Striking the appropriate balance between your power output and weight is critical for a sustainable and healthy approach to training and racing. Many athletes try to maintain as low a weight as possible while still being powerful on the bike — but it can be difficult to hit that sweet spot. If weight loss is taken too far, athletes can see a decrease in overall power, along with more serious health issues.

While watts per kilo are often top of mind, health and resiliency should ultimately be the key goals for any athlete. To illustrate this, consider that one pound of excess weight requires about two watts to pull up a hill.

On a bike, three kilograms of fat equates to around three seconds per kilometer on a climb. Tools such as BestBikeSplit can help illustrate the relationship between weight and power. Between the two figures below, every factor besides weight is identical, including FTP, bike weight, weather, and rolling resistance.

Depending on your goals, current level of fitness, and current weight, any of these three approaches could be the right one. They may also shift over time as fitness increases and weight loss goals are met. Have fun with DEAP, Marc-André.

Oh BTW, I forget that in my previous post : some of the selection operators say for instance a tournament will not care about those weights, because of the way fitness are compared.

For instance, if you have a set of weights : When the two individuals are compared, the default way to do so is to use a lexicographic approach : if the first value of an individual is better than the other, than the first is set to be better. If this first value is equal, then we go to the second value, and so on.

You have to use specific selection operators in order to make use of this weight information. In any other case, those values will be, in a way, ignored. I hope it's a bit more clear.

François-Michel De Rainville. Cheers, François-Michel. Félix-Antoine Fortin. Hi Jaime, If you plan to do multi-objective optimization and you do not have prior knowledge on the importance of each objective, you should stay away from aggregating function and instead use a multi-objective selection algorithm that is based on the concept of dominance, and most commonly Pareto dominance.

Regards, Félix-Antoine [1] Das, I. On Monday, March 31, AM UTC-4, Jaime RG wrote: Hi! Thanks for your time. That was quick! I think I've understood most of the concepts, but I would like to ask a couple of questions to confirm my thoughts. Since the score is compared lexicographically in the tournaments, would it be acceptable to clone that function and edit the comparison part to, for example, the weighted sum of the fitnesses?

Presumably important components first, then the rest? Thanks again, Jaime. Wow, totally missed this answer! Thanks for such a thorough explanation. I am so new to Python and GA, but I am slowly getting it! I will take a look at those papers and examples, but I think it's clear now.

Less handling effort, shorter waiting times, more independence are just a few of many reasons for an in-house simulation solution. Together with CADFEM Austria GmbH, a concept tailored to Schwarzmüller's individual requirements was developed. In addition to simulation-based topology optimizations for weight reduction of the support structures, vibration analyses turned out to be a topic with a need for action.

These and other tasks are meanwhile processed within the company's own small calculation team using Ansys simulation tools. In addition to the cost advantages, the development team benefits from the short time-saving paths and the better understanding of the physics in their own products.

Those who do the calculations themselves gain much deeper insights than those who "only" have to deal with the results. Optimization potentials are thus identified even faster and more precisely, ideas for new and further developments can be compared objectively in a short time without having to build real prototypes.

With Ansys Workbench, different analysis methods can be applied from a central platform. In topology optimization, a component is analyzed for the optimum structure by identifying the relevant load paths. For this purpose, Ansys determines the efficient material distribution within the defined design space.

In addition to the relevant mechanical loads, important design or manufacturing-related specifications are also included. With the information generated from this, we can derive optimal concepts in an even more targeted manner," confirms Harald Bruhns.

Until now, weight reductions have been repeatedly achieved at Schwarzmüller in various ways, which had to be validated by extensive real-world tests. Schwarzmüller products are analyzed in detail for such savings potentials which - very importantly - do not affect either safety or functionality.

Often, the well-founded simulations with Ansys not only lead to lighter, resource-saving solutions, but even contribute to increases in performance and an increase in transport capacity.

More performance. Less weight. Quality in trailers and vehicle bodies also means that they absorb vibrations during transport and keep them away from the cargo.

Vibrations caused by uneven road surfaces or strongly swinging tractor units can damage the transported goods. They can also generate disturbing noise for the environment.

