# Sample Paper on Training Optimization for the Decathlon

Summary: Training Optimization for the Decathlon
1. Introduction
The decathlon is a competition that comprises 10 track and field events spread over two days.
A significant challenge is to design an effective training schedule. The usual two phases are: first, time is assigned to each exercise and, second, the actual timing and the order of training is fixed.
2. Training Time Assignment
Training time assignment involves the distribution of the available exercising time over the training units.
2.1 Periodization and Time Budget
Budget is the amount of time, in minutes, availed for the actual training during the scheduling period. This paper’s scheduling period is one year. It is based on competition schedule. Periodization is the splitting of the year into training periods.

2.2 Exercises and Training Units
Each unit in a decathlon can have several exercises. The paper organized the set of exercises into 45 training units, where each unit represented a group of similar exercises. The coach can distribute time across specific exercises. The choice of exercise will depend on the athlete’s physical or mental state.
2.3 Performance; Event Condition
An expected performance in a fictitious competition was used to gauge the effectiveness of a training time assignment. Among the many factors that affect an athlete performance is his/her physical and mental condition.
Because the mental condition depends on competition participation and periodization; and because a training time assignment is being designed for the year about to begin, and form-on-the-day and external factors are mostly uncontrollable; the mental condition is taken for granted.
What is left-the physical condition-is measured by means of an event condition, which is defined as the expected performance for a given event at a given time, measured in meters or seconds. The mental condition can be implicitly accounted for by extending the set of training units, for example inclusion of confidence and perseverance training.
2.4. Initial and Maximal Event Condition
The expected event condition at the end of the scheduling period is determined by considering two extremes: the initial event condition and the maximal event condition.
The initial event condition measures how the athlete would perform in an that event in a decathlon at the beginning of the year, while the maximal event condition id the coaches approximation of how well the athlete can become if he/she were to specialize in it. Both estimates depend on the time span, which is chosen as four years for this paper.
The above represents the athlete’s position in his/her “development curve,” where it is assumed the closer the athlete gets to maximal event condition, the more difficult it becomes to improve, which in practice is difficult to reach.
2.5. Effects of Exercising
The crucial aspect of this paper’s model is the estimation of the effects of training on the athlete’s performance, called time/progress relations. The coach should provide these effects.
The first step is to specify the relevance of the training units for the various events. The training units can be divided into muscular group units, condition units, and technical units.
The next step is to specify for each event and for each relevant training unit the relation between the training time in minutes per year and the impact on the expected performance at the end of the year. Estimates provided by the coach are interpolated to obtain estimates for any sensible level.

3. Mathematical Model
This section presents a nonlinear optimization model for a one-year decathlon-training period, the solution to which is a time assignment to each training unit.

3.1. Events and Scores
Suppose an N-event competition. The score, Si depends on the athlete’s performance, Pi in the following way:

The total amount for the decathlon competition is,

Where α, β and γ are constants whose values , and depend on the event, i as determined by International Association of Athletic Federation (IAAF).
The absolute value operation is present because some events athletes have to be fast whereas others have to be far or high.
The assumptions are that if i is a time oriented event, and if i is distance oriented event.

3.2. Event Conditions; Relative Remaining Performance Gap
An athlete’s expected performance in a fictitious competition at the end of the year is the centre of interest.
Let t∈ [0, T] denote the time, where 0 and T are the beginning and the end of the year. For any event i, the athlete’s event condition, Ci at time, t is Ci (t).
Two points , which is Ci (t=0) and , which is Ci (t=max) are of particular interest. For time-oriented event, for all t. For distance oriented, for all t.
Ci (T) will implicitly appear in the objective function as the criterion for effectiveness of a training time assignment.
The performance gap, PG for event i at time t, PGi(t) is defined as,
PGi(t)=C
The Relative Remaining Performance Gap (RRPG) for event i at time t, RRPGi(t) is defined and denoted by,

From the equation, the smaller the , the better the athlete’s event condition. By rearranging the above equation, the event condition, Ci (t) is given by,
(3)
On account of the linearity of the function, for each event i on the interval [ ], the linearity can be applied to the event score corresponding to . The score gap at any time t is defined and denoted by,

Then, the score for event i at time t is given by,

This approximation will be used to estimate scores. The quality of this approximation depends on the size of the interval [ ], and is therefore athlete dependent.
3.3. Training Units; Training Time Assignment
An athlete’s performance dynamics can be modeled in a dynamic system of differential equations, where the athlete’s mental and physical states are the variables whose development are controlled by the athlete’s physical activity.
The difficulty of isolating individual exercises to obtain numerical data makes focusing on long term effects better.
If M is the number of training units (in this case M=49). The amount of time assigned for training unit j is , (j=1,.., M). From 2.1
Clearly,

