Fresh Smoothies$$ Prices and Quantities

Fresh Smoothies purchases fresh fruit and vegetables for their smoothies every week. Any produce left

over when a new shipment arrives is salvaged and usually used by other companies to produce jams,

jellies, and packaged juices. In the case that they don$$t have enough fresh fruit or vegetables to cover

the demand for the week, each store buys the necessary ingredients from local grocery stores at a price

higher than the negotiated price with the primary supplier.

Steve is in charge of the company$$s revenue management strategy and has recently completed a course

on data analytics that has caused him to question some company decisions. In particular, he wondered

why the company charges the prices they do and why they order the quantities of ingredients. Did they

know something he didn$$t know about the sensitivity of demand to changes in price and the distribution

of demand?

After significant debate with Charlie, whose approval was required for pricing changes, Steve finally got

approval to do a simple experiment at a single store with a single smoothie flavor: strawberry.

The approved plan included two weeks with a variation in price. The first week, the price would be

$$4\mathrm{.}00$; and the second week, the price would be $$6\mathrm{.}00.$ All other smoothie prices would remain constant

during the two weeks. Historically, the price of the strawberry smoothie had been $$5\mathrm{.}00$; and they had

always ordered the ingredients necessary to produce $410$ strawberry smoothies each week. To

complete the data acquisition, Steve tracked down the sales figures for the previous $30$ weeks. During

this $30-$week time frame, there hadn$$t been any store$-$wide promotional sales; and the prices of all

smoothie flavors had remained constant. The $30-$week sales figures are shown in the table below.

Week Sales Week Sales Week Sales

$14171141321352$

$23761236622398$

$34061339723404$

$43481442524405$

$54021541825468$

$64001641726379$

$74001743027408$

$83451838528443$

$93801942729359$

$104312040730432$

After waiting for what felt like an eternity, at last the experiment was complete and the data compiled.

The observed demand for the strawberry smoothie during the two experimental weeks is included in

the table below.

Price Observed

Demand

$$4\mathrm{.}00494$

$$6\mathrm{.}00344$

Although completing the experiment was satisfying, Steve$$s peers and superiors now expected some

results coming from the experiment. Steve hoped he could get enough evidence to support additional

price exploration throughout the company$$s stores and products. He felt like he had a decent amount of

information for the demand at a price of $$5\mathrm{.}00$; but two non$-$$$5\mathrm{.}00$ data points aren$$t a lot to go on$.$

After all, any model used to capture the variation in demand with price changes would have a lot of

uncertainty in the parameter estimates. This uncertainty in parameter estimates translates directly into

uncertainty about the demand. This demand uncertainty $($from model parameter estimates$)$ gets

amplified the further the price gets from tested values.

Steve gathered the necessary financial information related to the focal store$$s strawberry smoothies so

he could complete his analysis and make some recommendations. The data he collected are shown in

the table below.

Parameter Value

Ingredient salvage value $$0\mathrm{.}25/$ unit

Supplier ingredient cost $$2\mathrm{.}79/$ unit

Grocery store ingredient cost $$3\mathrm{.}95/$ unit

Part A

Develop a model to describe the demand, its randomness, and its dependence on the price.

Part B

What parameters of the model would make it so that a price of $$5\mathrm{.}00$ maximizes Fresh Smoothies$$

expected weekly profits for their strawberry smoothie at the focal store?

Part C

Estimate the parameters of the demand model and capture the information necessary to develop a

demand distribution for any given price.

Part D

Use the model to find the optimal: $($i$)$ price and $($ii$)$ quantity of ingredients to purchase from the primary

supplier. The optimal price and quantity are the ones that maximize Fresh Smoothies$$ expected weekly

profits for their strawberry smoothie at the focal store.

Part E

Describe your recommendations for a future course of action in pricing strawberry smoothies at the

focal store and for other stores and flavors. Also discuss issues not addressed with the analysis of the

experiment $($and the experiment itself$).$ What other information could Steve obtain from the data that

could influence the study$$s conclusions and recommendations? What analysis and experiments could

be completed to get a better picture of the effect of a price change?