3.3 - One More Example...

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1 c Kathryn Bollinger, September 28, One More Example... Ex: (from Tan) Solve the following LP problem using the Method of Corners. Kane Manufacturing has a division that produces two models of fireplace grates, Model A and Model B. To produce each Model A grate requires 3 lb. of cast iron and 6 min. of labor. To produce each Model B grate requires lb. of cast iron and 3 min. of labor. The profit for each Model A grate is $2.00 and the profit for each Model B grate is $1.50. If 1000 lbs. of cast iron and 20 labor-hours are available for the production of fireplace grates per day, how many grates of each model should the division produce in order to help maximize Kane s profits? What is the optimal profit?

2 c Kathryn Bollinger, September 28, Sensitivity Analysis Look at the fireplace grate example from the previous section. What if the company was still trying to maximize profits, but more or less iron was available? What if they hired or fired people and had a different number of labor-hours? What would happen if they changed the price (and thus the profit) of a model? Sensitivity analysis investigates how changes in the parameters of a LP problem will affect its optimal solution. Changes in the Coefficients of the Objective Function In the original objective function, P =2x+1.5y y = 2x + P 1.5 = ( ) ( ) 2 x + P. 3 3 The position of the line depends on how big P is, but the slope is always /3. Let s keep the profit on Model B fixed and see what happens if we let the profit on Model A vary. Suppose the profit on each Model A is $c. Then P = cx +1.5y. This new objective function will be 1.5y = cx + P y = ( 2 ) ( ) 2 3 c x + P. 3 Sothenewslopewillbe 2 3 c. What are the slopes of the constraints? IRON: LABOR: As long as the slope of the objective function is between these two values, we will end up with the optimal solution at the same corner point. Therefore, we can determine the value of c so that this occurs:

3 c Kathryn Bollinger, September 28, So, if the profit on each Model A grate is between $ and $,themaximum profit will occur at the same product mix, 120 Model A s and 160 Model B s. ** Of course, the maximum profit WILL change depending on the value of c. Notice if c =1.125, then the slope of the profit equation will be If the profit on Model A is smaller than $1.125, it means the location of the maximum profit will shift to the point (0,250), meaning it is not profitable to make any Model A s. Notice if c = 3, then the slope of the profit equation will be If the profit on Model A is larger than $3, it means the location of the maximum profit will shift to the point (200,0) and only Model A s will be made. You can perform a similar analysis on the profit of each Model B. Looking at a parametric solution, say when c =3 P =3x+y, using the Method of Corners we have:

4 c Kathryn Bollinger, September 28, 2005 What is happening at the point (120,160)? Iron Use = Labor Use = What is happening at the point (0,250)? Iron Use = Labor Use = If a resource is not fully used and there are leftovers at an optimal solution, then we say that the constraint is non-binding. A binding constraint is one in which the resource is fully used at the optimal solution. To find out if any resource is leftover, you can put your optimal solution (decision variable values) into the constraints and see if the constraints are fully used or not. Changes in Resources Let s say we have h pounds more or less of iron than our original 1000 pounds. Our constraint will then be 3x +y h. ( y 3x h y 3 ) ( x h ) Notice that the slope is unchanged from the original problem, but the y-intercept is now h. Therefore, the intersection of this iron line and the labor line (y = 2x + 00) will also shift. The intersection now becomes:

5 c Kathryn Bollinger, September 28, Since x cannot be negative: Since y cannot be negative: So, for the iron constraint to be meaningful (binding): Changing the amount of iron will, of course, change the profit as well. When we have 1000+h pounds of iron, we found the optimal solution to be at ( h, h) andsothe profit is: P =2x+1.5y= The amount by which the value of the objective function is improved if a constraining resource is increased by 1 unit is known as the shadow price for that resource.