Moreover, there are restrictions on time on individual training units: first, training benefits decline with time, eventually to zero, and, second, there are limits to what a coach can initially access. Consequently, is bound by the lower constraint, and the upper constraint, so that,

Obviously, these choices may always be made: and .
3.4 Training Time/Progress Relations
To formulate the relationship between the training time assignment and the development of the event conditions, the following assumptions are made:
1. When and in which order the training units are executed is not taken into account
2. Every training unit takes away a fraction of RRPG before exercising that unit
3. Only the event condition at time T is taken into account
Assuming that the training units are executed in the order 1, 2, …, M let be the time just after the training unit has been executed, and let denote the time at the beginning of the year. Assumptions 1 and 2 allow writing RRPG after executing training unit j as

Where is taken to be one and is defined as the expected ratio between the RRRPG before and after exercising training unit for minutes.
Since the interest is in the event condition at the end of the year, the above equation (7) implies,

Using equation (3) and (7), it can be derived for i and j, . This states that the RRPG after k training units acts as a multiplier for the progress achieved by training unit . That is, the smaller the RRPG (or, the better the event condition), the smaller the contribution of an additional unit.
3.5. Objective
The objective is to construct a training time assignment for which the expected performance at the end of the year is as high as possible. Using equations (4) and (8), the objectives read,
.
Because is a constant, it can be eliminated. Hence the objective is equivalent to

3.6. Construction of the Training Time/Progress Relations
For each i and j, the function is constructed by interpolating a set of data points. Suppose that the athlete will exercise training unit j for minutes during the year, while training all other units at neutral, the coach is asked by what amount, he or he/she expects the condition for event i to increase relative to the current situation.
The values for are interpreted as follows: if a training assignment is fixed ( such that for , then, satisfies

Hence specifying is equivalent to specifying because
4. Calculations
The model is solved using a sequential programming method that is based on the algorithm in subroutine NPSOL. The training time assignment and the corresponding estimated final event conditions are solve by it.
4.1. Time Budget
How the optimal training time changes with the value of is investigated.
Half of the training time is used for general training units, while the other half is used for technical training units.

4.2. Marginal Effects
If the upper and lower bounds are ignored for a moment, from Lagrangian formulation of the problem, the following first order optimality condition is obtained. For each j, it holds that

With

Where is the Langrange multiplier associated with the constraint,
Clearly, can be interpreted as the marginal effect of training unit j on the score for event i. This means that in the optimal solution, the marginal contribution to the expected final score is equal for every unit j.
Because was assumed to be a log convex for every i and j, its derivative is a decreasing function of
As a consequence, is a decreasing function of .
An interpretation is that if a training unit j positively affects a great deal of the events, the total marginal score has to be distributed over the per-event marginal scores . As a result, if observed individually, they are low, but because they are decreasing functions of , this means that has to be high.
In other words, training- units that affect a great deal of events in a positive way take a large part of the total training time, in an optimal solution.
4.3. Sensitivity Analysis with Respect to Maximal Event Condition
Because of the difficulty and uncertainty in the estimation of an athlete’s maximal event, , and the high sensitivity of the model to the same, , sensitivity analysis if performed in this form: for each event i, the value of is varied such that the corresponding event score varies from 30 points below to 30 points above the original estimates. It seems reasonable to assume that coaches are able to forecast short-term (in the paper’s case four years) maximal event conditions with such a precision.
The deduction is that the results depend on the value of the maximal event conditions. By varying them, it becomes clear which parts of the training time assignment are relatively stable, and which parts are more sensitive to the estimation of the athlete’s capabilities and talents. The coach may consider alternative scenarios with different values and make decisions accordingly.
4.4. Worst Case Benchmarking
Coaches either choose to bolster the athlete’s strong points or focus on training the weak points. The model in this paper balances both extremes by looking at the training units with the least effort and the highest impact on the on the total expected score.
5. Concluding Remarks
This paper has presented a model that distributes the total training time across the various training units in which the final objective is to reach the highest score. The total available training time depends on the periodization and the athlete’s competition schedule during the year.
The model presented allows the coach to use, among others, on a certain day to choose the actual training schedule for that day, and to keep track on what has been done.