Section A: Introduction to Linear Programming

1. What is the primary purpose of linear programming?
   a) To optimize a linear objective subject to linear constraints
   b) To solve only nonlinear equations
   c) To remove all resource limitations
   d) To forecast demand without a mathematical model
   Answer: a) To optimize a linear objective subject to linear constraints
2. In a linear programming problem, the quantities whose values must be determined are called:
   a) Constraints
   b) Decision variables
   c) Slack resources
   d) Objective coefficients
   Answer: b) Decision variables
3. The mathematical expression that is maximized or minimized is called the:
   a) Feasible region
   b) Resource equation
   c) Objective function
   d) Nonnegativity restriction
   Answer: c) Objective function
4. A constraint in an LPP represents:
   a) The final optimal answer
   b) The name of a decision variable
   c) The graphical scale
   d) A limitation or requirement affecting the decision
   Answer: d) A limitation or requirement affecting the decision
5. Which is an example of an objective in linear programming?
   a) Maximize total profit
   b) Eliminate all variables
   c) Draw every constraint vertically
   d) Make all coefficients equal
   Answer: a) Maximize total profit
6. Which of the following is a common minimization objective?
   a) Maximizing output
   b) Minimizing total cost
   c) Maximizing market share
   d) Maximizing resource use
   Answer: b) Minimizing total cost
7. Which problem is most suitable for linear programming?
   a) Predicting tomorrow’s weather
   b) Calculating compound interest only
   c) Selecting a product mix under limited resources
   d) Solving an equation containing (x_1x_2)
   Answer: c) Selecting a product mix under limited resources
8. Linear programming belongs to the field of:
   a) Financial accounting only
   b) Organizational behavior
   c) Descriptive writing
   d) Operations research
   Answer: d) Operations research
9. A feasible solution is one that:
   a) Satisfies all constraints and sign restrictions
   b) Gives the highest objective value automatically
   c) Uses all resources completely
   d) Contains no zero-valued variables
   Answer: a) Satisfies all constraints and sign restrictions
10. An optimal solution is:
    a) Any point on a constraint line
    b) The feasible solution giving the best objective value
    c) A point outside the feasible region
    d) A solution using the most variables
    Answer: b) The feasible solution giving the best objective value
11. The feasible region consists of:
    a) All points satisfying the objective function
    b) All points in the first quadrant
    c) All points satisfying every constraint simultaneously
    d) Only the corner points
    Answer: c) All points satisfying every constraint simultaneously
12. The nonnegativity restrictions usually require:
    a) Objective coefficients to be positive
    b) Right-hand sides to be equal
    c) Constraints to be binding
    d) Decision variables to be zero or positive
    Answer: d) Decision variables to be zero or positive
13. Which notation expresses nonnegativity for two variables?
    a) (x_1,x_2\geq0)
    b) (x_1+x_2=0)
    c) (x_1,x_2<0)
    d) (x_1=x_2)
    Answer: a) (x_1,x_2\geq0)
14. A linear programming model normally includes:
    a) Only decision variables
    b) An objective function, constraints and sign restrictions
    c) Only a graphical solution
    d) Only resource quantities
    Answer: b) An objective function, constraints and sign restrictions
15. Which expression is a linear objective function?
    a) (Z=x_1x_2)
    b) (Z=x_1^2+x_2)
    c) (Z=5x_1+3x_2)
    d) (Z=\sqrt{x_1}+x_2)
    Answer: c) (Z=5x_1+3x_2)
16. Which expression is nonlinear?
    a) (2x_1+5x_2)
    b) (7x_1-x_2)
    c) (4x_1+3x_2\leq20)
    d) (x_1x_2+2x_1)
    Answer: d) (x_1x_2+2x_1)
17. In a product-mix problem, decision variables generally represent:
    a) Quantities of products to produce
    b) Available machine hours
    c) Profit coefficients only
    d) Constraint signs
    Answer: a) Quantities of products to produce
18. In a resource constraint, the right-hand side usually represents:
    a) Unit profit
    b) Available resource capacity
    c) The optimal objective value
    d) The number of variables
    Answer: b) Available resource capacity
19. In (Z=8x_1+6x_2), the coefficient 8 represents:
    a) The maximum value of (x_1)
    b) Available resource capacity
    c) The objective contribution per unit of (x_1)
    d) The slope of every constraint
    Answer: c) The objective contribution per unit of (x_1)
20. In (3x_1+2x_2\leq100), the value 100 usually represents:
    a) Unit contribution
    b) Number of products
    c) Graphical slope
    d) Available amount of a resource
    Answer: d) Available amount of a resource
21. Which inequality commonly represents a limited resource?
    a) (2x_1+3x_2\leq60)
    b) (2x_1+3x_2\geq60) only
    c) (2x_1+3x_2\neq60)
    d) (x_1x_2\leq60)
    Answer: a) (2x_1+3x_2\leq60)
22. Which constraint commonly represents a minimum requirement?
    a) (x_1+x_2\leq20)
    b) (x_1+x_2\geq20)
    c) (x_1+x_2<0)
    d) (x_1x_2=20)
    Answer: b) (x_1+x_2\geq20)
23. An equality constraint requires:
    a) The left side to exceed the right side
    b) The right side to be zero
    c) The left side to equal the right side
    d) Both variables to be equal
    Answer: c) The left side to equal the right side
24. Which statement about an LPP is correct?
    a) It must have exactly two constraints
    b) It cannot contain equality constraints
    c) It must always maximize profit
    d) It may involve maximization or minimization
    Answer: d) It may involve maximization or minimization
25. The graphical method is most practical when the model contains:
    a) Two decision variables
    b) Ten decision variables
    c) No constraints
    d) Only integer variables
    Answer: a) Two decision variables
26. For models with many decision variables, the usual solution method is:
    a) Trial and error
    b) Simplex or optimization software
    c) A pie chart
    d) A frequency table
    Answer: b) Simplex or optimization software
27. A solution that violates even one required constraint is:
    a) Optimal
    b) Degenerate
    c) Infeasible
    d) Unbounded
    Answer: c) Infeasible
28. If no point satisfies all constraints, the LPP is:
    a) Bounded
    b) Redundant
    c) Alternately optimal
    d) Infeasible
    Answer: d) Infeasible
29. If the objective can improve indefinitely while feasibility is maintained, the model is:
    a) Unbounded
    b) Infeasible
    c) Redundant
    d) Degenerate
    Answer: a) Unbounded
30. If more than one feasible solution gives the same best value, the model has:
    a) No solution
    b) Multiple optimal solutions
    c) Only one corner point
    d) An inconsistent constraint
    Answer: b) Multiple optimal solutions

