Question

American Auto is evaluating its marketing plan for the sedans, SUVs, and trucks the company produces. A TV ad featuring this SUV has been developed. The company estimates each showing of this commercial will cost $500,000 and increase sales of SUVs by 3% but reduce sales of trucks by 1% and have no effect on the sales of sedans. The company also has a print ad campaign developed that it can run in various nationally distributed magazines at a cost of $750,000 per title. It is estimated that each magazine title the ad runs in will increase the sales of sedans, SUVs, and trucks by 2%, 1%, and 4%, respectively. The company desires to increase sales of sedans, SUVs, and trucks by at least 3%, 14%, and 4%, respectively, in the least costly manner.

a. Formulate an LP model for this problem.

b. Sketch the feasible region.

c. What is the optimal solution?

Answer 1

General guidance

Concepts and reason

The graphical solution is limited to linear programming models containing only two decision variables (can be used with three variables but only with great difficulty) and the shape of two curves is compared for comparison of dissolution pattern and the concentration of drug at each point is compared for extent of dissolution.

Fundamentals

The graphing an LP model helps provides insight into LP models and their solutions:

A straight line is plotted in place of each dis equation.

A convex and bounded set is generated and an ideal line, that is a family of parallel lines, is drawn to represent the objective function.

The optimum is found at the interception of the ideal line with a vertex.

Step-by-step

Step 1 of 3

Let us consider ${X_1}$ be the number of TV ads and ${X_2}$ be the number of prints ads. The cost of TV ad is $500,000 and the cost of print ad is 750000.

The company want to minimize its cost. Therefore, the objective function for this model is,

${\\rm{Minimiza}}\\left( Z \\right) = 500000{X_1} + 750000{X_2}$

Subject to the constraints:

$\\begin{array}{c}\\\\3{X_1} + {X_2} \\ge 14\\\\\\\\2{X_2} \\ge 3\\\\\\\\ - 1{X_1} + 4{X_2} \\ge 4\\\\\\\\{X_1},{X_2} \\ge 0\\\\\\end{array}$

Explanation | Hint for next step

The number of TV ads and ${X_2}$ be the number of prints ads. The cost of TV ad is $500,000 and the cost of print ad is 750000 and the constraints are a single TV ad increase the sale of SUVs by 3% and a single Magazine TV Ad increase the sale of SUVs by 1%. Also, the company want to increase the sale of SUVs by at least 14% and the company wanted to increase the same of sedan by at least 3% and its sale is only affected by the ad in magazine which is 2%.

Step 2 of 3

The procedure for creating these graph are quite simple. It is required to solve the three equations and plot the respective points.

It is parallel to $x -$axis and the feasible region is above this line. Similarly, the solutions of other two equations are provided as follows:

(0, 14) 16

In the above graph, light tomato colored region denotes the feasible region. There are only two corner points which are denoted by dark black.

Explanation | Hint for next step

The feasible region points of the graphical method is $\\left( {0,14} \\right){\\rm{ and }}\\left( {4,2} \\right).$

Step 3 of 3

The objection function of the problem is,$Z = 500000{X_1} + 750000{X_2}$

It is required to obtain the optimal solution of the problem defined in the first part. The minimum cost would be the optimal solution for this particular problem. Putting the first corner point in the above objective function:

$\\begin{array}{c}\\\\Z = 500000\\left( 0 \\right) + 750000\\left( {14} \\right)\\\\\\\\ = 10500000\\\\\\end{array}$

Similarly, for the other corner point:

$\\begin{array}{c}\\\\Z = 500000\\left( 4 \\right) + 750000\\left( 2 \\right)\\\\\\\\ = 3500000\\\\\\end{array}$

The optimum solution is, 3500000 at the point is (4, 2).

Explanation

The optimum solution of the two values, the minimum value of $Z$ is 3500000 at point $\\left( {4,2} \\right).$ Hence, the optimal number of TV ads is 4 and that of magazine ads is 2.

Answer

The optimum solution is, 3500000 at the point is (4, 2).

Answer only

The optimum solution is, 3500000 at the point is (4, 2).