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The Metropolitan police department had recently been 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 breaking,
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
had been assigned by the chief as head of a task force to find a way
to determine optimal patrol area(dimensions)for their cars that would
minimize the average
time it took 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
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 a car left immediately upon receiving the call) is simply t
he 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 manpower
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 were
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.
Develop a linear programming model for this problem,
and solve it using the graphical method.
大城市警察局最近被批評了在地方媒介為不反應警察電話在市中心足夠迅速地.
在幾個最近案件, 警報聽起來為打破. 但當警車到達了的時候. 犯人離開
並且在一個事例店主是射擊. 喬?迪維斯警官由長官分配作為工作小組的頭
發現方式確定優選的巡邏區域. 為使平均時間減到最小的他們的汽車它採ꠊ疃F對一個電話在市中心. 迪維斯警官被請求的幫助從Angela Maris. 一個
分析員在操作區域為警察局. 他們一起開始運作通過問題. 喬注意了對Angela,
正常巡邏區段被計劃在長方形以各個長方形包括一定數量的城市街區.
為說明目的他定義了區段的維度作為x 在水平的方向和作為y 在垂直的方向.
他解釋了對Angela,
汽車旅行了在或者水平的方向必須由旅行伴隨在一個垂直的方向的直線並且
總距離旅行了是水平和垂直的段的總和. 他進一步注意到,
通過對警察的研究巡邏在市區表示, 平均距離旅行了乘巡邏車反應
一個電話在或者方向是區段的維度的三分之一, 或x/3 和y/3
他並且解釋, 它採取反應電話的旅行時間(承擔汽車左立刻在接到電話)
是平均距離簡單地旅行由平均旅行速度劃分。 Angela 告訴喬即然她瞭解多麼
平均旅行時間對電話是堅定的, 她能看, 它與巡邏區域的大小緊密地相關。
她問喬如果有對汽車巡邏區域區段的大小的任何制約。 他反應了那為他們的城市,
部門相信, 巡邏區段的周長不應該是少於5 英哩或超出12 英哩。
他注意了要求這些規格的幾個政策問題和人力限制。
Angela 想知道如果任何另外的制約存在了, 並且喬表明,
距離在垂直的方向必須是至少50% 更比水平的距離為區段。
他解釋了那計劃區段方式意味巡邏區域會有一個更加巨大的傾向重疊另外住宅,
收入, 並且零售區域比如果他們跑了另一方式。 他說, 這些區域被分層堆積
了從北到南在城市。 如此如果扇區被計劃的東西向, 所有會傾向於是在一人口
統計的層數。 Angela 表明, 她有幾乎足夠的資訊開發模型,
除了她並且需要知道平均旅行速度巡邏車能旅行。 喬告訴了她,
汽車垂直行動旅行了平均15 英哩每小時, 但是汽車水平地旅行了平均20 英哩每小時。
他說, 區別歸結於不同的運輸流量。 開發一個線性規劃模型為這個問題,
並且解決它運用圖解方法。
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