Question

A company produces and sells bread. They make all their bread before 8:00 A.M. each day for sale between 9:00 A.M. and 6:00 P.M. They produce to a 90% service level. (200 POINTS)

1. How many loaves should be available for sale if the standard deviation from the forecast has been calculated to be 20 and the forecast is 200?

2. If the manager decides that 15 stockouts per is an acceptable expectation and they are open 360 days per year, how many loaves should now be available for sale?

Answer 1

a.Given : service level =90 %

Assuming normal distribution ,we have,

z=x-mean/standard deviation

But we have,z=NORMSINV(0.90)=1.28

1.28=x-200/20

x=200+1.28*20

=225.63

=226 loaves

Number of loaves that should be available for sale =226

b.Given that number of stockouts per year =15

Forecast =20

Number of stockouts=Forecast*probability of stockout

15=20*probability of stockout

Probability of stockout=15/20=0.75

Service level=1-Probability of stockout =1-0.75=0.25 or 25 %

Therefore,

z=NORMSINV(service level)=NORMSINV(0.25)=-0.674

z=x-mean/standard deviation

-0.674=x-200/20

x=200-0.674*20

=186.51

=187

Number of loaves that should be available for sale =187

Answer 2

a. Calculate the number of loaves available for sale

This question can be solved by using the safety stock formula based on the relationship between service level and normal distribution.

- **Step 1: Determine the Z value corresponding to the service level**

It is known that the service level reaches \(90\%\). In the normal distribution, by consulting the standard normal distribution table (or using relevant statistical software or functions), the corresponding \(Z\) value is approximately \(1.28\).

- **Step 2: Calculate safety stock**

The safety stock formula is \(SS = Z\times\sigma\), where \(SS\) represents safety stock, \(Z\) is the standard normal distribution coefficient corresponding to the service level, and \(\sigma\) is the predicted standard deviation.

Given the predicted standard deviation \(\sigma = 20\), \(Z = 1.28\), substituting it into the formula yields the safety stock \(SS=1.28\times20 = 25.6\).

- **Step 3: Calculate the number of breads available for sale**

The number of breads available for sale is equal to the forecast value plus the safety stock, that is, \(Q = \text{forecast value}+SS\).

Given that the forecast value is \(200\), substituting it and the safety stock value into the formula yields: \(Q = 200 + 25.6 = 225.6\approx226\) (the number of breads must be an integer).

b. Calculate the number of breads available for sale under the new circumstances

For this question, you can first calculate the service level based on the acceptable number of out-of-stocks, then determine the corresponding \(Z\) value, and finally calculate the number of breads available for sale.

- **Step 1: Calculate the service level**

It is known that the store is open \(360\) days a year, and the acceptable out-of-stock expectation is \(15\) times per day.

Then the acceptable number of out-of-stocks per year is \(15\times360 = 5400\) times.

Assume that the total annual demand is \(D\) (here it can be regarded as a relative concept and does not affect the calculation of the service level), and the service level is \(SL=1 - \frac{5400}{D}\).

For the convenience of calculation, we assume that the demand follows a uniform distribution (in the absence of more information), and do not consider the specific \(D\) value. From a probability perspective, the acceptable out-of-stock rate is \(= \frac{15}{D/360}\), and the service level is \(SL = 1-\frac{15}{D/360}\).

If the average daily demand is \(D/360 = 200\) (here a reasonable daily average demand value is assumed for calculation, which does not affect the principle), then the acceptable out-of-stock rate is \(=\frac{15}{200}= 0.075\), and the service level is \(SL = 1 - 0.075=0.925\).

Step 2: Determine the Z value corresponding to the service level

By consulting the standard normal distribution table (or using relevant statistical software or functions), when the service level is \(0.925\), the corresponding \(Z\) value is approximately \(1.44\).

- **Step 3: Calculate the safety stock and the number of loaves available for sale**

Also assuming that the forecast standard deviation \(\sigma = 20\) (the question does not mention the change, so the condition in question a is used), according to the safety stock formula \(SS = Z\times\sigma\), the safety stock \(SS = 1.44\times20 = 28.8\) can be obtained.

The number of loaves available for sale \(Q = \text{forecast value}+SS\), assuming that the forecast value is still \(200\), then \(Q = 200 + 28.8 = 228.8\approx229\) (the number of loaves must be an integer).

Therefore, a. there should be \(226\) loaves of bread available for sale; b. there should be \(229\) loaves of bread available for sale.