PhotoTech, Inc., a manufacturer of rechargeable batteries for digital cameras, signed a contract with a digital photography company to produce three models of lithium-ion battery packs for a new line of digital cameras. The contract calls for the following:

$$\begin{array}{cc} \text { Battery Pack } & \text { Production Quantity } \\ \text { PT-100 } & 200,000 \\ \text { PT-200 } & 100,000 \\ \text { PT-300 } & 150,000 \end{array}$$

PhotoTech can manufacture the battery packs at manufacturing plants located in the Philippines and Mexico. The unit cost of the battery packs differs at the two plants because of differences in production equipment and wage rates. The unit costs for each battery pack at each manufacturing plant are as follows:

$$\begin{array}{ccc} \text { Product } & \text { Philippines } & \text { Mexico } \\ \text { PT-100 } & \$ 0.95 & \$ 0.98 \\ \text { PT-200 } & \$ 0.98 & \$ 1.06 \\ \text { PT-300 } & \$ 1.34 & \$ 1.15 \end{array}$$

The PT-100 and PT-200 battery packs are produced using similar production equipment available at both plants. However, each plant has a limited capacity for the total number of PT-100 and PT-200 battery packs produced. The combined PT-100 and PT-200 production capacities are 175,000 units at the Philippines plant and 160,000 units at the Mexico plant. The PT-300 production capacities are 75,000 units at the Philippines plant and 100,000 units at the Mexico plant. The cost of shipping from the Philippines plant is $0.18 per unit, and the cost of shipping from the Mexico plant is$0.10 per unit.

a. Develop a linear program that PhotoTech can use to determine how many units of each battery pack to produce at each plant to minimize the total production and shipping cost associated with the new contract.

The profit p (in hundreds of dollars) that a company makes depends on the amount x (in hundreds of dollars) the company spends on advertising according to the model P =230 + 20x - 0.5x^2. What expenditure for advertising will yield a maximum profit?

"Vollmer Manufacturing makes three components for sale to refrigeration companies. The components are processed on two machines: a shaper and a grinder. The times (in minutes) required on each machine are as follows:

$$\begin{aligned} &\begin{array}{ccc} \text { Component } & \text { Shaper } & \text { Grinder } \\ 1 & 6 & 4 \\ 2 & 4 & 5 \\ 3 & 4 & 2 \end{array} \end{aligned}$$

The shaper is available for 120 hours, and the grinder is available for 110 hours. No more than 200 units of component 3 can be sold, but up to 1000 units of each of the other components can be sold. In fact, the company already has orders for 600 units of component 1 that must be satisfied. The profit contributions for components 1, 2, and 3 are $8,$6, and $9, respectively. a. Formulate and solve for the recommended production quantities.

If more units of component 3 can be sold by reducing the sales price by$4, should the company reduce the price?"

"Vollmer Manufacturing makes three components for sale to refrigeration companies. The components are processed on two machines: a shaper and a grinder. The times (in minutes) required on each machine are as follows:

$$\begin{aligned} &\begin{array}{ccc} \text { Component } & \text { Shaper } & \text { Grinder } \\ 1 & 6 & 4 \\ 2 & 4 & 5 \\ 3 & 4 & 2 \end{array} \end{aligned}$$

The shaper is available for 120 hours, and the grinder is available for 110 hours. No more than 200 units of component 3 can be sold, but up to 1000 units of each of the other components can be sold. In fact, the company already has orders for 600 units of component 1 that must be satisfied. The profit contributions for components 1, 2, and 3 are $8,$6, and $9, respectively. a. Formulate and solve for the recommended production quantities.

If more time could be made available on the grinder, how much would it be worth?"

