Diet Planning

This example demonstrates the use of a Leap hybrid Constrained Quadratic Models (CQM) solver on a simple mixed-integer linear-programming (MILP) type of optimization problem, which can be expressed as a linear objective and constraints with integer and real-valued variables. Other examples include quadratic objectives and constraints, which better make use of the strengths of D-Wave’s solvers.

The goal of this problem is to optimize the taste of a diet’s foods while keeping to the dieter’s budget and daily requirements on macro-nutrients.

Example Requirements

To run the code in this example, the following is required.

Solution Steps

Section Workflow Steps: Formulation and Sampling describes the problem-solving workflow as consisting of two main steps: (1) Formulate the problem as an objective function in a supported form of quadratic model (QM) and (2) Solve your QM with a D-Wave solver.

This example formulates this problem as a constrained quadratic model and uses the LeapHybridCQMSampler to find good solutions.

Formulate the Problem

The table below shows a selection of foods chosen by a dieter, ranked for the dieter’s taste on a scale of one to ten, with evaluations (not necessarily realistic) of nutrients and costs.

Nutrients, Cost, and Taste Rankings for Available Foods

Food

Calories

Protein

Fat

Carbs

Fiber

Taste

Cost

Rice

100

3

1

22

2

7

2.50

Tofu

140

17

9

3

2

2

4.0

Bananas

90

1

0

23

3

10

1.0

Lentils

150

9

0

25

4

3

1.30

Bread

270

9

3

50

3

5

0.25

Avocado

300

4

30

20

14

5

2.00

The following table shows the dieter’s daily requirements for selected nutrients.

Daily Required Nutrients

Nutrient

Calories

Protein

Fat

Carbs

Fiber

Daily Requirement

2000

50

30

130

30

For simplicity, store the contents of the two tables above as a dict. The dict also contains information on whether the portion of a particular food is best treated as continuous (for example, rice can be any weight), in which case it should be represented with real-valued variables, or as a discrete unit (such as fruit that is eaten whole), in which case it is best represented by integer-valued variables.

>>> foods = {
...   'rice': {'Calories': 100, 'Protein': 3, 'Fat': 1, 'Carbs': 22, 'Fiber': 2,
...            'Taste': 7, 'Cost': 2.5, 'Units': 'continuous'},
...   'tofu': {'Calories': 140, 'Protein': 17, 'Fat': 9, 'Carbs': 3, 'Fiber': 2,
...            'Taste': 2, 'Cost': 4.0, 'Units': 'continuous'},
...   'banana': {'Calories': 90, 'Protein': 1, 'Fat': 0, 'Carbs': 23, 'Fiber': 3,
...              'Taste': 10, 'Cost': 1.0, 'Units': 'discrete'},
...   'lentils': {'Calories': 150, 'Protein': 9, 'Fat': 0, 'Carbs': 25, 'Fiber': 4,
...               'Taste': 3, 'Cost': 1.3, 'Units': 'continuous'},
...   'bread': {'Calories': 270, 'Protein': 9, 'Fat': 3, 'Carbs': 50, 'Fiber': 3,
...             'Taste': 5, 'Cost': 0.25, 'Units': 'continuous'},
...   'avocado': {'Calories': 300, 'Protein': 4, 'Fat': 30, 'Carbs': 20, 'Fiber': 14,
...               'Taste': 5, 'Cost': 2.0, 'Units': 'discrete'}}
...
>>> min_nutrients = {"Protein": 50, "Fat": 30, "Carbs": 130, "Fiber": 30}
>>> max_calories = 2000

Variables

Instantiate some real and integer variables1 in a list, quantities, that in the solutions will be assigned values for the selected quantities of every available food.