With structural dynamic simulations in Ansys Mechanical, such effects, or their origin, are simulated and analyzed, and countermeasures are derived from the results. Simulation enables the Schwarzmüller technical team to prevent undesirable vibrations in advance in a targeted manner with considerably less effort and a higher, objective understanding of the details.

The result is literally "noticeable" for customers compared to many other products whose damping is based on the traditional and industry-standard approach of a purely experimental testing of the driving process.

When it comes to simulations with Ansys, CADFEM Austria GmbH is Schwarzmüller's partner. The cooperation began even before the introduction and has long since become a trusting and close exchange at eye level — from the training and further education of the users which at Schwarzmüller takes place primarily via the CADFEM eLearning formats, which can be used very flexibly to acute problem solving by user support to advice on all aspects of the future expansion of the Ansys infrastructure.

One thing is certain: the range of applications for detailed simulation at Schwarzmüller will be expanded in the near future following the successes already visible at an early stage in the areas of application described.

A first step will be the establishment of a dedicated calculation department, which will also deal with the virtual analysis and optimization of issues relating to operational strength and weld seam evaluations. Further ideas and plans are already underway and will be quickly brought to the table during the next discussions with CADFEM.

So far, it's been straight-forward experience and we're very glad with the results and performance. A couple of days with no prior knowledge of GA has been enough to achieve a 'beta-ish' status!

My scoring function has three components, so I am using a three-objective GA, with weights 1. However, we don't know how much each component contributes to the final behaviour in real life.

My guess is that, since you have chosen floats to represent the optimization type, can we tune them to find a better fitting score function? For example, if I feel that the third parameter won't be as important as the first two, could I use these weights: 1.

I have run a couple of tests, and DEAP doesn't complain about it, but I don't know if the change is having any effect. Anyway, if I am right, what would be the best way to find a better set of weights?

Marc-André Gardner. Hi Jaime, Thanks for the kind words, it's always great to see what people are able to achieve with DEAP. html fitness , including numbers higher than 1. DEAP will just use them as a multiplication factor to their relevant objective. Have fun with DEAP, Marc-André.

Oh BTW, I forget that in my previous post : some of the selection operators say for instance a tournament will not care about those weights, because of the way fitness are compared.

For instance, if you have a set of weights : When the two individuals are compared, the default way to do so is to use a lexicographic approach : if the first value of an individual is better than the other, than the first is set to be better. If this first value is equal, then we go to the second value, and so on.

You have to use specific selection operators in order to make use of this weight information. In any other case, those values will be, in a way, ignored. I hope it's a bit more clear.

François-Michel De Rainville. Cheers, François-Michel. Félix-Antoine Fortin. Hi Jaime, If you plan to do multi-objective optimization and you do not have prior knowledge on the importance of each objective, you should stay away from aggregating function and instead use a multi-objective selection algorithm that is based on the concept of dominance, and most commonly Pareto dominance.

Focus less on the actual caloric value of the food, and more on the nutrient value. Quality foods tend to have fewer calories, but are more nutrient dense, which will help your body stay healthier and help you feel fuller.

Unlike non-active individuals who attempt to lose weight, athletes have to balance calorie burn with calorie intake. Make sure to still get enough carbohydrates, fat and protein to fuel high-quality workouts. Taking in smaller, more frequent meals tends to help stabilize blood sugar and stave off overeating.

This awareness typically helps limit overall consumption. Balance is key! From a weight-loss perspective, using body fat percentage is a good gauge. These percentages can and will change during the course of focused training, but for most athletes, dropping below these ranges can negatively affect health and performance.

I prefer body fat as a measurement, rather Body Mass Index BMI , which is easy to calculate, but is calibrated based on the general population rather than athletes. In addition, computational resources are efficiently allocated to weighted optimization and the original problem optimization.

These improvements help make effective use of favorable information generated by the weighted optimization and save computational resources. The proposed LSWOA can reformulate the original problems into representative low-dimensional transformed ones and share the optimal weights to the whole population.

Through the application of the weighted optimization, LSWOA facilitates improving the performance of the algorithms for LSGO problems and helps tackle more complex problems better, e.

The rest of the paper is arranged as follows. In this section, the weighted optimization approach is reviewed. Then an essential concept, problem transformation, is presented to explain the transformation of optimization variable and the dimensionality reduction.