Section B: Assumptions of Linear Programming

31. The proportionality assumption means that:
    a) All variables must be equal
    b) All resources must be fully used
    c) Each variable’s contribution is proportional to its value
    d) Every constraint must have the same slope
    Answer: c) Each variable’s contribution is proportional to its value
32. Which relationship violates proportionality?
    a) (4x_1)
    b) (3x_1+2x_2)
    c) (5x_2)
    d) (x_1^2)
    Answer: d) (x_1^2)
33. The additivity assumption states that:
    a) Total effects are the sum of individual variable effects
    b) Decision variables must be integers
    c) Every coefficient is known with certainty
    d) Products must be produced in equal quantities
    Answer: a) Total effects are the sum of individual variable effects
34. Which expression violates additivity?
    a) (3x_1+4x_2)
    b) (x_1x_2)
    c) (5x_1-2x_2)
    d) (7x_1+x_2)
    Answer: b) (x_1x_2)
35. The divisibility assumption means that decision variables may:
    a) Take only integer values
    b) Take only zero or one
    c) Take fractional values
    d) Take only negative values
    Answer: c) Take fractional values
36. Which situation may violate divisibility?
    a) Producing liters of liquid
    b) Allocating hours of labor
    c) Blending kilograms of ingredients
    d) Choosing the number of indivisible aircraft to purchase
    Answer: d) Choosing the number of indivisible aircraft to purchase
37. The certainty assumption means that:
    a) Model coefficients are known and constant
    b) Every decision has no risk
    c) The optimum is always unique
    d) All constraints are binding
    Answer: a) Model coefficients are known and constant
38. Which situation most directly violates certainty?
    a) Known labor requirements
    b) Highly uncertain profit coefficients with no fixed estimates
    c) Fixed material usage
    d) Known resource capacities
    Answer: b) Highly uncertain profit coefficients with no fixed estimates
39. The nonnegativity assumption normally prevents:
    a) Positive production
    b) Zero production
    c) Negative values for decision quantities
    d) Fractional production
    Answer: c) Negative values for decision quantities
40. Which assumption permits production of 2.5 units mathematically?
    a) Certainty
    b) Additivity
    c) Proportionality
    d) Divisibility
    Answer: d) Divisibility
41. Under proportionality, doubling a decision variable should:
    a) Double its contribution to the objective and constraints
    b) Square its contribution
    c) Leave its contribution unchanged
    d) reduce its contribution by half
    Answer: a) Double its contribution to the objective and constraints
42. Under additivity, the total profit from two products is:
    a) Their average profit
    b) The sum of their individual profit contributions
    c) Their product
    d) The larger of the two profits only
    Answer: b) The sum of their individual profit contributions
43. Which LPP assumption is violated by quantity discounts?
    a) Divisibility only
    b) Certainty only
    c) Proportionality
    d) Nonnegativity
    Answer: c) Proportionality
44. Fixed setup costs usually violate:
    a) Nonnegativity
    b) Divisibility
    c) Certainty
    d) Proportionality and linearity
    Answer: d) Proportionality and linearity
45. If resource usage per unit changes at different production levels, which assumption fails?
    a) Proportionality
    b) Additivity
    c) Certainty
    d) Nonnegativity
    Answer: a) Proportionality
46. If the production of one product changes the profit of another, this may violate:
    a) Certainty
    b) Additivity
    c) Divisibility
    d) Nonnegativity
    Answer: b) Additivity
47. Which assumption allows separate product contributions to be added without interaction terms?
    a) Proportionality
    b) Certainty
    c) Additivity
    d) Divisibility
    Answer: c) Additivity
48. When decision variables must be whole numbers, the appropriate extension is:
    a) Goal programming
