1. A local pizzeria is facing severe competition and looking for measures to distinguish itself.
After going to a seminar on Service Management, the owner is considering offering a service guarantee.
Specifically, from a survey, she knows that customers would like to receive order delivery within 40 minutes of placing an order.
Therefore, “delivery within 40 minutes or $3 off” during rush hours sounds appealing.
Before offering it, the owner would like to perform a queueing analysis of the impact of such a guarantee.
From data collection, she believes:
• during rush hours, customer order arrivals follow Poisson distribution; on average an order arrives every 6 minutes
• almost all the orders are for one pizza only, so it’s safe to assume this
• although there are several steps in the entire production and delivery process, the pizza dough maker is the bottleneck
• there is only one pizza maker; he spends an exponential amount of time making each pizza, with an average of 4 minutes
• it’s safe to assume that all the other steps (e.g. baking, delivery) take a constant 15 minutes
1) Assuming the demand remains unchanged, how much would such a service guarantee cost per hour?
2) The owner hopes that the service guarantee will show customers that she is serious about making speedy delivery, hence increase customer demand.
If the guarantee is projected to increase demand by 20%, should it be offered (assuming an average order gives $3 in profit)?
(Hint: the MMm queueing spreadsheet gives you the probability of waiting longer than t in the queue.)
2. Jerry’s Cookie Company
Jerry and his roommate are preparing to launch a cookie company in their on-campus apartment to provide fresh cookies to hungry students late at night.
The idea is to bake fresh cookies to order, using any combination of ingredients that the buyer wants.
The cookies will be ready for pickup at the apartment.
Two factors distinguishes the new cookie company:
1) The cookies will be completely fresh and made to order.
2) There will be a variety of ingredients available to add to the basic dough, including chocolate chips, M&M's, chopped Heath bars, coconut, walnuts, and raisins.
Buyers will enter their orders online or in an app, and specify which of these ingredients they want in their cookies.
The Production Process
Baking cookies is simple: place all the ingredients in a mixing bowl and mix them; spoon the cookie dough onto a tray; put the cookies into the oven; bake them; take the tray of cookies out of the oven; let the cookies cool; and, finally, take the cookies off the tray and carefully pack them in a box.
A detailed examination of the production process follows.
Jerry and his roommate have carefully timed the necessary operations.
• Order processing The first step is to take the orders, which are processed on a first-come-first-served basis. Jerry has developed both an app and a website for taking orders.
• Mixing The next step is to place the specified ingredients in the electric mixer's bowl and turn on the mixer to mix the ingredients.
The electric mixer can hold and mix ingredients for up to three dozen cookies.
Jerry then spoon the cookies, one dozen at a time, onto a cookie tray.
Adding the ingredients to the bowl and mixing takes 6 minutes, regardless of how many cookies are being made in the batch.
That is, to mix enough dough and ingredients for three dozen cookies takes the same 6 minutes as for one dozen cookies.
However, spooning the cookies onto the tray takes 2 minutes per tray.
• Baking The next step, performed by Jerry’s roommate, is to put the cookies in the oven and set the timer.
The time to do this is negligible, and will be ignored in this analysis.
The cookies bake for 10 minutes.
Because the oven only holds one tray, a second dozen takes an additional 10 minutes to bake.
• Finishing Jerry’s roommate also performs the last steps of the process by first removing the cookies from the oven and putting them aside to cool for 5 minutes, then carefully packing them in a box and accepting payment.
Removing the cookies from the oven takes a negligible amount of time, but it must be done promptly.
It takes 2 minutes to pack each dozen and about 1 minute to accept payment for the order.
• As experienced bakers know, a few simplifications were made in describing the actual cookie production process.
For example, the first batch of cookies for the night requires preheating the oven.
However, such complexities will be put aside for now.
Begin your analysis by developing a process flow diagram of the cookie-making process
Given the wide variety of options and possibility for customization, assume that no two orders are exactly the same.
Hence different orders are always processed separately.
(To answer the following questions, you may find it better to draw a process diagram first.)
1) Assume all the orders are for one dozen cookies.
a. How long does it take to process a rush order of one dozen cookies, assuming that no other cookies are currently in process (i.e., what is the system flow time)?
b. Where is the bottleneck? What is the system capacity (that is, how many cookies can be made and sold each hour at the most)?
2) Answer the same questions as in 1) assuming all the orders are for two dozen cookies.
3) Answer the same questions as in 1) assuming all the orders are for three dozen cookies.
4) Jerry and his roommate are considering whether to add a second oven that’s identical to the current one. If all the orders are for one dozen cookies, how much would adding a new oven increase the system capacity?
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Target: “delivery within 40 minutes or $3 off” during rush hours
From data collection:
During rush hours, customer order arrivals follow Poisson distribution; on average an order arrives every 6 minutes
Almost all the orders are for one pizza only, so it’s safe to assume this
Although there are several steps in the entire production and delivery process, the pizza dough maker is the bottleneck
There is only one pizza maker; he spends an exponential amount of time making each pizza, with an average of 4 minutes
It’s safe to assume that all the other steps (e.g. baking, delivery) take a constant 15 minutes.
a. Assuming the demand remains unchanged, how much would such a service guarantee cost per hour?
Arrival rate = 6 min = 10 customers per hour
Process rate (maker) = 4 min per order = 15 orders per hour
Process rate (others) = 15 min per order = 4 orders per hour
System utilization = Arrival rate / Process rate = 10 / 15 = 66.67%
Since it says that maker is a bottleneck, process rate for maker will be used for calculations.
In order to have costs associated with service guarantee, we need to find the probability that customer will wait more than 40 minutes in system. For that, first we...
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