# Management Science

1- A USAID contractor produces two sizes of cans-regular and large- The cans are produced in 10,000-can lots- The cans are processed through
a stamping operation and a coating operation- The company has 30 days available for both stamping and coating- A lot of regular-size cans
requires 2 days to stamp and 4 days to coat, whereas a lot of large cans requires 4 days to stamp and 2 days to coat- A lot of regular-size cans
earns \$800 profit, and a lot of large-size cans earns \$900 profit- In order to fulfill its obligations under a shipping contract, the company must
produce at least nine lots- The company wants to determine the number of lots to produce of each size can (x1 and x2) in order to maximize profit-
a- Formulate a linear programming model for this problem-

b- Solve this model by using graphical analysis-

2- The admissions office at Tech wants to determine how many in-state and how many out-of-state students to accept for next fall’s entering
freshman class- Tuition for an in-state student is \$7,600 per year, whereas out-of-state tuition is \$22,500 per year- A total of 12,800 in-state and
8,100 out-of-state freshmen have applied for next fall, and Tech does not want to accept more than 3,500 students- However, because Tech is a
state institution, the state mandates that it can accept no more than 40% out-of-state students- From past experience the admissions office knows
that 12% of in-state students and 24% of out-of-state students will drop out during their first year- Tech wants to maximize total tuition while limiting
the total attrition to 600 first-year students-

a- Formulate a linear programming model for this problem-

b- Solve this model by using graphical analysis-

3- The Metropolitan Police Department was recently criticized in the local media for not responding to police calls in the downtown area rapidly
enough- In several recent cases, alarms had sounded for break-ins, but by the time the police car arrived, the perpetrators had left, and in one
instance a store owner had been shot- Sergeant Joe Davis was assigned by the chief as head of a task force to find a way to determine the optimal
patrol area (dimensions) for their cars that would minimize the average time it took to respond to a call in the downtown area-

Sergeant Davis solicited help from Angela Maris, an analyst in the operations area for the police department- Together they began to work through
the problem- Joe noted to Angela that normal patrol sectors are laid out in rectangles, with each rectangle including a number of city blocks- For
illustrative purposes he defined the dimensions of the sector as x in the horizontal direction and as y in the vertical direction- He explained to Angela
that cars traveled in straight lines either horizontally or vertically and turned at right angles- Travel in a horizontal direction must be accompanied by
travel in a vertical direction, and the total distance traveled is the sum of the horizontal and vertical segments- He further noted that past research
on police patrolling in urban areas had shown that the average distance traveled by a patrol car responding to a call in either direction was one-third
of the dimensions of the sector, or x>3 and y>3- He also explained that the travel time it took to respond to a call (assuming that a car left
immediately upon receiving the call) is simply the average distance traveled divided by the average travel speed-

Angela told Joe that now that she understood how average travel time to a call was determined, she could see that it was closely related to the size
of the patrol area- She asked Joe if there were any restrictions on the size of the area sectors that cars patrolled- He responded that for their city,
the department believed that the perimeter of a patrol sector should not be less than 5 miles or exceed 12 miles- He noted several policy issues and
staffing constraints that required these specifications- Angela wanted to know if any additional restrictions existed, and Joe indicated that the

distance in the vertical direction must be at least 50% more than the horizontal distance for the sector. He explained that laying out sectors in that
manner meant that the patrol areas would have a greater tendency to overlap different residential, income, and retail areas than if they ran the
other way. He said that these areas were layered from north to south in the city, so if a sector area was laid out east to west, all of it would tend to be
in one demographic layer.

Angela indicated that she had almost enough information to develop a model, except that she also needed to know the average travel speed the
patrol cars could travel- Joe told her that cars moving vertically traveled an average of 15 miles per hour, whereas cars traveled horizontally an
average of 20 miles per hour. He said that the difference was due to different traffic flows-

a- Formulate a linear programming model for this problem-

b- Solve this model by using graphical analysis-