Question

The Apex Television Company has to decide on the number of 65″ and 55″ sets to be produced at one of its factories. Market research indicates that at most 280 of the 65″ sets and 70 of the 55″ sets can be sold per month. The maximum number of work-hours available is 3,500 per month. A 65″ set requires 20 work-hours and a 55″ set requires 10 work-hours. Each 65″ set sold produces a profit of $270 and each 55″ set produces a profit of $180. A wholesaler has agreed to purchase all the television sets produced if the numbers do not exceed the maxima indicated by the market research.
a. Formulate and solve a linear programming model for this problem on a spreadsheet.

b. Formulate this same model algebraically.

c. Use the graphical method to solve this model.

please put solver steps

Answer 1

A) Formulation of the LPP in Spreadsheet :

The required table is :

On the spreadsheet, use the formula "Sumproduct" as shown for D6,D7, D8 and D11.

Now set B3, B4 to 0.

Then use "Solver" to optimze :

Step1 : Go to "Data" and click on "solver" and set the objective parameters.

Step 2 : Set objectives, select the cell for the total profit i.e D11 here.

Step 3 : The objective is to Maximaize, so choose "Max"

Step 3 : For "Changing variable cells" choose the cells B3 and C3 as we need to solve for these 2 values and hence we have choosen them to zero before running the colver.

Step 4 : For "Subject to constraints" , click on the "add" option in the right and choose the requied cells.

Step 5 : Finally Click on "Solve".

Then we will get the solution as below:

	65"	55"
Number to make	140	70
Required Input			Used		Available
Max Limit	1	0	140	<=	280
	0	1	70	<=	70
Work Hours	20	10	3500	<=	3500
			Total Profit
Unit Profit	270	180	50400

Therefore we get the maximum profit as $ 50400.

B) Formulation of the LPP algebraically :

Let  the number of 65″ sets to be produced = x

Let  the number of 55″ sets to be produced = y

The constraints are:

Given, that at most 280 of the 65″ sets can be sold per month i.e.

and  at most 70 of the 55″ sets can be sold per month i.e

A 65″ set requires 20 work-hours and a 55″ set requires 10 work-hours, while maximum number of work-hours available is 3,500 per month.

i.e.

The objective function is:

The profit is to be maximized and each 65″ set sold produces a profit of $270 and each 55″ set produces a profit of $180.

i.e.

The required LPP is:

C) Solving Graphically :

We get the following graph with the feasible region shaded:

Now to maximize the objective function, find the functional value at the corner points:

Therefore the maximum profit is $ 50400 , which is obtained for producing 140 set of 65" and 70 sets of 55".

This is same as spreadsheet solver.