At the same time, we introduce some terms of weighted optimization to pave the way for the theoretical analysis and algorithm improvement in the following sections. Weighted optimization is an approach which can reduces the difficulty of the problem by optimizing the low-dimensional transformed problem.

Here we present the detailed process of weighted optimization. This is the outline of weighted optimization. Assume the lower and upper boundaries of x are lb and ub , we have:. The concept of problem transformation is abstracted from weighted optimization, which is proposed in [ 16 ].

Problem transformation describes the change of objective function and decision variables. The objective function is transformed from the original problem to the low-dimension transformed problem with weight variables. Here, its mathematical expression is given.

Let Z be an optimization problem with n decision variables. Then, the original problem is. Then, the transformed problem can be formulated as. As a result, the transformed problem is much easier to solve.

Transformation function refers to the combination of weights and the candidate solution. By means of transformation function, the decision variable changes from x to w. Here, interval-intersection transformation is used, which is a canonical transformation function used in [ 16 ] and [ 25 ]:.

Every sub-vector of the candidate is associated with a weight and tuned by the weight. Then we optimize the weight vector as a new decision vector. Thus, the feasible space of the weight vector is obtained by all the constraints. In addition, there are other transformation functions, e.

Besides the transformation function, variable grouping is also an important factor affecting problem transformation. In LSGO, grouping strategy can be categorized into two types: grouping based on the variable interaction e.

Variable interaction refers to the coupling between variables. In general, variable grouping is related to how to assign weights to decision variables. Here is an example to illustrate the problem transformation. The above is all the details of the weighted optimization approach. Weighted optimization can reduce the dimension of the problem to reduce the difficulty of the problem effectively.

However, existing studies highlight the dimension reducing feature of adaptive weighting, yet they may not clearly figure out the underlying mechanism which leads to the failure when optimizing some LSGO problems.

In this paper, we conduct the theoretical analysis of weighted optimization, which is provided in the following section. In this section, we make a further analysis of weighted optimization first. Then, according to the analysis, we aim to determine what factors will affect the problem transformation and illustrate how they affect the problem transformation, which hence motivates the proposed algorithm LSWOA in this paper.

The analysis would help answer the following questions:. What are the factors that affect the problem transformation? How do these factors change the mathematical form of transformed problems and then influence the optimization process? To demonstrate the search behavior of weighted optimization better and explain the relationship between the transformed problem and the original one, we will show the convergence process of weighted optimization.

The optimization of one-dimensional and multi-dimensional transformed problems will be analyzed as examples successively for better understanding. Here, the dimension of the transformed problem is the number of weights, which is also equal to the group number.

Table 1 provides four examples for the following analysis. The original problems are provided in the second column. The third and fourth column provide the information about the candidate solutions and the variable grouping. The generated transformed problems are listed in the last column.

Among these examples, Examples 1 and 2 show two 1-D transformed problems generated by the same original problem with two different candidate solutions while Examples 3 and 4 show two 2-D transformed problems generated by the same original problem with two different variable grouping.

One-dimensional weighted optimization is simple and easy to understand. Example 1 and Example 2 in Table 1 are two instances of one-dimensional weighted optimization. We make a detailed explanation on Example 1. The optimization process is illustrated in Fig.

It demonstrates that adjusting the weight is to scale down the candidate solution vector. The other example is illustrated in Fig.

The problem transformation in Example 2 differs from Example 1 in the candidate solution, i. Accordingly, the optimization path and the optimal solution are also different. The optimization process of two 1-D transformed problems with different candidate solutions Examples 1 and 2 in Table 1.

By reflecting on the convergence process, we can generalize the search behaviour of one-dimensional weighted optimization: the algorithm searches for better solutions on a straight line connecting the initial candidate solution and the origin O. So, the search sub-space has become a 1-D straight line of the original search space.

The optimization process of the 2-D transformed problem Example 3 in Table 1 generated with non-ideal grouping. Ellipsoids of different sizes are the contour of the 3-D function. The original 3-D search space turns into a 2-D subspace the dark plane in the sub-figure a , and the candidate solution is optimized according to the red arrow.

The transformed problem is non-separable because of the non-ideal grouping. The optimization process of the 2-D transformed problem Example 4 in Table 1 generated with ideal grouping. The transformed problem is separable because of the ideal grouping.

