1) A professional football player is retiring, and he is thinking about going in

ID: 3109595 • Letter: 1

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:

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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?

Note: Do NOT solve the problem after formulating.

Explanation / Answer

Following are the data given

Policies

Profit

Cost

Efforts(Hours)

Home owners

Auto

18.5

3.2

Boat

Life

80000(Max Availability)

25000(Max Availability)

Let the number of Homeowners policies = x1

Let the number of Auto policies = x2

Let the number of Boat policies = x3

Let the number of Life policies = x4

All the decision variables are positive or zero i.e x1., x2, x3, x4 0

(b)Objective Function:- To obtain the maximum profit by selling different policies

Maximize z = 70 x1 +48 x2 +45 x3 +80 x4

( c ) Maximum available budget is 80,000 which means all the expenditure by all type of policies must be within 80000(i.e80,000)

The constraint is 25x1 + 18.5x2 +18x3 +32x4 80,000

Maximum effort hour 25,000 which means all the effort hour by all type of policies must be within 25,000(i.e 25,000)

The constraint is 6x1 + 3.2x2 +4x3 + 10x4 25,000

Twice as many auto policies as homeowners policies

The constraint is x1 - 2x2 = 0

The formulated LPP is

Maximize z = 70 x1 +48 x2 +45 x3 +80 x4

Subject to 25x1 + 18.5x2 +18x3 +32x4 80,000

6x1 + 3.2x2 +4x3 + 10x4 25,000

x1 - 2x2 = 0

x1., x2, x3, x4 0