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15УДК: 338.486.5
APPLICATION OF ADAPTIVE FORECASTING METHODS TO OPTIMIZE TANKER FLEET OPERATIONS
Moiseev G. V., Ph.D. tehn. Sciences
Financial University under the Government of the Russian Federation, Moscow, Russia
E-mail: [email protected]
Abstract. The article presents the problem of determining the optimal plan of trade routes for the tanker fleet in order to maximize their daily revenue named time-charter equivalence (TCE).
The key point of the task is forecasting the crude and oil products market for main international trade routes with a forecast horizon of up to 6 weeks. This market is difficult to predict, because statistic values does not have a clearly defined trend, are irregular and unsystematic in nature, and have low autocorrelation indicators.
It is proposed to use for solving this problem a model of exponential smoothing - hybrid adaptive combined model with a B-criterion (ACM-B), built on the Brown model. The work of the model consists in the simultaneous construction of three forecasting polynomials of 0, 1 and 2 orders, from which the main forecasting function is combined. The weighting coefficients of participation of each polynomial are calculated based on the current nature of the statistic time series.
To improve prediction accuracy the author proposes a modification of this model according to the Trigg-Leach method by introducing a tracking control signal to calculating the exponential smoothing parameter of the time series.
The adequacy of the model is confirmed by computational experiment series that confirm sufficient accuracy of the model in 80% of cases (stability market and medium fluctuations). In 20% of cases during periods of market chaos, the model required additional modifications.
Key words: exponential smoothing, adaptive forecasting, polynomials, tanker fleet, time-charter equivalence, Brown, Trigg, Leach.
ПРИМЕНЕНИЕ МЕТОДОВ АДАПТИВНОГО ПРОГНОЗИРОВАНИЯ ДЛЯ ОПТИМИЗАЦИИ ДЕЯТЕЛЬНОСТИ ТАНКЕРНОГО ФЛОТА
Моисеев Г.В., к.т.н.
Финансовый университет при Правительстве Российской Федерации, Москва, Россия
E-mail: [email protected]
Аннотация. В статье приводится постановка задачи определения оптимальных цепочек маршрутов танкерного флота с целью максимизации дневной выручки (TCE) каждого из них.
Ключевым моментом задачи является прогнозирование рынка нефти и нефтепродуктов по основным международным маршрутам перевозки с горизонтом прогнозирования до 6 недель. Данный рынок сложен для прогнозирования, так как характеризующий его временной ряд не имеет четко выраженного тренда, носит нерегулярный и несистематический характер, имеет низкие показатели автокорреляции.
Для решения задачи предлагается использовать модель экспоненциального сглаживания типа гибридная адаптивная комбинированная модель с В-критерием (АКМ-В), построенная на базе модели Брауна. Принцип работы модели заключается в одновременном построении трех прогностических полиномов 0, 1 и 2 порядков, из которых комбинируется основная прогностическая функция. Весовые коэффициенты участия каждого из полиномов вычисляются исходя из текущего характера временного ряда. Основными преимуществами модели является неприхотливость к характеру ряда, универсальность и низкая необходимость вмешательства аналитика.
Автором предлагается модификация указанной модели по методу Тригга-Лича, путем введения следящего контрольного сигнала при расчете параметра экспоненциального сглаживания временного ряда для повышения точности прогнозирования.
Работоспособность модели подтверждается серией вычислительных экспериментов, которые показывают достаточную точность работы модели в 80% случаев (периоды стабильности и колебания рынка). В 20% случаев (периоды хаоса рынка) требуется применение дополнительных модификаций модели.
Ключевые слова: экспоненциальное сглаживание, адаптивное прогнозирование, танкерный флот, таймчартерный эквивалент, Браун, Тригг, Лич.
The goal was to maximize the net revenue of the tanker fleet. As a criterion for optimization, the Time Charter Equivalent (TCE) was chosen - net vessel revenue per day.
