Linear Programming Models: Graphical and Computer Methods l CHAPTER 7TRUE/FALSE 7.1 Management resources that need control include machinery usage, labor volume, money spent, time used, warehouse space used, and material usage. ANSWER: TRUE 7.2 In the term linear programming, the word programming comes from the phrase computer programming. ANSWER: FALSE 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. ANSWER: FALSE 7.4 Any linear programming problem can be solved using the graphical solution procedure. ANSWER: FALSE 7.5 Linear programming is designed to allow some constraints to be maximized. ANSWER: FALSE 7.6 A typical LP involves maximizing an objective function while simultaneously optimizing resource constraint usage. ANSWER: TRUE 7.7 Resource restrictions are called constraints. ANSWER: TRUE 7.8 Industrial applications of linear programming might involve several thousand variables and constraints. ANSWER: TRUE 7.9 An important assumption in linear programming is to allow the existence of negative decision variables. ANSWER: FALSE 7.10 7.11 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. ANSWER: TRUE An objective function is necessary in a maximization problem but is not required in a minimization problem. ANSWER: FALSE 173 Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7.12 In some instances, an infeasible solution may be the optimum found by the corner-point method. ANSWER: FALSE 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. ANSWER: TRUE 7.14 The solution to a linear programming problem must always lie on a constraint. ANSWER: TRUE 7.15 In a linear program, the constraints must be linear, but the objective function may be nonlinear. ANSWER: FALSE 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. ANSWER: FALSE 7.17 One can employ the same algorithm to solve both maximization and minimization problems. ANSWER: TRUE 7.18 One converts a minimization problem to a maximization problem by reversing the direction of all constraints. ANSWER: FALSE 7.19 The graphical method of solution illustrates that the only restriction on a solution is that the solution must lie along a constraint. ANSWER: FALSE 7.20 Anytime we have an iso-profit line which is parallel to a constraint, we have the possibility of multiple solutions. ANSWER: TRUE 7.21 If the iso-profit line is not parallel to a constraint, then the solution must be unique. ANSWER: TRUE 174 Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7.22 The iso-profit solution method and the corner-point solution method always give the same result. ANSWER: TRUE 7.23 When two or more constraints conflict with one another, we have a condition called unboundedness. ANSWER: FALSE 7.24 The addition of a redundant constraint lowers the iso-profit line. ANSWER: FALSE 7.25 Sensitivity analysis enables us to look only at the effects of changing the coefficients in the objective function. ANSWER: FALSE *7.26 All linear programming problems require that we maximize some quantity. ANSWER: FALSE *7.27 If we do not have multiple constraints, we do not have a linear programming problem. ANSWER: FALSE *7.28 Inequality constraints are mathematically easier to handle than equality constraints. ANSWER: TRUE *7.29 Every solution to a linear programming problem lies at a “corner point.” ANSWER: FALSE *7.30 A linear programming problem can have, at most one, solution. ANSWER: FALSE *7.31 A linear programming approach can be used to solve any problem for which the objective is to maximize some quantity. ANSWER: FALSE 175 ANSWER: e 7. allowing non-integer solutions. warehouse space utilization. need not satisfy all of the constraints. Solutions or variables may take values from to + . Additivity exists for the activities.34 Which of the following is not a basic assumption of linear programming? (a) (b) (c) (d) (e) The condition of certainty exists. only the non-negativity constraints.35 A feasible solution to a linear programming problem (a) (b) (c) (d) must satisfy all of the problem's constraints simultaneously. Proportionality exists in the objective function and constraints. must be a corner point of the feasible region. all of the above ANSWER: e 7. must give the maximum possible profit.33 Which of the following is not a property of all linear programming problems? (a) (b) (c) (d) (e) the presence of restrictions optimization of some objective a computer program alternate courses of action to choose from usage of only linear equations and inequalities ANSWER: c 7.32 Typical management resources include (a) (b) (c) (d) (e) machinery usage. Divisibility exists. ANSWER: a 176 . labor volume.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 MULTIPLE CHOICE 7. raw material usage. will always be unique (only one optimal solution possible for any one problem). there is no solution that satisfies all the constraints given.38 In a maximization problem. an unbounded solution. ANSWER: a 7. must always be in whole numbers (integers). then the linear program has (a) (b) (c) (d) (e) an infeasible solution. will always include at least some of each product or variable. ANSWER: e 7. could be any point in the feasible region of the problem. none of the above ANSWER: b 7. alternate optimal solutions. more than one solution is optimal. the feasible region is unbounded. a constraint is redundant.37 Infeasibility in a linear programming problem occurs when (a) (b) (c) (d) (e) there is an infinite solution.36 An optimal solution to a linear program (a) (b) (c) (d) (e) will always lie at an extreme point of the feasible region. a redundant constraint. when one or more of the solution variables and the profit can be made infinitely large without violating any constraints.39 Which of the following is not a part of every linear programming problem formulation? (a) (b) (c) (d) (e) an objective function a set of constraints non-negativity constraints a redundant constraint maximization or minimization of a linear function ANSWER: d 177 .Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7. can only be used if no inequalities exist.41 Which of the following is not true about product mix linear programming problems? (a) (b) (c) (d) (e) Two or more products are produced. It is the area satisfying all of the problem's resource restrictions.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. The feasible region cannot include negative areas. none of the above ANSWER: c 7. All possible solutions to the problem lie in this region. (b) moving the iso-profit lines towards the origin in a parallel fashion until the last point in the feasible region is encountered. but is useful as a learning tool. Limited resources are involved. is the most difficult approach. (d) none of the above (e) all of the above ANSWER: a 7. They always have integer (whole number) solutions. is useful for four or fewer decision variables. is inappropriate for more than two constraints.43 Which of the following about the feasible region is false? (a) (b) (c) (d) (e) It is only found in product mix problems. all of the above ANSWER: a 178 .42 The graphical solution to a linear programming problem (a) (b) (c) (d) (e) includes the corner-point method and the iso-profit line solution method. It is also called the area of feasible solutions.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7. ANSWER: a 7. (c) locating the point which is highest on the graph. 0) (60. and only one.44 The corner-point solution method: (a) (b) (c) (d) (e) will yield different results from the iso-profit line solution method.Y) is not a feasible corner point? (a) (b) (c) (d) (e) (0. useful only in minimization methods.47 Consider the following linear programming problem: Maximize Subject to: 12X + 10Y 4X + 3Y 480 2X + 3Y 360 all variables 0 Which of the following points (X. the problem involves redundancy. (a) (b) (c) (d) (e) there may be more than one optimum solution. an algebraic means for solving the intersection of two constraint equations. an error has been made in the problem formulation. requires that all corners created by all constraints be compared. none of the above ANSWER: a 7. optimum.80) none of the above ANSWER: c 179 . ANSWER: b 7. a condition of infeasibility exists.46 The simultaneous equation method is (a) (b) (c) (d) (e) an alternative to the corner-point method. none of the above ANSWER: c 7. will provide one.120) (120. useful only when more than two product variables exist in a product mix problem.45 When a constraint line bounding a feasible region has the same slope as an iso-profit line.0) (180. will not provide a solution at an intersection or corner where a non-negativity constraint is involved.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7. requires that the profit from all corners of the feasible region be compared. 49 Consider the following linear programming problem: Maximize Subject to: 12X + 10Y 4X + 3Y 480 2X + 3Y 360 all variables 0 Which of the following points (X. 1520. 1560. 480.Y) is not feasible? (a) (b) (c) (d) (e) (0.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7. none of the above ANSWER: c 7.10) (20.90) (60.120) (100.90) none of the above ANSWER: d 7.50 Consider the following linear programming problem: Maximize Subject to: 4X + 10Y 3X + 4Y 480 4X + 2Y 360 all variables 0 180 .48 Consider the following linear programming problem: Maximize Subject to: 12X + 10Y 4X + 3Y 480 2X + 3Y 360 all variables 0 The maximum possible value for the objective function is (a) (b) (c) (d) (e) 360. 84).0) (120. 525.120).0).0).10) none of the above ANSWER: c 7. 360.51 Consider the following linear programming problem: Maximize Subject to: 5X + 6Y 4X + 2Y 420 1X + 2Y 120 all variables 0 Which of the following points (X.0) (100. none of the above ANSWER: d 181 . (0.60) (105.Y) is not a feasible corner point? (a) (b) (c) (d) (e) (0.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 The feasible corner points are (48. (0. What is the maximum possible value for the objective function? (a) (b) (c) (d) (e) 1032 1200 360 1600 none of the above ANSWER: b 7. (90.52 Consider the following linear programming problem: Maximize Subject to: 5X + 6Y 4X + 2Y 420 1X + 2Y 120 all variables 0 The maximum possible value for the objective function is (a) (b) (c) (d) (e) 640. 560. 10) none of the above ANSWER: a 7. (0.100). (90. A linear programming model is used to determine the production schedule.54 Consider the following linear programming problem: Maximize Subject to: 20X + 8Y 4X + 2Y 360 1X + 2Y 200 all variables 0 The optimum solution occurs at the point (X. none of the above ANSWER: b 7.55 Two models of a product – Regular (X) and Deluxe (Y) – are produced by a company. 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 0 The optimal solution is X = 100 Y = 0.50) (60.30) (90.Y) is not feasible? (a) (b) (c) (d) (e) (50.40) (20.0).20).Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7.0).Y) (a) (b) (c) (d) (e) (100. (80.53 Consider the following linear programming problem: Maximize Subject to: 5X + 6Y 4X + 2Y 420 1X + 2Y 120 all variables 0 Which of the following points (X. 182 . 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. 183 . 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 0 The optimal solution is X = 100 Y = 0. Y=0. Y 0 The optimal solution is X=100.57 Two models of a product – Regular (X) and Deluxe (Y) – are produced by a company.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 How many units of the regular model would be produced based on this solution? (a) (b) (c) (d) (e) 0 100 50 120 none of the above ANSWER: b 7. A linear programming model is used to determine the production schedule. A linear programming model is used to determine the production schedule.56 Two models of a product – Regular (X) and Deluxe (Y) – are produced by a company. How many units of the raw materials would be used to produce this number of units? (a) (b) (c) (d) (e) 400 200 500 120 none of the above ANSWER: a 7. (50.0). there is a redundant constraint.0). none of the above ANSWER: b 7.0).59 Consider the following linear programming problem. Maximize Subject to: 20X + 30Y X + Y 80 6X + 12Y 600 X. (400.