Our work is based on multi-dimensional problem transformation. To illustrate multi-dimensional transformed problem, we take a 2-D transformed problem, Example 3, as an instance.

The process of weighted optimization is presented in Fig. The ellipsoids of different sizes in Fig. The subspace is illustrated in Fig.

The other 2-D transformed problem in Example 4 is illustrated in Fig. Without loss of generality, we give a proposition of the optimization of multi-dimensional transformed problems:.

Therefore, we obtain a constrained form. Besides the original range constraint on x i. The constrained form 7 illustrates the reduction of dimension clearly. In the constraints formed by each variable group, i. Proposition 1 and the constrained form 7 are the key points of our theoretical analysis, which help us understand weighted optimization thoroughly.

The essence of dimension reduction in weighted optimization is reducing the number of variables by equality constraints, i. According to Proposition 1 and the constrained form, we can figure out the influence factors and the limitation of weighted optimization.

With regard to the constrained form 7 and Proposition 1 , we can obtain three enlightenments:. The origin satisfies all the constraints, i. Therefore, the global optimum can be easily found by weighted optimization.

This explains why weighted optimization usually works in the experiments using early benchmark problems that the global optima are at the origin. But there is no special effect for those problems where the global optima are shifted. Some transformed problems are similar as long as their candidate solutions are close to each other and the variable grouping keeps unchanged.

Thus, it is feasible to reuse the optimized weights of a transformed problem to similar ones. Following this idea, we can construct similar transformed problems and share weights to the similar ones for saving computational resources.

Weighted optimization has its limitations when searching for optimal solutions, i. Because weighted optimization provides a potential subspace, but the global optimum is often not in the subspace. Especially in the later stage of the original problem optimization, the candidate solution may be attracted by the local optimum, so the optimum of the transformed problem often becomes the local optimum of the original problem.

Therefore, in the later stage of the whole optimization process, weighted optimization may not work. For generating a transformed problem, candidate solution, grouping strategy, and transformation function are required.

These three factors will affect problem transformation. By changing these factors, the search subspace will become different, which explains why problem transformation diversifies. The following analysis elaborates these three factors. However, if we investigate the question in terms of the constrained problem shown in 7 that the coefficient of constraints will change if the candidate solutions are different.

Accordingly, the two transformed problems have different optima. However, when the positions of candidate solutions are close, and the variable grouping is unchanged, their transformed problems can be regarded as similar because the coefficients of these transformed problems will not change much.

Therefore, if we can find some specific candidate solutions close to each part of the population, the weights from optimizing these transformed problems can be reused to weigh the population. Variable grouping is related to how decision variables are divided into groups and cooperate with weights.

Two aspects the grouping strategy and the grouping number of variable grouping are taken into account. The grouping strategy refers to the basis of grouping variables.

In this part, the grouping strategy we study can be classified into ideal grouping and non-ideal grouping e. Ideal grouping divides all variables entirely based on interaction, which means that all variables in a group are interdependent while no interaction exists between each variable group.

Non-ideal grouping does not exactly following the interaction. Here, we figure out the influence of these two types of grouping strategies on problem transformation, which is illustrated in Proposition 2.

In 9 , the decision variable has been transformed to w. Correspondingly, a non-ideal grouping leads to a non-separable transformed problem.

It is more challenging to solve non-separable problems than separable problems. Figure 2 Example 3 in Table 1 and Fig. If variables are divided by ideal grouping, the subspace will section the original contour based on the separability of Z properly. Therefore, the sub-contour in Fig. However, in Fig.

So the sub-contour becomes a rotated ellipse with a non-separability feature. The grouping number determines the dimension of the transformed problem.

It involves the fineness of weighted optimization, that is, the more variable groups are divided beforehand, the higher optimization accuracy we achieve. However, it means a lower convergence speed.

Even when we divide each variable into a group, the transformed problem is equivalent to the original problem. This factor reminds us to adjust and balance the level of optimization between coarseness and fineness. Figure 4 shows the convergence characteristic of two transformed problems with different scales D and D.

The D transformed problem can get better fitness, but it is harder to optimize than the D one. The above analysis about grouping numbers enlightens us that it is necessary to allocate appropriate computational resources to a transformed problem according to its dimension.