It was found that the ship-owner operates several hundred vessels in the oceans of various classes, designed to transport various types of oil and oil products. As a result of the analysis of the fleet, it
was revealed that at each moment (for example, one day) only for a small number of vessels (up to 10) it is necessary to determine the next route. Thus, the task of optimal routing of ships around the world can be simplified to find the optimal chain of routes for one ship.
Due to the fact that the average duration of tanker fleet flights is 10-40 days, it was decided to establish a forecast horizon of 7-42 days (1-6 weeks).
As the initial data at this stage of the evolution of the algorithm, only historical data on the main transportation routes (TDx and TCx Baltic International Trade Routes) were used. At the next stages of development, it is planned to include in the model an additional set of initial data that increases the accuracy of forecasting.
As a result of the analysis of the behaviour of the TCE values over time, it was revealed that the time series do not have a pronounced trend, are irregular and unsystematic in nature, and have low indicators of autocorrelation and covariance.
Based on the conditions of the statement of the problem and the nature of the source data from the whole variety of forecasting methods, a family of adaptive forecasting methods based on exponential smoothing was selected [6]. It was experimentally found that the most adequate forecasting results were shown by the hybrid adaptive combined model with the B-criterion (ACM-B) [6], based on the Brown model and modified by the Trigg-Leach method. Its main advantage is unpretentiousness to the nature of the series and versatility, as well as a high degree of automation.
The family of these methods was widespread in the 60-70s in the USA and was used to forecast IBM stocks, gold prices on the London Currency Exchange, prices on the London Metal Exchange, in the USA and Western Europe for the tasks of reducing investment risks, in the USSR in the interests of various spheres of the national economy [6]. The author of the modernized model used it in the 90s at the Central Bank of the Russian Federation, Vnesheconombank, and the Central Clearing House. At the Moscow Interbank Currency Exchange, when forecasting currency auctions, a
forecast accuracy of about 5 rubbles was achieved. and about 2% for growth and decrease [5]. The indicated indicators satisfy the requirements of the problem under consideration.
The indicated group of methods was combined to solve the problem as follows. The initial data set in the form of TCE values for a specific route per day is a time series [3], consisting of the trend which is a characteristic of the market at the time, and noise, reflecting market fluctuations around the trend:
*t = tt + £t Our task is to highlight the trend, remove noise and continue the trend into the future in order to make a forecast.
For this, as suggested by [1], 3 polynomials of degree zero, first and second degree are distinguished, respectively:
1) fr = ai (1) 2) tt = ai + a2t (2)
3) %t = ai + a2t + -a3t2
(3)
The zero degree polynomial is the most "calm", smooths out sharp emissions and stably holds the average value, however, it reacts rather slowly to market changes that are taking place.
The second-degree polynomial can be called "sharp" - it reacts most quickly to market changes in the form of the initial series, seeks to reduce the discrepancy as quickly as possible, however, it is very unstable and fluctuates strongly around the initial series.
The polynomial of the first degree is a compromise among the above.
All three polynomials lead a separate forecast model and do not exchange data with each other.
In order to determine which of the polynomials at the current moment is the closest to the true value of the forecast, the following control signal [9] is applied, which is calculated for all three polynomials in the same way:
1) n = 0 2) n = 1 3) n = 2
Kt= + =
(1 - Y)et-i + yet
et (1 - Y)et-i + y^1 The tracking signal is the ratio of the exponentially smoothed prediction error Gf- to its absolute value et, is calculated separately for each of the polynomials and varies in the interval [-1; 1].
Here y is the smoothing parameter, et is the polynomial prediction error at time t.
The values of the tracking pilot signal are used when calculating the parameter of smoothing time series as a function of [10]:
<*t = f(Kt).
For a polynomial of degree zero, single smoothing is performed, for the first degree, double smoothing, for the second degree, triple [1]. The smoothed average value of St is the sum of the previous value of St-1 and the new value of the series xt, multiplied by the smoothing coefficients at and ¡3t = 1 — at.
