Section 3.4 A Standard Maximization Problem has the following properties: The objective function is to be maximized. All variables are non-negative. Fall 2017 Math 140 Week-in-Review #5 courtesy: Kendra Kilmer (covering Sections 3.4 and 4.1) Each constraint can be written as variables non-negative constant. The Simplex Method 1. Set up the initial simplex tableau. (a) Convert each constraint into an equation by introducing a slack variable. (b) Rewrite the objective function so that: the coefficent of P is 1. all variables and P are on the same side of the equal sign. (c) Place the constraints and the objective function in the initial simplex tableau. 2. Determine whether or not the optimal solution has been reached. The optimal solution has been reached if all entries in the last row to the left of the vertical line are non-negative. If the optimal solution has been reached, skip to step 4. If the optimal solution has not been reached, proceed to step 3. 3. Perform Pivot Operations (a) Locate pivot element. Pivot Column: The column where the most negative entry to the left of the vertical line in the last row lies. Pivot Row: For each row, except the last, divide the entry in the last column by the corresponding entry in the pivot column. Select the row with the smallest (non-negative) ratio. (Ignore negative ratios, the ratios that come from dividing by zero, and the zero ratios that are divided by a negative number.) Pivot Element: The element shared by the pivot row and pivot column. (b) Pivot about the pivot element. We can use what we learned in Section 1.3 but to save time we can use the following webpage: http://www.zweigmedia.com/realworld/tutorialsf1/scriptpivot2.html (c) Return to step 2. 4. Determine the solution: The value of the variable heading each unit column is given by the entry lying in the column of constants in the row containing a 1. The variables heading columns that are not in unit form are assigned the value 0. 1. Solve the following problem using the simplex method. Maximize P = 2x + 3y Subject to 3x + 5y 20 4x + 2y 15 x 0,y 0 1
2. Solve the following problem using the simplex method. Maximize R = 2x + 7y z Subject to y 2(x + z) 2x + 4y + 5z 10 x 2y + z 9 x 0,y 0,z 0 3. Melanie has decided that she wants to start a candle business. She has discovered that a small candle requires 5 oz of wax, 1 oz of scent, and a 1 inch wick. A medium candle requires 7 oz of wax, 2 oz of scent, and a 2 inch wick. A large candle requires 9 oz of wax, 3 oz of scent, and a 3 inch wick. She has 1300 oz of wax, 380 oz of scent, and 380 inches of wick. If she can sell the small candles for $6, medium candles for $8, and large candles for $10, how many candles of each size should she make to maximize her revenue? Are there any resources leftover? 2
4. What is the initial simplex tableau for the problem below? Maximize R =.06x +.1y +.15z Subject to x + y + z 250000 z.25(x + y + z) y.5(x + y + z) x 0,y 0,z 0 5. Decide whether or not the following simplex tableaus are in final form. If the tableau IS in final form, what is the solution? If the tableau IS NOT in final form, determine the next pivot element. (a) x y z s 1 s 2 s 3 P 3 4 5 1 0 0 0 624 1 0.5.75 0 1 0 0 130 1 0 0 0 0 1 0 70 4 5 6 0 0 0 1 0 (b) x y z s 1 s 2 P 0 1 0 1 1 0 20 0 5 1 2 0 0 5 1 0 0 4 0 0 8 0 1 0 3 0 1 60 Section 4.1 An experiment is an activity with an observable result. The outcome is the result of an experiment. The sample space is the set of all possible outcomes of an experiment. There are two ways in which we can represent a set. 1. Roster Notation: Lists each element between curly braces. 2. Set-builder Notation: A rule is given that describes the property an object x must satisfy to qualify for membership in the set. An event is a subset of the sample space. (Note: An event E is said to occur whenever E contains the observed outcome.) An event which consists of exactly one outcome is called a simple (elementary) event of the experiment. We can use tree diagrams to help us determine all of the possible outcomes in an experiment. If S is the sample space and E is an event of the experiment, then E c consists of all of the outcomes that are in S but not in E and is called the complement of E. The intersection of events E and F, E F, is the set of all outcomes that belong to both E and F. The union of events E and F, E F, is the set of all outcomes that belong to E or F. We can use Venn Diagrams to visually represent set operations. The Sample Space, S, is denoted by a rectangle. Events of the experiment are represented by circles inside the rectangle. The empty set, denoted by /0 or {}, is called the impossible event. Let S be the sample space. The event S is called the certain event. E and F are mutually exclusive if E F = /0. 3
