求教一道business case interview的问题

貌似是capital one 的一道老题

Now we would like to compare two approaches:

1） Tel-marketing (make direct phone calls to potential customers)

2） Direct Mail (send mails to potential customers)

Suppose the cost will be $1.00 per call, $0.50 per mail.

Suppose the contact rate will be 50% for phone calls (assume that each call cost $1.00 no matter the customer is contacted or not),

and 100% for direct mail.

Suppose the successful rate for those made through phone calls is 6%, for direct mail is 2%.

Suppose the profit for bringing in each customer is $100.

Which approach is better, and why?

我知道这个要分情况，fixed budget 和fixed # of customers 没什么问题，我想问下的是：

1. fixed budget + fixed # of customers: say $ 100 and 150 customers     这里的意思应该指phone call 和mail两种方法一起用是吧？用线性规划画个图来解？

2. What is the exact limited funds where you should mail the same number of customers as you call?  这个我实在是没弄明白。。

3. given unlimited budget    这个是不是指可以不用管cost, 找有较高成功率的那个就成？也就是phone call?

4. given unlimited customer list    这个是用cost比较低的？

5. unlimited budget & customer list .

囧。。把自己都绕晕了。。求教各位指点，先在此谢过！

(1)条件如下 x #mails, y #calls
x+y<=150
0.5x+y<=100
x>=0
y>=0
收益是 z = 100*(100%*2%*x + 50%*6%*y) = 2x + 3y

线性规划是正确的的解法，按照你给的数据，我算了算最大效益是$350。

(2)
如果你画出图来的话，其实就是问你怎么上下平移y=-0.5x+z这个直线（z是limited fund），使得这个直行经过y=x(#mail = #call)和y=-x+150的焦点。
然后再求解z，我算出来是$122.5.

(3)
all calls, no mail

(4)
. ----all mails, no call

(5)
all calls, no call
但是收益最大值是无穷。
可以考虑为，当资金非常充分时，选call，这个结论也match (3)

最后这三个问，你把图画出来就能很明了了。
(3)和(5)都是当x=0时取最大值。