Depending on their size, the smoothing is sharper or smoother:
1) St = atxt + ptSt-i
at = 0,01/Kt
2) St = atxt + ptSt-i
St[2] = atSt + ptSx
at = 0,05
3) St = atxt + ptSt.
Sl2] = atSt + ptS.
[2] t-i
[2] J t-1
= atSn+fksW
<Xi
= 0,1/Kt
The form of function at = f(Kt) in this case is selected empirically as the most appropriate for the conditions of the problem.
According to [1], estimates of the coefficients of polynomials (1) - (3) are calculated as follows: 1) = St (4)
2) âit = 2St - St Û2
[2]
= al(St-Sl2])
[2] , cp]
(5)
3) aH = 3St - 3S[2] + St a
2@t
â2t = —[(.6- 5at)St - 2(5 - 4at)St
[2]
+ (4- 3at)St
[3]
(6)
2
Û3t = aMSt-2Sl2]+SP)
Predicted functions take the form of polynomials (1) - (3) with estimates of the coefficients (4) - (6) [1]:
1) XT(t) = ait (7)
2)XT(t) = a1t + Ta2t (8)
3) XT( t) = ait + Tci2t + 1T2a3t (9)
Here parameter displays the forecast depth (forecast horizon). The forecast for these formulas is made z days in advance.
Further, from three forecast polynomials (7) -(9), one forecast function is combined that has an accuracy higher than that of each of the polynomials individually.
For this, 5-criterion [6] is calculated, which reflects the deviation of each polynomial from the true value of the series by smoothing the square of the forecast error e 2:
1) B(V = (1
, (2) _
P)B(-{
(2)
+ Pet
2) B(2) = (1 — p)B(2_[ + pe2
3) B(3) = (1 — p)B(-\ + pe2 where p is the smoothing parameter.
Since the values of 5-criterion can be of different orders, the weighting coefficients are calculated for the combination of polynomials, which are the normalized values of the 5-criterion [6]:
1) (
(1)
b(2)b(3)
B(1)B(2)+B(2)B(3)+B(1)B(3)
2) (
(2)
B(1)B(3)
B(1)B(2)+B(2)B(3)+B(1)B(3) fl(1)fl(2)
, ,(3) _ _Bt Bt_
3) °t = d(1)d(2)^d(2)d(3)^d(1)d(3)
(10)
(11)
(12)
The resulting forecast function is compiled as the sum of the forecast polynomials of the 0th, 1st and 2nd orders (7) - (9) multiplied by the weighting coefficients (10) - (12) [6].
fx(t) = «¿^(t) + ^px^tt)
(13)
The form of the resulting forecast polynomial fT(t) and the initial series xt is shown in Fig. 1. The figure shows a polynomial with a forecast horizon of t = 28 days.
120 000 100 000 80 000 60 000 40 000 20 000 0
Statistical data ii
_ii
+30 days forecast ----Confidence interval
01.04.2015
01.07.2015
01.10.2015
01.01.2016
01.04.2016
Fig. 1. Resulting Prediction Function
In general, the forecast function may contain arbitrary members:
=
2nt
2nt
a1 + a2 sin--+ a3 cos-,
1 2 12 3 12
fr(t)
— /.\y -vv J
(t) + v(2)42)(o + + M(n)4n)(t)
exponential growth models
X.
(l)(t) = a1 + a2et,
= !■
i=l
Xi)a(i)
(t)
or corrective actions of the analytic operator to increase the accuracy of forecasting.
taking into account, for example, seasonal variations [2]
5% 0%
+-1000 +-3000 +-5000 +-8000 +-10000+-15000+-20000+-25000+-50000 E^e
100% 80% 60% 40% 20% 0%
+-1000 +-3000 +-5000 +-8000 +-10000+-15000+-20000+-25000+-50000 E^e
Deviation
Fig. 2.Histogram of prediction accuracy a) +14, b) +28, c)+42 days
For a detailed assessment of the accuracy of the method, a series of computational experiments was conducted on a typical TD7 route from the BITR structure. TCE data for 2015-2018 was taken as initial data [4, 7, 8]. Prediction was carried out with the horizons +7, +14, +21, +28, +35 and +42 days.
