Many applications of mathematics involve systems of inequalities rather than systems of equations. We will discuss solving (graphing) a single linear inequality in two variables and a system of linear inequalities in two variables. Graphing Linear Inequalities in Two Variables. Terminology. A line divides the plane into two regions called half-planes. A vertical line divides the plane into left half-plane and right half-plane. A non-vertical line divides the plane into upper half-plane and lower half-plane. In either case, the dividing line is called the boundary line of each half-plane. Problem #. Make sketch of the following line and label half-planes. a) x = 3 b) y = 5 c) y = 3x 4 d) y = x + 3
Problem #. Graph the following inequality. a) y > 5 b) y 5 c) y < 5 d) y 5
e) x 4y f ) 3x y g) 3x+ y 5 h) x y > 3 3
Solving Systems of Linear Inequalities (graphically). Terminology. To solve a System of Linear Inequalities means to find the xy, that graph of all ordered pairs of real numbers ( ) simultaneously satisfy all the inequalities in the system. The graph is called the solution region for the system, or feasible region. Problem #3. Solve the following system of linear inequalities graphically. 3x + y 4 x y 4
Problem #4. Solve the following system of line inequalities graphically. 3x+ y 4 0.5x+ y x+ y 5 x 0 y 0 y = 3x + 4 y = 0.5x + y = x + 5 Small shaded triangle is the feasible region for the system of inequalities. P P 5
Corner points. A corner point of a solution region is a point in the solution region that is the intersection of two boundary lines. Problem #5. Find coordinates of corner points for solution region in Problem #4. Corner points of the feasible region are P ( ) 0, - y-intercept of the line 0.5x+ y=, 6 0 P, - point of intersection of the lines 7 7 3x + y = 4.5x+ y= P 3 (0, 4) - y intercept of the line 3x+ y = 4 Bounded and Unbounded Solution Region. A solution region of a system of linear inequalities is bounded if it can be enclosed within a circle. If it cannot be enclosed within a circle, it is unbounded. 6
Applications. Resource allocation. Problem #6. A manufacturing plant makes two types of inflatable boats, a two-person boat and a four-person boat. Each two-person boat requires 0.9 labor-hour of the cutting department and 0.8 labor-hour in the assembly department. Each four-person boat requires.8 labor-hours of the cutting department and. labor-hours in the assembly department. The maximum labor-hours available each month in the cutting and assembly departments are 864 and 67 respectively. If x two-person boats and y four-person boats are manufactured each month, write a system of linear inequalities that reflect the conditions indicated. Find the feasible set graphically. 0.9x +.8y 864 x 0 0.8x+.y 67 y 0 7
Linear programming is a mathematical process that has been developed to help management in decision making. We will continue work with the Problem #6. Conditions of the manufacturing are translated into the following system of linear inequalities. Variables x and y were changed to and x. x 0.9x+.8x 864 0.8x+.x 67 x 0 x 0 Feasible region for this system is shown on the right. Additional INFO. The company makes a profit of $5 on each two-person boat and $40 on each four-person boat. Q uestion: How many boats of each type should be manufactured each month to maximize the total monthly profit, assuming that all boats can be sold? 8
The objective of management is to maximize profit. Since the profits for two-person boat and four-person boat differ, management must decide how many of each type of boat to manufacture. x = number of two-person boats and x = number of two-person boats are called decision variables. We can form the objective function P = 5x + 40x The objective is to find values of the decision variables that pr oduce the optimal value (in this case, maximum value) of the objective function. Following inequalities are problem constrains. 0.9x +.8x 864 0.8x +.x 67 Nonnegative constrains are x 0 x 0 9
Mathematical formulation of the problem. Maximize P = 5x + 40x (objective function) subject to 0.9x +.8x 864 0.8x +.x 67 x 0 x 0 Problem constraints Nonnegative constraints By choosing a production schedule x, x from the ( ) feasible region, a profit can be determined using the objective function P = 5x + 40x For example, if x = 00 and x = 300 the monthly profit would be ( ) ( ) P = 5 00 + 40 300 = $7,000 If x = 00 and x = 400 the monthly profit would be ( ) ( ) P = 5 00 + 40 400 = $8,500 0
But the question is, out of all possible production schedules x, x from the feasible region, which schedule(s) ( ) produces the maximum profit? We have a maximization problem. Point-by-point checking is not possible. Fundamental Theorem of Linear Programming. If the optimal value of the objective function in a linear programming problem exists, then the value must occur at one (or more) of the corner points of feasible region. Existence of Optimal Solutions.. If the feasible region for a linear programming problem is bounded, then both the maximum value and minimum value of the objective function always exist.. If the feasible region is unbounded and the coefficients of the objective function are positive, then the minimum value of the objective function exists, but the maximum value does not. 3. If the feasible region is empty, then both the maximum value and minimum value of the objective function do not exist.
Problem of boats manufacturing (finish). Feasible region is bounded. Solution of the optimization problem exists. C Corner points of the feasible region. C To find corner points we need work with equations of boundary lines. 0.9x+.8x = 864 0.8x +.x = 67 ( ) ( 0, 0 ) C : 0, 480 is the intersection of the line 0.9x+.8x = 864 with the vertical axis x = 0. C 3 C : ( 480, 40 ) is the intersection of two straight lines, solution of the system of two equations. ( ) : 840, 0 3 is the intersection of the line 0.8x +.x = 67 with the horizontal axis x = 0. C
x x P = 5( x ) + 40( x ) 0 480 9,00 480 40,600 840 0,000 Answer. Schedule production that maximizes profit: 480 two-person boats and 40 four-person boats per month. Maximum profit per month $,600. 3
Linear Programming: General Description. Constructing the Model for an Applied Linear Programming Problem.. Introduce decision variables.. Summarize relevant material (in table form). 3. Determine the objective and write a linear objective function. 4. Write problem constraints using linear equations and inequalities. 5. Write nonnegative constraints. Geometric Solution of Linear Programming Problems.. Graph the feasible region. If an optimal solution exists (according to the formulated criteria), find the coordinates of each corner point of feasible region.. Construct a corner point table listing the values of the objective function at each corner point. 3. Determine the optimal solution from the table constructed. 4. For an applied problem, interpret the optimal solution(s) in terms of the original problem. 4