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 Which of these constraints is redundant? (a) (b) (c) (d) (e) the first constraint the second constraint the third constraint all of the above none of the above ANSWER: a 7. (0.Y) (a) (b) (c) (d) (e) (0. Y 0 This is a special case of a linear programming problem in which (a) (b) (c) (d) (e) there is no feasible solution. Y 0 The optimum solution to this problem occurs at the point (X. Minimize Subject to 20X + 30Y 2X + 4Y 800 6X + 3Y 300 X. this cannot be solved graphically.100). none of the above ANSWER: c 184 .58 Consider the following linear programming problem. there are multiple optimal solutions. either (c) or (a) depending on the constraint. no change to the objective function.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7. Maximize Subject to 20X + 30Y X + Y 80 8X + 9Y 600 3X + 2Y 400 X. either (c) or (b) depending on the constraint. none of the above ANSWER: a 7. either (c) or (b) depending on the constraint. ANSWER: e 185 . ANSWER: d 7. there is a redundant constraint. an increase in the value of the objective function. an increase in the value of the objective function. Y 0 This is a special case of a linear programming problem in which (a) (b) (c) (d) (e) there is no feasible solution. either (c) or (a) depending on the constraint. there are multiple optimal solutions.62 Deleting a constraint from a linear programming (maximization) problem may result in (a) (b) (c) (d) (e) a decrease in the value of the objective function.60 Consider the following linear programming problem.61 Adding a constraint to a linear programming (maximization) problem may result in (a) (b) (c) (d) (e) a decrease in the value of the objective function. this cannot be solved graphically. no change to the objective function. all of the above none of the above ANSWER: b 7. none of the above ANSWER: b 7. can produce a significant change in the shape of the feasible solution region. have no effect on the objective function of the linear program.63 Which of the following is not acceptable as a constraint in a linear programming problem (maximization)? Constraint 1 Constraint 2 Constraint 3 Constraint 4 (a) (b) (c) (d) (e) X + XY + Y 12 X 2Y 20 X + 3Y = 48 X + Y + Z 150 Constraint 1 Constraint 2 Constraint 3 Constraint 4 none of the above ANSWER: a 7.65 If one changes the contribution rates in the objective function of an LP.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7. the problem is unbounded.66 Changes in the technological coefficients of an LP problem (a) (b) (c) (d) (e) often reflect changes in the state of technology. then (a) (b) (c) (d) (e) the solution is infeasible. the optimal solution to the LP will no longer be optimal. the problem is degenerate. (a) (b) (c) (d) (e) the feasible region will change.64 If two corner points tie for the best value of the objective function. all of the above none of the above ANSWER: d 186 . there are an infinite number of optimal solutions. the slope of the iso-profit or iso-cost line will change. Independence exists for the activities.70 Which of the following is a basic assumption of linear programming? (a) (b) (c) (d) (e) The condition of uncertainty exists. Proportionality exists in the objective function and constraints. allowing only integer solutions. all of the above none of the above ANSWER: d 7. ANSWER: b 7.68 Sensitivity analyses are used to examine the effects of changes in (a) (b) (c) (d) (e) contribution rates for each variable. all of the above none of the above ANSWER: d 7. none of the above ANSWER: a 7.72 The condition when there is no solution which satisfies all the constraints is called: (a) boundedness 187 .Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7.69 Which of the following is a property of all linear programming problems? (a) (b) (c) (d) (e) alternate courses of action to choose from minimization of some objective a computer program usage of graphs in the solution usage of linear and nonlinear equations and inequalities ANSWER: a 7. parametric programming. corner point. intersection of the profit line and a constraint. intersection of two or more constraints.67 Sensitivity analysis may also be called (a) (b) (c) (d) (e) postoptimality analysis. optimality analysis.71 A point which satisfies all of a problem's constraints simultaneously is a(n) (a) (b) (c) (d) (e) feasible point. Solutions or variables may take values from to + . Divisibility exists. technological coefficients. available resources. redundant.74 If the addition of a constraint to a linear programming problem does not change the solution. They never have integer (whole number) solutions. ANSWER: d 7. none of the above ANSWER: e 7. the constraint is said to be (a) (b) (c) (d) (e) unbounded. then the linear program has (a) (b) (c) (d) (e) an infeasible solution. infeasible. alternate optimal solutions. a redundant constraint. when one or more of the solution variables and the cost can be made infinitely large without violating any constraints. none of the above ANSWER: b 188 . bounded. Cost is always to be minimized. an unbounded solution.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 (b) (c) (d) (e) redundancy optimality dependency none of the above ANSWER: e 7. non-negative. Individual resources are used in only a single product.75 The following is not true about product mix linear programming problems: (a) (b) (c) (d) (e) Two or more products are produced.73 In a minimization problem. on a constraint parallel to the objective function.120) (30. at the intersection of three or more constraints. none of the above ANSWER: b 7.