The obtained accuracy is presented in Fig. 2. It is most convenient for analysis to represent accuracy in the form of a distribution of deviations.
For convenience, the results are summarized in fig. 3. From it we see that when forecasting for +14 days, 62% of all time, high accuracy with deviations
High precision ±1,000 $ A
+ 14 days
+ 28 days
62% of the time
of ± $ 1,000 ... ± $ 3,000 was observed, another 22% of the accuracy time worsened to ± $ 5,000 ... ± 8,000 $, the remaining 16% of the deviation time excessively high and exceeded $ ± 10,000. With an increase in the forecast horizon by +28 and 42 days, the plots decreased with high accuracy, but increased with a low accuracy.
To answer the question of why, in some areas, the accuracy of forecasting the TCE was ± $ 1,000 ... ± $ 3,000, and in some areas exceeded the values of ± $ 10,000, we will construct confidence intervals of forecast values.
Average
Low
±3,000 $
±5,000$ ±8,00^ > ±$ 10,000,
J $
22% of the time
16% of the time
High precision ±1,000 $ ^ ±3,000 $
Average Low
5,000 $ 8,000 ^ > ±$ 10,000
56% of the time
W
25% of the time
19% of the time
+ 42 days
High precision ±1,000 $ A ±3,000 $
49% of the time
Average 5,000 $ 8,000 :
J J
22% of the time
Fig. 3. Distribution of prediction accuracy +14, +28 and +42 days
Low > ±$ 10,000
29% of the time
In fig. 1 dashed lines show the dynamic confidence interval ±2a, reflecting the smoothed value of the standard deviation a for the last 28 days.
As we can see, there are periods of market chaos in which the confidence interval is excessively
widening and the forecast becomes irrelevant, and periods of fluctuation and relative stability of the market, in which the confidence interval does not exceed acceptable values.
Highlighting these three periods of the market, we obtain the following accuracy indicators.
Table 1. Forecasting accuracy in market periods
Market periods Duration (% of total time) Accuracy of forecasting
GP +14 days GP +28 days GP +42 days
Calm 35% ± 2,000 $ ± $ 4,000 ± 6,000 $
Oscillations 45% ± $ 4,000 ± 6,000 $ ± $ 8,000
Chaos 20% ± 16,000 $ ± $ 20,000 ± 24,000 $
The behavior of forecasting functions during these periods is shown in Fig. 4.
SUMMARY
1. The accuracy of predicting TCE for 2-6 weeks for 80% of the entire time does not exceed ± $ 2,000 ... ± $ 8,000, which is an acceptable value with the existing source data.
2. The proposed mathematical model is applicable for solving this problem and gives acceptable results in practice.
3. To increase accuracy, it is necessary to build a "multiple regression" with additional initial data: number of competitor vessels, volumes of oil production and refining of petroleum products, external information from analytical agencies, geopolitical and natural factors.
4. To increase the forecast horizon to medium-term (3-6 months) and long-term forecasts (up to 1 year) it is needed to it is necessary to consider other mathematical techniques.
100 000 80 000 60 000 40 000 20 000 0
01.10.2015
Statistical data Prediction accuracy ±2000$...±6000$
_ - -1 - - - _
---'-^ - ----
Calm market
01.11.2015
01.12.2015
100 000 80 000 60 000 40 000 20 000 0
Statistical data Prediction accuracy ±4000$...±8000$
✓ 1 ~ **"_^^^^^^^
Oscillated market
01.08.2015
01.09.2015
01.10.2015
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