>>> quantities = [dimod.Real(f"{food}") if foods[food]["Units"] == "continuous"
...                                     else dimod.Integer(f"{food}")
...                                     for food in foods.keys()]
1

Always keep in mind that such “variables” are actually QuadraticModel objects,

>>> quantities[0]
QuadraticModel({'rice': 1.0}, {}, 0.0, {'rice': 'REAL'}, dtype='float64')

with a single variable with the requested label; e.g., rice. This means, for example, that multiplying by two doubles the linear bias,

>>> 2*quantities[0]
QuadraticModel({'rice': 2.0}, {}, 0.0, {'rice': 'REAL'}, dtype='float64')

multiplying two such “variables” creates a quadratic bias,

>>> quantities[0] * quantities[1]          
QuadraticModel({'rice': 0.0, 'tofu': 0.0}, {('tofu', 'rice'): 1.0}, 0.0,
...            {'rice': 'REAL', 'tofu': 'REAL'}, dtype='float64')

but multiplying three such quadratic models requires a non-quadratic term and so quantities[0] * quantities[1] * quantities[2] cannot generate a quadratic model and results in an error.

Bounds on the range of values for non-binary variables shrink the solution space the solver must search, so it is helpful to set such bounds; for many problems, you can find bounds from your knowledge of the problem. In this case, no food by itself should be assigned a quantity that exceeds max_calories.

>>> for ind, food in enumerate(foods.keys()):
...   ub = max_calories / foods[food]["Calories"]
...   quantities[ind].set_upper_bound(food, ub)

The maximum quantity of rice, for example, which here has 100 calories per portion, is 20 portions because \(20*100 = 2000\).

>>> quantities[0].upper_bound("rice")
20.0

Lower bounds are actually required in this problem for correct formulation: a valid mathematical solution might be to offset the calories of gorging on large numbers of tasty bananas by eating a negative amount of high-in-calories bread, so the formulation must include the impossibility of consuming negative quantities of food. Because Ocean sets a default value of zero for ~dimod.quadratic.Real variables, no explicit configuration is needed.

You can now formulate an objective function and any constraints feasible solutions must meet, and set these in your CQM.

Objective Function

The objective function must maximize taste of the diet’s foods while minimizing purchase cost.

To maximize taste and minimize cost is to assign values to the variables that represent quantities of each food, \(q_i\), such that when multiplied by coefficients representing the cost, \(c_i\), or taste, \(t_i\), of each food, form the linear terms of the following summations to be optimized:

\[ \begin{align}\begin{aligned} \min \sum_i q_i c_i \qquad \text{Minimize cost}\\\max \sum_i q_i t_i \qquad \text{Maximize taste}\end{aligned}\end{align} \]

To optimize two different objectives, taste and cost, requires weighing one against the other. A simple way to do this, is to set priority weights; for example,

\[\text{objective} = \alpha \text{(objective 1)} + \beta \text{(objective 2)}\]

By setting, for example \(\alpha=2, \beta=1\), you double the priority of the first objective compared to the second.

Instantiate a CQM.

>>> cqm = dimod.ConstrainedQuadraticModel()

You can define a utility function, total_mix, to calculate the summations for any given category such as calories.

>>> def total_mix(quantity, category):
...   return sum(q * c for q, c in zip(quantity, (foods[food][category] for food in foods.keys())))

Set the objective2. Because Ocean solvers minimize objectives, to maximize taste, Taste is multiplied by -1 and minimized.

>>> cqm.set_objective(-total_mix(quantities, "Taste") + 6*total_mix(quantities, "Cost"))
2

Section Tuning the Solution belows shows how the priority weights \(\alpha, \beta\) above were chosen.

Constraints

The problem has the following constraints of the Daily Required Nutrients table:

  1. Calories: no more than 2000

  2. Protein: at least 50

  3. Fat: at least 30

  4. Carbs: at least 130

  5. Fiber: at least 30

Constrain the diet’s calorie intake to the required daily maximum.

>>> cqm.add_constraint(total_mix(quantities, "Calories") <= max_calories, label="Calories")
'Calories'

Require that the daily minimum of each nutrient is met or exceeded.