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7.0) (180. is the most difficult approach. at the intersection of a non-negativity constraint and a resource constraint. is useful primarily in helping one understand the linear programming solution process.78 In order for a linear programming problem to have a multiple solutions.Y) could be a feasible corner point? (a) (b) (c) (d) (e) (40. at the intersection of the objective function and a constraint. the solution must exist (a) (b) (c) (d) (e) at the intersection of the non-negativity constraints. can only be used in a maximization problem.48) (120. at the intersection of the objective function and a constraint. at the intersection of two or more constraints.77 In order for a linear programming problem to have a unique solution.79 Consider the following linear programming problem: Maximize Subject to: 12X + 10Y 4X + 3Y 480 2X + 3Y 360 all variables 0 Which of the following points (X.36) none of the above ANSWER: b 189 . ANSWER: c 7. none of the above ANSWER: d 7.76 The graphical solution to a linear programming problem (a) (b) (c) (d) (e) is a useful tool for solving practical problems. the solution must exist (a) (b) (c) (d) (e) at the intersection of the non-negativity constraints. is useful for four or less decision variables. 81 Consider the following linear programming problem.82 Consider the following linear programming problem.100) (60.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7. Maximize Subject to: 6X +8Y 3X + 4Y 480 4X + 2Y 360 all variables 0 190 . Maximize Subject to: 12X + 10Y 4X + 3Y 480 2X + 3Y 360 all variables 0 Which of the following points (X.Y) is feasible? (a) (b) (c) (d) (e) (10.10) (30. 1560.90) none of the above ANSWER: c 7. 1520.120) (120. none of the above ANSWER: e 7.80 Consider the following linear programming problem: Maximize Subject to: 12X + 10Y 2X + 3Y 480 4X + 3Y 360 all variables 0 The maximum possible value for the objective function is (a) (b) (c) (d) (e) 360. 480. 120). 560. (0.0).210) (100. What is the maximum possible value for the objective function? (a) (b) (c) (d) (e) 540 1200 360 960 none of the above ANSWER: d 7.83 Consider the following linear programming problem.60) (105. (0.5) (0. (90.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 The feasible corner points are (48.84). Maximize Subject to: 6X +5Y 4X + 2Y 420 1X + 2Y 120 all variables 0 The maximum possible value for the objective function is (a) (b) (c) (d) (e) 530. Maximize Subject to: 5X + 6Y 4X + 2Y 420 1X + 2Y 120 all variables 0 Which of the following points (X.0).Y) is in the feasible region? (a) (b) (c) (d) (e) (30. 525.84 Consider the following linear programming problem. none of the above ANSWER: a 191 .10) none of the above ANSWER: d 7. 360. 85 Consider the following linear programming problem.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7. Maximize Subject to: 5X + 6Y 4X + 2Y 420 1X + 2Y 120 all variables 0 Which of the following points (X.20) none of the above ANSWER: e 7.40) (30. How many units of the labor hours would be used to produce this number of units? 192 .50) (60.30) (90. A linear programming model is used to determine the production schedule.Y) is feasible? (a) (b) (c) (d) (e) (50.86 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.87 Two models of a product – Regular (X) and Deluxe (Y) – are produced by a company. 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 0 The optimal solution is X = 100 Y = 0. How many units of the Deluxe model would be produced based on this solution? (a) (b) (c) (d) (e) 0 100 50 120 none of the above ANSWER: a 7. 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 0 The optimal solution is X = 100 Y = 0. The formulation is as follows: Maximize profit = 50X + 60 Y Subject to: 8X + 10Y 500 (labor hours) X + Y 120 (total units demanded) 4X + 5Y 800 (raw materials) X. A linear programming model is used to determine the production schedule. Y 0 This is a special case of a linear programming problem in which (a) (b) (c) (d) (e) there is no feasible solution. Y 0 Which of the constraints is active in determining the solution? (a) (b) (c) (d) (d) the first constraint the second constraint the third constraint constraints (a) and (b) none of the above ANSWER: a 7.89 Consider the following linear programming problem. this cannot be solved graphically.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 (a) (b) (c) (d) (e) 400 200 500 120 none of the above ANSWER: e 7.88 Two models of a product – Regular (X) and Deluxe (Y) – are produced by a company. there are multiple optimal solutions. none of the above ANSWER: c 193 . Maximize Subject to: 18X + 36Y X + Y 80 6X + 12Y 600 X. there is a redundant constraint. this cannot be solved graphically. no change in the value of the objective function. there are multiple optimal solutions. Maximize Subject to: 20X + 30Y X + Y 80 12X + 12Y 600 3X + 2Y 400 X. either an increase or decrease in the value of the objective function. none of the above ANSWER: b 7. an increase in the value of the objective function. no change in the value of the objective function. ANSWER: e 194 . there is a redundant constraint. either an increase or decrease in the value of the objective function. either (b) or (d) ANSWER: e 7. either a decrease or no change in the value of the objective function.92 Adding a constraint to a linear programming (maximization) problem may result in (a) (b) (c) (d) (e) a decrease in the value of the objective function.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7.90 Consider the following linear programming problem. an increase in the value of the objective function.91 Removing a constraint from a linear programming (maximization) problem may result in (a) (b) (c) (d) (e) a decrease in the value of the objective function. Y 0 This is a special case of a linear programming problem in which (a) (b) (c) (d) (e) there is no feasible solution. changes in the raw materials used. changes in government rules and regulations.96 Consider the following linear programming problem.93 Which of the following is not acceptable as a constraint in a linear programming problem (minimization)? Constraint 1 Constraint 2 Constraint 3 Constraint 4 Constraint 5 (a) (b) (c) (d) (e) X + Y 12 X . Maximize Subject to 10X + 30Y X + 2Y 80 8X + 16Y 640 4X + 2Y 100 X.2Y 20 X + 3Y = 48 X + Y + Z 150 2X .94 Changes in the contribution rates in the objective function of an LP may represent (a) (b) (c) (d) (e) changes in the technology used to produce the good. changes in the amount of resources used for a product. Y 0 195 .Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7. none of the above ANSWER: c *7. changes in the value of the resources used. changes in the degree to which a resource contributes to the cost of a product. none of the above ANSWER: b 7. changes in the price for which the product can be sold.3Y + Z > 75 Constraint 1 Constraint 2 Constraint 3 Constraint 4 Constraint 5 ANSWER: e 7.95 Changes in the technological coefficients of an LP problem may represent (a) (b) (c) (d) (e) changes in the price for which the product can be sold. 98 Consider the following linear programming problem.0) (180. there is a redundant constraint.36) none of the above ANSWER: b 196 .97 Which of the following is not acceptable as a constraint in a linear programming problem (maximization)? Constraint 1 Constraint 2 Constraint 3 Constraint 4 (a) (b) (c) (d) (e) X + Y 12 X 2Y 20 X + 3Y = 48 X2 + Y + Z 150 Constraint 1 Constraint 2 Constraint 3 Constraint 4 none of the above ANSWER: d *7.Y) could be a feasible corner point? (a) (b) (c) (d) (e) (40.48) (120.120) (30. there are multiple optimal solutions.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 This is a special case of a linear programming problem in which (a) (b) (c) (d) (e) there is no feasible solution. Maximize Subject to: 12X + 10Y 4X + 3Y 480 2X + 3Y 360 all variables 0 Which of the following points (X. none of the above ANSWER: b *7. this cannot be solved graphically. Maximize Subject to: 12X + 10Y 4X + 3Y 480 2X + 3Y 360 all variables 0 Which of the following points (X.10) (30. 480.99 Consider the following linear programming problem.120) (120.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 *7.100) (60.Y) is feasible? (a) (b) (c) (d) (e) (10. 1520. Maximize Subject to: 12X + 10Y 2X + 3Y 480 4X + 3Y 360 all variables 0 The maximum possible value for the objective function is (a) (b) (c) (d) (e) 360. 1560. none of the above ANSWER: e 7. Maximize Subject to: 5X + 6Y 4X + 2Y 420 1X + 2Y 120 all variables 0 197 .90) none of the above ANSWER: c *7.101 Consider the following linear programming problem.100 Consider the following linear programming problem. The formulation is as follows: Maximize profit = 60X + 50 Y Subject to: 6X +10Y 500 (labor hours) X + Y 120 (total units demanded) 6X + 5Y 800 (raw materials) X.Y) is not in the feasible region? (a) (b) (c) (d) (e) (30.5) (20. The formulation is as follows: Minimize cost Subject to: = 60X + 50 Y 8X + 10Y 800 (labor hours) X + Y 120 (total units demanded) 4X + 5Y 500 (raw materials) all variables 0 How many units of the Deluxe model would be produced based on this solution? (a) (b) (c) (d) (e) 0 100 50 120 none of the above ANSWER: a *7.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 Which of the following points (X. Y 0 Which of the constraints is active in determining the solution? (a) (b) (c) (d) (d) the first constraint the first and third constraints the third constraint the second constraint none of the above ANSWER: a 198 .40) (100.40) none of the above ANSWER: b *7. A linear programming model is used to determine the production schedule.30) (60.103 Two models of a product – Regular (X) and Deluxe (Y) – are produced by a company.102 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. (a) Formulate this as a linear programming problem. none of the above ANSWER: e PROBLEMS 7.4X1 + 0. Maximize Subject to: 15X + 36Y X + Y 80 27.5X + 55Y 1200 X. The liver flavored biscuits contain 1 unit of nutrient A and 2 units of nutrient B. The profit on the deluxe model is $12 per unit and the special's profit is $10. and there are only 100 man-hours available daily at the construction stage and only 80 man-hours available at the finishing and inspection stage. X2 = 400 7. the deluxe and the special.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 *7. The company has also decided that the special model must comprise at least 40 percent of the production total. there are multiple optimal solutions.000 The Fido Dog Food Company wishes to introduce a new brand of dog biscuits (composed of chicken and liver flavored biscuits) that meets certain nutritional requirements. this cannot be solved graphically. there is a redundant constraint.6X2 0 X1. Each model goes through two phases in the production process. X2 0 (b) Optimal solution: X1 = 0. Y 0 This is a special case of a linear programming problem in which (a) (b) (c) (d) (e) there is no feasible solution.106 Profit = $4.104 Consider the following linear programming problem. ANSWER: (a) Let X1 = number of deluxe models produced X2 = number of special models produced Maximize 12X1 + 10X2 Subject to: 1/3X1 + 1/4X2 100 1/6X1 + 1/4X2 80 0.105 As a supervisor of a production department. you must decide the daily production totals of a certain product that has two models. Each deluxe model requires 20 minutes of construction time and 10 minutes of finishing and inspection time. (b) Find the solution that gives the maximum profit. while the chicken flavored ones 199 . Each special model requires 15 minutes of construction time and 15 minutes of finishing and inspection time. 25) Optimal solution is (15.25) with cost of 65. giving the optimal values of all variables. what is the optimal product mix for a package of the biscuits in order to minimize the firm's cost? (a) (b) (c) (d) Formulate this as a linear programming problem. X2 0 (b) Corner points (0.107 Consider the following linear program. (c) 2X1 + 4X2 60 is redundant (d) minimum cost = 65 cents 7. X2 0 (a) Solve the problem graphically. Are any constraints redundant? If so.