QUESTION THREE
(a) The actual demand of a product for 8 months are summarized as follows:
Month Demand
1 20
2 26
3 32
4 44
5 66
6 88
7 96
8 112
Clearly show your work step by step:
i. Find three months moving averages
ii. Compute forecast error
iii. Compute mean square error (MSE)
iv. Compute mean absolute deviation (MAD)
(b) A food product company is contemplating the introduction of a revolutionary new product
with new packaging to replace the existing product at a large increase in price (S1 ) or a
moderate change in composition of the existing product with a new packaging at a small
increase in price (S2 ) or a small change in the composition of the existing product with a
negligible increase in price (S3 ) . The three states of nature are: i. high increase in sales
(Ν1 ), ii. No change in sales (Ν2 ) and iii. Decrease in sales (Ν3 ). The marketing
department of the company worked out the payoffs in terms of yearly net profits for each
course of action for these events. This is represented in the following table: 4
States of Nature Courses of Action
S1 S2 S3
Ν1 K700,000 K500,000 K300,000
Ν2 300,000 450,000 300,000
Ν3 150,000 0 300,000
Clearly show your work step by step
i. Maximin criterion?
ii. Maximax criterion?
iii. Minimax regret criterion?
iv. Laplace criterion?
v. Hurwicz criterion? Use α = 0.8
(c) The forecast of a product for the first week of July was 3800 units, whereas the actual
demand turned out to be 4850 units.
i. Find the forecast for the second week of July by assuming the smoothing constant
(α ) as 0.25.
ii. Find the forecast for the third week of July if the actual demand of the second week
is 750 units.
(a)
i. Three months moving averages:
Month Demand 3-Month MA
1 20 (20+26+32)/3 = 26
2 26 (26+32+44)/3 = 34
3 32 (32+44+66)/3 = 47.33
4 44 (44+66+88)/3 = 66
5 66 (66+88+96)/3 = 83.33
6 88 (88+96+112)/3 = 98.67
7 96 (96+112)/2 = 104
8 112 112
ii. Forecast error:
Month Demand 3-Month MA Forecast Error
1 20 26 -6
2 26 34 -8
3 32 47.33 -15.33
4 44 66 -22
5 66 83.33 -17.33
6 88 98.67 -10.67
7 96 104 -8
8 112 112 0
iii. Mean square error (MSE):
MSE = ( (-6)^2 + (-8)^2 + (-15.33)^2 + (-22)^2 + (-17.33)^2 + (-10.67)^2 + (-8)^2 + 0^2 ) / 8
MSE = ( 36 + 64 + 236.28 + 484 + 300.52 + 113.77 + 64 + 0 ) / 8
MSE = 129.07
iv. Mean absolute deviation (MAD):
MAD = ( | -6 | + | -8 | + | -15.33 | + | -22 | + | -17.33 | + | -10.67 | + | -8 | + | 0 | ) / 8
MAD = ( 6 + 8 + 15.33 + 22 + 17.33 + 10.67 + 8 + 0 ) / 8
MAD = 11.33
(b)
i. Maximin criterion:
The minimum payoff for each course of action is:
S1: K300,000
S2: K300,000
S3: K0
Therefore, the maximin criterion is to choose S3 as it has the highest minimum payoff of K0.
ii. Maximax criterion:
The maximum payoff for each course of action is:
S1: K700,000
S2: K450,000
S3: K300,000
Therefore, the maximax criterion is to choose S1 as it has the highest maximum payoff of K700,000.
iii. Minimax regret criterion:
Regret table
States of Nature Courses of Action
S1 S2 S3
Ν1 400,000 200,000
Ν2 0 150,000 0
Ν3 550,000 300,000 0
Minimum regret for each course of action is:
S1: K400,000
S2: K0
S3: K0
Therefore, the minimax regret criterion is to choose S2 as it has the smallest minimum regret.
iv. Laplace criterion:
Expected payoff for each course of action is:
S1: (700,000 + 500,000 + 300,000) / 3 = 500,000
S2: (300,000 + 450,000 + 300,000) / 3 = 350,000
S3: (150,000 + 0 + 300,000) / 3 = 150,000
Therefore, the Laplace criterion is to choose S1 as it has the highest expected payoff of K500,000.
v. Hurwicz criterion:
Using α = 0.8, the Hurwicz criterion is to choose the course of action that maximizes the weighted sum of the maximum payoff and minimum payoff.
S1: 0.8 * 700,000 + 0.2 * 300,000 = 620,000
S2: 0.8 * 500,000 + 0.2 * 300,000 = 460,000
S3: 0.8 * 300,000 + 0.2 * 300,000 = 300,000
Therefore, according to the Hurwicz criterion, S1 should be chosen.
(c)
i. Forecast for the second week of July using exponential smoothing:
Forecast = α * Actual Demand + (1 - α) * Previous Forecast
Forecast = 0.25 * 4850 + 0.75 * 3800
Forecast = 1212.5 + 2850
Forecast = 4062.5 units
ii. Forecast for the third week of July using the updated actual demand of the second week:
Forecast = 0.25 * 750 + 0.75 * 4062.5
Forecast = 187.5 + 3046.875
Forecast = 3234.375 units