Build your optimal circuit training workout via linear programming

The days are getting longer and it is time to prepare for summer again. All the months spent indoors, waiting for the warm weather, eating cookies, and drinking hot chocolate could make you nervous going back to the beach. Imagine a turbocharged workout routine that mixes cardio and strength training and let you train everywhere. Plus, you don’t even need expensive equipment and it’s easily customized to help you to get your best Beach Body ever. Sound too good to be true? It’s not! It’s called circuit training.

Scheveningen, Netherlands

Circuit Training

Circuit training lets you alternate between 4 to 10 exercises that target different muscle groups. The whole idea is to train different muscles all at the same time in a minimum amount of rest. Usually, you train between 15 and 40 minutes. During this time, you will make multiple circuits, with each course containing all exercises. It is a good idea to make 2-3 minute breaks per circuit, 30-90 seconds per exercise, and a 3-5 minute warm-up. The exercises should be arranged so that different muscle groups are trained alternately.

First tries to make a human flag, Germany

To create your perfect circuit training routine you could hire a personal trainer. But let’s be honest: trainers are very expensive. For this task, we will just replace the trainer with an artificial intelligence agent. The agent will receive our training goals (e.g. a bigger biceps or a six-pack) as input and then tell us our personalized circuit training routine.

Linear Programming

To solve this kind of optimization problem we will use linear programming. Linear programming finds an optimum of a linear function that is subjected to various constraints. You can model a lot of problems as linear functions, finding the perfect circuit training is one of them. To solve our problem we have to do the following steps of the optimization workflow:

  1. Identify the exact problem
  2. Model the problem
  3. Choose a tool to deal with the model
  4. Retrieve the solution
  5. Analysis
One of the first applications of linear programming was in the second world war. But it can also be used for diet optimization, in sports and much more.

In our case, we want to build a circuit training that is as compact as possible and trains all of our muscles. So we want to build a linear function/objective function that describes the total amount of time for the workout. We want to optimize this objective function by modifying the design variables \(x_{i} \in \{0,1\}\).

minimize: \( \sum_{i=1}^{n} t_i \cdot x_{i} \)

The design variable \(x_{i} \in \{0,1\}\) is an exercise and the given coefficents \(t_i\) is the needed time for exercise \(x_i\). \(n\) is the number of exercises.

Each exercise has got its own intensity per muscle group. So pull-ups train for example the biceps, the back and also a little bit abs. But how can we measure the intensity? I define intensity as a value between 0 and 1. 0 means it doesn’t train the specific muscle group at all. 1 means it totally trains the specific muscle group and you don’t need to train this muscle group anymore in the current circle. So I collected a bunch of exercises and defined for each muscle group its intensity and saved it into a CSV-file as a table:

pull ups0.51010.50.5000
push ups00100.50.25000

I recommend that you define your own intensities for each exercise. Currently, these intensity values are a little bit relative, if you know a better measurement unit please let me know.

So now we can choose for every circle training how much we want to train at least the specific muscle group. The only thing which is left for our LP-solver is to define these intensities as constraints so that the LP-solver takes the intensities into account. For that reason we create for each muscle group a constraint:

\( \sum_{i=1}^{n} y_i \cdot x_{i} \ge intensity_{shoulders}\)

\(y_i \in [0,1]\) is the intensity for exercise \(x_i\).


To solve LP-problems you can use different kinds of LP-solvers. A very nice Open Source LP-solver is PULP. I personally prefer the Gurobi LP-solver, it is a very powerful solver that has got a huge variety of features. If you are in academic you can receive an academic license for free. You can find the full code in my GitHub-Repository.

if __name__ == "__main__":
    # Define your intensities manually
    categories, min_intensity, max_intensity = gp.multidict({
        'shoulders' : [0,GRB.INFINITY],
        'back' : [0,GRB.INFINITY],
        'breast' : [0,GRB.INFINITY],
        'biceps' : [0,GRB.INFINITY],
        'triceps' : [0,GRB.INFINITY],
        'abs' : [0,GRB.INFINITY],
        'butt' : [0.5,0.5],
        'legs' : [2,2]

    # Read exercises and their intensity
    exercises_intensities = build_dict_from_csv_file("exercises.csv", "name")
    # Read exercises and their needed time
    exercises, time = gp.multidict(exercise_and_time_dict(exercises_intensities))
    # Build model
    m = gp.Model("circle_training")
    # Create trainings variables (each exercise is a decision variable)
    training = m.addVars(exercises, vtype=GRB.BINARY, name="training")
    # Objective Function
    m.setObjective(, GRB.MINIMIZE)
    # Constraint:
    # shoulders * push ups + shoulders * biceps + ...
    # others * push ups + ....
    m.addConstrs((gp.quicksum(exercises_intensities[e, c] * training[e] for e in exercises)
        == [min_intensity[c], max_intensity[c]]
        for c in categories), "_"
    # Find Solution
    # if there is a solution
    if m.status == GRB.OPTIMAL:
        print("Your training plan:")
        print('\nTime: %g' % m.objVal)
        trainingx = m.getAttr('x', training)
        for f in exercises:
            if trainingx[f] > 0:
                print('%s %g' % (f, trainingx[f]))
        print('No solution')


If you want to learn more:

Optimize your diet via SMT

Are you struggling with what to eat today? Sometimes it is tough to decide, especially if you are hungry and find out there is nothing left in your fridge. I want to show you how it is possible to develop your dish planner and let the computer automatically optimizes your meals according to your daily nutrition intake.

For this program, we use SMT solvers. SMT solvers try to figure out which variable assignment satisfies a given logic formula. If a logic formula is satisfiable, it returns the assigned variables (in our case, your meals). PySMT is a very user-friendly interface for Python developers to encode and solve these kinds of SMT problems. In this post, I give you a small introduction into the field of Satisfiability Modulo Theories by providing you an SMT solver for figuring out what you could eat today.

