A company sells seven types of boxes, ranging in volume from 17 to 33 cubic feet. The demand and size of each box is given in Table 7. 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 a shortest-path problem whose solution will minimize the cost of meeting the demand for boxes. TABLE 7:

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

A library must build shelving to shelve 200 4-inch high books, 100 8-inch high books, and 80 12-inch high books. Each book is 0.5 inch thick. The library has several ways to store the books. For example, an 8-inch high shelf may be built to store all books of height less than or equal to 8 inches, and a 12-inch high shelf may be built for the 12-inch books. Alternatively, a 12-inch high shelf might be built to store all books. The library believes it costs $2,300 to build a shelf and that a cost of$5 per square inch is incurred for book storage. (Assume that the area required to store a book is given by height of storage area times book’s thickness.) Formulate and solve a shortest-path problem that could be used to help the library determine how to shelve the books at minimum cost.

At the beginning of year 1, a new machine must be purchased. The cost of maintaining a machine i years old is given in Table 5. TABLE 5:

$$\begin{matrix} \text{Age at Beginning of Year} & \text{Maintenance Cost for Next Year (\$)}\\ \text{0} & \text{38,000}\\ \text{1} & \text{50,000}\\ \text{2} & \text{97,000}\\ \text{3} & \text{182,000}\\ \text{4} & \text{304,000}\\ \end{matrix}$$

The cost of purchasing a machine at the beginning of each year is given in Table 6. TABLE 6:

$$\begin{matrix} \text{Year} & \text{Purchase Cost (\$)}\\ \text{1} & \text{170,000}\\ \text{2} & \text{190,000}\\ \text{3} & \text{210,000}\\ \text{4} & \text{250,000}\\ \text{5} & \text{300,000}\\ \end{matrix}$$

There is no trade-in value when a machine is replaced. Your goal is to minimize the total cost (purchase plus maintenance) of having a machine for five years. Determine the years in which a new machine should be purchased.

Candy Kane Cosmetics (CKC) produces Leslie Perfume, which requires chemicals and labor. Two production processes are available: Process 1 transforms 1 unit of labour and 2 units of chemicals into 3 oz of perfume. Process 2 transforms 2 units of labor and 3 units of chemicals into 5 oz of perfume. It costs CKC $3 to purchase a unit of labor and$2 to purchase a unit of chemicals. Each year, up to 20,000 units of labor and 35,000 units of chemicals can be purchased. In the absence of advertising, CKC believes it can sell 1,000 oz of perfume. To stimulate demand for Leslie, CKC can hire the lovely model Jenny Nelson. Jenny is paid $100/hour. Each hour Jenny worksfor the company is estimated to increase the demand for Leslie Perfume by 200 oz. Each ounce of Leslie Perfume sells for$5. Use linear programming to determine how CKC can maximize profits.

During each 4-hour period, the Smalltown police force requires the following number of on-duty police officers 12 midnight to 4 a.m. — 8; 4 to 8 a.m. —7; 8 a.m. to 12 noon—6; 12 noon to 4 p.m. —6; 4 to 8 p.m.—5; 8 p.m. to 12 midnight—4. Each police officer works two consecutive 4-hour shifts Formulate an LP that can be used to minimize the number of police officers needed to meet Smalltown’s daily requirements