4.10 连接:线性规划,以最大限度地提高营养水平

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• 选择节连接: 最大限度地提高营养度的线性编程

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  连接: 最大限度地提高营养度的线性编程

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  Linear programming is a technique used to maximize or minimize the values in a mathematical model. It was developed during World War II to try to address the costs and planning of the armed forces. In 1975, Leonid Kantorovich and T.C. Koopmans received the Nobel prize in economics for transforming economics problems into linear programming problems ¹ .
  ::线性编程是一种在数学模型中最大限度地或尽量减少价值的技术,在二战期间开发,试图解决武装部队的费用和规划问题,1975年Leonid Kantorovich和T.C.Koopmans获得经济学诺贝尔经济学奖,将经济问题转化为线性编程问题1。

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  Today, we can use linear programming in planning, transportation, scheduling, technology, and business. Let's consider a problem for planning for proper nutrition. 
  ::今天,我们可以在规划、运输、日程安排、技术和商业方面使用线性编程。让我们考虑一个规划适当营养的问题。

  

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  Say you want to eat yogurt and bananas for breakfast. You want your breakfast to contain at least 15 grams of protein and at most 15 grams of fat. You also want your breakfast to be 400 grams. Yogurt contains 4 grams of fat and 8 grams of protein per 100 grams  serving and bananas contain 0.3 grams  of fat and 1 grams  of protein per 100 grams  serving ^2,3 . How many 100- gram  servings of yogurt and bananas should you eat to minimize the amount of fat? 
  ::说你想吃酸奶和香蕉做早餐。你希望你的早餐至少含有15克蛋白质,最多含有15克脂肪。你还希望你的早餐为400克。Yogurt每服务100克含有4克脂肪和8克蛋白质,香蕉每服务100克含有0.3克脂肪和1克蛋白质。3你应该吃多少100克酸奶和香蕉来尽量减少脂肪的数量?

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  The first step in a linear programming problem is to define variables. Let  y be the number of 100- gram  servings of yogurt and let  b be the number of 100- gram  servings of bananas. 
  ::线性编程问题的第一步是定义变量。 以酸奶100克服务数为依次, 以香蕉100克服务数为依次。

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  Next, we need to determine our objective function , the quantity we are trying to maximize or minimize. We are trying to minimize the fat given a certain amount of yogurt and bananas. We want the amount of our breakfast to be 400 grams , or  $\begin{align*}y+b=4\end{align*}$  (recall  y and  b are 100- gram  servings). This is graphed below as a blue line.
  ::接下来,我们需要确定我们的目标功能, 我们试图最大化或最小化的数量。 我们正在尝试将脂肪最小化, 给定一定数量的酸奶和香蕉。 我们希望我们的早餐数量为400克, 或y+b=4( recall y and b are 100 gram services) 。 下图显示为“ 蓝线 ” 。

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  We also need to determine the constraints or restrictions. Two assumed constraints in this situation are that the amount of either yogurt or bananas is positive or zero. We also have constraints on the amount of fat and the amount of protein. 
  ::我们还需要确定制约或限制,两种假定的制约是酸奶或香蕉的数量是正的或零的,我们对脂肪和蛋白质的数量也有限制。

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  Fat (green):  $\begin{align*}4y+0.3b \le 15\end{align*}$  
  ::脂肪(绿色):4y+0.3b15

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  Protein (red):  $\begin{align*}8y+b \ge 15\end{align*}$  
  ::蛋白(红):8y+b}15

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  Next, we graph the line and the inequalities on the same graph. Where all of the solutions overlap, which is usually shaped like a polygon, creates a feasible region . The image below shows what a feasible region can look like. 
  ::接下来,我们用同一图表绘制线条和不平等。当所有解决方案重叠时(通常形成为多边形),创造出一个可行的区域。下面的图像显示一个可行的区域会是什么样子。

  

   

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  The graph of our situation is below. The feasible region exists between the blue line and the horizontal axis, in this case, the yogurt axis, where both inequalities are shaded. 
  ::我们的情况图如下:在蓝线和横向轴(此处指酸奶轴轴)之间,存在着可行的区域,两者的不平等都有阴影。

  

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  The beauty in this technique is that the values that will maximize or minimize the model are located at the vertices of the feasible region. In the image above, only the two points on the blue line are highlighted because they satisfy the objective function. Testing these values out, we have
  ::这种技术的美之处在于将模型最大化或最小化的值位于可行区域的顶端。在以上图像中,仅突出显示蓝线上的两点,因为它们符合客观功能。测试这些值,我们发现这些值。

  Point	Fat	Protein
  (1.571, 2.429)	$\begin{align*}(4 \times 1.571)+ (0.3 \times 2.429) = 7.0127\end{align*}$	$\begin{align*}(8 \times 1.571) +2.429 =15\end{align*}$
  (3.73, 0.27)	$\begin{align*}(4 \times 3.73) + (0.3 \times 0.27) =15\end{align*}$	$\begin{align*}(8 \times 3.73)+ 0.27 =30.11\end{align*}$

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  So these points present the two extremes. You can eat a 400- gram  breakfast with only 7  grams of fat and 15 grams  of protein or you can eat a 400- gram  breakfast with 15 grams  of fat but as much as 30.11 grams  of protein.
  ::因此,这两个点代表了两个极端。你可以吃一个400克的早餐, 只有7克的脂肪和15克的蛋白质, 或者你可以吃一个400克的早餐, 15克的脂肪,但多达30.11克的蛋白质。