The convergence characteristic of two transformed problems with different grouping numbers dimensions. The candidate solution and the grouping strategy random grouping keep unchanged. The two transformed problems are optimized by DE with a population size of The data in the figure are averaged after 25 independent tests.

The transformation function can be considered as a combination operator applied to weights and the candidate solution. The transformation function we used in this work is the interval-intersection transformation, which is introduced in 5.

In addition, an improvement transformation function is proposed in [ 39 ] aiming to uncouple the relationship between absolute variable values and the boundaries of the original variable:.

Different transformation functions change the structure of constraints and make the search subspace vary. For example, every grouped variable using transformation function 5 leads to a line constraint connecting the origin to the candidate solution, while transformation function 10 lead to two lines connecting the candidate solution to the upper and lower bound points.

The pseudo-code of the proposed LSWOA is presented in Algorithm 1, which works with the cooperation of weighted optimization Algorithm 2 , weights sharing Algorithm 3 , and transformation function Algorithm 4. In LSWOA, the weighted optimization is utilized in two stages: population initialization and integrated optimization.

The former refers to weighting the initial population at the beginning, and the latter refers to integrating the optimizations of the original problem and the transformed problem. The main framework of LSWOA is shown in Algorithm 1. Lines demonstrate the stage of population initialization with weighting.

First, the algorithm creates an initial population randomly Line 2 and divides the decision variables into some groups stored in a cell g Line 3. Then, q individuals are selected from the initial population, which can be seen as q reference candidate solutions preparing for weighted optimization Line 4.

These beneficial weight vectors are shared with the population See Algorithm by an interval-intersection transformation function Line 7. By repeating this process q times, an improved population is produced for the next optimization stage. Lines 9—17 of Algorithm 1 demonstrate the integrated optimization stage.

We allocate half of the total evaluation resources to the integrated optimization stage. In this stage, the optimizations of the original problem and the transformed problem are conducted in turn. Finally, the weighted optimization is banned after the integrated optimization stage if half of the computational resources are used to keep the diversity.

The process of the weighted optimization is provided in Algorithm 2. Line 2 demonstrates the initialization of the weight population. Note that the weight individual cannot be evaluated directly. Finally, the best weight vector is obtained according to the fitness of the weight population WP.

Algorithm 3 shows the weight-sharing strategy, which enables the population to be improved by a weight vector. Finally, a selection procedure is started to create an improved population.

Algorithm 4 presents the details of the transformation function. For an input solution x and a weight vector w , each variable group is multiplied by the corresponding weight. In this subsection, we illustrate why LSWOA is effective. Three main improvements we make are presented, and the contributions that these improvements make are analyzed.

Compared with adaptive weighting, which only optimizes few selected individuals at the end of each CC cycle, LSWOA integrates the weighted optimization into the initialization and optimization stages of the algorithm. In these two stages, the weighted optimization helps to improve the quality of the whole population.

For better utilizing the weights, a weight-sharing strategy with the construction of reference candidate solutions is proposed. The best weight vectors are obtained by the weighted optimization of some specific reference candidate solutions. Then these weight vectors can be shared with the whole population.

Meanwhile, a candidate solution inheriting strategy is proposed to take the candidate solution as the initial information in each weighted optimization, which ensures a better performance of each weighted optimization. Furthermore, we design a computational resources allocation scheme for striking a balance between the weighted optimization and the original optimization.

Weight-sharing refers to reusing the best weight vector of a transformed problem to other transformed problems.

It will cost too many computational resources if we want to solve transformed problems generated by all individuals in population. Thus, in the proposed LSWOA, we designed a weight-sharing method to evolve the population approximately. The condition of weight-sharing is that the two transformed problems are similar.

In the process of problem transformation, similar transformed problems can be generated as long as the candidate solutions with similar positions are selected when the variable grouping is unchanged. After sharing weights with all solution in the population, the next population is selected from the weighted population and the original population.

In general, weight-sharing can be summarized as optimizing reference transformed problems, then directly improving the whole population through the experience information the optimized weight vectors generated in the weighted optimization process.

We select or construct reference solutions which can represent the population to some extent. In the initialization stage, we select q random individuals as the reference candidate solutions; and in the integrated stage, we create a representative solution by averaging all individuals in the population.

It is designed because we simply treat the population as a cluster in the decision space along with the optimization.

