Stochastic Demand Forecasting Method using Bootstrap Algorithm

The previous method for forecasting stochastic demand assumed that within three time periods, the increase in demand for all products are fully based on the parameter tk. Instead of having fixed tk for each product, the method for forecasting the demand proposed in this chapter applied the concept of bootstrapping to find an alternate approach for forecasting the demand.

There are two main differences between the previous stochastic demand forecasting method and the new method using a bootstrap algorithm.

The first difference is the method of finding the percentage increase in demand for each product produced by the chemical plants in the chemical complex. Instead of assigning fixed values for the percentage increase in demand for all products, the bootstrap method is applied. The new percentage increase in demand derived from the bootstrap method is represented by parameter BID(s).

Due to the financial crisis in Thailand between the year 1990 and 1999, the changes in historical demands of each product between those years fluctuated significantly. The annual historical demands of each product are only available between the years 1989 and 2007. In order to compensate for the fluctuation in demand of each product, the values of the percentage increase and decrease in demand of each product between the years 2000 and 2007 were weighted. This is done by adding another two sets of the changes in historical demands between the year 2000 and 2007 to the original samples. This implies that the original historical demand samples consist of one set of the percentage increase or decrease in demand between the year 1989 and 1999, and three sets of the percentage increase or decrease in demand between the year 2000 and

2007. Note that the percentage increase or decrease in demand was used instead of the actual demand.

The bootstrap algorithm to find the parameter BID(s) is as follow:

 Select B number of independent bootstrap samples (B = 200) from the weighted historical percentage increase or decrease in demand. Each of the bootstrap samples consists of n samples drawn with replacement from the original samples (n = 10).

 Evaluate mean, median and standard deviation of each of the 200 bootstrap samples.

 The average percentage increase in demand using bootstrap method (BID(s)) is the average of the mean of all 200 bootstrap samples

The second difference is the method to find the percentage changes in demand for “each time period of each scenario”. As mentioned above, in the previous method the changes are based solely on the parameter tk. For the new method, the concept of using the bootstrap standard deviation was introduced. As the standard deviation is the measure of how spread out the data are demand volatility, it is sensible to adjust the changes in percentage increase in demand for each time period of each scenario according to the bootstrap standard deviation.

The bootstrap algorithm for finding bootstrap standard deviation is the same as the bootstrap algorithm described above but instead of averaging the mean, the standard deviation of the mean of each 200 bootstrap samples was calculated to give the bootstrap standard deviation of the percentage increase in demand of each product s (Bstd(s)).

The formulas for calculating the bootstrap demand for each time period of each scenario are given in Table 5.1 and Figure 5.1.

Table 5.1: Formulas for calculating future demand of each scenario in each time period Time Period t (Year) Scenario (k) Bootstrap Stochastic stk D t1 (2008 – 2010) 1 to 9 D0 ּ BID(s) D1 t2 (2011 – 2017) 1 to 3 D1 ּ (BID(s) - ( ּ Bstd(s))) D2L t2 (2011 – 2017) 4 to 6 D1 ּ BID(s) D2 t2 (2011 – 2017) 7 to 9 D1 ּ (BID(s) + ( ּ Bstd(s))) D2H t3 (2018 – 2032) 1 D2L ּ (BID(s) - ( ּ Bstd(s))) t3 (2018 – 2032) 2 D2 ּ BID(s) t3 (2018 – 2032) 3 D2H ּ (BID(s) + ( ּ Bstd(s))) t3 (2018 – 2032) 4 D2L ּ (BID(s) - ( ּ Bstd(s))) t3 (2018 – 2032) 5 D2 ּ BID(s) t3 (2018 – 2032) 6 D2H ּ (BID(s) + ( ּ Bstd(s))) t3 (2018 – 2032) 7 D2L ּ (BID(s) - ( ּ Bstd(s))) t3 (2018 – 2032) 8 D2 ּ BID(s) t3 (2018 – 2032) 9 D2H ּ (BID(s) + ( ּ Bstd(s)))

It can be seen that during the first time period of all scenarios the bootstrap demand is the latest known demand (D0) multiplied by the new percentage increase in demand derived by the bootstrap method (BID(s)). The new demand value becomes the starting demand (D1) for the next time period.

In the second time period, the nine scenarios were divided in to 3 sets of 3 scenarios. The demand for the first set which represents a lower growth in demand, is calculated using the following formula:

stk

D = D1 ּ (BID(s) - ( ּ Bstd(s))) for t = 2, k = 1,2,3 (5.1)

where D1 is the demand of the previous period

BID(s) is the bootstrap percentage increase in demand for product s

Bstd(s) is the bootstrap standard deviation of the percentage increase in demand for product s

 is a number usually between zero and two

The formula implies that the percentage increase in demand for this group of scenarios will be less than the one calculated in the previous time period. The difference is determined by the function of the bootstrap standard deviation of each product.

The demand for the second set which represents a steady growth in demand, is derived using the following formula:

stk

D = D1 ּ BID(s) for t = 2, k = 4,5,6 (5.2)

This formula implies that the rate of increase in demand for this group of scenarios is the same as the previous time period.

The final set represents a higher growth in demand. The formula for the demand is:

stk

D = D1 ּ (BID(s) + ( ּ Bstd(s))) for t = 2, k = 7,8,9 (5.3)

The increase in growth of the demand is also determined by the function of the bootstrap standard deviation of each product. A similar approach is then applied to the demand of each product in the third time period. Furthermore, the value of  between zero and two will be investigated.

Figure 5.1: The scenario tree for the future demand of each scenario in each time period deriving from bootstrap method.

As mentioned earlier in this thesis, the models developed in the previous chapters were formulated and solved using GAMS and the CPLEX solver. However, an Excel spreadsheet was used as a tool for forecasting demand of all chemical products based on the bootstrap algorithm described above. The mathematical formulation the stochastic chemical complex optimisation model under bootstrap demand is described in the next section.

5.2 Mathematical Formulation

The mathematical formulation of the model in this chapter is similar to the previous model in Chapter 4. The only modification to the formulation is the use of “bootstrap” demand instead of the simple scenario-based demand used in previous model

5.2.1

Notation

Only the newly formulated notation is summarised below. However, a summary of the formulation for this model is also provided in this section for ease of future reference.

Parameters

bs stk

D = Bootstrap Demand of product s in year t at scenario k