Estimation of Exponential Smoothing Parameter on Pesticide Characteristic Forecast using Ant Colony Optimization ( ACO ) ( Dinita Rahmalia ) 56 Estimation of Exponential Smoothing Parameter on Pesticide Characteristic Forecast using Ant Colony Optimization ( ACO )

Pest in agriculture can raise plant disease and fail to harvest. The pest problem in agriculture can be solved by using pesticide. Pesticide usage must be done proportionally. So, the manufacturer should fix standard pesticide active ingredient in pesticide production. Forecast is a prediction of some future evens. In forecast problem, there are any parameters which should be determined. Parameters can be estimated by exact method or heuristic method. Ant Colony Optimization (ACO) is inspired from the cooperative behavior of ant colonies, which can find the shortest path from their nest to a food source. In this research, we use heuristic method like ACO to estimate exponential smoothing parameter on pesticide active ingredient forecast and pesticide sample weight forecast. From the simulation, on the first iteration, all ants choose parameter randomly. At the optimization process, we update pheromone until all ants choose the similar parameter so that process converges and variance approaches to zero. The optimal exponential smoothing parameter can be applied in forecasting with minimum sum of squared error (SSE).

Generally, parameter of forecasting equation can be applied or choosen by trial and error and parameter estimation using the exact method such as least square estimation have been applied (Montgomery, 2015).Beside exact method, heuristic method can be used in estimating parameter of forecasting (Elvural et al., 2016;Wu and Chang, 2002).Heuristic method used in this research is Ant Colony Optimization (ACO).
Ant Colony Optimization (ACO) is inspired from the cooperative behavior of ant colonies, which can find the shortest path from their nest to a food source.The method was developed by Dorigo in 1990 (Dorigo andStutzle, 2004).In previous research, ACO method have been applied on optimization problem (Rao, 2012;Rahmalia and Herlambang, 2017).In this research, we use ACO to estimate the exponential smoothing parameter on pesticides active ingredient and sample weight forecasts.
In simulation, at the first iteration, all ants choose parameter randomly.At the optimization process, we update the pheromone until all ants choose the similar parameter such that the process converges and the variance approaches to zero.After that, we will obtain the optimal exponential smoothing parameter.The optimal exponential smoothing parameter can be applied in forecasting where the sum of squared error (SSE) will be minimum.

First Order Exponential Smoothing
The simple exponential smoothing is represented as in equation (1) (Montgomery, 2015) : and the equation can be explained : t f : forecast of the time series for period t t y : actual value of the time series for period t The objective is to minimize the sum of squared error (SSE), subject to the smoothing parameter requirement ) 1 0 ( ≤ ≤ λ as the decision variable in equation ( 3) (Anderson, 2012).In other words The optimization problem above can be solved

Ant Searching Behavior
In equation ( 4), an ant k , when located at node i , uses the pheromone trail ij τ to compute the probability of choosing j as the next node (4) Where α denotes the degree of importance of the pheromone and

Pheromone Trail Evaporation
When an ant k moves to the next node, the pheromone evaporates from all the arcs ij according to equation ( 5) : Where ] 1 , 0 ( ∈ ρ is a parameter and A denotes the segments or arcs traveled by ant k in its path from home to destination. After all the ants return to the home (nest), the pheromone information is updated according to the equation ( 6) : is the evaporation rate (pheromone decay factor) and is the amount of pheromone deposited on arc ij by the best ant k .
The pheromone deposited on arc ij by the best ant is taken as in equation ( 7) where Q is a constant

Path Retracing and Pheromone Updating
The pheromone value ij τ on the arc ) , ( j i traversed is updated.Because of the increasing in the pheromone, the probability of this arc being selected by the forthcoming ants will increase .

Overall Algorithm
The algorithm of determining optimal parameter using ACO can be constructed as follows : 1. Set the number of ants N and the pheromone decay factor ρ .
2. In equation ( 8 11. Repeat step 3-10 until all ants choose the simular parameter and such that the process converges and the variance approaches to zero.

Materials and Methods
Data are obtained from one of the pesticide manufacturer company in Gresik, East Java.Data used in this research are pesticide active ingredient data (%) and pesticide sample weight data (miligram) during January-July 2009 (Rahmalia, 2009).
Simulations are applied by Matlab.From each data, ACO will be applied for estimating exponential smoothing parameter.

Simulation and Discussion
The ACO parameters such as the number of ants, the range of parameter, pheromone decay factor, and maximum iteration of estimating exponential smoothing parameter on pesticide active ingredient forecast are given in Table 1.
Figure 2 shows the time series of pesticide active ingredient (%) during January-July 2009.From actual value of time series, we will forecast using exponential smoothing method.
Figure 3 shows optimization process ACO in estimating exponential smoothing parameter on pesticide active ingredient forecast.On the first iteration, all ants choose parameter randomly so that variance is relatively high.At the optimization process, we update pheromone until all ants choose the similar parameter so that process converges and variance approaches to zero.
After ACO is applied until maximum iteration, we obtain the optimal exponential smoothing parameter is 0.9.The optimal exponential parameter is applied in forecasting such that the simulation can be seen in Figure 4 with sum of the squared error (SSE) is 0.77.
Table 1.ACO parameter of estimating exponential smoothing parameter on pesticide active ingredient forecast The number of ants 100 The range of parameter (0-1) Pheromone decay factor 0.9    After ACO is applied until maximum iteration, we obtain the optimal exponential smoothing parameter is 0.9.The optimal exponential parameter is applied in forecasting such that the simulation can be seen in Figure 7 with sum of the squared error (SSE) is 194.34.

Maximum iteration 100
the sum of the squared error (SSE) ACO process can be explained as follows : the ants start at the home node, travel through the various layers from the first layer to the last or final layer, and stop at the destination node in each cycle or iteration.In the beginning of the optimization process, all edges are initialized with an equal amount of pheromone.All the ants start from the home node and stop at the destination node by randomly selecting a node in each layer.In general, at the optimum solution, all ants travel along the same best (converged) path.There are three steps in ACO process : ant searching behavior, pheromone trail evaporation, and path retracing and pheromone updating (Rao, 2009).

Figure 2 .
Figure 2. Time series of pesticide active ingredient

Figure 3 .Figure 4 .
Figure 3. Simulation ACO in estimating exponential smoothing parameter on pesticide active ingredient forecast

Figure 5
Figure 5 shows the time series of pesticide sample weight (miligram) during January-July 2009.From actual value of time series, we will forecast using exponential smoothing method.

Figure 6
Figure6shows optimization process ACO in estimating exponential smoothing parameter on pesticide sample weight forecast.On the first iteration, all ants choose parameter randomly so that variance is relatively high.At the optimization process, we update pheromone until all ants choose the similar parameter so that process converges and variance approaches to zero.

Figure 5
Figure 5 Time series of pesticide sample weight

Figure 6 .Figure 7
Figure 6.Simulation ACO in estimating exponential smoothing parameter on pesticide sample weight forecast