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:

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

Ashland Oil buys its crude oil in the market. Larger oil refiners, such as Texaco, drill for their own crude oil. Why do some oil companies drill for their own crude oil and others buy crude oil in the market?

Chemco produces two chemicals: A and B. These chemicals are produced via two manufacturing processes. Process 1 requites 2 hours of labor and 1 lb of raw material to produce 2 oz of A and 1 oz of B. Process 2 requires 3 hours of labor and 2 lb of raw material to produce 3 oz of A and 2 oz of B. Sixty hours of labor and 40 lb of raw material are available. Demand for A is unlimited, but only 20 oz of B can be sold. A sells for $16/oz, and B sells for$14/oz. Any B that is unsold must be disposed of at a cost of $2/oz. Formulate an LP to maximize Chemco’s revenue less disposal costs.

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.

Lizzie’s Dairy produces cream cheese and cottage cheese. Milk and cream are blended to produce these two products. Both high-fat and low-fat milk can be used to produce cream cheese and cottage cheese. High-fat milk is 60% fat; low-fat milk is 30% fat. The milk used to produce cream cheese must average at least 50% fat and that for cottage cheese, at least 35% fat. At least 40% (by weight) of the inputs to cream cheese and at least 20% (by weight) of the inputs to cottage cheese must be cream. Both cottage cheese and cream cheese are produced by putting milk and cream through the cheese machine. It costs 400 to process 1 lb of inputs into a pound of cream cheese. It costs 400 to produce 1 lb of cottage cheese, but every pound of input for cottage cheese yields 0.9 lb of cottage cheese and 0.1 lb of waste. Cream can be produced by evaporating high-fat and low-fat milk. It costs 400 to evaporate 1 lb of high-fat milk. Each pound of high-fat milk that is evaporated yields 0.6 lb of cream. It costs 400 to evaporate 1 lb of low-fat milk. Each pound of low-fat milk that is evaporated yields 0.3 lb of cream. Each day, up to 3,000 lb of input may be sent through the cheese machine. Each day, at least 1,000 lb of cottage cheese and 1,000 lb of cream cheese must be produced. Up to 1,500 lb of cream cheese and 2,000 lb of cottage cheese can be sold each day. Cottage cheese is sold for $1.20/lb and cream cheese for$1.50/lb. High-fat milk is purchased for 800/lb and low-fat milk for 400/lb. The evaporator can process at most 2,000 lb of milk daily. Formulate an LP that can be used to maximize Lizzie’s daily profit.

A company faces the following demands during the next three periods: period 1, 20 units; period 2, 10 units; period 3, 15 units. The unit production cost during each period is as follows: period 1—$13; period 2—$14; period 3—$15. A holding cost of$2 per unit is assessed against each period’s ending inventory. At the beginning of period 1, the company has 5 units on hand. In reality, not all goods produced during a month can be used to meet the current month’s demand. To model this fact, we assume that only one half of the goods produced during a period can be used to meet the current period’s demands. Formulate an LP to minimize the cost of meeting the demand for the next three periods. (Hint: Constraints such as i1 = x1 + 5 – 20 are certainly needed. Unlike our example, however, the constraint $i 1 \geq 0$ will not ensure that period 1’s demand is met. For example, if x1 = 20, then $i 1 \geq 0$ will hold, bur because only $\frac{1}{2}(20)=10$ units of period 1 production can be used to meet period 1’s demand, x1 = 20 would not be feasible. Try to think of a type of constraint that will ensure that what is available to meet each period’s demand is at least as large as that period’s demand.)

Each week Chemco can purchase unlimited quantities of raw material at $6/lb. Each pound of purchased raw material can be used to produce either input 1 or input 2. Each pound of raw material can yield 2 oz of input 1, requiring 2 hours of processing time and incurring$2 in processing costs. Each pound of raw material can yield 3 oz of input 2, requiring 2 hours of processing time and incurring $4 in processing costs. Two production processes arc available. It takes 2 hours to run process 1, requiring 2 oz of input 1 and 1 oz of input 2. It costs$1 to run process 1. Each time process 1 is run 1 oz of product A and 1 oz of liquid waste arc produced. Each time process 2 is run requires 3 hours of processing time, 2 oz of input 2 and 1 oz of input 1. Process 2 yields 1 oz of product B and .8 oz of liquid waste. Process 2 incurs $8 in costs. Chemco can dispose of liquid waste in the Port Charles River or use the waste to produce product C or product D. Government regulations limit the amount of waste Chemco is allowed to dump into the river to 1,000 oz/week. One ounce of product C costs$4 to produce and sells for $11. One hour of processing time, 2 oz of input 1, and .8 oz of liquid waste arc needed to produce an ounce of product C. One unit of product D costs$5 to produce and sells for $7. One hour of processing lime, 2 oz of input 2, and 1.2 oz of liquid waste arc needed to produce an ounce of product D.At most 5,000 oz of product A and 5,000 oz of product B can be sold each week, but weekly demand for products C and D is unlimited.Product A sells for$18/oz and product B sells for $24/oz. Each week 6,000 hours of processing time is available. Formulate an LP whose solution will tell Chemco how to maximize weekly profit.

The mayor of Llanview is trying to determine the number of judges needed to handle the judicial caseload. Table 84:

$$\begin{matrix} \text{Month} & \text{Hours}\\ \text{January} & \text{400}\\ \text{February} & \text{300}\\ \text{March} & \text{200}\\ \text{April} & \text{600}\\ \text{May} & \text{800}\\ \text{June} & \text{300}\\ \text{July} & \text{200}\\ \text{August} & \text{400}\\ \text{September} & \text{300}\\ \text{October} & \text{200}\\ \text{November} & \text{100}\\ \text{December} & \text{300}\\ \end{matrix}$$

During each month of the year it is estimated that the number of judicial hours needed is as given in Table 84. a. Each judge works all 12 months and can handle as many as 120 hours per month of casework. To avoid creating a backlog, all cases must be handled by the end of December. Formulate an LP whose solution will determine how many judges Llanview needs. b. If each judge received one month of vacation each year, how would year answer change?

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:

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

Furnco manufactures tables and chairs. A table requires 40 board ft of wood, and a chair requires 30 board ft of wood. Wood may be purchased at a cost of $1 per board ft, and 40,000 board ft of wood are available for purchase. It takes 2 hours of skilled labor to manufacture an unfinished table or an unfinished chair. Three more hours of skilled labor will turn an unfinished table into a finished table, and 2 more hours of skilled labor will turn an unfinished chair into a finished chair. A total of 6,000 hours of skilled labor are available (and have already been paid for). All furniture produced can be sold at the following unit prices: unfinished table,$70; finished tabic, $140; unfinished chair,$60; finished chair, $110. Formulate an LP that will maximize the contribution to profit from manufacturing tables and chairs.

A small logo is embedded in a thick block of crown glass (n = 1.52), 3.20 cm beneath the top surface of the glass. The block is put under water, so there is 1.50 cm of water above the top surface of the block. The logo is viewed from directly above by an observer in air. How far beneath the top surface of the water does the logo appear to be?

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.

Bullco blends silicon and nitrogen to produce two types of fertilizers. Fertilizer 1 must be at least 40% nitrogen and sells for $70/lb. Fertilizer 2 must be at least 70% silicon and sells for$40/lb. Bullco can purchase up to 80 lb of nitrogen at $15/lb and up to 100 lb of silicon at$10/lb. Assuming that all fertilizer produced can be sold, formulate an LP to help Bullco maximize profits.