a) The actual demand of a product for 8 months are summarized as follows:
Month Demand
1
2
3
4
5
6
7
8
20
26
32
44
66
88
96
112
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
1
S 2
S 3
S
Ν1
Ν2
Ν3
K700,000
300,000
150,000
K500,000
450,000
0
K300,000
300,000
300,000
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 Moving Average
1 20 -
2 26 -
3 32 -
4 44 (20+26+32)/3 = 26
5 66 (26+32+44)/3 = 34
6 88 (32+44+66)/3 = 47.33
7 96 (44+66+88)/3 = 66
8 112 (66+88+96)/3 = 83.33
ii. Forecast error:
Month Demand Moving Average Error
1 20 - -
2 26 - -
3 32 - -
4 44 26 18
5 66 34 32
6 88 47.33 40.67
7 96 66 30
8 112 83.33 28.67
iii. Mean square error (MSE):
MSE = Σ(Error^2) / n
= (18^2 + 32^2 + 40.67^2 + 30^2 + 28.67^2) / 5
= (324 + 1024 + 1645.4889 + 900 + 823.0569) / 5
= 4716.5458 / 5
= 943.31
iv. Mean absolute deviation (MAD):
MAD = Σ|Error| / n
= (|18| + |32| + |40.67| + |30| + |28.67|) / 5
= (18 + 32 + 40.67 + 30 + 28.67) / 5
= 149.34 / 5
= 29.87
b)
i. Maximin criterion: Choose the minimum payoff in each state of nature
Maximin = min(700,000, 300,000, 150,000) = 150,000
ii. Maximax criterion: Choose the maximum payoff in each state of nature
Maximax = max(700,000, 500,000, 300,000) = 700,000
iii. Minimax regret criterion: Find the maximum regret in each course of action and choose the minimum
Regret(S1) = max(700,000-500,000, 700,000-300,000) = 400,000
Regret(S2) = max(300,000-300,000, 500,000-300,000) = 200,000
Regret(S3) = max(150,000-0, 300,000-0) = 300,000
Minimax regret = min(max(400,000, 200,000, 300,000), max(200,000, 0, 0)) = min(400,000, 200,000) = 200,000
iv. Laplace criterion: Find the average payoff for each course of action
Laplace(S1) = (700,000 + 500,000 + 300,000) / 3 = 500,000
Laplace(S2) = (300,000 + 450,000 + 300,000) / 3 = 350,000
Laplace(S3) = (150,000 + 0 + 300,000) / 3 = 150,000
v. Hurwicz criterion: Use α = 0.8 to balance between optimistic and pessimistic forecast
Hurwicz(S1) = 0.8 * max(700,000, 500,000) + 0.2 * min(700,000, 500,000) = 0.8 * 700,000 + 0.2 * 500,000 = 660,000
Hurwicz(S2) = 0.8 * max(300,000, 450,000) + 0.2 * min(300,000, 450,000) = 0.8 * 450,000 + 0.2 * 300,000 = 420,000
Hurwicz(S3) = 0.8 * max(150,000, 0) + 0.2 * min(150,000, 0) = 0.8 * 150,000 + 0.2 * 0 = 120,000
c)
i. Forecast for the second week of July:
Ft = α * At-1 + (1-α) * Ft-1
Ft = 0.25 * 4850 + 0.75 * 3800
Ft = 1212.5 + 2850
Ft = 4062.5 units
ii. Forecast for the third week of July:
Ft = 0.25 * 750 + 0.75 * 4062.5
Ft = 187.5 + 3046.875
Ft = 3234.375 units