Ford has four automobile plants. Each is capable of producing the Taurus, Lincoln, or Escort, but it can only produce one of these cars. The fixed cost of operating each plant for a year and the variable cost of producing a car of each type at each plant are given in Table 51. Ford faces the following restrictions: a. Each plant can produce only one type of car. b. The total production of each type of car must be at a single plant; that is, for example, if any Tauruses are made at plant 1, then all Tauruses must be made there. c. If plants 3 and 4 are used, then plant 1 must also be used. Each year, Ford must produce 500,000 of each type of car. Formulate an IP whose solution will tell Ford how to minimize the annual cost of producing cars. TABLE 51:

$$\begin{matrix} \text{Plant} & \text{Fixed Cost (\$)} & \text{Variable Cost (\$)}\\ \text{ } & \text{ } & \text{Taurus} & \text{Lincoln} & \text{Escort}\\ \text{1} & \text{7 billion} & \text{12,000} & \text{16,000} & \text{9,000}\\ \text{2} & \text{6 billion} & \text{15,000} & \text{18,000} & \text{11,000}\\ \text{3} & \text{4 billion} & \text{17,000} & \text{19,000} & \text{12,000}\\ \text{4} & \text{2 billion} & \text{19,000} & \text{22,000} & \text{14,000}\\ \end{matrix}$$

Assume that females have pulse rates that are normally distributed with a mean of 7 4. 0 beats per minute and a standard deviation of 12.5 beats per minute. If 4 adult females are randomly selected, find the probability that they have pulse rates with a mean between 72 beats per minute and 76 beats per minute.

New York City has 10 trash districts and is trying to determine which of the districts should be a site for dumping trash. It costs $1,000 to haul one ton of trash one mile. The location of each district, the number of tons of trash produced per year by the district, the annual fixed cost (in millions of dollars) of running a dumping site, and the variable cost (per ton) of processing a ton of trash at a site are shown in Table 112. For example, district 3 is located at coordinates (10,8). District 3 produces 555 tons of trash a year, and it costs$1 million per year in fixed costs to operate a dump site in district 3. Each ton of trash processed at site 3 incurs $51 in variable costs. Each dump site can handle at most 1,500 tons of trash. Each district must send all its trash to a single site. Determine how to locate the dump sites in order to minimize total cost per year. TABLE 112:

$$\begin{matrix} \text{District} & \text{Coordinates} & \text{ } & \text{Cost (\$ Millions)}\\ \text{ } & \text{x} & \text{y} & \text{Tons} & \text{Fixed} & \text{Variable}\\ \text{1} & \text{4} & \text{3} & \text{49} & \text{2} & \text{310}\\ \text{2} & \text{2} & \text{5} & \text{874} & \text{1} & \text{40}\\ \text{3} & \text{10} & \text{8} & \text{555} & \text{1} & \text{51}\\ \text{4} & \text{2} & \text{8} & \text{352} & \text{1} & \text{341}\\ \text{5} & \text{5} & \text{3} & \text{381} & \text{3} & \text{131}\\\text{6} & \text{4} & \text{5} & \text{428} & \text{2} & \text{182}\\ \text{7} & \text{10} & \text{5} & \text{985} & \text{1} & \text{20}\\ \text{8} & \text{5} & \text{1} & \text{105} & \text{2} & \text{40}\\ \text{9} & \text{5} & \text{8} & \text{258} & \text{4} & \text{177}\\\text{10} & \text{1} & \text{7} & \text{210} & \text{2} & \text{75}\\ \end{matrix}$$

Cousin Bruzie of radio station WABC schedules radio commercials in 60-second blocks. This hour, the station has sold commercial time for commercials of 15, 16, 20, 25, 30, 35, 40, and 50 seconds. Formulate an integer programming model that can be used to determine the minimum number of 60-second blocks of commercials that must be scheduled to fit in all the current hour’s commercials.