The Clark County Sheriff’s Department schedules police officers for 8-hour shifts. The beginning times for the shifts are 8:00 a.m., noon, 4:00 p.m., 8:00 p.m., midnight, and 4:00 a.m. An officer beginning a shift at one of these times works for the next 8 hours. During normal weekday operations, the number of officers needed varies depending on the time of day. The department staffing guidelines require the following minimum number of officers on duty:

$$\begin{aligned} &\text { Minimum Officers }\\ &\begin{array}{lc} \text { Time of Day } & \text { On Duty } \\ \text { 8:00 a.m.-Noon } & 5 \\ \text { Noon-4:00 p.m. } & 6 \\ \text { 4:00 p.m.-8:00 p.m. } & 10 \\ \text { 8:00 p.m.-Midnight } & 7 \\ \text { Midnight-4:00 a.m. } & 4 \\ \text { 4:00 a.m.-8:00 a.m. } & 6 \end{array} \end{aligned}$$

Determine the number of police officers that should be scheduled to begin the 8-hour shifts at each of the six times (8:00 a.m., noon, 4:00 p.m., 8:00 p.m., midnight, and 4:00 a.m.) to minimize the total number of officers required. (Hint: Let $x_1$ = the number of officers beginning work at 8:00 a.m., $x_2$ = the number of officers beginning work at noon, and so on.)

"Vollmer Manufacturing makes three components for sale to refrigeration companies. The components are processed on two machines: a shaper and a grinder. The times (in minutes) required on each machine are as follows:

$$\begin{aligned} &\begin{array}{ccc} \text { Component } & \text { Shaper } & \text { Grinder } \\ 1 & 6 & 4 \\ 2 & 4 & 5 \\ 3 & 4 & 2 \end{array} \end{aligned}$$

The shaper is available for 120 hours, and the grinder is available for 110 hours. No more than 200 units of component 3 can be sold, but up to 1000 units of each of the other components can be sold. In fact, the company already has orders for 600 units of component 1 that must be satisfied. The profit contributions for components 1, 2, and 3 are $8,$6, and $9, respectively. a. Formulate and solve for the recommended production quantities.

What are the right-hand-side ranges? Interpret these ranges for company management."

"Adirondack Savings Bank (ASB) has $1 million in new funds that must be allocated to home loans, personal loans, and automobile loans. The annual rates of return for the three types of loans are 7 percent for home loans, 12 percent for personal loans, and 9 percent for automobile loans. The bank’s planning committee has decided that at least 40 percent of the new funds must be allocated to home loans. In addition, the planning committee has specified that the amount allocated to personal loans cannot exceed 60 percent of the amount allocated to automobile loans.

Assume that ASB has the original$1 million in new funds available and that the planning committee has agreed to relax the requirement that at least 40 percent of the new funds must be allocated to home loans by 1 percent. How much would the annual return change? How much would the annual percentage return change?"

Kelson Sporting Equipment, Inc., makes two types of baseball gloves: a regular model and a catcher’s model. The firm has 900 hours of production time available in its cutting and sewing department, 300 hours available in its finishing department, and 100 hours available in its packaging and shipping department. The production time requirements and the profit contribution per glove are given in the following table:

$$\begin{array}{lcccc} \text { Model } & \begin{array}{c} \text { Cutting } \\ \text { and Sewing } \end{array} & \text { Finishing } & \begin{array}{c} \text { Packaging } \\ \text { and Shipping } \end{array} & \text { Profit/Glove } \\ \text { Regular model } & 1 & 1 / 2 & 1 / 8 & \$ 5 \\ \text { Catcher's model } & 3 / 2 & 1 / 3 & 1 / 4 & \$ 8 \end{array}$$

What is the linear programming model for this problem?

"Vollmer Manufacturing makes three components for sale to refrigeration companies. The components are processed on two machines: a shaper and a grinder. The times (in minutes) required on each machine are as follows:

$$\begin{aligned} &\begin{array}{ccc} \text { Component } & \text { Shaper } & \text { Grinder } \\ 1 & 6 & 4 \\ 2 & 4 & 5 \\ 3 & 4 & 2 \end{array} \end{aligned}$$

The shaper is available for 120 hours, and the grinder is available for 110 hours. No more than 200 units of component 3 can be sold, but up to 1000 units of each of the other components can be sold. In fact, the company already has orders for 600 units of component 1 that must be satisfied. The profit contributions for components 1, 2, and 3 are $8,$6, and $9, respectively. a. Formulate and solve for the recommended production quantities.

