The IRS has determined that during each of the next 12 months it will need the number of supercomputers given in Table 45. To meet these requirements, the IRS rents supercomputers for a period of one, two, or three months. It costs $100 to rent a supercomputer for one month,$180 for two months, and $250 for three months. At the beginning of month 1, the IRS has no supercomputers. Determine the rental plan that meets the next 12 months’ requirements at minimum cost. Note: You may assume that fractional rentals are okay, so if your solution says to rent 140.6 computers for one month we can round this up or down (to 141 or 140) without having much effect on the total cost. Table 45:

$$\begin{matrix} \text{Month} & \text{Computer Requirements}\\ \text{1} & \text{800}\\ \text{2} & \text{1,000}\\ \text{3} & \text{600}\\ \text{4} & \text{500}\\ \text{5} & \text{1,200}\\ \text{6} & \text{400}\\ \text{7} & \text{800}\\ \text{8} & \text{600}\\ \text{9} & \text{400}\\ \text{10} & \text{500}\\ \text{11} & \text{800}\\ \text{12} & \text{600}\\ \end{matrix}$$

Each hour from 10 a.m. to 7 p.m., Bank One receives checks and must process them. Its goal is to process all the checks the same day they are received The bank has 13 check-processing machines, each of which can process up to 500 checks per hour. It takes one worker to operate each machine. Bank One hires both full-time and part-time workers. Full-time workers work 10 a.m.–6 p.m., 11 a.m.– 7 p.m., or Noon–8 p.m. and are paid $160 per day. Part-time workers work either 2 p.m.–7 p.m. or 3 p.m.– 8 p.m. and are paid$75 per day. The number of checks received each hour is given in Table 6. In the interest of maintaining continuity. Bank One believes it must have at least three full-time workers under contract. Develop a cost-minimizing work schedule that processes all checks by 8 p.m. TABLE 6:

$$\begin{matrix} \text{Time} & \text{Checks Received}\\ \text{10 a.m.} & \text{5,000}\\ \text{11 a.m.} & \text{4,000}\\ \text{Noon} & \text{3,000}\\ \text{1 p.m.} & \text{4,000}\\ \text{2 p.m.} & \text{2,500}\\ \text{3 p.m.} & \text{3,000}\\ \text{4 p.m.} & \text{4,000}\\ \text{5 p.m.} & \text{4,500}\\ \text{6 p.m.} & \text{3,500}\\ \text{7 p.m.} & \text{3,000}\\ \end{matrix}$$

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 $0.50 per barrel. Purification eliminates half the impurities in the crude oil. Determine how to maximize profit. Table 31:$

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

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A small toy store, Toyco projects the monthly cash flows (in thousands of dollars) in Table 42 during the year 2003. A negative cash flow means that cash outflows exceed cash inflows to the business. To pay its bills, Toyco will need to borrow money early in the year. Money can be borrowed in two ways: a. Taking out a long-term one-year loan in January. Interest of 1% is charged each month, and the loan must be paid back at the end of December. b. Each month money can be borrowed from a short-term bank line of credit. Here, a monthly interest rate of 1.5% is charged. All short-term loans must be paid off at the end of December. At the end of each month, excess cash earns 0.4% interest. Formulate an LP whose solution will help Toyco maximize its cash position at the beginning of January, 2004. Table 42:

$$\begin{matrix} \text{Month} & \text{Cash Row} & \text{Month} & \text{Cash Flow}\\\text{January} & \text{−12} & \text{July} & \text{−7}\\ \text{February} & \text{−10} & \text{August} & \text{−2}\\ \text{March} & \text{−8} & \text{September} & \text{15}\\ \text{April} & \text{−10} & \text{October} & \text{12}\\ \text{May} & \text{−4} & \text{November} & \text{−7}\\ \text{June} & \text{5} & \text{December} & \text{45}\\ \end{matrix}$$

Par, Inc., produces a standard golf bag and a deluxe golf bag on a weekly basis. Each golf bag requires time for cutting and dyeing and time for sewing and finishing, as shown in the following table:

	HOURS REQUIRED PER BAG
PRODUCT	CUTTING AND DYEING	SEWING AND FINISHING
Standard bag	1/2	1
Deluxe bag	1	2/3

The profits per bag and weekly hours available for cutting and dyeing and for sewing and finishing are as follows:

PRODUCT	PROFIT PER UNIT (S)
Standard bag	10
Deluxe bag	8

ACTIVITY	WEEKLY HOURS AVAILABLE
Cutting and dyeing	300
Sewing and finishing	360

Par, Inc., will sell whatever quantities it produces of these two products.

What is the value of the profit?

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:

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

A company must meet (on time) the following demands: quarter 1—30 units; quarter 2—20 units; quarter 3—40 units. Each quarter, up to 27 units can be produced with regular-time labor, at a cost of $40 per unit. During each quarter, an unlimited number of units can be produced with overtime labor, at a cost of$60 per unit. Of all units produced, 20% arc unsuitable and cannot be used to meet demand. Also, at the end of each quarter, 10% of all units on hand spoil and cannot be used to meet any future demands. After each quarter’s demand is satisfied and spoilage is accounted for, a cost of $ 15 per unit is assessed against the quarter’s ending inventory. Formulate an LP that can be used to minimize the total cost of meeting the next three quarters’ demands. Assume that 20 usable units are available at the beginning of quarter 1.

