Chapter 2, Problem 2E Show all steps ON Boo Problemm A baker is trying to figure

Question

Chapter 2, Problem 2E Show all steps ON Boo Problemm A baker is trying to figure out how many dozens of bagels to bake each day. The probability distribution of the number of bagel customers is as follows: Number of Customers/Day 810 12 14 0.35 0.30 0.25 0.10 Customers order 1, 2, 3, or 4 dozen bagels according to the following probability distribution. 2 3 0.4 0.3 0.2 0.1 Number of Dozen Ordered/Customer Bagels sell for $8.40 per dozen. They cost $5.80 per dozen to make. All bagels not sold at the end of the day are sold at half-price to a local grocery store. Based on 5 days of simulation, how many dozen (to the nearest 5 dozen) bagels should be baked each day? Step-by-step solution

Explanation / Answer

We expect 24 dozen bagels to maximize profit. To do this, we can run simulation of the number of bagels made to get estimated profit. Details are given below on the simulation run.

First, we estimate the number of customers expected per day.

From the probability distribution, we get:

E(number of customers/day) = 8*0.35 + 10*0.30 + 12*0.25 + 14*0.10 = 2.8 + 3.0 + 3.0 + 1.4 = 10.2

Similary, we can estimate bagels per customer.

E(number of dozens of bagels/customer) = 1*0.4 + 2*0.3 + 3*0.2 + 4*0.1 = 0.4 + 0.6 + 0.6 + 0.4 = 2

This gives E(number of dozens of bagels/day)

= E(number of dozens of bagels/customer) * E(number of customers/day)

= 2 * 10.2

= 20.4

Though this is not the answer, it gives an estimate of how many dozens of bagels can be safely made everyday.

Now, we can run simulations on the profitability for different number of bagels.

Running simulations requires first choosing number of bagels made that day. Then, using a random number generator, get a random number and map it to the probability distribution described in the problem. This will give an estimate of how many orders came that day. If this number is more than the number of bagels actually made, then there is no wastage of bagels. But profit will be less if number of orders is much more than the number of bagels actually made. If this number is less than the number of bagels made that day, then there will be bagels sold at throwaway price. This too will impact profit.

Here is a sample of the estimates that were made:

When number of orders is more, extra orders go waste.

Running simulation for different number of bagel dozens gave this sort of output:

So, we can see that the peak is around 24 dozen bagels with reasonable standard deviation.

Sample Sales # Trial # of Dozens of Bagels Made # of Orders Profit 1 20 36 $52 2 20 40 $52

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