Linear Programming Models: Graphical and Computer Methods l CHAPTER 7
TRUE/FALSE
7.1 Management resources that need control include machinery usage, labor volume, money spent, time used, warehouse space used, and material usage.
7.2 In the term linear programming, the word programming comes from the phrase computer programming.
7.3 Linear programming has few applications in the real world due to the assumption of certainty in the data and relationships of a problem.
7.4 Any linear programming problem can be solved using the graphical solution procedure.
7.5 Linear programming is designed to allow some constraints to be maximized.
7.6 A typical LP involves maximizing an objective function while simultaneously optimizing resource constraint usage.
7.7 Resource restrictions are called constraints.
7.8 Industrial applications of linear programming might involve several thousand variables and constraints.
7.9 An important assumption in linear programming is to allow the existence of negative decision variables.
7.10 The set of solution points that satisfies all of a linear programming problem's constraints simultaneously is defined as the feasible region in graphical linear programming.
7.11 An objective function is necessary in a maximization problem but is not required in a minimization problem.
7.12 In some instances, an infeasible solution may be the optimum found by the corner-point method.
7.13 The analytic postoptimality method attempts to determine a range of changes in problem parameters that will not affect the optimal solution or change the variables in the basis.
7.14 The solution to a linear programming problem must always lie on a constraint.
7.15 In a linear program, the constraints must be linear, but the objective function may be nonlinear.
7.16 Early applications of linear programming were primarily industrial in nature, later the technique was adopted by the military for scheduling and resource management.
7.17 One can employ the same algorithm to solve both maximization and minimization problems.
7.18 One converts a minimization problem to a maximization problem by reversing the direction of all constraints.
7.19 The graphical method of solution illustrates that the only restriction on a solution is that the solution must lie along a constraint.
7.20 Anytime we have an iso-profit line which is parallel to a constraint, we have the possibility of multiple solutions.
7.21 If the iso-profit line is not parallel to a constraint, then the solution must be unique.
7.22 The iso-profit solution method and the corner-point solution method always give the same result.
7.23 When two or more constraints conflict with one another, we have a condition called unboundedness.
7.24 The addition of a redundant constraint lowers the iso-profit line.
7.25 Sensitivity analysis enables us to look only at the effects of changing the coefficients in the objective function.
*7.26 All linear programming problems require that we maximize some quantity.
*7.27 If we do not have multiple constraints, we do not have a linear programming problem.
*7.28 Inequality constraints are mathematically easier to handle than equality constraints.
*7.29 Every solution to a linear programming problem lies at a “corner point.”
*7.30 A linear programming problem can have, at most one, solution.
*7.31 A linear programming approach can be used to solve any problem for which the objective is to maximize some quantity.
MULTIPLE CHOICE
7.32 Typical management resources include
(a) machinery usage.
(b) labor volume.
(c) warehouse space utilization.
(d) raw material usage.
(e) all of the above
7.33 Which of the following is not a property of all linear programming problems?
(a) the presence of restrictions
(b) optimization of some objective
(c) a computer program
(d) alternate courses of action to choose from
(e) usage of only linear equations and inequalities
7.34 Which of the following is not a basic assumption of linear programming?
(a) The condition of certainty exists.
(b) Proportionality exists in the objective function and constraints.
(c) Additivity exists for the activities.
(d) Divisibility exists, allowing non-integer solutions.
(e) Solutions or variables may take values from - ¥ to + ¥.
7.35 A feasible solution to a linear programming problem
(a) must satisfy all of the problem's constraints simultaneously.
(b) need not satisfy all of the constraints, only the non-negativity constraints.
(c) must be a corner point of the feasible region.
(d) must give the maximum possible profit.
7.36 An optimal solution to a linear program
(a) will always lie at an extreme point of the feasible region.
(b) could be any point in the feasible region of the problem.
(c) will always be unique (only one optimal solution possible for any one problem).
(d) will always include at least some of each product or variable.
(e) must always be in whole numbers (integers).
7.37 Infeasibility in a linear programming problem occurs when
(a) there is an infinite solution.
(b) a constraint is redundant.
(c) more than one solution is optimal.
(d) the feasible region is unbounded.
(e) there is no solution that satisfies all the constraints given.
7.38 In a maximization problem, when one or more of the solution variables and the profit can be made infinitely large without violating any constraints, then the linear program has
(a) an infeasible solution.