Donovan Enterprises produces electric mixers. During the next four quarters, the following demands for mixers must be met on time: quarter 1—4,000; quarter 2—2,000; quarter 3—3,000; quarter 4—10,000. Each of Donovan’s workers works three quarters of the year and gets one quarter off. Thus, a worker may work during quarters 1, 2, and 4 and get quarter 3 off. Each worker is paid $30,000 per year and (if working) can produce up to 500 mixers during a quarter. At the end of each quarter, Donovan incurs a holding cost of$30 per mixer on each mixer in inventory. Formulate an LP to help Donovan minimize the cost (labor and inventory) of meeting the next year’s demand (on time). At the beginning of quarter 1, 600 mixers are available.

If all life were destroyed on Earth by a large impact, would new life eventually form to take its place? Explain how conditions would have to change for life to start again on our planet.

Solve each equation for exact solutions over the interval $[0,2 \pi)$. cos 2x+cos x=0

You arc a CFA (chartered financial analyst). Madonna has come to you because she needs help paying off her credit card bills. She owes the amounts on her credit cards shown in Table 43. Madonna is willing to allocate up to $5,000 per month to pay off these credit cards. All cards must be paid off within 36 months. Madonna’s goal is to minimize the total of all her payments. To solve this problem, you must understand how interest on a loan works. To illustrate, suppose Madonna pays$5,000 on Saks during month 1. Then her Saks balance at the beginning of month 2 is 20,000 - (5,000 - .005(20,000)) This follows because during month 1 Madonna incurs .005(20,000) in interest charges on her Saks card. Help Madonna solve her problems! Table 43: $$ \begin{matrix} \text{Card} & \text{Balance }($) & \text{Monthly Rate (\\%)}\\ \text{Saks Fifth Avenue} & \text{20,000} & \text{.5}\\ \text{Bloomingdale’s} & \text{50,000} & \text{1}\\ \text{Macys} & \text{40,000} & \text{1.5}\\ \end{matrix} $$

Sunco Oil has three different processes that can be used to manufacture various types of gasoline. Each process involves blending oils in the company’s catalytic cracker. Running proccss 1 for an hour costs $5 and requires 2 barrels of crude oil 1 and 3 barrels of crude oil 2. The output from running process 1 for an hour is 2 barrels of gas 1 and 1 barrel of gas 2. Running process 2 for an hour costs$4 and requires 1 barrel of crude 1 and 3 barrels of crude 2. The output from running process 2 for an hour is 3 barrels of gas 2. Running process 3 for an hour costs $1 and requires 2 barrels of crude 2 and 3 barrels of gas 2. The output from running process 3 for an hour is 2 barrels of gas 3. Each week, 200 barrels of crude 1, at$2/barrel, and 300 barrels of crude 2, at $3/barrel, may be purchased. All gas produced can be sold at the following per-barrel prices: gas 1,$9; gas 2, $10; gas 3,$24. Formulate an LP whose solution will maximize revenues less costs. Assume that only 100 hours of time on the catalytic cracker are available each week.

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: $$ \begin{matrix}\text{ } & \text{Month 1} & \text{ } & \text{Month 2} & \text{ } & \text{Month 3} & \text{ }\\\text{Item} & \text{Demand} & \text{Cost/Cake }($) & \text{Demand} & \text{Cost/Cake }($) & \text{Demand} & \text{Cost/Cake }($) \\\text{Cheesecake} & \text{40} & \text{3.00} & \text{30} & \text{3.40} & \text{20} & \text{3.80}\\\text{Black Forest} & \text{20} & \text{2.50} & \text{30} & \text{2.80} & \text{10} & \text{3.40}\\\end{matrix} $$

Broker Steve Johnson is currently trying to maximize his profit in the bond market. Four bonds are available for purchase and sale, with the bid and ask pricc of each bond as shown in Table 40. Steve can buy up to 1,000 units of each bond at the ask price or sell up to 1,000 units of each bond at the bid price. During each of the next three years, the person who sells a bond will pay the owner of the bond the cash payments shown in Table 41. Steve’s goal is to maximize his revenue from selling bonds less his payment for buying bonds, subject to the constraint that after each year’s payments are received, his current cash position (due only to cash payments from bonds and not purchases or sale of bonds) is nonnegative. Assume that cash payments are discounted, with a payment of $1 one year from now being equivalent to a payment of 90¢ now. Formulate an LP to maximize net profit from buying and selling bonds, subject to the arbitrage constraints previously described. Why do you think we limit the number of units of each bond that can be bought or sold? Table 40:

$$\begin{matrix} \text{Bond} & \text{Bid Price} & \text{Ask Price}\\ \text{1} & \text{980} & \text{990}\\ \text{2} & \text{970} & \text{985}\\ \text{3} & \text{960} & \text{972}\\ \text{4} & \text{940} & \text{954}\\ \end{matrix}$$

Table 41:

$$\begin{matrix} \text{Year} & \text{Bond 1} & \text{Bond 2} & \text{Bond 3} & \text{Bond 4}\\ \text{1} & \text{100} & \text{80} & \text{70} & \text{60}\\ \text{2} & \text{110} & \text{90} & \text{80} & \text{50}\\ \text{3} & \text{1,100} & \text{1,120} & \text{1,090} & \text{1,110}\\ \end{matrix}$$