Solving linear programming (From Last week s Introduction) Consider a manufacturer of tables and chairs. They want to maximize profits. They sell tables for a profit of $30 per table and a profit of $10 per chair. Let x= number of tables and y = the number of chairs. Then the profit equations is: There are 5 employees working 40 hours a week for a total of 200 total hours of labor. A table requires 10 hours of labor. Chairs require 2 hour of labor. Then we have the condition on making tables and chairs that the total time spent must be less than 200 hours and the following equation must be true. A second condition is that there are two employees that stain the tables and chairs. Thus, there is at most 80 hours to stain the tables and chairs. It takes 1 hour to stain a chair and 3 hours to stain a table. Thus there is a second conditional equation We also have the conditions that the number of tables and chairs created must be greater or equal to zero. Giving us two more conditions: With this information we can set up a mathematical process to solve called linear programming. The way this equation would be stated is to Maximize the profit equation Subject to the four constraint equations These four conditions stated above are called constraint equations. Today our goal will be to examine the area where all constraints are valid. This area is called the feasible region, as it represents the area where only possible answers occur.
Graph the four constraint equations. Since the constraint equations and imply that the answer must be in the first quadrant. We have already shaded this region. Please graph and correctly shade the other two constraint equations The region you have shaded is called the feasible region as only points in the shaded region have values that meet all constraints. Thus, these points are the only valid points for checking for determining maximum profits. Solving the Linear Programming Problem Using EXCEL to solve the Linear Programming problem (SIMPLEX Method). SIMPLEX method is the most common method for solving a linear system. This is the method used by solve in EXCEL. In practice EXCEL uses Matrices to solve these problems.
Let s solve our existing problem: Maximize profit Subject to the constraints These four conditions stated above are called constraint equations. Open an EXCEL Spread sheet Labels in A3, we have labeled x= In A4, we have labeled y= The values for x will actually be in B3 and for y will be in B4. They are currently left blank which defaults to zero. Profit is what we wish to maximize is in A6 and in B6 we typed the equation =30*B3 + 10*B4. Constraints is the label in A8. In B8, we have placed the just the equation side of the constraint =10*B3+2*B4 In B9, we have placed the second constraint, =3*B3+B4 We will show how to create an inequality in the next step. We will also see that we do not need to enter the last two constraints Once this is entered, click on the Data tab and click on solver
In the Solver Window, the first step is to change the pull down window to read Simplex LP At this time, also make sure the Make all unconstrained variables Non-Negative We can now enter the equations Begin by entering the cell you wish to maximize $B$6 in set objective. Click on Max And enter the cells you will allow to change separated by commas. $B$3,$B$4 Next step (below) we can enter constraints Start this process by clicking on Add
When you have clicked on Add a new window appears to add a constraint. Enter the cell of the first constraint Pick <= And enter 200 When this is done, click ok and the constraint will be added. Repeat For the second constraint The solver parameters will now have the constraints entered. Remember the constraints that x and y have to both be greater than equal to zero are covered by clicking the non-negative box. Click solve
You will now have a new window open stating a solution has been found and asking if you wish to keep the solution. Click OK You have a solution that you can maximize profit at $800.00 when the number of tables created is 10 and the number of chairs created is 50. Challenge problem. For this challenge problem we are going to produce four items In cell A1 enter the label Item 1 and B1 will represent the number of item 1 s created In cell A2 enter the label Item 2 and B2 will represent the number of item 2 s created In cell A3 enter the label Item 3 and B3 will represent the number of item 3 s created In cell A4 enter the label Item 4 and B4 will represent the number of item 4 s created The profit equation that will be maximized is
The constraints are that B1, B2, B3, B4 >= 0 and will be entered by checking the Non-Negative Box. Also subjet to. Note: These equations are all less than Note: These four equations are all greater than The answer is? The number of item 1 = The number of item 2 = The number of item 3 = The number of item 4 = The maximum profit is