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 contain 1 unit of nutrient A and 4 units of nutrient B. In addition. there must be at least 40 units of nutrient A and 60 units of nutrient B in a package of the new mix.40) and (15. Is there more than one optimal solution? Explain. According to federal requirements. Solve this problem graphically. the company has decided that there can be no more than 15 liver flavored biscuits in a package. If it costs 1 cent to make a liver flavored biscuit and 2 cents to make a chicken flavored one. Maximize Subject to: 30X1 + 10X2 3X1 + X2 300 X1 + X2 200 X1 100 X2 50 X1 X2 0 X1. which one or ones? What is the total cost of a package of dog biscuits using the optimal mix? ANSWER: (a) Let X1 = number of liver flavored biscuits in a package X2 = number of chicken flavored biscuits in a package Minimize Subject to: X1 + 2X2 X1 + X2 40 2X1 + 4X2 60 X1 15 X1. (b) Are there any redundant constraints? 200 . 3). (75.000 type #1 or 2. 201 .000 b) * Produce 133. Besides these shared constraints. There is enough skilled labor to make either 1.8.33 type #1 lamps and 1. of which the type #1 lamp requires 3 and the type #2 lamp requires 4.75) and (50.108 The No-Glare Company is making two types of antique-style lamps.000 0 Profit($) 0 5.400 133.000.600 6. There are only 6.10) (2.150) Optimum solutions: (75. there are only enough fancy switches to make 1. (b) The constraint X1 100 is redundant since 3X1 + X2 300 also means that X1 cannot exceed 100.50). Both yield profit of $3.000 * 3. etc.200).400 400 1.3) (0.000 * 6. (a) Identify each corner point bounding the feasible region and find the total variable profit at each point. type #1 and type #2.400 of the type #2 lamps per day. 7. (b) How many type #1 and type #2 lamps should be produced? What is the maximum possible profit? ANSWER: (a) Corner points X1 X2 0 0 0 1.109 Solve the following linear programming problem using the corner point method.150).000 inserts available per day.75.000 type #2 lamps per day.75. (50.000 or produce 400 type #1 lamps and 1. Let X1 = the hundreds of type #1 lamps per day.33 1.3) Maximum profit 70. 7. Y 36 40 3 0 ANSWER: Feasible corner points (X.400 type #2 lamps for a profit of $6.200 1. Marginal profit (contribution) is $3 per type #1 lamp and $4 per type #2 lamp. (0.50).5 at (6.Y): (0.200 type #2 lamps for a profit of $6.000. Maximize Subject to: 10X + 1Y 4X + 3Y 2X + 4Y Y X. (50.8) (6.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 ANSWER: (a) Corner points (0.75).4. Y 0 Maximum profit is $780 by producing 9 standard and 8 deluxe models. Each newspaper ad reaches 6. The standard model requires 20 minutes of assembly time. How many ads of each type should be placed? 202 . The company must fill an order for 6 deluxe models. How many units of each product should be manufactured to maximize profits? ANSWER: Let X = number of standard model to produce Y = number of deluxe model to produce Maximize Subject to: 60X + 40Y 20X + 35Y 450 10X + 15Y 180 X6 X. The profit per unit on the standard model is $60. while the deluxe model requires 15 minutes of inspection time. 7. with at least 2 of each type.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7. while the profit per unit on the deluxe model is $40.Y): (0. while the other is the deluxe model. There are 450 minutes of assembly time and 180 minutes of inspection time available each day.000 people.8) (10. The company wishes to reach as many people as possible while meeting all the constraints stated.200 per week.110 Solve the following linear programming problem using the corner point method. The total budget is $7. while the deluxe model requires 35 minutes of assembly time. Y 0 ANSWER: Feasible corner points (X.112 Two advertising media are being considered for promotion of a product.10) (4. 7. The total number of ads should be at least 15. The standard model requires 10 minutes of inspection time.8). Maximize Subject to: 3 X + 5Y 4X + 4Y 48 1X + 2Y 20 Y 2 X. while newspaper ads cost $600 each.2) Maximum profit is 52 at (4.2) (0.000 people.111 Billy Penny is trying to determine how many units of two types of lawn mowers to produce each day. Radio ads cost $400 each. One of these is the standard model. while each radio ad reaches 2. 67 newspaper ads. Even though his electric saw overheats. N 0 Feasible corner points (R. the drying operation limits him to 16 Wuns or 8 Toos per day.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 ANSWER: Let R = number of radio ads placed N = number of newspaper ads placed Maximize Subject to: 2000R + 6000N R + N 15 400R + 600N 7200 R2 N2 R.2) (9.2) (2.N): (2. it will remain the optimal solution. he can make 7 Wuns or 14 Toos each day.114 Upon retirement. Suppose an additional constraint is added to this problem. Mr. If the original solution is still feasible.67) Maximum exposure 68.10. Since he doesn't have equipment for drying the lacquer finish he puts on the toys. 7.113 Suppose a linear programming (maximization) problem has been solved and the optimal value of the objective function is $300. therefore.020 with 2 radio and 10.6) (13. 7. Explain how this might affect each of the following: (a) the feasible region (b) the optimal value of the objective function ANSWER: (a) Adding a new constraint will reduce the size of the feasible region unless it is a redundant constraint. Wuns and Toos. Klaws started to make two types of children’s wooden toys in his shop. Wuns yield a variable profit of $9 each and Toos have a contribution margin of $8 apiece. (b) For what profit ratios would the optimum solution remain the optimum solution? 203 . the value of the objective function will either decrease or remain the same. (b) A new constraint can only reduce the size of the feasible region. (a) Solve this problem using the corner point method. It can never make the feasible region any larger. Each special model requires 15 minutes of construction time and 15 minutes of finishing and inspection time. The company has also decided that the special model must comprise at most 60 percent of the production total. Formulate this as a linear programming problem.