Understanding Feasible Regions in Linear Programming Problems

J.B. Hunt Business Feasible region is the overlapping area common to all of the constraints of the problem. This area may be a closed region called bounded feasible region , or it can be open from at least one side and is called unbounded region . In the closed feasible region, you always have at least one optimal solution if not more. The optimal solution(s) can only happen at the intersection of any two constraints called node or corner . There are cases that the optimal solution has infinite number of same value solution. In these cases, the objective function is parallel to at least one of the given constraints. In this section, you will be presented an original scenario. You will then revise the linear programming model to a new example. Original Scenario Consider the following scenario: You are in a business that assembles either bicycles or tricycles. You can sell each bicycle for $150 and each tricycle for $200. The time taken to assemble a bicycle is 4 hours and that of a tricycle is 5 hours. You have a total of 200 hours to spend assembling the bikes. You have necessary parts for assembling at most 25 bicycles and at most 32 tricycles. The first part of developing the mathematical modeling is to formulate the objective function and constraints. To properly formulate the problem, you need to assign a variable such as x to the number of bicycles assembled and another variable such as y to the number of tricycles assembled. The objective function is written as: Maximizing revenue = $150x + $200y subject to the following constraints 4x + 5y ≤ 200 hours x ≤ 25 bicycles y ≤ 32 tricycles x ≥ 0 bicycle y ≥ 0 tricycle Graphing the constraints, you see the first constraint has the y-intercept at (0, 40) and the x-intercept at (50, 0). The second constraint is a vertical line with the x-intercept of (25, 0). The third constraint is a horizontal line with the y-intercept of (0, 32). The fourth and the fifth constraints are the x and y

J.B. Hunt Business coordinates. If you solve the first and second constraint equations, you get the intersection (25, 20). If you solve the first and the third constraint equations, you get the intersection (10, 32). The feasible region is bounded between these five segment lines and has corners of (0, 0), (0, 32), (10, 32), (25, 20), and (25, 0). Any point within the feasible region is a solution. This means that each point inside the region satisfies all five constraints. There are infinite numbers of solution for this problem. However, you are trying to maximize revenue. The maximum or minimum values always happen at the corners. If you apply the five corners coordinates into the objective function, you get the revenue for $0 at (0, 0), $6,400 at (0, 32), $7,900 at (10, 32) $7,750 at (25, 20), and $3.750 at (25, 0). Comparing these prices, you can conclude that the maximum revenue happens at the corner with coordinate of (10, 32). This means that you can achieve maximum revenue if you assemble only 10 bicycles and all 32 tricycles. Therefore, the optimal solution is to assemble 10 bicycles and 32 tricycles. There is no other corner(s) or along any of the five segment lines to achieve either the maximum $7,900 or more. Figure 1 shows the original scenario feasible region. Figure 1: Original Scenario Feasible Region Revised Scenario The objective function is as follows: Maximizing revenue = $150x + $200y subject to the following constraints 4x + 5y ≤ 200 hours x ≤ 25 bicycles y ≤ 32 tricycles x ≥ 10 bicycles y ≥ 28 tricycles