What are the objective coefficient ranges for the three components? Interpret these ranges for company management."

Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows:

$$\begin{aligned} &\text { Rental Class }\\ &\begin{array}{lccc} \text { Room } & \text { Super Saver } & \text { Deluxe } & \text { Business } \\ \text { Type I } & \$ 30 & \$ 35 & - \\ \text { Type II } & \$ 20 & \$ 30 & \$ 40 \end{array} \end{aligned}$$

Type I rooms do not have wireless Internet access and are not available for the Business rental class. Round Tree’s management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is 130 rentals in the Super Saver class, 60 rentals in the Deluxe class, and 50 rentals in the Business class. Round Tree has 100 Type I rooms and 120 Type II rooms.

Management is considering offering a free breakfast to anyone upgrading from a Super Saver reservation to Deluxe class. If the cost of the breakfast to Round Tree is $5, should this incentive be offered?

Vollmer Manufacturing makes three components for sale to refrigeration companies. The components are processed on two machines: a shaper and a grinder. The times (in minutes) required on each machine are as follows:

$$\begin{aligned} &\begin{array}{ccc} \text { Component } & \text { Shaper } & \text { Grinder } \\ 1 & 6 & 4 \\ 2 & 4 & 5 \\ 3 & 4 & 2 \end{array} \end{aligned}$$

The shaper is available for 120 hours, and the grinder is available for 110 hours. No more than 200 units of component 3 can be sold, but up to 1000 units of each of the other components can be sold. In fact, the company already has orders for 600 units of component 1 that must be satisfied. The profit contributions for components 1, 2, and 3 are $8,$6, and $9, respectively. a. Formulate and solve for the recommended production quantities.

Formulate and solve for the recommended production quantities.

The University of Cincinnati Center for Business Analytics is an outreach center that collaborates with industry partners on applied research and continuing education in business analytics. One of the programs offered by the Center is a quarterly Business Intelligence Symposium. Each symposium features three speakers on the real-world use of analytics. Each corporate member of the center (there are currently ten) receives five free seats to each symposium. Nonmembers wishing to attend must pay $75 per person. Each attendee receives breakfast, lunch, and free parking. The following are the costs incurred for putting on this event:

Rental cost for the auditorium$150

Registration processing $8.50 per person

Speaker costs 3@$800 = $2,400

Continental breakfast$4.00 per person

Lunch $7.00 per person

Parking$5.00 per person

Use Goal Seek to find the number of nonmember registrants that will make the event break even.

The University of Cincinnati Center for Business Analytics is an outreach center that collaborates with industry partners on applied research and continuing education in business analytics. One of the programs offered by the Center is a quarterly Business Intelligence Symposium. Each symposium features three speakers on the real-world use of analytics. Each corporate member of the center (there are currently ten) receives five free seats to each symposium. Nonmembers wishing to attend must pay $75 per person. Each attendee receives breakfast, lunch, and free parking. The following are the costs incurred for putting on this event:

Rental cost for the auditorium$150

Registration processing $8.50 per person

Speaker costs 3@$800 = $2,400

Continental breakfast$4.00 per person

Lunch $7.00 per person

Parking$5.00 per person

Build a spreadsheet model that calculates a profit or loss based on the number of nonmember registrants.

The Transylvania Olympic Gymnastics Team consists of six people. Transylvania must choose three people to enter both the balance beam and floor exercises. They must also enter a total of four people in each event. The score that each individual gymnast can attain in each event is shown in Table 89. Formulate an IP to maximize the total score attained by the Transylvania gymnasts. TABLE 89:

$$\begin{matrix} \text{Gymnast} & \text{Balance Beam} & \text{Floor Exercise}\\ \text{1} & \text{8.8} & \text{7.9}\\ \text{2} & \text{9.4} & \text{8.3}\\ \text{3} & \text{9.2} & \text{8.5}\\ \text{4} & \text{7.5} & \text{8.7}\\ \text{5} & \text{8.7} & \text{8.1}\\ \text{6} & \text{9.1} & \text{8.6}\\ \end{matrix}$$