﻿ 1 A professional football player is retiring and he is thinking about going into the insurance bus
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Question

1) A professional football player is retiring, and he is thinking about going into the insurance business. He plans to sell four types of policies— homeowner’s insurance, auto insurance, boat insurance and life insurance. The average amount of profit returned per year by each type of insurance policy is as follows:

Policy                            Yearly Profit/Policy

Homeowner’s               \$70

Auto                 48

Boat                 45

Life                  80

Each homeowner’s policy will cost \$25, each auto policy will cost \$18.50, each boat policy will cost \$18 and each life insurance policy will cost \$32 to sell and maintain. He has a budget of \$80,000 per year. In addition, the sale of a homeowner’s policy will require 6 hours of effort; the sale of an auto policy will require 3.2 hours of effort, the sale of a boat policy will require 4 hours of effort and the sale of a life insurance policy will require 10 hours of effort. There are a total of 25,000 hours of working time available per year from himself and his employees.

He wants to sell at least twice as many auto policies as homeowner’s policies.

Formulate a linear programming model to maximize his profit by determining

(a) The decision variables.

(b) The objective function. What does it represent?

(c) All the constraints. Briefly describe what each constraint represents.

Note: Do NOT solve the problem after formulating.

Solution

xxxxxxxxx are xxx xxxx xxxxx

Policies

xxxxxx

Cost

xxxxxxx(xxxxx) xxxx xxxxxx

70

xx

6

xxxx xx xx.x

3.2

xxxx

45

xx x xxxx

80

xx

10

xxxxx(xxx xxxxxxxxxxxy) xxxxx(xxx Availability)

xxx the xxxxxx xx xxxxxxxxxx policies = x1

xxx xxx xxxxxx of xxxx policies = xx xxx the xxxxxx of xxxx xxxxxxxx = x3

xxx the xxxxxx xx xxxx policies = x4

xxx xxx xxxxxxxx variables xxx positive xx zxxx x.x x1., xx, x3, xx             0

(x)xxxxxxxxx Function:- xx xxxxxx xxx maximum xxxxxx by xxxxxxx xxxxxxxxx xxxxxxxx

Maximize z = xx xx +xx x2 +xx x3 +xx xx ( c ) Maximum xxxxxxxxx xxxxxx xx 80,000 xxxxx means xxx xxx xxxxxxxxxxx by xxx type xx xxxxxxxx xxxx be xxxxxx 80000(i.e80,000)

xxx xxxxxxxxxx xx 25x1 + 18.5x2 +xxxx +xxxx                 80,000

xxxxxxx effort xxxx xx,xxx xxxxx means xxx the xxxxxx xxxx xy all xyxx of xxxxxxxx xxxx xx within xx,xxx(x.x 25,000)

xxx xxxxxxxxxx xx 6x1 + 3.2x2 +xxx + xxxx                       25,000

xxxxx xx xxxy auto xxxxxxxx as xxxxxxxxxx xxxxxxxx xxx constraint xx x1 -           xxx     =          0

xxx xxxxxxxxxx xxx is

xxxxxxzx     z = xx x1 +xx x2 +xx xx +xx x4

xxxxxxx to     xxxx + xx.xxx +18x3 +xxxx                     xx,xxx                         xxx + x.xxx +xxx + 10x4           25,000

xx -     xxx =          x      x1., xx, x3, xx             0

xxxxxxxx

Profit

xxxx

Efforts(Hours)

xxxx xxxxxx
xx

25

x

Auto

xx
xx.x x.x

Boat

xx

18

x
xxxx xx

32

xx
xxxxx(xxx xxxxxxxxxxxy)

25000(Max xxxxxxxxxxxy)