J.B. Hunt Business Figure 2 shows the new feasible region. Figure 2: New Feasible Region As you see, the lower limits for decision-making variables x and y rise from 0 to 10 and 28 respectively. You will then graph the feasible region by plotting the constraints lines. You may notice that the only difference between the old and new feasible regions is the movement of x- and y-axis to their new positions at x = 10 and y = 28. The intersection of x = 10 and 4x + 5y = 200 produces the corner of (10, 32) as you had the corner in the old feasible region. However, the intersection of y = 28 and 4x + 5y = 200 creates a new corner of (15, 28). Therefore, the new feasible region is now bounded by corners of (10, 28), (10, 32), and (15, 28). Figure 2 shows the new feasible region. Comparing the Feasible Regions As you compare these two feasible regions, you can certainly see the shape and the amount of area of the two feasible regions have changed substantially. Calculating the value of the objective function at these new nodes provides the answer of $7,100 at (10, 28), $7,900 at (10, 32), and $7,850 at (15, 28). The problem has only one optimal solution. In the maximization problems, the optimal solution is usually on the node farther from the origin of the coordinate system. In the minimization problems, the optimal solution is usually on the node closest to the origin. Where is the optimal solution and how much is it? You are in the business of making shirts and skirts. The profit on each shirt is $40 and on each skirt is $60. The number of skirts should not exceed 50. It takes 1 hour to make a shirt and 2 hours for a skirt. Both shirt and skirt utilize 1 square yard of fabric each. You have a total of 125 hours of labor and 100 square yards of fabric. At (100, 0) and $4,000 At (25, 50) and $4,000 At (0, 50) and $3,000 At (75, 25) and $4,500 Which of the following is true about the feasible region in a linear programming model?

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

J.B. Hunt Business It will always be bounded. It is not affected by the constraints. Every point in the feasible region satisfies all constraints. It is the solution to the objective function. A constraint does not represent the intersection of any two constraints. Which of the following is the correct constraint for the given problem? You are in the business of making shirts and skirts. The profit on each shirt is $40 and on each skirt is $60. The number of skirts should not exceed 50. It takes 1 hour to make a shirt and 2 hours for a skirt. Both shirt and skirt utilize 1 square yard of fabric each. You have a total of 125 hours of labor and 100 square yards of fabric. 40x + 60y ≥ 0 x ≥ 0 and y ≥ 0 x + y > 100 Maximizing profit = $40x + $60y Write the objective function for the following problem. You are in the business of making shirts and skirts. The profit on each shirt is $40 and on each skirt is $60. The number of skirts should not exceed 50. It takes 1 hour to make a shirt and 2 hours for a skirt. Both shirt and skirt utilize 1 square yard of fabric each. You have a total of 125 hours of labor and 100 square yards of fabric. x + y ≤ 100 Maximizing profit = 40x + 60y y ≤ 50 x + 2y ≤ 125 If the feasible region is open from at least one side, it is unbounded . Which of the following is the definition of the feasible region? It is the overlapping area that is common to all of the constraints. It is the non-overlapping area of the graphs of the individual constraints. It is an unbounded region. It is the vertex that contains the optimal solution. A company produces gizmos and gadgets. Each gizmo sells for $15, and each gadget sells for $18. It takes 8 hours to produce a gizmo, and it takes 6 hours to produce a gadget. The company has a maximum of 240 hours each week for the production of

J.B. Hunt Business gizmos and gadgets. The company has the equipment to produce at most 24 gizmos and at most 30 gadgets. If x represents the number of gizmos produced and y represents the number of gadgets produced, which of the following represents the objective function? f(x, y) = 8x + 6y f(x, y) = 15x + 18y f(x, y) = 24x + 30y 8x + 6y = 240 Which constraint is redundant? You are in the business of making shirts and skirts. The profit on each shirt is $40 and on each skirt is $60. The number of skirts should not exceed 50. It takes 1 hour to make a shirt and 2 hours for a skirt. Both shirt and skirt utilize 1 square yard of fabric each. You have a total of 125 hours of labor and 100 square yards of fabric. x + 2y ≤ 125 x ≥ 0 x + y ≤ 100 Limitation on the number of skirts The constraints for a given linear programming model are as follows: 4b + 5t ≤ 100 b ≤ 20 bicycles t ≤ 25 tricycles b ≥ 0 bicycle t ≥ 0 tricycle Which of the following is the intersection of the equations for the first and second constraints? (25, 0) (0, 20) (20, 4) (0, 0) Which resources do you exhaust at the optimal solution?

