Brenda’s Ski Shop . Brenda Kelley runs a specialty ski clothing shop outside of
ID: 3041592 • Letter: B
Question
Brenda’s Ski Shop. Brenda Kelley runs a specialty ski clothing shop outside of Baltimore. She must place a large, medium, or small order for ski parkas well in advance of ski season because the manufacturer produces them in the summer months. The number she will sell depends whether the area receives a heavy, normal, or light amount of snow during the ski season. The table below summarizes the net profit Brenda will receive under each snow level. Brenda estimates the probability of heavy, normal, and light snowfalls as 0.25, 0.6, and 0.15, respectively.
(a) What size order should she choose under the Maximax rule?
(b) What size order should she choose under the Maximin rule?
(c) What size order should she choose under the Minimax regret rule?
(d) What size order should she choose under the Expected value rule?
(e) In terms of dominance, why is the Small order option always a poor choice?
(f) In terms of variance, why might the Small order option be a good choice?
Order Size Option
Heavy Snow
Moderate Snow
Light Snow
Large
$10,000
$7,000
$3,000
Medium
$8,000
$8,000
$6,000
Small
$4,000
$4,000
$4,000
Order Size Option
Heavy Snow
Moderate Snow
Light Snow
Large
$10,000
$7,000
$3,000
Medium
$8,000
$8,000
$6,000
Small
$4,000
$4,000
$4,000
Explanation / Answer
a) By the maximax rule , we need to choose that profit option which is maximum out of all the maximum profit choices ,
Hence
1)maximum profit under heavy snow is : max{10000,8000,4000} =10,000 (i.e large order size)
2)maximum profit under moderate snow is : max{7000,8000,4000} =8000 (i.e Medium order size)
3)maximum profit under light snow is : max{3000,6000,4000} =6000 (i.e Medium order size)
Maximum (1,2,3)=1 i.e under maximax rule ,the maximum profit would be earned under the Large order size option
b)Under the maximin rule , we choose maximum out of the minimum profit earned :
Hence
1)Minimum profit under heavy snow is : min{10000,8000,4000} =4000(i.e small order size)
2)Minimum profit under moderate snow is : min{7000,8000,4000} =4000(i.e small order size)
3)Minimum profit under light snow is : max{3000,6000,4000} =3000 (i.e Large order size)
The maximum out of these gives maximin criteria for obtaining profit as the small order size.
c) Under the minimax regret rule, it is a rule which minimizes the maximum regret.
To solve this a table showing the size of the regret needs to be constructed. This means we need to find the biggest pay-off for each demand row, then subtract all other numbers in this row from the largest number.
Order Size Option
Heavy Snow
Moderate Snow
Light Snow
Large
$0
$3000
$7000
Medium
$0
$0
$2000
Small
$0
$0
$0
As for a large order size maximum profit would have been $ 10000, hence we subtract it from each profit earned under that large order size and obtain the first row of regret matrix.
Hence under the minimax regret criteria ,minimum regret would occur when we take up the small order ($0 regret)
d) Under the Expected value rule ,
P[Large order]=10000*0.25+7000*0.6+3000*0.15=$7150
P[Medium order]=8000*0.25+8000*0.6+6000*0.15=$7700
P[Small order]=4000*0.25+4000*0.6+4000*0.15=$4000
As the Expected value of profit under the medium order is maximum , WE CHOOSE THE MEDIUM ORDER.
e) In terms of dominance , if we see under the heavy ,moderate and light snow option , the small order never dominates the other order size opton i.e profit earned (if we go columnwise ) would always be higher for heavy snow or moderate snow , hence we end up eliminating the small order size out of our pay off matrix to obtain a dominant option.Hence small order size is poor choice for a dominant strategy.
f) In terms of variance , As the small size order option earns same amount of profit under all the three scenarios unlike it's other counterparts that have very low profit in light snow as compared to heavy and moderate snow ,hence small size order would be a good choice under the variance category as it has zero deviation from the mean , hence it has 0 variance .
Order Size Option
Heavy Snow
Moderate Snow
Light Snow
Large
$0
$3000
$7000
Medium
$0
$0
$2000
Small
$0
$0
$0