1. Exponential Smoothing
a. To prepare a forecast for September, we will use the exponential smoothing formula:
Forecast for September = (Previous Month’s Forecast) + (Smoothing Constant * (Actual Usage – Previous Month’s Forecast))
Given:
Previous Month’s Forecast (August) = 53% of capacity
Actual Usage (August) = 52% of capacity
Smoothing Constant = 0.05
Forecast for September = (0.53) + (0.05 * (0.52 – 0.53))
Forecast for September = (0.53) + (0.05 * (-0.01))
Forecast for September = 0.53 – 0.0005
Forecast for September = 0.5295 or 52.95%
b. To prepare a forecast for October, we will use the same exponential smoothing formula as above:
Forecast for October = (Previous Month’s Forecast) + (Smoothing Constant * (Actual Usage – Previous Month’s Forecast))
Given:
Previous Month’s Forecast (September) = 52.95% of capacity
Actual Usage (September) = 60% of capacity
Smoothing Constant = 0.05
Forecast for October = (0.5295) + (0.05 * (0.60 – 0.5295))
Forecast for October = (0.5295) + (0.05 * 0.0705)
Forecast for October = 0.5295 + 0.003525
Forecast for October = 0.533025 or 53.30%
2. Linear Model for Insurance
Using the equation y = 157 – 0.50x, where y is the insurance needed ($000) and x is the current age of the head of household.
b. To determine the amount of term life insurance to recommend for a family of four with a head of household age of 47 years old:
y = 157 – 0.50x
y = 157 – 0.50(47)
y = 157 – 23.5
y = 133.5 or $133,500
The amount of term life insurance to recommend is $133,500.
3. Forecasting Job Requests
a. Naive method:
The naive method assumes that the forecast for the next period will be equal to the actual value of the current period.
For week 6, the forecast would be equal to the number of requests in week 5, which is 30.
b. Four-period moving average:
To calculate the four-period moving average, we take the average of the last four weeks’ requests.
Week 6 forecast = (28 + 24 + 22 + 23) / 4
Week 6 forecast = 97 / 4
Week 6 forecast = 24.25 or 24 requests
c. Exponential smoothing with α = 0.25:
To calculate the exponential smoothing forecast, we use the formula:
Forecast for week t = α * Actual value for week t + (1 – α) * Forecast for week t-1
Given:
α = 0.25
Week 2 forecast = 27
Forecast for week 3 = 0.25 * 24 + (1 – 0.25) * 27
Forecast for week 3 = 6 + 20.25
Forecast for week 3 = 26.25 or 26 requests
Forecast for week 4 = 0.25 * 22 + (1 – 0.25) * 26.25
Forecast for week 4 = 5.5 + 19.6875
Forecast for week 4 = 25.1875 or 25 requests
Forecast for week 5 = 0.25 * 23 + (1 – 0.25) * 25.1875
Forecast for week 5 = 5.75 + 18.890625
Forecast for week 5 = 24.640625 or approximately 24.64 requests
Forecast for week 6 = 0.25 * 30 + (1 – 0.25) * 24.640625
Forecast for week 6 = 7.5 + 18.480469
Forecast for week 6 = 25.980469 or approximately 25.98 requests
4. Seasonal Relatives
To describe the situation where the restaurant does about 30% of its business on Friday night, 30% on Saturday night, and 20% on Thursday night, we can calculate the seasonal relatives:
Seasonal Relative for Friday night = Friday night business / Total business
Seasonal Relative for Saturday night = Saturday night business / Total business
Seasonal Relative for Thursday night = Thursday night business / Total business
Given:
Friday night business: 30%
Saturday night business: 30%
Thursday night business:20%
Total business seasonal relative = Friday night seasonal relative + Saturday night seasonal relative + Thursday night seasonal relative
Total business seasonal relative = (30/100) + (30/100) + (20/100)
Total business seasonal relative = (0.3) + (0.3) + (0.2)
Total business seasonal relative = 0.8 or 80%
The seasonal relatives that describe this situation are: Friday night – 30%, Saturday night -30%, Thursday night -20%, and Total business -80%.
5. Seasonal Relatives Using Simple Averaging Method
To compute seasonal relatives using the simple averaging method, we calculate the average value for each quarter and then divide each value by the average of all quarters.
Given:
Quarter | Year 1 | Year 2 | Year 3 | Year 4
1 | 2 | 1 | 1 | 0
2 | 6 | 6 | 7 | 6
3 | 1 | 4 | 5 | 8
4 | 4 | 4 | 6 | 8
Quarter averages:
Quarter 1 average = (2 +1 +1 +0) /4
Quarter 1 average =4/4
Quarter 1 average=1
Quarter 2 average= (6+6+7+6)/4
Quarter2 average=25/4
Quarter2 average=6.25
Quarter3 average=(1+4+5+8)/4
Quarter3 average=18/4
Quarter3 average=4.5
Quarter4 average=(4+4+6+8)/4
Quarter4 average=22/4
Quarter4 average=5.5
Calculating seasonal relatives:
Seasonal relative Quarter1=1/((1+6.25+4.5+5.5)/4)
Seasonal relative Quarter1=1/(17.25/4)
Seasonal relative Quarter1=1/(4.3125)
Seasonal relative Quarter1=0.231 or approximately equal to 0.231
Seasonal relative Quarter2=6.25/((1+6.25+4.5+5.5)/4)
Seasonal relative Quarter2=6.25/(17.25/4)
Seasonal relative Quarter2=6.25/4.3125
Seasonal relative Quarter2=1.448 or approximately equal to 1.448
Seasonal relative Quarter3=4.5/((1+6.25+4.5+5.5)/4)
Seasonal relative Quarter3=4.5/(17/25/4)
Seasonal relative Quarter3=4/5/4/3125
Seasonal relative Quarter3=1/061 or approximately equal to 1/061
Seasonal relative Quarter4=5.5/((1+6/25+4/5+5/50)/4)
Seasonal relative Quarter4=5/50/(17/50)/4)
Seasonal relative Quarter4=5/50/4/3125
Seasonal relative Quarter4=1/237 or approximately equal to 1/237
The seasonal relatives using the simple averaging method are as follows:
Quarter1: 0.231
Quarter2: 1.448
Quarter3: 1/061
Quarter4: 1/237
