Modified Hybrid Grey Model (1,1) to Forecast Cellular Subscribers

This study develops MHGM (1,1) (Modified Hybrid Grey Model) which is the combination of two models first one is improved GM (1,1), this model consists of optimization of initial and background values and other is concave EDDGM (1,1) (Dynamic Discrete Grey Model) termed, in this model equal division technique is applied to fit the concavity of cumulative sequence and after that created dynamic average value and on the basis of that dynamic average value dynamic discrete GM (1,1) model is established and by the gradual heuristics method or the dichotomy approach the initial equal division number is obtained. We have fixed equal division number ‘n’ between 0 and 1in MHGM (1,1). For forecasting of starting half years we use y(m) as initial condition of model in time restored function and also multiply by a factor e 1 to adjust the model. This model has applied without solving by heuristics or dichotomy method. Subscribers of cellular networks increase day by day in Pakistan; cellular industry has total five networks in Pakistan. In this paper data of three cellular networks subscribers that are Mobilink, Ufone and Zong have taken as application of models and it has been proved by using mean absolute percentage error that the forecast accuracy of MHGM (1,1) is better than GM (1,1) (Grey Model) and improved grey model (1,1).

Warid. The subscriber of cellular network increases day by day in Pakistan so in these situations the questions produce that: How many cellular subscribers will increase in future? How can the decision maker make the future plan based on small set of data? This leads us to grey theory which was given by Chinese professor Deng Julong in 1982. The Grey theory has been used in many areas as in electric engineering, transportation, economics, education etc. through Grey theory any one can encounter easy calculation and better results. In grey theory forecast is done on K e" 4 points. GM (1,1) has a number of advantages but its boundaries are limited. GM (1,1) is used for short term forecasting and it is best fit for the development coefficient a d" 0.5 and for little bit variation.
For better forecasting; many methods have developed.
The grey theory has two aspect of information one is known and other information is not known. There are many works has presented on the accuracy of said model.

Erdal Kayacan, Baris Ulutas, Okyay Kaynak gave MGM
(1,1) by using Fourier series in time series [1]. An optimized nonlinear GBM (1,1) (Grey Bernoulli Model) was given by which parameters optimization of that model is suggested a combination of optimization problem and is performed on LINGO software and the model is used to forecast the per year industrial waste water rate of 31 state of China's [2]. The idea of omni-direction forecasting of non-equip gap data was given by improving the inverse accumulating generator operator [3]. It was forecasted and DDGM (1,1) in that study of four estimate approaches of stepwise ratio in generalize GM are given [5]. It was studied AGM (1,1) (Adaptive Grey Model)for the solution of wafer level packaging process and showed that this process give effective results with small data and it can be improved wafer level packaging process [6], and it was also explained that through perturbation bound parameters of GM (1,1) will change large when sample size of sample is large, so if sequence greater than or equal to zero, checking of quasi smooth and exponential condition is satisfied by input sequence then the original GM can achieve good prediction although the samples will be large [7]. It was told in study that what kind of factors affecting the tourist flow in china and an improved GM (1,1) was given which has a high prediction accuracy then the traditional GM (1,1) [8]. Huang CY, LU CY and Chen Cl this work is on GM (1,1) and nonlinear GBM and it is suggested that GM (1,1) is good for slow rate of change of data whereas nonlinear GBM (1,1) is good for drastically rate of change of data [9]. Ersi Liu, Qiangqiang Wang, Xinran Ge and Wei Zhou has shown two models that is the concave and convex DDGM (1,1) in these models equal division numbers are used by using two kind of approaches first is gradually heuristics method and other one is dichotomy method and eventually through conflict events in the urbanization process in china it was proved that DDGM (1,1) has higher accuracy than Discrete, optimal and GM (1,1) [10]. In another work that was put to forecast the growth rate of renewable energy consumption in China, in this work three model that is GM (1,1), nonlinear grey Bernoulli (1,1) and grey verhulst model is compared with each other and it can be show that grey verhulst model has greater accuracy than the two model the accuracy and fitness of models are also compared by regression analysis [11].

METHODOLOGY
First GM (1,1) and improved GM (1,1) will be applied and then compare results with MHGM (1,1). Following are steps of these three models.
Time responded formula is Where 'b 1 ' stands for developing coefficient and 'b 0 ' stands for grey input action.
Time response function is,  Construction of grey differential equation: Where 'b 1 ' stands for developing coefficient and 'b 0 ' stands for grey input action.
The above differential equation can be written: is background value and is represented: where n is number of step size taken, r is number of years which will be forecasted.
Then MAPE is computed to compare three models in Table 3. Microsoft excel is used to solve these calculations.
Further Fig. 2 shows graphically comparison between three models and actual data of model.

Discussion
Forecasting results of these three models in Table 2 it can be shown that MHGM (1,1) gives better results in comparison of other two models. MAPE in Table 3  So; this method can be applied to other real-world growth problem when only small data is in hand.