J.B. Hunt Business You are in the business of making shirts and skirts. The profit on each shirt is $40 and on each skirt is $60. The number of skirts should not exceed 50. It takes 1 hour to make a shirt and 2 hours for a skirt. Both shirt and skirt utilize 1 square yard of fabric each. You have a total of 125 hours of labor and 100 square yards of fabric. Square yard of fabric Hours of labor Both the hours of labor and square yard of fabric Neither the hours of labor nor the square yard of fabric Which of the following is true about the feasible region in a linear programming model? It will always be bounded. It is the solution to the objective function. Every point in the feasible region is an optimal solution. It is bounded by the constraints. Which of the following refers to the intersection of any two constraints? Unbounded point Interior point Node Feasible region Which of the following is true when the constraints are changed for a linear programming model? The objective function will change. The optimal solution may change. The feasible region will not be affected. The optimal solution will always be different. Which of the following is true when the constraint 6x + 3y ≤ 84 is graphed? The x-intercept is (0, 28). There will be no y-intercept. The y-intercept is (14, 0). The line created will be a boundary of the feasible region. Which of the following refers to the intersection of any two constraints? Feasible region Interior point Vertex Unbounded point Which of the following could be true when some constraints are changed in a linear programming model? The feasible region is totally revised, and the optimal solution is revised. The feasible region will always be bounded. The feasible region will be undefined.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

J.B. Hunt Business The optimal solution will always be changed. Which of the following is true about the feasible region in a linear programming model? It is the solution to the objective function. Every point in the feasible region is an optimal solution. It is not affected by the constraints. It can be bounded or unbounded. Which vertex will minimize the objective function f(x, y) = 12x + 15y? (5, 5) (8, 1) (1, 8) (7, 3) Which of the following is a possible result when one or more constraints are changed in the linear programming model? The optimal solution is at the same vertex. The feasible region will be the same. The objective function is revised. The feasible region will become unbounded. Which of the following is true about an optimization problem? The optimal solution could change even if the parameters of the constraints are unchanged. For a minimization problem, the optimal solution is usually at the vertex that is closest to the origin. The optimal solution is always unchanged even if the constraints are changed. For a minimization problem, the optimal solution is usually at the vertex that is farthest from the origin. What are the corners of the feasible region? You are in the business of making shirts and skirts. The profit on each shirt is $40 and on each skirt is $60. The number of skirts should not exceed 50. It takes 1 hour to make a shirt and 2 hours for a skirt. Both shirt and skirt utilize 1 square yard of fabric each. You have a total of 125 hours of labor and 100 square yards of fabric. (0, 0), (0, 50), (25, 50), (75, 25), and (100, 0) (0, 0), (0, 62.5), and (125, 0) (0, 0), (0, 50), (50, 50), and (100, 0) (0, 0), (0, 100), and (100, 0) How much should you raise the profit of skirt to have two optimal solutions?

J.B. Hunt Business You are in the business of making shirts and skirts. The profit on each shirt is $40 and on each skirt is $60. The number of skirts should not exceed 50. It takes 1 hour to make a shirt and 2 hours for a skirt. Both shirt and skirt utilize 1 square yard of fabric each. You have a total of 125 hours of labor and 100 square yards of fabric. You cannot have two optimal solutions $140 $80 $20 How many constraint inequalities do you have for the given problem? You are in the business of making shirts and skirts. The profit on each shirt is $40 and on each skirt is $60. The number of skirts should not exceed 50. It takes 1 hour to make a shirt and 2 hours for a skirt. Both shirt and skirt utilize 1 square yard of fabric each. You have a total of 125 hours of labor and 100 square yards of fabric. 3 2 5 4 Which of the following is true when the constraints are changed for a linear programming model? The optimal solution may stay the same. The optimal solution will always be different. The objective function will change. The optimal solution will never change. If a constraint in a linear programming model must be changed, it is possible that the optimal solution will be the same .

J.B. Hunt Business

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

MGMT455 unit 2 intellipath

Related Documents