------------------------------------------- # Solver Information # ---------------------------------------------------------- Solver: - Status: ok Termination condition: optimal Statistics: Branch and bound: Number of bounded subproblems: 89 Number of created subproblems: 89 Error rc: 0 Time: 0.00977396965027 # ---------------------------------------------------------- # Solution Information # ---------------------------------------------------------- Solution: - number of solutions: 1 number of solutions displayed: 1 - Gap: 0.0 Status: optimal Message: None Objective: cost: Value: 15.05 Variable: x[Cheeseburger]: Value: 4 x[Fries]: Value: 5 x[Fish Sandwich]: Value: 1 x[Lowfat Milk]: Value: 4 Constraint: No values
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.