As the optimization runs, individuals distribute widely in the beginning and gather gradually afterwards. Thus, we select q random individuals to simulate the widely distributed population initially and use a mean vector to approximate the gathered population later.

This strategy is called candidate solution inheriting. In fact, the transformed problem is still with a medium scale and it is not easy to deal with by an ordinary EA. The candidate solution can provide a prior information, which greatly accelerates the convergence of the weighted optimization.

In the early stage of the algorithm, the quality of candidate solutions is poor, which may not provide good initial information to guide the weighted optimization.

But when the algorithm runs for a period of time, the fitness values of the initial candidate solutions are often better than that of the candidate solutions with random weights. Therefore, it is particularly important to apply this strategy in the integrated optimization stage.

In previous study on adaptive weighting, weighted optimization is considered a waste of computational resources through their experiment [ 34 ]. So, it is necessary to allocate appropriate computational resources to weighted optimization and ordinary optimization.

Weighting optimization have a high convergence speed, but the trade-off is reducing the optimization accuracy. The solution after the weighted optimization hardly reaches the global optimum of the original problem.

To begin with, the weighted optimization and weight-sharing are implemented after population initialization. When entering the integrated optimization, we still use these two strategies in the first half of the whole optimization process, so that weighted optimization can speed up convergence and the applied EA helps to keep diversity.

However, weighting optimization is banned in the later stage because it is hard to find a better solution than the candidate solution by optimizing its weights.

Thus, some fitness evaluations are saved for optimizing the original problem. Furthermore, the fitness evaluations of the weighted optimization should be assigned depends on the number of groups. In this section, we integrate LSWOA into two typical LSGO algorithms, DECC-G and SaNSDE, to verify its effectiveness first.

In particular, we pay attention to the performance of the algorithms on non-separable and multi-modal problems. Then, we test the scalability of the proposed LSWOA. The number of variables is scaled to to investigate whether LSWOA is more helpful than D problems.

Finally, we explore the impact of grouping strategy on the performance of weighted optimization. DECC-G [ 25 ]: It is a cooperative co-evolution differential evolution with a random grouping that belongs to the decomposition approach.

It is one of the most classic CCEAs.

We recommended a BIA at the beginning and end of a program to compare results. Nutritional needs vary. Starting with a diet diary can be helpful for some. energy output exercise. It is often surprising to see how many calories are in a single serving of our favorite foods, such as pizza, and tracking what we eat and drink improves awareness about what we are putting into our bodies.

Food should be seen as providing sufficient energy to function and to provide essential nutrients and vitamins to help your body repair, heal and grow.

For additional support, we offer a specialized nutrient injections that work synergistically to support fat and carbohydrate metabolism, while also boosting energy and motivation. If weight loss is taken too far, athletes can see a decrease in overall power, along with more serious health issues.

While watts per kilo are often top of mind, health and resiliency should ultimately be the key goals for any athlete. To illustrate this, consider that one pound of excess weight requires about two watts to pull up a hill.

On a bike, three kilograms of fat equates to around three seconds per kilometer on a climb. Tools such as BestBikeSplit can help illustrate the relationship between weight and power.

Between the two figures below, every factor besides weight is identical, including FTP, bike weight, weather, and rolling resistance. Depending on your goals, current level of fitness, and current weight, any of these three approaches could be the right one. They may also shift over time as fitness increases and weight loss goals are met.

The four standard durations are 5 second, 1-minute, 5-minute, and FTP Functional Threshold Power. For the purposes of power profiling, these are the ranges that best reflect neuromuscular power, anaerobic capacity, maximal oxygen uptake VO2 Max , and lactate threshold LT respectively.

Focused workouts with supplemental strength work will help replace fat with lean muscle, as well as increase your power along the power duration curve. Increased efficiency in oxygen delivery helps your body to more quickly buffer lactic acid, letting you spend less time anaerobic when you begin your next sprint or big climb.

A healthy diet is important for any athlete interested in maximizing their performance and looking to reach their full potential.

However, there are a few guidelines to help increase the likelihood of achieving any weight loss goal. Focus less on the actual caloric value of the food, and more on the nutrient value. Quality foods tend to have fewer calories, but are more nutrient dense, which will help your body stay healthier and help you feel fuller.