Subject: management science METROPOLITAN POLICE PATROLtime it took to respond to a call (assuming that a car leftimmediately upon receiving the call) is simply the averagedistance traveled divided by the average travel speed.Angela told Joe that now that she understood how aver-I he Metropolitan Police Department was recently criti-cized in the local media for not responding to police calls age travel time to a call was determined, she could see thatin the downtown area rapidly enough. In several recent it was closely related to the size of the patrol area. Shecases, alarms had sounded for break-ins, but by the time asked Joe if there were any restrictions on the size of thethe police car arrived, the perpetrators had left, and in one area sectors that cars patrolled. He responded that for theirinstance a store owner had been shot. Sergeant Joe Davis city, the department believed that the perimeter of a patrolwas assigned by the chief as head of a task force to find a sector should not be less than 5 miles or exceed 12 miles.way to determine the optimal patrol area (dimensions) for He noted several policy issues and staffing constraints thattheir cars that would minimize the average time it took to required these specifications. Angela wanted to know if anyrespond to a call in the downtown area.Sergeant Davis solicited help from Angela Maris, an distance in the vertical direction must be at least 50% moreanalyst in the operations area for the police department. than the horizontal distance for the sector. He explainedTogether they began to work through the problem.Joe noted to Angela that normal patrol sectors are laid areas would have a greater tendency to overlap differentout in rectangles, with each rectangle including a number residential, income, and retail areas than if they ran theof city blocks. For illustrative purposes he defined the di- other way. He said that these areas were layered from northmensions of the sector as x in the horizontal direction and to south in the city, so if a sector area was laid out east toas y in the vertical direction. He explained to Angela that west, all of it would tend to be in one demographic layer.cars traveled in straight lines either horizontally or verti-cally and turned at right angles. Travel in a horizontal di- tion to develop a model, except that she also needed torection must be accompanied by travel in a vertical know the average travel speed the patrol cars could travel.direction, and the total distance traveled is the sum of the Joe told her that cars moving vertically traveled an averagehorizontal and vertical segments. He further noted that past of 15 miles per hour, whereas cars traveled horizontally anresearch on police patrolling in urban areas had shown that average of 20 miles per hour. He said that the differencethe average distance traveled by a patrol car responding to was due to different traffic flows.a call in either direction was one-third of the dimensions ofthe sector, or x/3 and y/3. He also explained that the travel and solve it by using the graphical method.additional restrictions existed, and Joe indicated that thethat laying out sectors in that manner meant that the patrolAngela indicated that she had almost enough informa-Develop a linear programming model for this problem

Question

Accepted Answer

The horizontal & vertical distance of the respective sector is measured to be x & y,…