A power plant has three boilers. If a given boiler is operated, it can be used to produce a quantity of steam (in tons) between the minimum and maximum given in Table 34. The cost of producing a ton of steam on each boiler is also given. Steam from the boilers is used to produce power on three turbines. If operated, each turbine can process an amount of steam (in tons) between the minimum and maximum given in Table 35. Thecost of processing a ton of steam and the power produced by each turbine is also given. Formulate an IP that can be used to minimize the cost of producing 8,000 kwh of power. TABLE 34:

$$\begin{matrix} \text{Boiler Number} & \text{Minimum Steam} & \text{Maximum Steam} & \text{Cost/Ton (\$)}\\ \text{1} & \text{500} & \text{1,000} & \text{10}\\ \text{2} & \text{300} & \text{900} & \text{8}\\\text{3} & \text{400} & \text{800} & \text{6}\\ \end{matrix}$$

TABLE 35:

$$\begin{matrix} \text{Turbine Number} & \text{Minimum} & \text{Maximum} & \text{Kwh per Ton of Steam} & \text{Processing Cost per Ton (\$)}\\ \text{1} & \text{300} & \text{600} & \text{4} & \text{2}\\ \text{2} & \text{500} & \text{800} & \text{5} & \text{3}\\ \text{3} & \text{600} & \text{900} & \text{6} & \text{4}\\ \end{matrix}$$

A consulting company has 10 employees, each of whom can work on at most two team projects. Six projects are under consideration. Each project requires 4 of our 10 workers. The required workers and the revenue earned from each project are shown in Table 109. Each worker who is used on any project must be paid the retainer shown in Table 110. Finally, each worker on a project is paid the project fee shown in Table 111. How can we maximize our profit? TABLE 109:

$$\begin{matrix} \text{Project} & \text{Required Workers} & \text{Revenue (\$)}\\ \text{1} & \text{1,4,5,8} & \text{10,000}\\ \text{2} & \text{2,3,7,10} & \text{15,000}\\ \text{3} & \text{1,6,8,9} & \text{6,000}\\\text{4} & \text{2,3,5,10} & \text{8,000}\\ \text{5} & \text{1,6,7,9} & \text{12,000}\\ \text{6} & \text{2,4,8,10} & \text{9,000}\\ \end{matrix}$$

TABLE 110:

$$\begin{matrix} \text{ } & \text{Worker}\\ \text{ } & \text{1} & \text{2} & \text{3} & \text{4} & \text{5} & \text{6} & \text{7} & \text{8} & \text{9} & \text{10}\\\text{Retainer (\$)} & \text{800} & \text{500} & \text{600} & \text{700} & \text{800} & \text{600} & \text{400} & \text{500} & \text{400} & \text{500}\\ \end{matrix}$$

TABLE 111:

$$\begin{matrix} \text{ } & \text{Project}\\\text{ } & \text{1} & \text{2} & \text{3} & \text{4} & \text{5} & \text{6}\\ \text{Fee (\$)} & \text{250} & \text{300} & \text{250} & \text{300} & \text{175} & \text{180}\\ \end{matrix}$$

A company sells seven types of boxes, ranging in volume from 17 to 33 cubic feet. The demand and size of each box are given in Table 39. The variable cost (in dollars) of producing each box is equal to the box’s volume. A fixed cost of $1,000 is incurred to produce any of a particular box. If the company desires, demand for a box may be satisfied by a box of larger size. Formulate and solve (with LINDO, LINGO, or Excel Solver) an IP whose solution will minimize the cost of meeting the demand for boxes. TABLE 39:

$$\begin{matrix} \text{ } & \text{Box}\\ \text{ } & \text{1} & \text{2} & \text{3} & \text{4} & \text{5} & \text{6} & \text{7}\\ \text{Size} & \text{33} & \text{30} & \text{26} & \text{24} & \text{19} & \text{18} & \text{17}\\ \text{Demand} & \text{400} & \text{300} & \text{500} & \text{700} & \text{200} & \text{400} & \text{200}\\ \end{matrix}$$

Four trucks are available to deliver milk to five groceries. The capacity and daily operating cost of each truck are shown in Table 107. The demand of each grocery store can be supplied by only one truck, but a truck may deliver to more than one grocery. The daily demands of each grocery are as follows: grocery 1, 100 gallons; grocery 2,200 gallons; grocery 3, 300 gallons; grocery 4, 500 gallons; grocery 5, 800 gallons. Formulate an IP that can he used to minimize the daily cost of meeting the demands of the four groceries. TABLE 107:

$$\begin{matrix} \text{Truck} & \text{Capacity (Gallons)} & \text{Daily Operating Cost (\$)}\\ \text{1} & \text{400} & \text{45}\\ \text{2} & \text{500} & \text{50}\\\text{3} & \text{600} & \text{55}\\ \text{4} & \text{1,100} & \text{60}\\\end{matrix}$$

Hallco runs a day shift and a night shift. No matter how many units are produced, the only production cost during a shift is a setup cost. It costs $8,000 to run the day shift and$4,500 to run the night shift. Demand for the next two days is as follows: day 1, 2,000; night 1, 3,000; day 2, 2,000; night 2, 3,000. It costs $ 1 per unit to hold a unit in inventory for a shift. Determine a production schedule that minimizes the sum of setup and inventory costs. All demand must be met on time.