>>> for nutrient, amount in min_nutrients.items():
...   cqm.add_constraint(total_mix(quantities, nutrient) >= amount, label=nutrient)
'Protein'
'Fat'
'Carbs'
'Fiber'

You can access these constraints as a dict with the labels as keys:

>>> list(cqm.constraints.keys())
['Calories', 'Protein', 'Fat', 'Carbs', 'Fiber']
>>> print(cqm.constraints["Calories"].to_polystring())
100*rice + 140*tofu + 90*banana + 150*lentils + 270*bread + 300*avocado <= 2000
>>> print(cqm.constraints["Protein"].to_polystring())
3*rice + 17*tofu + banana + 9*lentils + 9*bread + 4*avocado >= 50

Solve the Problem by Sampling

D-Wave’s quantum cloud service provides cloud-based hybrid solvers you can submit arbitrary QMs to. These solvers, which implement state-of-the-art classical algorithms together with intelligent allocation of the quantum processing unit (QPU) to parts of the problem where it benefits most, are designed to accommodate even very large problems. Leap’s solvers can relieve you of the burden of any current and future development and optimization of hybrid algorithms that best solve your problem.

Ocean software’s dwave-system LeapHybridCQMSampler class enables you to easily incorporate Leap’s hybrid CQM solvers into your application:

>>> from dwave.system import LeapHybridCQMSampler
>>> sampler = LeapHybridCQMSampler()                        

Submit the CQM to the selected solver. For one particular execution, the CQM hybrid sampler returned 49 samples, out of which 25 were solutions that met all the constraints.

>>> sampleset = sampler.sample_cqm(cqm)                    
>>> feasible_sampleset = sampleset.filter(lambda row: row.is_feasible)   
>>> print("{} feasible solutions of {}.".format(len(feasible_sampleset), len(sampleset)))    
25 feasible solutions of 49.

You can define a utility function, print_diet, to display returned solutions in an intuitive format.

>>> def print_diet(sample):
...    diet = {food: round(quantity, 1) for food, quantity in sample.items()}
...    print(f"Diet: {diet}")
...    taste_total = sum(foods[food]["Taste"] * amount for food, amount in sample.items())
...    cost_total = sum(foods[food]["Cost"] * amount for food, amount in sample.items())
...    print(f"Total taste of {round(taste_total, 2)} at cost {round(cost_total, 2)}")
...    for constraint in cqm.iter_constraint_data(sample):
...       print(f"{constraint.label} (nominal: {constraint.rhs_energy}): {round(constraint.lhs_energy)}")

The best solution found in this current execution was a diet of bread and bananas, with avocado completing the required fiber and fat portions:

>>> best = feasible_sampleset.first.sample                       
>>> print_diet(best)                                             
Diet: {'avocado': 1.0, 'banana': 6.0, 'bread': 4.1, 'lentils': 0.3, 'rice': 0.0, 'tofu': 0.0}
Total taste of 86.56 at cost 9.46
Calories (nominal: 2000): 2000
Protein (nominal: 50): 50
Fat (nominal: 30): 42
Carbs (nominal: 130): 372
Fiber (nominal: 30): 46

Tuning the Solution

Consider sampling each part of the combined objective on its own (i.e., \(\alpha=1, \beta=0\) and \(\alpha=0, \beta=1\)), and comparing the best solutions. Start with taste:

>>> cqm.set_objective(-total_mix(quantities, "Taste"))
>>> sampleset_taste = sampler.sample_cqm(cqm)                     
>>> feasible_sampleset_taste = sampleset_taste.filter(lambda row: row.is_feasible)  
>>> best_taste = feasible_sampleset_taste.first                   
>>> print(round(best_taste.energy))                               
-177
>>> print_diet(best_taste.sample)                                 
Diet: {'avocado': 0.0, 'banana': 17.0, 'bread': 0.0, 'lentils': 0.0, 'rice': 0.0, 'tofu': 3.3}
Total taste of 176.93 at cost 30.41
Calories (nominal: 2000): 2000
Protein (nominal: 50): 74
Fat (nominal: 30): 30
Carbs (nominal: 130): 402
Fiber (nominal: 30): 58

You can see that this diet is high in bananas, the tastiest food, and makes up for that food’s low levels of protein and fat with tofu.

