Math 117 Chapter 3 Linear Programming

Transcription

1 Math 117 Chapter 3 Flathead Valley Community College Page 1 of 12

2 1. Recall a linear equation in standard form is Ax + By = C. All solutions to the equation can be represented by graphing the line. In the first chapter we looked at two ways to graph lines: 1. Rewrite the equation in slope-intercept form, y = mx+b. Plot the y intercept and use the slope=rise/run to find one more point on the line. Draw the line through the two points. 2. Find the x intercept, y = 0, and the y intercept, x = 0. Plot the intercepts and draw the line through the two points For the remainder of this chapter method 2 will be used. Method 2 is quick, simple and involves very little algebra. It is important to remember that every point on the graph of the line satisfies the equations. Solutions to inequalities consist of all points that satisfy the inequality. The task at hand now is to find all of the points in the plane that satisfy the inequality. Page 2 of 12 Linear Inequality A linear inequality is of the form or ax + by c, ax + by < c, ax + by c, ax + by > c

3 for real numbers a, b, and c, with a and b not both 0. The key to graphing a linear inequality is to understand that the linear equality ax + by = c divides the xy plane into two parts called half-planes. The inequality is satisfied on one side of the line, but not the other. Now one only has to check one point on one side of the line, usually (0, 0). If the inequality holds, all of that portion of the half-plane will satisfy the linear inequality. If not, the other side of the line satisfies the linear inequality Graphing Linear Equalities To graph the linear inequality first graph the equality, this is called the boundary of the solution region. If the inequality is a strict inequality, < or >, the boundary line is not included in the solution and is expressed with a dashed line on the graph. If the inequality is or the boundary line is drawn in solid on the graph. Once the boundary is set test one point, not on the line, to decide which half-plane is to be shaded. This process can be summarized in one simple table. Inequalty x intercept, y = 0 y intercept, x = 0 Test (0, 0) Page 3 of 12 For example, graph 4x 3y 12. Inequalty x intercept (y = 0) y intercept (x = 0) Test (0, 0) 4x 3y 12 (3, 0) (0, 4) 0 12, False

4 Plot the intercepts (3, 0) and (0, 4). Draw the solid line through the intercepts. Since (0, 0) does not satisfy the inequality, shade the region that does not contain the origin. The resulting graph should look similar the one below. y 5 5 (3, 0) 5 x (0, 4) 5 Page 4 of Systems of Like systems of equations, a system of linear inequalities is a collection of two or more inequalities. Just as with systems of linear equations, the solution to a

5 system of linear inequalities is the set of points in the xy plane that satisfy all of the inequalities. This solution is called the feasible region. To find the feasible region for a system of linear inequalities, graph the solution to each inequality and keep the region that satisfies all of the inequalities. Shading many different regions can get quite messy. One technique is to shade the regions that do not satisfy the inequality. Here the solution is the region that is not shaded. Another method is to draw arrows on the boundaries to indicate the appropriate regions. In the end shade the region included by all of the arrows. The table can be very useful when finding the feasible region. For example, graph the feasible region for the system of inequalities 3x 2y 6, x + y 5, y < 4. Use a table to organize your work Inequalty x intercept (y = 0) y intercept (x = 0) Test (0, 0) 3x 2y 6 (2, 0) (0, 3) 0 6, False x + y 5 ( 5, 0) (0, 5) 0 5, True y < 4 None, Vertical (0, 4) 0 < 4, True Now plot the intercepts, draw the lines and shade the appropriate half-planes to find the feasible region. As will be seen in the next section, it is important to find all corner points in feasible region. Some of these points will be determined from our table while others must be solved as systems of linear equations. For Page 5 of 12

6 example to find the corner point formed by the lines 3x 2y = 6 and x + y = 5 we could use the echelon method to solve the system or use an augmented matrix and reduced-row echelon form or even use the inverse matrix to find the corner points. Using the calculator yields rref [ 3 2 ] = [ 1 0 ] 4/ /5 So the corner point is at ( 4/5, 21/5). The other two corner points can be found by substituting y = 4 into the other equations to get points at ( 9, 4) and (14/3, 4). Whenever possible leave your answers as fractions. Some of your homework will ask you to round off your answers, read the instructions carefully. Once the lines and corner points have been plotted, the test point from the table will indicate the proper shading to satisfy the system of inequalities. Your final graph should look similar to the graph below. Page 6 of 12