(b) an unbounded solution.
(c) a redundant constraint.
(d) alternate optimal solutions.
(e) none of the above
7.39 Which of the following is not a part of every linear programming problem formulation?
(a) an objective function
(b) a set of constraints
(c) non-negativity constraints
(d) a redundant constraint
(e) maximization or minimization of a linear function
7.40 The optimal solution to a maximization linear programming problem can be found by graphing the feasible region and
(a) finding the profit at every corner point of the feasible region to see which one gives the highest value.
(b) moving the iso-profit lines towards the origin in a parallel fashion until the last point in the feasible region is encountered.
(c) locating the point which is highest on the graph.
(d) none of the above
(e) all of the above
7.41 Which of the following is not true about product mix linear programming problems?
(a) Two or more products are produced.
(b) Limited resources are involved.
(c) They always have integer (whole number) solutions.
(d) The feasible region cannot include negative areas.
(e) none of the above
7.42 The graphical solution to a linear programming problem
(a) includes the corner-point method and the iso-profit line solution method.
(b) is useful for four or fewer decision variables.
(c) is inappropriate for more than two constraints.
(d) is the most difficult approach, but is useful as a learning tool.
(e) can only be used if no inequalities exist.
7.43 Which of the following about the feasible region is false?
(a) It is only found in product mix problems.
(b) It is also called the area of feasible solutions.
(c) It is the area satisfying all of the problem's resource restrictions.
(d) All possible solutions to the problem lie in this region.
(e) all of the above
7.44 The corner-point solution method:
(a) will yield different results from the iso-profit line solution method.
(b) requires that the profit from all corners of the feasible region be compared.
(c) will provide one, and only one, optimum.
(d) requires that all corners created by all constraints be compared.
(e) will not provide a solution at an intersection or corner where a non-negativity constraint is involved.
7.45 When a constraint line bounding a feasible region has the same slope as an iso-profit line,
(a) there may be more than one optimum solution.
(b) the problem involves redundancy.
(c) an error has been made in the problem formulation.
(d) a condition of infeasibility exists.
(e) none of the above
7.46 The simultaneous equation method is
(a) an alternative to the corner-point method.
(b) useful only in minimization methods.
(c) an algebraic means for solving the intersection of two constraint equations.
(d) useful only when more than two product variables exist in a product mix problem.
(e) none of the above
7.47 Consider the following linear programming problem:
Maximize 12X + 10Y
Subject to: 4X + 3Y £ 480
2X + 3Y £ 360
all variables ³ 0
Which of the following points (X,Y) is not a feasible corner point?
(a) (0,120)
(b) (120,0)
(c) (180,0)
(d) (60,80)
(e) none of the above
7.48 Consider the following linear programming problem:
Maximize 12X + 10Y
Subject to: 4X + 3Y £ 480
2X + 3Y £ 360
all variables ³ 0
The maximum possible value for the objective function is
(a) 360.
(b) 480.
(c) 1520.
(d) 1560.
(e) none of the above
7.49 Consider the following linear programming problem:
Maximize 12X + 10Y
Subject to: 4X + 3Y £ 480
2X + 3Y £ 360
all variables ³ 0
Which of the following points (X,Y) is not feasible?
(a) (0,120)
(b) (100,10)
(c) (20,90)
(d) (60,90)
(e) none of the above
7.50 Consider the following linear programming problem:
Maximize 4X + 10Y
Subject to: 3X + 4Y £ 480
4X + 2Y £ 360
all variables ³ 0
The feasible corner points are (48,84), (0,120), (0,0), (90,0). What is the maximum possible value for the objective function?
(a) 1032
(b) 1200
(c) 360
(d) 1600
(e) none of the above
7.51 Consider the following linear programming problem:
Maximize 5X + 6Y
Subject to: 4X + 2Y £ 420
1X + 2Y £ 120
all variables ³ 0
Which of the following points (X,Y) is not a feasible corner point?
(a) (0,60)
(b) (105,0)
(c) (120,0)
(d) (100,10)
(e) none of the above
7.52 Consider the following linear programming problem:
Maximize 5X + 6Y
Subject to: 4X + 2Y £ 420
1X + 2Y £ 120
all variables ³ 0
The maximum possible value for the objective function is
(a) 640.
(b) 360.
(c) 525.
(d) 560.