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 ANSWER: Let X1 = numbers of wuns/day X2 = number of toos/day Maximize Subject to: 9X1 + 8X2 2X1 + 1X2 14 1X1 + 2X2 16 X1.8). The company produces tables (X) and chairs (Y). and Y = 120. you must decide the daily production totals of a certain product that has two models. The following linear programming problem represents this situation.6) Optimum profit $84 at (4. (b) 360.115 Susanna Nanna is the production manager for a furniture manufacturing company. the deluxe and the special. what would happen to the maximum possible profit? ANSWER: (a) 6000. 204 . Each table generates a profit of $80 and requires 3 hours of assembly time and 4 hours of finishing time. There are 360 hours of assembly time and 240 hours of finishing time available each month. Each chair generates $50 of profit and requires 3 hours of assembly time and 2 hours of finishing time. were added.6). Each model goes through two phases in the production process. X2 0 Corner points (0. (4. Each deluxe model requires 20 minutes of construction time and 10 minutes of finishing and inspection time. 7. Y 0 The optimal solution is X = 0.0). Maximize Subject to: 80X + 50Y 3X + 3Y 360 4X + 2Y 240 X.0). (a) What would the maximum possible profit be? (b) How many hours of assembly time would be used to maximize profit? (c) If a new constraint. (0. (7. (c) it would not change 7. and there are only 100 man-hours available daily at the construction stage and only 80 man-hours available at the finishing and inspection stage.116 As a supervisor of a production department. and the special's profit is $10. 2X + 2Y 400. The profit on the deluxe model is $12 per unit. 5X1 + X2 0 X1. 7. 3X1 + 7X2 0. while the chicken flavored ones contain 1 unit of nutrient A and 4 units of nutrient B.117 Maximize 12X1 + 10X2 Subject to: 1/3X1 + 1/4X2 100 1/6X1 + 1/4X2 80 1. (b) What constraint may be unrealistic? ANSWER: 205 . Marginal profit (contribution) is $3 per type #1 lamp and $4 per type #2 lamp.000 type #2 lamps per day.000 inserts available per day. According to federal requirements. of which the type #1 requires 3 and the type #2 requires 4.118 The No-Glare Company is making two types of antique-style lamps. however.000 type #1 or 2. is impossible to achieve. what is the optimal product mix for a package of the biscuits in order to minimize the firm's cost? (a) Formulate this as a linear programming problem. they do not believe that they can sell more than 25 percent more type #2 lamps than type #1 lamps. Besides these shared constraints. X2 0 The Fido Dog Food Company wishes to introduce a new brand of dog biscuits (composed of chicken and liver flavored biscuits) that meets certain nutritional requirements. There are only 6. and at least 10 chicken flavored biscuits in a package. (a) Formulate this as a linear program. If it costs 1 cent to make a liver flavored biscuit and 2 cents to make a chicken flavored one. (b) Are any constraints impossible to achieve? If so which one(s)? ANSWER: (a) Let X1 = number of liver flavored biscuits in a package X2 = number of chicken flavored biscuits in a package Minimize Subject to: X1 + 2X2 3X1 + 7X2 0 X1 15 X2 X1. In addition. There is enough skilled labor to make either 1.400 of the type #2 lamps per day. there are only enough fancy switches to make 1. there must be at least twice as many units of nutrient A as of nutrient B in a package of the new mix.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 ANSWER: Let X1 = number of deluxe models produced X2 = number of special models produced 7. type #1 and type #2. The liver flavored biscuits contain 1 unit of nutrient A and 2 units of nutrient B. the company has decided that there can be no more than 15 liver flavored biscuits. X2 0 Ratio of A to B Maximum liver Minimum chicken Non-negativity (b) The constraint. Management would like to make at least 10 percent more type #2 lamps than type #1 lamps. Maximize: 300X1 + 400 X2 Subject to: 0. N 0 206 . 20 percent of whom will respond.50*6000N 500R + 3000N R + N 15 R + N 19 400R + 600N 7200 1R . The company wishes to reach as many respondents as possible while meeting all the constraints stated. 7. Develop the appropriate LP model for determining the number of ads of each type that should be placed? ANSWER: Let R = number of radio ads placed N = number of newspaper ads placed Maximize: or Maximize: Subject to: 0. and there should be no more than 19 ads in total.10X1 + 0.119 Two advertising media are being considered for promotion of a product.200 per week.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 (a) Let X1 = the hundreds of type #1 lamps per day. X2 0 Labor Inserts Fancy switches Minimum Type #2 to Type #1 ratio Maximum Type #2 to Type #1 ratio (b) The labor constraint may be unrealistic because it assumes a continuous tradeoff between labor required for the Type #1 and Type #2 lamps.1X1 – X2 0 1. etc.25X1 – X2 0 X1 . with at least 2 of each type. while newspaper ads cost $600 each. The total number of ads should be at least 15.000 people.N R2 N2 R.20*2000R + 0. Radio ads cost $400 each.05X2 < 1 3X1 + 4X2 60 X2 14 1. Each newspaper ad reaches 6.000 people. 50 percent of whom will respond. The company does not want the number of newspaper ads to exceed the number of radio ads by more than 25 percent. The total budget is $7. while each radio ad reaches 2. the company has decided that there can be no more than 10 liver flavored biscuits. therefore. According to federal requirements. Explain how this might affect each of the following: (a) the feasible region (b) the optimal value of the objective function ANSWER: (a) Adding a new constraint will either.120 Suppose a linear programming (maximization) problem has been solved and the optimal value of the objective function is $300. *7. the value of the objective function will either stay the same or be lowered. there must be a ratio of 3 units of A to 2 of B in the new mix. If it costs 3 cents to make a liver flavored biscuit and 2 cents to make a chicken flavored one. and that there must be least 10 chicken flavored biscuits in a package. (b) A new constraint can only reduce the size of the feasible region. *7. The liver flavored biscuits contain 2 units of nutrient A and 1 unit of nutrient B. Therefore.121 Suppose a linear programming (maximization) problem has been solved and the optimal value of the objective function is $300. the value of the objective function will either increase or remain the same. leave the feasible region as it was. removing it will also result in a new optimal solution. or make it smaller. In addition. (b) Removal of a constraint can only increase or leave the same the size of the feasible region. what is the optimal product mix for a package of the biscuits in order to minimize the firm's cost? (a) Formulate this as a linear programming problem. (b) What is the solution? 