The Gotham City Police Department employs 30 police officers. Each officer works 5 days per week. The crime rate fluctuates with the day of the week, so the number of police officers required each day depends on which day of the week it is: Saturday, 28; Sunday, 18; Monday, 18; Tuesday, 24. Wednesday, 25. Thursday, 16; Friday, 21. The police department wants to schedule police officers 10 minimize the number whose days off are not consecutive Formulate an LP that will accomplish this goal. (Hint Have a constraint for each day of the week that ensures that the proper number of officers are not working on the given day.)

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.

A company has nine projects under consideration. The NPV added by each project and the capital required by each project during the next two years is given in Table 9. All figures are in millions. For example. Project 1 will add $14 million in NPV and require expenditures of$12 million during year 1 and $3 million during year 2. Fifty million is available for projects during year 1 and$20 million is available during year 2. Assuming we may undertake a fraction of each project, how can we maximize NPV? TABLE 9:

$$\begin{matrix} \text{ } & \text{Project}\\ & \text{1} & \text{2} & \text{3} & \text{4} & \text{5} & \text{6} & \text{7} & \text{8} & \text{9}\\ \text{Year 1 Outflow} & \text{12} & \text{54} & \text{6} & \text{6} & \text{30} & \text{6} & \text{48} & \text{36} & \text{18}\\ \text{Year 2 Outflow} & \text{3} & \text{7} & \text{6} & \text{2} & \text{35} & \text{6} & \text{4} & \text{3} & \text{3}\\ \text{NPV} & \text{14} & \text{17} & \text{17} & \text{15} & \text{40} & \text{12} & \text{14} & \text{10} & \text{12}\\ \end{matrix}$$