An insurance company believes that it will require the following numbers of personal computers during the next six months: January, 9; February, 5; March, 7; April, 9; May, 10; June, 5. Computers can be rented for a period of one, two, or three months at the following unit rates: one-month rate, $\$200$; two-month rate, $\$350$; three-month rate, $\$450$. Formulate an LP that can be used to minimize the cost of renting the required computers. You may assume that if a machine is rented for a period of time extending beyond June, the cost of the rental should be prorated. For example, if a computer is rented for three months at the beginning of May, then a rental fee of $\frac{2}{3}(450)=\$ 300$, not $\$450$, should be assessed in the objective function.

Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that we win $2 for each black ball selected and we lose$1 for each white ball selected. Let X denote our winnings. What are the possible values of X, and what are the probabilities associated with each value?

Farmer Jones bakes two types of cake (chocolate and vanilla) to supplement his income. Each chocolate cake can be sold for $1, and each vanilla cake can be sold for 50¢. Each chocolate cake requires 20 minutes of baking time and uses 4 eggs. Each vanilla cake requires 40 minutes of baking time and uses 1 egg. Eight hours of baking time and 30 eggs are available Formulate an LP to maximize Farmer Jones’s revenue, then graphically solve the LP (A fractional number of cakes is okay.)

Carrington Oil produces two types of gasoline, gas 1 and gas 2, from two types of crude oil, crude 1 and crude 2. Gas 1 is allowed to contain up to 4% impurities, and gas 2 is allowed to contain up to 3% impurities. Gas 1 sells for $8 per barrel, whereas gas 2 sells for $12 per barrel. Up to 4,200 barrels of gas 1 and up to 4,300 barrels of gas 2 can be sold. The cost per barrel of each crude, availability, and the level of impurities in each crude arc as shown in Table 31. Before blending the crude oil into gas, any amount of each crude can be “purified” for a cost of $0.50 per barrel. Purification eliminates half the impurities in the crude oil. Determine how to maximize profit. Table 31:$

$$\begin{matrix} \text{Oil} & \text{Cost per Barrel}(\$) & \text{Impurity Level (\\\%)} & \text{Availability (Barrels)}\\ \text{Crude 1} & \text{6} & \text{10\\\%} & \text{5,000}\\ \text{Crude 2} & \text{8} & \text{2\\\%} & \text{4,500}\\ \end{matrix}$$

$

Each item produced by a certain manufacturer is independently, of acceptable quality with probability .95. Approximate the probability that at most 10 of the next 150 items produced are unacceptable.

Consider a collection of n individuals. Assume that each person’s birthday is equally likely to be any of the 365 days of the year and also that the birthdays are independent. Let

$$A _ { i , j } , i \neq j$$

, denote the event that persons i and j have the same birthday. Show that these events are pairwise independent, but not independent. That is, show that

$$A _ { i , j }$$

and

$$A _ { r , s }$$

are independent, but the

$$\left( \begin{array} { l } { n } \\ { 2 } \end{array} \right)$$

events

$$A _ { i , j } , i \neq j$$

are not independent.

Leary Chemical manufactures three chemicals: A, B, and C. These chemicals are produced via two production processes: 1 and 2. Running process 1 for an hour costs $4 and yields 3 units of A, 1 of B, and 1 of C. Running process 2 for an hour costs$1 and produces 1 unit of A and 1 of B. To meet customer demands, at least 10 units of A, 5 of B. and 3 of C must be produced daily. Graphically determine a daily production plan that minimizes the cost of meeting Leary Chemical’s daily demands.

Solve and check. How many square meters of contact paper are needed to cover a poster measuring 10 meters by 20 meters?

O.J. Juice Company sells bags of oranges and cartons of orange juice. O.J. grades oranges on a scale of 1 (poor) to 10 (excellent). O.J. now has on hand 100,000 lb of grade 9 oranges and 120,000 lb of grade 6 oranges. The average quality of oranges sold in bags must be at least 7, and the average quality of the oranges used to produce orange juice must be at least 8. Each pound of oranges that is used for juice yields a revenue of $1.50 and incurs a variable cost (consisting of labor costs, variable overhead costs, inventory costs, and so on) of$1.05. Each pound of oranges sold in bags yields a revenue of 50¢ and incurs a variable cost of 20¢. Formulate an LP to help O.J. maximize profit.