207 . while the chicken flavored ones contain 3 units of nutrient A and 4 units of nutrient B.122 The Dog Food Company wishes to introduce a new brand of dog biscuits (composed of chicken and liver flavored biscuits) that meets certain nutritional requirements. If the constraint was active in the solution.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7. Explain how this might affect each of the following: (a) the feasible region (b) the optimal value of the objective function ANSWER: (a) Removing a constraint may. Suppose a constraint is removed from this problem. It can never make the feasible region any smaller. increase the size of the feasible region. Suppose an additional constraint () is added to this problem. if the constraint is not redundant. 6X2 0 X1 10 X2 0 X1. Besides these shared constraints. 30 minutes. Marginal profit (contribution) is $4 per type #1 lamp and $6 per type #2 lamp. There are 525 minutes of assembly time and 220 minutes of inspection time available each day.800 of the type #2 lamps per day.124 Billy Penny is trying to determine how many units of two types of lawnmowers to produce each day. (a) Identify each corner point bounding the feasible region and find the total variable profit at each point. the deluxe model.000 type #2 lamps per day. 30 minutes of assembly time. The standard model requires 15 minutes of inspection time. The profit per unit on the standard model is $60.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 ANSWER: (a) Let X1 = number of liver flavored biscuits in a package X2 = number of chicken flavored biscuits in a package Minimize Subject to: 3X1 + 2X2 X1 . (b) How many type #1 and type #2 lamps should be produced? What is the maximum possible profit? ANSWER: (a) Corner points X1 X2 0 0 0 15 20 0 Profit($) 0 9. The standard model requires 20 and the deluxe model. X2 = 10: Liver: 12 biscuits Chicken: 10 biscuits *7. X2 0 Ratio of A to B Maximum liver Minimum chicken Non-negativity (b) X1 = 12.000 *7.000 inserts available per day. type #1 and type #2.000 type #1 or 4. The company must fill an order for 12 deluxe models. there are only enough fancy switches to make 2. There is enough skilled labor to make either 2. while the other is the deluxe model.000 b) * Produce 0 type #1 lamps and 1500 type #2 lamps for a profit of $9.000 * 8. of which the type #1 lamp requires 6 and the type #2 lamp requires 8. How many units of each product should be manufactured to maximize profits? 208 . Let X1 = the hundreds of type #1 lamps per day. while the profit per unit on the deluxe model is $40. There are only 12. One of these is the standard model. etc.123 The No-Glare Company is making two types of automobile headlights. time used.5 standard and 12 deluxe models.125 List at least three typical management resources that warrant control. Explain its need.127 One basic assumption of linear programming is proportionality. 7.128 One basic assumption of linear programming is additivity. 209 . labor volume. ANSWER: The total of all activities equals the sum of individual activities. ANSWER: Solutions need not be whole numbers. ANSWER: machinery usage. 7. dollars spent. e.130 One basic assumption of linear programming is non-negativity. Explain its need. raw material usage 7. ANSWER: Only solution values of zero or positive values are allowed. if the production of 1 unit requires 4 units of a resource. 40 units of the resource are required. ANSWER: When there is no solution that can satisfy all constraints simultaneously. ANSWER: Rates of consumption exist. 7. SHORT ANSWER/ESSAY 7. then if 10 units are produced. Y 0 Maximum profit is $870 by producing 6. warehouse space usage.129 One basic assumption of linear programming is divisibility. 7. Explain its need.g.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 ANSWER: Let X1 = number of standard model to produce X2 = number of deluxe model to produce Maximize Subject to: 60X + 40Y 20X + 30Y 525 15X + 30Y < 220 X 12 X. Explain its need. ANSWER: Objective function rates and resource consumption are known and do not change during the analyzed time period.126 The basic assumption of linear programming is certainty.131 Define infeasibility with respect to an LP solution. 7. Explain its need.. ANSWER: In a bounded solution. ANSWER: the presence of one or more constraints that have no effect on the feasible solution area 7. briefly. In an unbounded solution.132 Define unboundedness with respect to an LP solution. the objective function must be parallel to an active constraint. *7. the difference between an unbounded solution and a bounded solution.135 Explain. 210 . what are the requirements for multiple solutions? ANSWER: For multiple solutions to occur.Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 7. *7. ANSWER: More than one optimal solution point exists. the solution is not restricted.133 Define redundancy with respect to an LP solution.134 Define alternate optimal solution with respect to an LP solution.136 Mathematically. ANSWER: a solution variable that is allowed to increase without limit while satisfying all constraints 7. one or more of the constraints restricts the solution.