Next, for cost:

>>> cqm.set_objective(total_mix(quantities, "Cost"))
>>> sampleset_cost = sampler.sample_cqm(cqm)                     
>>> feasible_sampleset_cost = sampleset_cost.filter(lambda row: row.is_feasible)  
>>> best_cost = feasible_sampleset_cost.first                    
>>> print(round(best_cost.energy))                               
3
>>> print_diet(best_cost.sample)                                 
Diet: {'avocado': 1.0, 'banana': 0.0, 'bread': 5.3, 'lentils': 0.0, 'rice': 0.0, 'tofu': 0.0}
Total taste of 31.67 at cost 3.33
Calories (nominal: 2000): 1740
Protein (nominal: 50): 52
Fat (nominal: 30): 46
Carbs (nominal: 130): 287
Fiber (nominal: 30): 30

This diet is ranked as less tasty than the previous but much cheaper. It relies mainly on bread and uses avocado to add fat and fiber.

Because of the differences in energy scale between the two parts of the combined objective, \(177 \gg 3\), if you do not multiply the part representing cost by some positive factor, optimal solutions will maximize taste and neglect cost. That is, if in \(\text{objective} = \alpha \text{(objective 1)} + \beta \text{(objective 2)}\) you set set \(\alpha = \beta = 1\), solutions will likely be close or identical to those found when optimizing for taste alone.

>>> cqm.set_objective(-total_mix(quantities, "Taste") + 1 * total_mix(quantities, "Cost"))
>>> sampleset = sampler.sample_cqm(cqm)                        
>>> feasible_sampleset = sampleset.filter(lambda row: row.is_feasible)  
>>> best = feasible_sampleset.first.sample                     
>>> print_diet(best)                                           
Diet: {'avocado': 0.0, 'banana': 17.0, 'bread': 0.0, 'lentils': 0.0, 'rice': 0.0, 'tofu': 3.3}
Total taste of 176.93 at cost 30.41
Calories (nominal: 2000): 2000
Protein (nominal: 50): 74
Fat (nominal: 30): 30
Carbs (nominal: 130): 402
Fiber (nominal: 30): 58

Compare the best solutions found when optimizing for taste and cost alone. Notice that to reduce 27 units of cost (\(30 - 3\)) in the latter solution, taste was decreased by 145 (\(177 - 32\)), for a ratio of \(145/27 \approx 5.5\). To give each part of the combined objective a similar weighting, the Objective Function section above multiplied the part of the objective that minimizes cost by a factor of \(\beta=6\).

The graphic below shows the solution energies of the combined objective and both of its parts for \(\alpha = 1, \beta = \{1, 2, 3, ... 19, 20\}\).

image

Energy of the objective for various multipliers of the cost.

For low (\(1-5\)) ratios of \(\frac{\beta}{\alpha}\) solutions are optimized for taste alone; for high ratios (\(> 15\)) solutions are optimized for cost. The relationship between this ratio and the weightings of the two parts of the combined optimization is non-linear, so while you can use such reasoning as was done above to find a starting point for “good” relative weightings, typically you need to experiment.

Notice that in all the previous solutions, the resulting diet relied on only two or three foods. If the dieter wants a more diverse diet, you can enforce that by setting appropriate bounds on the variables (or, equivalently, adding constraints on minimum/maximum quantities of each food).

>>> cqm.set_objective(-total_mix(quantities, "Taste") + 6*total_mix(quantities, "Cost"))
>>> for variable in cqm.variables:
...    cqm.set_lower_bound(variable, 1)
>>> sampleset_diverse = sampler.sample_cqm(cqm)                   
>>> feasible_sampleset_diverse = sampleset_diverse.filter(lambda row: row.is_feasible)  
>>> best_diverse = feasible_sampleset_diverse.first.sample        
>>> print_diet(best_diverse)                                      
Diet: {'avocado': 1.0, 'banana': 11.0, 'bread': 1.2, 'lentils': 1.0, 'rice': 1.0, 'tofu': 1.0}
Total taste of 132.93 at cost 21.1
Calories (nominal: 2000): 2000
Protein (nominal: 50): 55
Fat (nominal: 30): 44
Carbs (nominal: 130): 382
Fiber (nominal: 30): 59