Emma wrote $8.4 x+6.3 x+12.6$as an equivalent expression for $4.2(2 x+1.5 x+3)$ . She said that her equivalent expression is simplified. Do you agree? Explain.

You have decided to enter the candy business. You are considering producing two types of candies: Slugger Candy and Easy Out Candy, both of which consist solely of sugar, nuts, and chocolate. At present, you have in stock 100 oz of sugar, 20 oz of nuts, and 30 oz of chocolate. The mixture used to make Easy Out Candy must contain at least 20% nuts. The mixture used to make Slugger Candy must contain at least 10% nuts and 10% chocolate. Each ounce of Easy Out Candy can be sold for 250¢, and each ounce of Slugger Candy for 20¢. Formulate an LP that will enable you to maximize your revenue from candy sales.

James Beerd bakes cheesecakes and Black Forest cakes. During any month, he can bake at most 65 cakes. The costs per cake and the demands for cakes, which must be met on time, are listed in Table 33. Itcosts500 to hold a cheesecake, and 40¢ to hold a Black Forest cake, in inventory for a month. Formulate an LP to minimize the total cost of meeting the next three months’ demands. Table 33: $$ \begin{matrix}\text{ } & \text{Month 1} & \text{ } & \text{Month 2} & \text{ } & \text{Month 3} & \text{ }\\\text{Item} & \text{Demand} & \text{Cost/Cake }($) & \text{Demand} & \text{Cost/Cake }($) & \text{Demand} & \text{Cost/Cake }($) \\\text{Cheesecake} & \text{40} & \text{3.00} & \text{30} & \text{3.40} & \text{20} & \text{3.80}\\\text{Black Forest} & \text{20} & \text{2.50} & \text{30} & \text{2.80} & \text{10} & \text{3.40}\\\end{matrix} $$

Each year, Paynothing Shoes faces demands (which must be met on time) for pairs of shoes as shown in Table 37. Workers work three consecutive quarters and then receive one quarter off. For example, a worker may work during quarters 3 and 4 of one year and quarter 1 of the next year. During a quarter in which a worker works, he or she can produce up to 50 pairs of shoes. Each worker is paid $500 per quarter. At the end of each quarter, a holding cost of$50 per pair of shoes is assessed. Formulate an LP that can be used to minimize the cost per year (labor + holding) of meeting the demands for shoes. To simplify matters, assume that at the end of each year, the ending inventory is zero. (Hint: It is allowable to assume that a given worker will get the same quarter off during each year.) Table 37:

$$\begin{matrix} \text{Quarter 1} & \text{Quarter 2} & \text{Quarter 3} & \text{Quarter 4}\\ \text{600} & \text{300} & \text{800} & \text{100}\\\end{matrix}$$

Winstonco is considering investing in three projects. If we fully invest in a project, the realized cash flows (in millions of dollars) will be as shown in Table 44. For example, project 1 requires cash outflow of $3 million today and returns$5.5 million 3 years from now. Today we have $2 million in cash. At each time point (0, .5, 1, 1.5, 2, and 2.5 years from today) we may, if desired, borrow up to$2 million at 3.5% (per 6 months) interest. Leftover cash earns 3% (per 6 months) interest. For example, if after borrowing and investing at time 0 we have $1 million we would receive$30,000 in interest at time .5 years. Winstonco’s goal is to maximize cash on hand after it accounts for time 3 cash flows. What investment and borrowing strategy should be used? Remember that we may invest in a fraction of a project. For example, if we invest in .5 of project 3, then we have cash outflows of −$1 million at time 0 and .5. Table 44:

$$\begin{matrix} \text{ } & \text{Cash Flow}\\ \text{Time (Years)} & \text{Project 1} & \text{Project 2} & \text{Project 3}\\ \text{0} & \text{−3} & \text{−2} & \text{−2}\\ \text{.5} & \text{−1} & \text{−.5} & \text{−2}\\ \text{1} & \text{+1.8} & \text{1.5} & \text{−1.8}\\ \text{1.5} & \text{1.4} & \text{1.5} & \text{1}\\ \text{2} & \text{1.8} & \text{1.5} & \text{1}\\ \text{2.5} & \text{1.8} & \text{.2} & \text{1}\\ \text{3} & \text{5.5} & \text{−1} & \text{6}\\ \end{matrix}$$