Eastinghouse ships 12,000 capacitors per month to their customers. The capacitors may be produced at three different plants. The production capacity, fixed monthly cost of operation, and variable cost of producing a capacitor at each plant are given in Table 96. The fixed cost for a plant is incurred only if the plant is used to make any capacitors Formulate an integer programming model whose solution will tell Eastinghouse how to minimize their monthly costs of meeting their customers’ demands. TABLE 96:

$$\begin{matrix} \text{Plant} & \text{Fixed Cost (in \$ Thousands)} & \text{Variable Cost (\$)} & \text{Production Capacity}\\ \text{1} & \text{80} & \text{20} & \text{6,000}\\ \text{2} & \text{40} & \text{25} & \text{7,000}\\ \text{3} & \text{30} & \text{30} & \text{6,000}\\ \end{matrix}$$

Harold sorted his jellybeans into two jars. He likes the purple ones best and the black ones next best, so these are both in one jar. His next favorites are yellow, orange , and white, and these are in another jar. He gave all the rest to his little sister. Harold allows himself to eat only one jellybean from each jar per day. He wears a blindfold when he selects his jellybeans so he cannot choose his favorites first. Make an area model that represents the probabilities. What is the probability that Harold gets one black jellybean and one orange jellybean if the first jar has 60% black and 40% purple jellybeans, and the second jar has 30% yellow, 50% orange, and 20% white jellybeans?

At Blair General Hospital, six types of surgical operations are performed. The types of operations each surgeon is qualified to perform (indicated by an X) are given in Table 95. Suppose that surgeon 1 and surgeon 2 dislike each other and cannot be on duty at the same time. Formulate an IP whose solution will determine the minimum number of surgeons required so that the hospital can perform all types of surgery. TABLE 95:

$$\begin{matrix} \text{Surgeon} & \text{Operation}\\ \text{ } & \text{1} & \text{2} & \text{3} & \text{4} & \text{5} & \text{6}\\ \text{1} & \text{x} & \text{x} & \text{ } & \text{X} & \text{ } & \text{ }\\ \text{2} & \text{ } & \text{ } & \text{x} & \text{ } & \text{x} & \text{x}\\ \text{3} & \text{ } & \text{ } & \text{x} & \text{ } & \text{x} & \text{ }\\ \text{4} & \text{x} & \text{ } & \text{ } & \text{ } & \text{ } & \text{x}\\ \text{5} & \text{ } & \text{x} & \text{ } & \text{ } & \text{ } & \text{ }\\ \text{6} & \text{ } & \text{ } & \text{ } & \text{X} & \text{x} & \text{ }\\ \end{matrix}$$

You have been assigned to arrange the songs on the cassette version of Madonna’s latest album. A cassette tape has two sides (1 and 2). The songs on each side of the cassette must total between 14 and 16 minutes in length. The length and type of each song are given in Table 28. The assignment of songs to the tape must satisfy the following conditions: 1. Each side must have exactly two ballads. 2. Side 1 must have at least three hit songs. 3. Either song 5 or song 6 must be on side 1. 4. If songs 2 and 4 are on side 1, then song 5 must be on side 2. Explain how you could use an integer programming formulation to determine whether there is an arrangement of songs satisfying these restrictions. Table 28:

$$\begin{matrix} \text{Song} & \text{Type} & \text{Length (in minutes)}\\ \text{1} & \text{Ballad} & \text{4}\\ \text{2} & \text{Hit} & \text{5}\\ \text{3} & \text{Ballad} & \text{3}\\ \text{4} & \text{Hit} & \text{2}\\\text{5} & \text{Ballad} & \text{4}\\ \text{6} & \text{Hit} & \text{3}\\ \text{7} & \text{ } & \text{5}\\ \text{8} & \text{Ballad and hit} & \text{4}\\ \end{matrix}$$

Let

$$f ( x ) = x ^ { 2 } + 3 x.$$

Find a formula for

$$L_N$$

for f on [0, 2] and compute

$$\int _ { 0 } ^ { 2 } f ( x ) d x$$

by taking the limit.