1 GIAPETTO S WOODCARVING PROBLEM

Transcription

1 1 GIAPETTO S WOODCARVING PROBLEM EZGİ ÇALLI OBJECTIVES CCSS.MATH.CONTENT.HSA.REI.D.12: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. MoNE: MATERIALS Graphing display calculator MS Excel 1

2 Graphing Calculator Activities: For Teachers by Student Teachers ACTIVITY 1 Introduction The following problem appears in Winston s operations research book (2004). Giapetto s Woodcarving, Inc., manufactures two types of wooden toys: soldiers and trains. A soldier sells for $27 and uses $10 worth of raw materials. Each soldier that is manufactured increases Giapetto s variable labor and overhead costs by $14. A train sells for $21 and uses $9 worth of raw materials. Each train built increases Giapetto s variable labor and overhead costs by $10. The manufacture of wooden soldiers and trains requires two types of skilled labor: carpentry and finishing. A soldier requires 2 hours of finishing labor and 1 hour of carpentry labor. A train requires 1 hour of finishing labor and 1 hour of carpentry labor. Each week, Giapetto can acquire all of the needed raw material, but he is only allotted 100 finishing hours and 80 carpentry hours. There is an unlimited demand for trains. However, at most, 40 soldiers are sold each week. Giapetto wants to maximize his weekly profit (Revenues - Costs). Formulate a mathematical model for Giapetto s situation that can be used to maximize Giapetto s weekly profit. (Winston, 2004). 2

3 Instructions Step 1: Understand the problem Help Giapetto with his decision making by defining a scientific approach using mathematical models. In order to do this, you first need to understand your objective. Questions 1) What is your objective in solving this problem? Next, you need to collect data on Giapetto s problem. 2) What factors influence Giapetto s weekly profit? 3) Use your intuition to guess the optimal solution. Instructions Step 2: Formulate the problem The variables for which the decision maker can control the values and that have an influence on your objective are called decision variables. Questions 4) Define the decision variables: X= 3

4 Graphing Calculator Activities: For Teachers by Student Teachers Y= Now, write a function using the decision variables. The function to be minimized or maximized is called the objective function. Giapetto s objective is to choose X and Y to maximize the objective function. 5) Write the objective function in terms of X and Y: Hint : Weekly revenues $ soldier $ trains soldier week train week Weekly raw material cost $ soldier $ trains soldier week train week Weekly variable cost $ soldier $ trains soldier week train week Remember there are always economic tradeoffs. (A tradeoff is a situation that involves forfeiting a quality or aspect of a product in return for gaining another quality or aspect). The decision maker needs to determine which decision variable values are possible and which are not. The restrictions on the decision variable values are called constraints. Constraints can be expressed as algebraic inequalities. 6) Determine the constraints in terms of X and Y: Constraint 1: Each week, no more than 100 hours of finishing time may be used. 4

5 Constraint 2: Each week, no more than 80 hours of carpentry time may be used. Constraint 3: Due to limited demand, a maximum of 40 soldiers should be produced each week. Constraints 4 & 5: Can the decision variables be negative? (Warning: Be mindful of the terms units in a constraint. They should all be the same. Otherwise, the constraint will be meaningless). Instructions Step 3: Solve the problem. As you can observe, we have a linear objective function. Similarly, our constraints are linear inequalities. The above methodology that we applied to Giapetto s problem is called Linear Programming. A linear programming problem is an optimization problem. Now, we need to find the values of X and Y, so that the objective function is maximized. 5

6 Graphing Calculator Activities: For Teachers by Student Teachers Instructions ACTIVITY 2 Any linear program with only two variables can be solved graphically. Use your TI calculators to solve Giapetto s problem graphically. You will need to find the intersection set of all of the constraints, also known as the feasible region. First run the inequalities application Inequalz. Press [APPS]. Select Inequalz. At the bottom of the screen, you will see the inequality signs. In order to choose an inequality sign, use ALPHA and then select whichever of the following buttons corresponds to the related sign. For example, in order to select the sign, press [ALPHA] and then [ZOOM] (i.e. F3). In order to type in the calculator, you will need to rewrite your constraints so that Y is alone on one side of the inequality. Before you begin the solution, adjust the scales of the graph. Press [WINDOW] and adjust the min, max, and scl values as follows. 6

