Decision making using perfect information.
Linear Programming Background Scenario West Abbey Winery
Bill Yeager, a member of the partnership that owned West Abbey Winery, had to make a decision: should the harvest the Riesling grapes immediately or leave them on the vines in spite of the approaching storm? A storm just before harvest is typically detrimental, often ruining the crop. A warm light rain, however, will sometimes cause a beneficial mold, botrytis cineria, to form on the grape skins. The result is a luscious, complex sweet wine, highly valued by connoisseurs.
West Abbey is located in northern Napa Valley. The winery produced only premium wines from the best grape varieties. Of the 25,000 cases of wines bottled each year, most are Cabernet and Chardonnay. About 1,000 cases of Riesling are also typically bottled. (A case contains 12 bottles of wine.)
The Napa Valley extends for 30 miles to the north of San Francisco. The average temperature decreases as one moves south, closer to San Francisco Bay and the cold ocean waters. West Abbey grapes come from an ideal climate in the central and southern parts of the valley.
Wine is produced when fruit sugar, which is of course present in the juice of grapes, is converted by yeast, through fermentation, into approximately equal molecular quantities of alcohol and carbon dioxide. Sparkling wines accepted, the carbon dioxide is allowed to bubble up and dissipate. The wine then ages in barrels for one or more years until it is ready for bottling.
By a variety of decisions all through vinification (special ingredients or secrets in making the wine) the winemaker influences the style of wine produced. The style adopted by a particular winery depends mainly on the owners' preferences, although it is influenced by marketing considerations. Usually, as the grapes ripen, the sugar levels increase and acidity levels decrease. The winemaker tries to harvest the grapes when they have achieved the proper balance of sugar and acidity for the style of wine sought. The ripening procedure is variable, however, and, if the weather is not favourable, the proper balance might not occur.
Several different styles of Riesling are on the market. If the grapes are harvested at 20% sugar, the wine is fermented "dry" (all the sugar is converted to alcohol and carbon dioxide). The resulting wine, which is about 10 percent alcohol, is light bodied. If the grapes are harvested at 25% sugar, the wine maker can produce a wine with the same 10 percent alcohol but with 5 percent residual sugar; this wine is sweet and relatively full bodied.
A third and rare style results when almost-ripe Riesling grapes are attacked by the botrytis mold. The skins of the grapes become porous, allowing water to evaporate while the sugar remains. Thus, the sugar concentration increases greatly, sometimes to 35% or more. The resulting wine, with about 11 percent alcohol and 13% residual sugar, has extraordinary concentration, and the botrytis itself adds to the wine's complexity. The last time West Abbey winery produced a botrytised Riesling was three years ago.
The Problem
From the weather reports, Yeager concluded that there was a 50/50 chance that the rainstorm could hit the Napa Valley. As the storm had originated over the warm waters of Mexico, he thought there was a 40 percent chance that, if the storm did strike, it would lead to the development of the botrytis mold. If the mold did not form, however, the rainwater, which would be absorbed into the grapes through the roots of the vines, would merely swell the grapes by 10%, which would make more juice while decreasing the sugar concentration. This generally means crop failure which would yield a thin wine that would sell wholesale for about $1 per bottle, about $1.85 per bottle less than Yeager could obtain by harvesting the not-so-ripe grapes now and eliminating the risk.
If Yeager decided not to harvest the grapes immediately in anticipation of the storm, and the storm did not strike, Yeager would probably leave the grapes to ripen more fully. With luck the grapes would reach 25 percent sugar, resulting in a wine selling for around $3.50 per bottle. Even with less favorable weather, the sugar levels would probably top 20 percent, yielding a lighter wine selling at around $3.00. Yeager thought these possibilities were equally likely. In the past, sugar levels occasionally failed to rise about 19 percent. Furthermore, while waiting for sugar levels to rise, the acidity levels must be monitored. When the acidity drops below .7%, the grapes must be harvested whatever the sugar level. If this happened, the wine would be priced at only about $2.50. Yeager felt that this event had only about a .2 probability.
The wholesale price for a botrytised Riesling would be about $8 per bottle. Unfortunately, the same process that resulted in increased sugar concentration also caused a 30% reduction in the total juice. The higher price was, therefore, partly offset by a reduction in quantity. Although fewer bottles would be produced, there would be essentially no savings in vinification costs. The costs to the winery were about the same for each of the possible styles of wine and were relative to the wholesale price.
Deliverables:
Write a short executive summary to Mr. Yeager which would include your answers to questions 2 and 3 below. Following the executive summary, show the work and solutions for the 3 questions/problems below.
1) Utilize a decision tree to illustrate the possible strategies and the consequences (or payoffs) of each. Show your decision tree.
2) Using the expected value approach, what would your recommendation be to Yeager? Pick the grapes now or wait? Validate your answer using the results of the decision tree, i.e., what is the expected value of picking the grapes now versus waiting?
3) What is the expected value of perfect information on whether or not the mold will form, if the storm hits? Show your tree and your calculation of the Expected Value Of Perfect Information.