7 y 5 (14/3, 4) ( 5, 0) 5 (2, 0) (0, 3) 5 x ( 4/5, 21/5) 5 (0, 5) Page 7 of 12

8 2. One of the most powerful uses of mathematics to optimize a given objective. This may be finding the minimum cost to produce some objects or maximizing the output of several products in manufacturing. In a linear programming problem one must find the maximum or minimum of a function, called the objective function, subject to a set of restrictions, called constraints, given as a system linear inequalities Optimizing the Objective Trying to optimize the objective over the entire xy plane seem like a very daunting task. The system of linear inequalities yields the region which greatly reduces the area we must search for our solution. While the feasible region has narrowed the search area in which to find a solution, there are still a great number of points to check. The number of points is greatly reduced if we apply a nice theorem (that will not be proven here). Page 8 of 12 Corner-Point Theorem If an optimum value (either a maximum or a minimum) of the objective function exists, it will occur at one or more of the corner points of the feasible region. Now, this really narrows down our search. Once the corner points of the feasible region have been found, one need only check the points in the objec-

9 tive function and choose the largest (maximum) or smallest (minimum) value to optimize the objective. Again a table will be most useful. Corner Point Value of Objective Function The procedure to solve a linear programming problem is summarized below. Solving a Problem 1. Write the objective function and all necessary constraints. 2. Graph the feasible region. Use the intercept/test table to assist in graphing and speed up your work. 3. Identify the corner points. You must show your work in finding the linear system solutions. Looking at the graph is not enough. 4. Find the value of the objective function at each corner point. Again, a table will greatly help your efforts. 5. Choose the optimal value of the objective needed to solve the problem. For unbounded feasible regions it is important to verify that a solution exists before picking the appropriate corner point. Page 9 of 12

10 3. Applications of After some practice finding solutions to linear programming problems is fairly straight forward. The real challenge comes when the objective and constraints are not given, but must be pulled out of an application problem. These problems are often called word problems and usually tend to be more troublesome than the mathematics used to solve the problem. Usually the biggest mistake people make is trying to solve the problem before the problem is completely understood. Here are a few guidelines to assist in your problem solving. Four Step to Problem Solving 1. Understand the Problem Most students will try to jump to step 2 or 3 before really understanding what the problem is asking and what information is given. It often takes two or three reading to totally understand the problem. There are two main items that must be established to truly begin to understand a word problem (a) What exactly is the problem asking for? For each problem you need to complete the following sentence To solve this problem I need to find... It will do no good to copy the original text. You must be precise and finish the sentence in your own words. (b) What information am I given? Once the goal is understood it is time to record all the information given in the problem. First it is a good idea to define your variables, the unknowns, needed to solve the prob- Page 10 of 12

11 lem. These definitions need to be as precise as possible even including the units of the quantity in question. After the statement of the problem the variables should be defined by a statement Let x =..., y =... Next organized the given data. Tables can be a very efficient tool to clearly lay out all of the given information Understanding a problem is probably the most important, yet most time consuming part of problem solving. Just remember, the time invested in this step will save a great deal of frustration and wasted time from jumping directly to steps 2 or Write the Mathematics Once the problem has been stated, variables defined and data organized, translating the word problem into a math problem should be relatively easy. For linear programming clearly write the objective function (as an equation) and the constraints (as inequalities). 3. Solve the Problem Now it is time to actually solve the problem. A major source of frustration for most math students is not actually solving the problem, but the fact that do not want to take the time to work through the first two steps. They try to solve the problem before the equations have been developed. Don t fall into this trap. If you have completed the previous two sections, you can do the mathematics. Now go back and totally understand what it is you are solving. 4. Check your Solution Before moving to the next word problem be sure you ask yourself if your solution makes sense. First go back to make sure Page 11 of 12

12 you have answered the original question. Finally, and most important, make sure your answer is reasonable and within the constraints of the problem. Page 12 of 12