7 Now, write the first inequality (Constraint 1) and graph it. Press GRAPH. You can see the graph of the first inequality as an example. The restricted area for the first constraint is shaded. Now, enter the other constraints. 7

8 Graphing Calculator Activities: For Teachers by Student Teachers Note: You have inequalities with no Y value. How can you enter them? Look at the upper left corner of the screen. Move the cursor to X= sign and press [ENTER]. In this screen, you can easily enter your constraints that do not have a Y. Don t worry; your previous constraints are still saved. Now press [GRAPH] to view the shaded inequalities. You will see the shaded region for all five inequalities is interwoven. In order to easily view the feasible region, you will need to make minor adjustments to the shading. To select Shades on the screen, press F1 (ALPHA and Y=). This will provide you with some options for shading. 8

9 To answer this question, we need to find the feasible region, i.e. the intersection of all constraints. Therefore, Press 1. Congratulations! You have found the feasible region. This very useful theorem will help you to find the exact solution for which you are looking. Fundamental Theorem of Linear Programming: The minima and maxima of a linear function on a convex feasible region occur at the vertices of the region. Warning: Be aware of whether or not the vertices have integer coordinates. If all of the vertices have integer coordinates, we only need to consider these points when determining the maximum or minimum. However, if the vertices are not all integer points and we are only accounting for integer points, we need to be more careful. As noted in this question, the feasible region has five vertices. How will you determine the exact coordinates of these points? 9

10 Graphing Calculator Activities: For Teachers by Student Teachers At the bottom of the screen, you will see the PoI-Trace button. To select this button, press F3 (ALPHA and ZOOM). Here you will find the first intersection point, which is vertex #1, the intersection of Y1 and Y2. You can find the other intersection points by using the up or down arrow keys. Questions 7) Write down the coordinates of all five vertices below. ( X, Y ) Vertex #1: (, ) Vertex #2: (, ) Vertex #3: (, ) Vertex #4: (, ) Vertex #5: (, ) 10

11 8) Now that you have identified the possible optimal solutions, you can find the objective function value for each of them. Find the maximum value of the objective function and the corresponding decision variable values. Vertex #1: (, ) weekly profit = Vertex #2: (, ) weekly profit = Vertex #3: (, ) weekly profit = Vertex #4: (, ) weekly profit = Vertex #5: (, ) weekly profit = 9) OPTIMAL SOLUTION: 11

12 Graphing Calculator Activities: For Teachers by Student Teachers Instructions ACTIVITY 3 You can solve the same question by using MS Excel Solver Add-in. First, enter the constraint coefficients and objective function coefficients as follows. Reserve empty cells for the decision variables. Reserve a cell for the maximum value of the objective function. Soldiers/week (X) Trains /week (Y) Decision variables Constraint 1 - Finishing constraint Constraint 2- Carpentry constraint Constraint 3 - Constraint on demand Constraint 4 - Sign restriction Constraint 5 - Sign restriction Objective Function - Weekly profit 3 2 RED For each constraint, enter the SUMPRODUCT of the decision variables and the constraint coefficients in column D. Adjust the decision variables by using the $ sign. Similarly, insert the 12

13 SUMPRODUCT of the decision variables and the objective function coefficients in cell D8. Now, from the Data tab, select Solver. (If you don t have Excel Solver, you may need to download it). 13

14 Graphing Calculator Activities: For Teachers by Student Teachers In the figure shown below, you will see the necessary cells are selected. Select Simplex LP for the Solving Method. When you click on Solve, you will see the optimal solution in the red-coded cell. Questions Step 4: Review 10) Compare the results you found by using TI calculator and Excel. Are they the same? (Hint: You should arrive at the same result using either method.) 14

15 11) Compare the results you found 3 with your intuitive answer. To what degree do these results differ? For what reasons do you believe these results are different? REFERENCES Winston, W. L. (2004). Operations research: Applications and algorithms. Belmont, CA: Thomson Learning. 15