Data Optimization 2 - Linear Programming Test With Verified Correct Complete Answers/2023.

What is the objective in all linear programming problems? - correct answer The maximization or minimization of some quantity A linear programming model is a mathematical model with the following characteristics: (3) - correct answer 1. A linear objective function that is to be maximized or minimized 2. A set of linear constraints 3. Variables that are all restricted to nonnegative values Steps for developing a mathematical model: (7) - correct answer 1. Understand the problem thoroughly 2. Describe the objective -- maximize/minimize? 3. Describe each constraint 4. Define the decision variables i.e. the controllable inputs 5. Write the objective in terms of the decision variables -- *Objective function* e.g. where is the profit contribution coming from 6. Write the constraints in terms of the decision variables (+ the *nonegativity constraint*) e.g. 1S + 2D = 630 (capacity) 7. Write the mathematical statement, structured like: - MAX or MIN - subject to (s.t.) - constraints - nonnegativity constraint Steps of the graphical solution procedure for a maximization problem (5) - correct answer 1. Prepare a graph of the feasible solutions for each of the constraints. 2. Determine the feasible region by identifying the solutions that satisfy all the constraints simultaneously. 3. Draw an objective function line showing the values of the decision variables that yield a specified value of the objective function. - Set the objective function equal to an arbitrary constant - Solve for 2 points by equating each variable to 0 and then solving for the other variable - Draw line 4. Move parallel objective function lines (using arbitrary value for the equation solution) toward larger objective function values until further movement would take the line completely outside the feasible region. 5. Any feasible solution on the objective function line with the largest value is an optimal solution. What is a feasible solution point? - correct answer any solution point on the boundary of the feasible region or within the feasible region How to find the exact values of the decision variables? - correct answer The optimal solution point is where at least 2 lines connect, so solve for the point where they connect! What is slack in terms of constraints? How can it be made better (into an equal constraint)? - correct answer any unused/idle capacity for a = constraint have objective function coefficients of 0 can be turned into a equal constraint (in order for the extreme points to be on the line) by adding a non-negative variable to the LHS Why are slack or surplus variables added to the formulation of a linear programming problem? - correct answer to represent the slack (i.e. idle capacity) for slacks. to represent the surplus for surpluses Unused capacity makes no contribution to profit; thus, slack variables have coefficients of zero in the objective function. What is a redundant constraint? - correct answer The constraint does not affect the feasible region and thus cannot affect the optimal solution i.e. the feasible region would be the same whether the particular capacity constraint were included or not, which tells us that enough time is available to accommodate any production level that can be achieved by the other departments. What is surplus in terms of constraints? - correct answer any excess quantity corresponding to a = constraint have objective function coefficients = 0 a surplus variable can be subtracted from the left-hand side of the inequality to convert the constraint to equality form 3 special cases when we attempt to solve linear programming problems - correct answer 1. Alternative Optimal Solutions 2. Infeasibility 3. Unbounded Alternative optimal solutions - correct answer The case in which more than one solution provides the optimal value for the objective function. if the objective function line is parallel to a boundary constraint in the direction of optimization, there are alternate optimal solutions, with all points on this line segment being optimal Infeasibility - correct answer No solution to the linear programming problem satisfies all the constraints. Graphically, infeasibility means that a feasible region does not exist; that is, no points satisfy all the constraints and the nonnegativity conditions simultaneously. Will need more resources most likely. In some situations, the only reasonable approach is to drop one or more constraints and re-solve the problem. Unbounded - correct answer having or appearing to have no limits - For a maximization problem: the value of the solution may be made infinitely large without violating any of the constraint