(e) none of the above
7.53 Consider the following linear programming problem:
Maximize 5X + 6Y
Subject to: 4X + 2Y £ 420
1X + 2Y £ 120
all variables ³ 0
Which of the following points (X,Y) is not feasible?
(a) (50,40)
(b) (20,50)
(c) (60,30)
(d) (90,10)
(e) none of the above
7.54 Consider the following linear programming problem:
Maximize 20X + 8Y
Subject to: 4X + 2Y £ 360
1X + 2Y £ 200
all variables ³ 0
The optimum solution occurs at the point (X,Y)
(a) (100,0).
(b) (90,0).
(c) (80,20).
(d) (0,100).
(e) none of the above
7.55 Two models of a product – Regular (X) and Deluxe (Y) – are produced by a company. A linear programming model is used to determine the production schedule. The formulation is as follows:
Maximize profit = 50X + 60 Y
Subject to: 8X + 10Y £ 800 (labor hours)
X + Y £ 120 (total units demanded)
4X + 5Y £ 500 (raw materials)
all variables ³ 0
The optimal solution is X = 100 Y = 0.
How many units of the regular model would be produced based on this solution?
(a) 0
(b) 100
(c) 50
(d) 120
(e) none of the above
7.56 Two models of a product – Regular (X) and Deluxe (Y) – are produced by a company. A linear programming model is used to determine the production schedule. The formulation is as follows:
Maximize profit = 50X + 60 Y
Subject to: 8X + 10Y £ 800 (labor hours)
X + Y £ 120 (total units demanded)
4X + 5Y £ 500 (raw materials)
all variables ³ 0
The optimal solution is X = 100 Y = 0.
How many units of the raw materials would be used to produce this number of units?
(a) 400
(b) 200
(c) 500
(d) 120
(e) none of the above
7.57 Two models of a product – Regular (X) and Deluxe (Y) – are produced by a company. A linear programming model is used to determine the production schedule. The formulation is as follows:
Maximize profit = 50X + 60 Y
Subject to: 8X + 10Y £ 800 (labor hours)
X + Y £ 120 (total units demanded)
4X + 5Y £ 500 (raw materials)
X, Y ³ 0
The optimal solution is X=100, Y=0.
Which of these constraints is redundant?
(a) the first constraint
(b) the second constraint
(c) the third constraint
(d) all of the above
(e) none of the above
7.58 Consider the following linear programming problem.
Minimize 20X + 30Y
Subject to 2X + 4Y £ 800
6X + 3Y ³ 300
X, Y ³ 0
The optimum solution to this problem occurs at the point (X,Y)
(a) (0,0).
(b) (50,0).
(c) (0,100).
(d) (400,0).
(e) none of the above
7.59 Consider the following linear programming problem.
Maximize 20X + 30Y
Subject to: X + Y £ 80
6X + 12Y £ 600
X, Y ³ 0
This is a special case of a linear programming problem in which
(a) there is no feasible solution.
(b) there is a redundant constraint.
(c) there are multiple optimal solutions.
(d) this cannot be solved graphically.
(e) none of the above
7.60 Consider the following linear programming problem.
Maximize 20X + 30Y
Subject to X + Y £ 80
8X + 9Y £ 600
3X + 2Y ³ 400
X, Y ³ 0
This is a special case of a linear programming problem in which
(a) there is no feasible solution.
(b) there is a redundant constraint.
(c) there are multiple optimal solutions.
(d) this cannot be solved graphically.
(e) none of the above
7.61 Adding a constraint to a linear programming (maximization) problem may result in
(a) a decrease in the value of the objective function.
(b) an increase in the value of the objective function.
(c) no change to the objective function.
(d) either (c) or (a) depending on the constraint.
(e) either (c) or (b) depending on the constraint.
7.62 Deleting a constraint from a linear programming (maximization) problem may result in
(a) a decrease in the value of the objective function.
(b) an increase in the value of the objective function.
(c) no change to the objective function.
(d) either (c) or (a) depending on the constraint.
(e) either (c) or (b) depending on the constraint.
7.63 Which of the following is not acceptable as a constraint in a linear programming problem (maximization)?
Constraint 1 X + XY + Y ³ 12
Constraint 2 X - 2Y £ 20
Constraint 3 X + 3Y = 48
Constraint 4 X + Y + Z £ 150
(a) Constraint 1
(b) Constraint 2
(c) Constraint 3