Exponential Smoothing

by | Jan 12, 2024 | Operations management

1. A dry cleaner uses exponential smoothing to forecast equipment usage at its main plant. August usage was forecasted to be 53 percent of capacity; actual usage was 52 percent of capacity. A smoothing constant of .05 is used.
a. Prepare a forecast for September. (Round your final answer to 2 decimal places.)
b. Assuming actual September usage of 60 percent, prepare a forecast for October usage. (Round your answer to 2 decimal places.)
2. Long-Life Insurance has developed a linear model that it uses to determine the amount of term life insurance a family of four should have, based on the current age of the head of the household. The equation is:

y = 157 -0.50x

where

y = Insurance needed ($000)
x = Current age of head of household

b. Use the equation to determine the amount of term life insurance to recommend for a family of four if the head of the household is 47 years old. (Round your answer to 2 decimal places.)
3. An electrical contractor’s records during the last five weeks indicate the number of job requests:

Week: 1 2 3 4 5
Requests: 28 24 22 23 30

Predict the number of requests for week 6 using each of these methods:
a. Naive.
b. A four-period moving average. (Round your answer to 2 decimal places.)
c. c. Exponential smoothing with α = 0.25. Use 27 for week 2 forecast. (Round your intermediate forecast values and final answers to 2 decimal places.)
4. The manager of a fashionable restaurant open Wednesday through Saturday says that the restaurant does about 30 percent of its business on Friday night, 30 percent on Saturday night, and 20 percent on Thursday night. What seasonal relatives would describe this situation? (Round your answers to 2 decimal places.)
5. Compute seasonal relatives for this data using the simple averaging (SA) method:

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
________________________________________
(Round all your answers to three decimal points.)

Sample solution

Dante Alighieri played a critical role in the literature world through his poem Divine Comedy that was written in the 14th century. The poem contains Inferno, Purgatorio, and Paradiso. The Inferno is a description of the nine circles of torment that are found on the earth. It depicts the realms of the people that have gone against the spiritual values and who, instead, have chosen bestial appetite, violence, or fraud and malice. The nine circles of hell are limbo, lust, gluttony, greed and wrath. Others are heresy, violence, fraud, and treachery. The purpose of this paper is to examine the Dante’s Inferno in the perspective of its portrayal of God’s image and the justification of hell. 

In this epic poem, God is portrayed as a super being guilty of multiple weaknesses including being egotistic, unjust, and hypocritical. Dante, in this poem, depicts God as being more human than divine by challenging God’s omnipotence. Additionally, the manner in which Dante describes Hell is in full contradiction to the morals of God as written in the Bible. When god arranges Hell to flatter Himself, He commits egotism, a sin that is common among human beings (Cheney, 2016). The weakness is depicted in Limbo and on the Gate of Hell where, for instance, God sends those who do not worship Him to Hell. This implies that failure to worship Him is a sin.


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God is also depicted as lacking justice in His actions thus removing the godly image. The injustice is portrayed by the manner in which the sodomites and opportunists are treated. The opportunists are subjected to banner chasing in their lives after death followed by being stung by insects and maggots. They are known to having done neither good nor bad during their lifetimes and, therefore, justice could have demanded that they be granted a neutral punishment having lived a neutral life. The sodomites are also punished unfairly by God when Brunetto Lattini is condemned to hell despite being a good leader (Babor, T. F., McGovern, T., & Robaina, K. (2017). While he commited sodomy, God chooses to ignore all the other good deeds that Brunetto did.

Finally, God is also portrayed as being hypocritical in His actions, a sin that further diminishes His godliness and makes Him more human. A case in point is when God condemns the sin of egotism and goes ahead to commit it repeatedly. Proverbs 29:23 states that “arrogance will bring your downfall, but if you are humble, you will be respected.” When Slattery condemns Dante’s human state as being weak, doubtful, and limited, he is proving God’s hypocrisy because He is also human (Verdicchio, 2015). The actions of God in Hell as portrayed by Dante are inconsistent with the Biblical literature. Both Dante and God are prone to making mistakes, something common among human beings thus making God more human.

To wrap it up, Dante portrays God is more human since He commits the same sins that humans commit: egotism, hypocrisy, and injustice. Hell is justified as being a destination for victims of the mistakes committed by God. The Hell is presented as being a totally different place as compared to what is written about it in the Bible. As a result, reading through the text gives an image of God who is prone to the very mistakes common to humans thus ripping Him off His lofty status of divine and, instead, making Him a mere human. Whether or not Dante did it intentionally is subject to debate but one thing is clear in the poem: the misconstrued notion of God is revealed to future generations.

 

References

Babor, T. F., McGovern, T., & Robaina, K. (2017). Dante’s inferno: Seven deadly sins in scientific publishing and how to avoid them. Addiction Science: A Guide for the Perplexed, 267.

Cheney, L. D. G. (2016). Illustrations for Dante’s Inferno: A Comparative Study of Sandro Botticelli, Giovanni Stradano, and Federico Zuccaro. Cultural and Religious Studies, 4(8), 487.

Verdicchio, M. (2015). Irony and Desire in Dante’s” Inferno” 27. Italica, 285-297.

Sample Answer

Sample Answer

 

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

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