Ethiopian cuisine


To solve this problem we have to do the following steps:

  1. Create a data set with your favorite dishes (dish name, number of calories, etc. etc.)
  2. Research your daily nutrition intake
  3. Encode a propositional logic formula based on this data set
  4. Let the SMT-solver decide what you should eat

1. Your personalized dish data set

To get your personalized data set, you must figure out how much nutritions are in your dishes. Thanks to the internet there are a variety of websites that provide you with a lot of recipes and the number of nutritions. My favorite ones are:

If you don’t have time to create your data set, I prepared a data set for you. There are many tastes out there, and so I just looked for the most famous food in the world. I ended up with a subset of dishes from CNN travel, and I hope you like it (a lot of them I didn’t know before).

2. Your daily nutrition intake

So how many calories should you eat today? Well, if you’re like me and have got already eaten 3 dishes, much chocolate and haven’t done any sport today, you should probably eat anything. But OK, let’s say the SMT solver should tell you how much you should eat tomorrow. For finding out the exact number of calories you can google for something like “calories calculator” and you may end up on:

On these websites, you can easily calculate your needed number of calories.

3. Formula Encoding

So how can we tell our overweight/optimization problem to our computer? Just with these 4 simple constraints:

(I) \(\sum_{i=1}^{|D|} x_i \cdot d_{i2} \leq (C + \alpha)\\\)

(II) \(\sum_{i=1}^{|D|} x_i \cdot d_{i2} \leq (C – \alpha)\\\)

(III) \(\sum_{i=1}^{|D|} x_i = k\)

(IV) \(\bigwedge\limits_{i}^E x_i = 0\)

\( D \) is a set of dishes. \( d_i \in D \) is a tuple \((dish, calories)\) and represents a single dish \(i\).

\(x \in \{ 0, 1 \}\) is an SMT variable that gets automatically assigned by the SMT solver. If \(x_i = 1\), then we will eat this dish the next day. \(k \in \mathbb{N}\) is the number of dishes per day.

\( C \in \mathbb{R} \) is the allowed number of daily calories. \( \alpha \in \mathbb{R} \) is the allowed deviation of \(C\).

\(B\subseteq \mathbb{N}\) is a set of disabled dish indices for the next calculation.

(I) and (II) make sure that our dishes give us the predefined number of calories. (III) makes sure that we get \(k\) dishes per day. (IV) allows us to exclude specific dishes for the next day.

4. Coding

You can find the full code in my GitHub-Repository. This method builds from lines 15 – 53, all the constraints from Section 3. The SMT solving happens at line 55. If a solution is found, it returns the dishes for cooking.

def nutrition_calculator(dishes, number_of_calories, number_of_dishes, disabled_dishes = [], alpha = 100):
        This method creates a dish plan based on the daily allowed calories
        and the number of dishes.
            dishes, pandas data frame. Each row looks like this: (dish name, calories, proteins).
            number_of_calories, the daily number of calories.
            number_of_dishes, the daily number of dishes.
            disabled_dishes, list of row indizes from dishes which are not allowed in the current calculation.
            alpha, the allowed deviation of calories
            row indizes from dishes data frame. These dishes can be eaten today.
    # Upper and lower calory boundaries
    calories_upper_boundary = Int(int(number_of_calories + alpha))
    calories_lower_boundary = Int(int(number_of_calories - alpha))
    # List of SMT variables
    x = []
    # List of all x_i * d_i2
    x_times_calories = []
    # List of all x_i € {0,1}
    x_zero_or_one = []
    # List of disabled dishes
    x_disabled_sum = []
    for index, row in dishes.iterrows():
        x.append(Symbol('x' + str(index), INT))
        # x_i * d_i2
        x_times_calories.append(Times(x[-1], Int(row.calories)))
        # x_i € {0,1}
        x_zero_or_one.append(Or(Equals(x[-1], Int(0)), Equals(x[-1], Int(1))))
        # Disable potential dishes
        if index in disabled_dishes:
    x_times_calories_sum = Plus(x_times_calories)
    x_sum = Plus(x)
    if len(x_disabled_sum) == 0:
        x_disabled_sum = Int(0)
        x_disabled_sum = Plus(x_disabled_sum)
    formula = And(
            # Makes sure that our calories are above the lower boundary
            GE(x_times_calories_sum, calories_lower_boundary),
            # Makes sure that our calories are below the upper boundary
            LE(x_times_calories_sum, calories_upper_boundary),
            # Makes sure that we get number_of_dishes dishes per day
            Equals(x_sum, Int(int(number_of_dishes))),
            # Makes sure that we don't use the disabled dishes
            Equals(x_disabled_sum, Int(0)),
            # Makes sure that each dish is maximal used once
    # SMT solving
    model = get_model(formula)
    # Get indizes
    if model:
        result_indizes = []
        for i in range(len(x)):
            if(model.get_py_value(x[i]) == 1):
        return result_indizes
        return None

Extensions & More

We can easily add two further constraints to support e.g. proteins in our encoding:

(V) \(\sum_{i=1}^{|D|} x_i \cdot d_{i3} \leq (P + \beta)\\\)

(VI) \(\sum_{i=1}^{|D|} x_i \cdot d_{i3} \leq (P – \beta)\\\)

\( P \in \mathbb{R} \) is the allowed number of proteins. \( \beta \in \mathbb{R} \) is the allowed deviation of \(P\). Don’t forget to update our set of dishes \(D\) and extend our tuple \(d_i\) to \((dish, calories, proteins)\).

If you want to learn more about SMT solving, check out the following links: