Elizabeth Bailey is the owner and general manager of Princess Brides, which provides a wedding planning service in Southwest Louisiana.    She uses radio advertising to market her business. Two types of ads are available - those during prime time hours and those at other times. Eachprime time ad costs $390 and reaches 8,200 people, while the off-peak ads each cost $240 and reach 5,100 people. Bailey has budgeted 1,800 per week for advertising.  Based on comments from her customers,Bailey wants to have at least 2 primetime ads and not more than 6 off-peak ads.

Solve this as an integer programming problem using a computer.

Answer:-

Formulation of LP

Step-1: Decision Variables

Let x1 and x2 be the decision variables denoting the number of ads in prime hours and off-peak hours.

Step-2 Optimization Function

The objective here is to maximize the reach of the Ads

Maximize: Z = 8200 x1 + 5100 x2

Step-3 Constraints

1. x1>=2

2. x2<=6

3. 390x1 + 240 x2 <= 1800

4. x1, x2 >=0

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Solving the LP

To draw constraint x1≥2→(1)

Treat it as x1=2

Here the line is parallel to Y-axis

To draw constraint x2≤6→(2)

Treat it as x2=6

Here the line is parallel to X-axis

To draw constraint 390x1+240x2≤1800→(3)

Treat it as 390x1+240x2=1800

When x1=0 then x2=?

390(0)+240x2=1800

240x2=1800

x2=1800/240=7.5

When x2=0 then x1=?

390x1+240(0)=1800

390x1=1800

x1=1800/390=4.62

Graphical Representation:

In the graph, the green line represents the equation: x2=6, the red line represents the equation: x1=2, the blue line represents the equation: 390x1+240x2=1800.

The value of the objective function at each of these extreme points is as follows:

The maximum value of the objective function z = 38075 occurs at the extreme point (2,4.25).

Hence, the optimal solution to the given LP problem is x1=2, x2=4.25, and max z=38075.

Since the number of Ads can't be in fraction and should be a whole number, therefore an educated guess will be

x1 = 2, x2 = 4, then the Z = 36800.

If I take x1= 2 and x2 = 5 then it violates the budget constraint.

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Solving the Integer Programming Problem using Excel Solver

Objective Function: Maximize  Z = 8200 x1 + 5100 x2

Subject to Constraints:

1. x1>=2

2. x2<=6

3. 390x1 + 240 x2 <= 1800

4. x1, x2 >=0 and x1, x2 = integer

On solving, We get

x1 = 4

x2 = 1

Z = 37900

Please find below the screenshots of the Excel Sheet:

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