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  The first combination minimizes the amount of fat in the breakfast. 157.1 grams  is about 1 small container of yogurt and 242.9 grams  of banana is about two small bananas. 
  ::第一个组合将早餐中的脂肪含量降到最低。 157.1克是约1小箱酸奶,242.9克香蕉是约2小香蕉。

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     To Solve a Linear Programming Problem
  ::解决线性编程问题

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  1. Define variables. 
  ::1. 界定变量。

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  2. Determine the objective function.
  ::2. 确定目标职能。

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  3. Determine the constraints, including assumed ones.
  ::3. 确定制约因素,包括假定的制约因素。

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  4. Graph all of the models on one graph. Identify the feasible region. 
  ::4. 用一个图绘制所有模型图,确定可行的区域。

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  5. Find the maximum or minimum values by considering the values at the vertices of the feasible region. 
  ::5. 通过考虑可行区域顶端的数值,确定最大值或最低值。

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  1. You are making trail mix out of peanuts and sunflower seeds for your next hike. You want your trail mix to have less than 60 grams  of fat and 28 grams  of protein, and contain 0.75 grams  of sodium. Salted peanuts have 49 grams  of fat, 26 grams  of protein, and 0.75 grams  of sodium per 100- gram  serving. Sunflower seeds have 51 grams  of fat, 21 grams  of protein, and 0.45 grams  of sodium per 100- gram  serving ^4,5 . What combination of 100- gram  servings of peanuts and sunflower seeds will maximize the amount of sodium in your trail mix? 
  ::1. 为下次远足,你正在从花生和日葵种子中做小路混合。你希望你的小路混合能有不到60克的脂肪和28克的蛋白质,并含有0.75克的钠。盐花生有49克的脂肪、26克的蛋白质和0.75克每100克的钠。太阳花种子有51克的脂肪、21克的蛋白质和0.45克每100克的钠4,5。 花生和日葵种子的100克的混合配方是什么?

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  2. You are trying to limit your caffeine intake. You decide you are going to drink at most 5 cups of coffee and tea a day. Coffee contains 95 milli grams and tea contains 26 milligrams of caffeine. It is recommended you get at most 400 milligrams of caffeine a day. You would also like to control the expense of your caffeine habit. Coffee costs $1.50 a cup and tea costs $1.25 ^5,6 . You would like to spend at most $7 a day. What combination of cups of coffee and tea will minimize your caffeine intake? 
  ::2. 限制咖啡因摄入量,决定每天最多喝5杯咖啡和茶,咖啡含95毫克,茶含26毫克咖啡因,建议每天最多喝400毫克咖啡因,还想控制咖啡因消费费用,咖啡每杯150美元,茶费1.255美元,每天最多喝7美元。咖啡和茶的组合可以尽量减少咖啡因摄入量。

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  3. James is trying to expand his pastry business to include cupcakes and personal cakes. He has 40 hours available to decorate the new items and can use no more than 22 pounds of cake mix. Each personal cake requires 2 pounds of cake mix and 2 hours to decorate. Each cupcake order requires one pound of cake mix and 4 hours to decorate. If he can sell each personal cake for $14.99 and each cupcake order for $16.99, how many personal cakes and cupcake orders should James make to make the most revenue?
  ::3. James试图扩大他的糕点业务,以包括蛋糕和个人蛋糕,他有40小时可以装饰新的物品,不能使用超过22磅的蛋糕混合,每个个人蛋糕需要2磅的蛋糕混合和2小时的装饰,每个蛋糕订单需要1磅的蛋糕混合和4小时的装饰,如果他能卖出每个个人蛋糕14.99美元,每个蛋糕订单16.99美元,James应该做多少个人蛋糕和蛋糕订单来赚取最大的收入?

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  4. Create a linear programming problem for a situation with your own menu planning. 
  ::4. 造成线性编程问题,以应对你自己的菜单规划情况。

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  References 
  ::参考参考资料

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  1. "Linear Programming," last edited May 1, 2017, https://en.wikipedia.org/wiki/Linear_programming.
  ::1. “Linear programming”, 2017年5月1日编辑, https://en.wikipedia.org/wiki/Linear_programming。

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  2. "Yogurt," last edited May 5, 2017, https://en.wikipedia.org/wiki/Yogurt.
  ::2. " Yogurt " , 上一期于2017年5月5日编辑,https://en.wikipedia.org/wiki/Yogurt。

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  3. "Bananas," last edited May 16, 2017, https://en.wikipedia.org/wiki/Banana.
  ::3. “香蕉”, 2017年5月16日上一期编辑, https://en.wikipedia. org/wiki/Banana。

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  4. "The Peanut Institute-Nutritional Breakdown," last accessed May 17, 2017, http://www.peanut-institute.org/peanut-facts/nutritional-breakdown.asp.
  ::4. “花生协会-营养崩溃”, 2017年5月17日最后一次访问, http://www.peanut-institut.org/peanut-facts/national-breaknow.

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  5. "Sunflower Seed," last edited May 11, 2017, https://en.wikipedia.org/wiki/Sunflower_seed.
  ::5. “日葵种子”, 2017年5月11日上一期编辑, https://en.wikipedia.org/wiki/Sunflower_seed。

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  6. "Caffeine Content for Coffee, Tea, Soda, and More," last accessed May 17, 2017, http://www.mayoclinic.org/healthy-lifestyle/nutrition-and-healthy-eating/in-depth/caffeine/art-2004937  
  ::6. “咖啡、茶、索达和更多咖啡的咖啡内容”,2017年5月17日最后一次查阅,http://www.mayoclinic.org/healthy-lifestem/nicky-and-healthy-eating/in-explocation/caffeine/art-2004937。