Exercise
The Comida Sana Company makes a feed mix from four ingredients—oats, corn, soybeans,
and a vitamin supplement. The company has 300 pounds of oats, 400 pounds of corn, 200 pounds
of soybeans, and 100 pounds of vitamin supplement available for the mix. The company has the
following requirements for the mix:
 At least 30% of the mix must be soybeans.
 At least 20% of the mix must be the vitamin supplement.
 The ratio of corn to oats cannot exceed 2 to 1.
 The amount of oats cannot exceed the amount of soybeans.
 The mix must be at least 500 pounds.
A pound of oats costs $0.50; a pound of corn, $1.20; a pound of soybeans, $0.60; and a pound
of vitamin supplement, $2.00. The feed company wants to know the number of pounds of each
ingredient to put in the mix in order to minimize cost.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
[Ingridients: Oats = 200 lbs; corn = 0 lbs; soybeans = 200 lbs; vitamins = 100 lbs. Cost = $420]
The Oriental Coffee Shop at the Forks Market serves two coffee blends it brews on a daily basis,
Pomona and Coastal. Each is a blend of three high-quality coffees from Colombia, Kenya, and
Indonesia. The coffee shop has 6 pounds of each of these coffees available each day. Each pound
of coffee will produce sixteen 16-ounce cups of coffee. The shop has enough brewing capacity to
brew 30 gallons of these two coffee blends each day. Pomona is a blend of 20% Colombian, 35%
Kenyan, and 45% Indonesian, while Coastal is a blend of 60% Colombian, 10% Kenyan, and
30% Indonesian. The shop sells 1.5 times more Pomona than Coastal each day. Pomona sells for
$2.05 per cup, and Coastal sells for $1.85 per cup. The manager wants to know how many cups
of each blend to sell each day in order to maximize sales.
a. Formulate a linear programming model for this problem.
b. Solve this model by using the computer.
A European oil company has to decide whether to optimize its existing wells in the North Sea or
to invest in the Arctic as many companies now scramble for these remote resources. However,
with the discovery of shale drilling, going into the Arctic is only sustainable with high oil prices.
Optimizing its wells in the North Sea will cost the company $1,000,000. Once that decision is
made, there is a 60% chance that oil prices will rise. The expected profit is $3,000,000. If oil
prices fall, the expected profit will be $200,000.
The other decision is to explore in the Arctic for $2,000,000. There is then a 60% chance oil
prices will rise. If they rise, the company can either decide to drill for $7,000,000 or abandon the
project and develop shale drilling in another location for $5,000,000 with an expected profit of
$6,000,000. If the company continues in the Arctic, there is an 80% chance oil prices will rise,
with an expected profit of $10,000,000. If oil prices fall, the loss will be $3,000,000.
To finish the tree, let’s go back to the “explore in Arctic” decision node at the beginning of
the tree. We have elaborated what will happen if oil prices rise, but there is a 40% chance they
will fall. If they fall, the company can decide to drill in the Arctic for $7,000,000. If subsequently
the prices rise (80% chance), the expected profit will be $10,000,000. If prices fall, the loss will
be $3,000,000. The company can also decide to abandon the project and develop shale drilling
somewhere else for $5,000,000 with an expected profit of $6,000,000. Construct a sequential
decision tree. According to your results, what should the company do?
[Optimizing its existing wells]
Visit one or more local car dealers and select four new models of cars you might like to purchase.
Using AHP and your own preferences, rank your selections according to the following criteria: price,
style/appearance, reliability/maintenance, engine size, gas mileage, safety, and features/options.
(You may need to access some additional references, such as Consumer Reports, to facilitate your
judgments.) After you have ranked your car selections using AHP, develop a scoring model to
perform the same analysis. Then compare the results of the two models and discuss which method
you prefer.
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