6. A group of students is asked the number of home football games that they have attended (there have been four) and whether they live in a dorm or apartment. (a) What is the sample space for this experiment? (b) Find the event, E, that they have attended fewer than three games. (c) Find the event, F, that they live in an apartment. (d) Find the event, E F c, and describe it in words. (e) Find the event, E c F c, and describe it in words. 7. For each event below, draw a two circle Venn Diagram and shade the region(s) that represent the event. (a) E c (b) E F c (c) E F c (d) (E c F) c 8. The numbers 1, 4, 7, and 12 are written on separate pieces of paper and placed in a container. An experiment consists of drawing two pieces of paper out of the box and observing the sum of the numbers. (a) What is the sample space for this experiment? (b) Find the event, E, that the piece of paper with a 1 on it is drawn. (c) Find the event, F, that the piece of paper with a 7 on it is not drawn. 4
(d) Find the event, G, that the sum of the numbers is odd. (e) Are F and G mutually exclusive? (f) Find (E F c ) G. (g) Find (F G c ) c E 9. An experiment consists of drawing two marbles out of a box that contains five maroon and six white marbles and observing the colors of the marbles in your hand. (a) What is the sample space? (b) Find the event, E, that at least one white marble is drawn. (c) Find the event, F, that at least three maroon marbles are drawn. (d) Find the event, G, that at most one maroon marble is drawn. (e) Are the events E and G mutually exclusive? (f) Are the events E and F mutually exclusive? (g) Find all events of this experiment. Which of these events are simple events? 10. A pair of fair four-sided dice is rolled (one red and one green) and the number that falls uppermost on each die is observed. (a) How many outcomes are in the event the sum of the numbers is at least six? (b) How many outcomes are in the event the red die shows a three and the green die shows a number less than three? (c) How many outcomes are in the event at least one die lands on a two or the sum is at most four? (d) How many outcomes are in the event the red die does not show a one or the green die shows a number less than three? 5
Multiple Choice and True/False Questions: 11. Determine whether each of the following statments is True or False. Use the following information for the first four statements: Jar A contains 3 red marbles, 2 yellow marbles, and 4 green marbles. Jar B contains 3 slips of paper numbered 1 through 3. An experiment consists of drawing one marble out of Jar A and observing the color and drawing one slip of paper out of Jar B and observing the number. (a) The sample space for the experiment described above has 9 outcomes. (b) If E is the event that a green marble is selected, then there are three outcomes in E. (c) If E is the event that a green marble is selected and F is the event that the number 1 is selected, then E and F are mutually exclusive events. (d) The experiment described above has 512 events. (e) In the simplex method, it is possible that the pivot row has a ratio of 1/2. 12. Given the final Simplex table, what is the solution? (a) x = 2,y = 3,z = 4,u = 0,v = 0,w = 0,P = 20 (b) x = 2,y = 0,z = 3,u = 0,v = 4,w = 0,P = 20 (c) x = 4,y = 2,z = 3,u = 0,v = 0,w = 0,P = 20 (d) x = 4,y = 0,z = 2,u = 0,v = 3,w = 0,P = 20 (e) None of the above 13. Given the following simplex tableau: Which of the following is a TRUE statement? (a) The optimal solution has been reached. (b) The next pivot element is in row 2 column 3. (c) The next pivot element is in row 4 column 1. (d) The next pivot element is in row 1 column 3. (e) The next pivot element is in row 4 column 3. 14. Which of the following is a standard maximization problem? (i) Maximize P = 4x + 2y + z Subject to 3x + y + 2z 8 x + y + 3z 6 x 0,y 0,z 0 (ii) Maximize P = 3x + 9y Subject to 3x + 8y 86 3x + 7y 62 2y + x y x 0,y 0 (iii) Maximize P = 3x + y Subject to 3x + 9y 78 5x + 3y 16 x 0,y 0 x y z u v w P 0 5/8 1 2 0 1/3 0 2 0 2/5 0 3 1 2/3 0 3 1 1/3 0 4 0 1/3 0 4 0 2 0 5 0 3 1 20 3 4 5 1 0 0 0 624 1 2 10 0 0 0 0 725 1 0.5 0.75 0 1 0 0 130 1 0 0 0 0 1 0 70 4 5 6 0 0 0 1 0 (a) Problem (i) and (iii) only (b) Problem (ii) only (c) Problem (iii) only (d) Problems (ii) and (iii) only (e) None of the above 6