    b) Dynamic programming
    c) Transportation programming
    d) Integer programming
    Answer: d) Integer programming
49. Which assumption supports using a single constant profit per unit?
    a) Proportionality
    b) Nonnegativity
    c) Divisibility
    d) Certainty only
    Answer: a) Proportionality
50. If a model permits half a worker, this results from the:
    a) Certainty assumption
    b) Divisibility assumption
    c) Additivity assumption
    d) proportionality assumption
    Answer: b) Divisibility assumption
51. If all coefficients are treated as fixed during analysis, the model follows:
    a) Divisibility
    b) Nonnegativity
    c) Certainty
    d) Additivity only
    Answer: c) Certainty
52. Which assumption is violated by (Z=5x_1+3x_2+2x_1x_2)?
    a) Nonnegativity
    b) Certainty
    c) Divisibility
    d) Additivity
    Answer: d) Additivity
53. Which expression satisfies linearity?
    a) (6x_1+4x_2)
    b) (6x_1^2+4x_2)
    c) (6x_1x_2)
    d) (6/x_1+4x_2)
    Answer: a) (6x_1+4x_2)
54. The assumption that resources can be divided among activities is related to:
    a) Additivity
    b) Divisibility
    c) Certainty
    d) Proportionality
    Answer: b) Divisibility
55. Which assumption means there are no synergistic effects among decision variables?
    a) Certainty
    b) Divisibility
    c) Additivity
    d) Nonnegativity
    Answer: c) Additivity
56. If demand values are random rather than known, ordinary deterministic LPP may violate:
    a) Additivity
    b) Proportionality
    c) Divisibility
    d) Certainty
    Answer: d) Certainty
57. Which model remains linear?
    a) (Z=3x_1-4x_2)
    b) (Z=3x_1^2-4x_2)
    c) (Z=3\sqrt{x_1}-4x_2)
    d) (Z=3x_1x_2)
    Answer: a) (Z=3x_1-4x_2)
58. Which assumption is most relevant when a solution gives (x_1=7.25)?
    a) Additivity
    b) Divisibility
    c) Certainty
    d) Proportionality
    Answer: b) Divisibility
59. If one additional unit always consumes three labor hours, this reflects:
    a) Additivity
    b) Certainty
    c) Proportionality
    d) Nonnegativity
    Answer: c) Proportionality
60. Which statement best summarizes LPP assumptions?
    a) All variables must be integers
    b) All constraints must be equalities
    c) Every resource must be fully consumed
    d) Relationships must be linear, additive, divisible and based on known coefficients
    Answer: d) Relationships must be linear, additive, divisible and based on known coefficients

Section C: Mathematical Model Formulation

61. What is usually the first step in formulating an LPP?
    a) Define the decision variables
    b) Draw the feasible region
    c) calculate corner-point values
    d) Add slack variables
    Answer: a) Define the decision variables
62. After defining decision variables, the next major step is to:
    a) Identify shadow prices
    b) formulate the objective function
    c) Solve the dual
    d) choose a pivot element
    Answer: b) Formulate the objective function
63. Resource limitations are expressed as:
    a) Objective coefficients
    b) Graph scales
    c) Constraints
    d) Decision-variable names
    Answer: c) Constraints
64. The final formulation step usually includes:
    a) Calculating sensitivity ranges
    b) Finding reduced costs
    c) Adding artificial variables
    d) Stating nonnegativity restrictions
    Answer: d) Stating nonnegativity restrictions
65. A company produces products A and B. Which variables are appropriate?
    a) (x_1=) units of A and (x_2=) units of B
    b) (x_1=) total profit and (x_2=) total cost
    c) (x_1=) labor capacity and (x_2=) machine capacity
    d) (x_1=) first constraint and (x_2=) second constraint
    Answer: a) (x_1=) units of A and (x_2=) units of B
66. If profits are $6 for A and $4 for B, the objective is:
    a) Minimize (Z=6x_1+4x_2)
    b) Maximize (Z=6x_1+4x_2)
    c) Maximize (Z=x_1+x_2)
    d) Minimize (Z=4x_1+6x_2)
    Answer: b) Maximize (Z=6x_1+4x_2)
67. If A uses two labor hours and B uses three, with 60 hours available, the labor constraint is:
    a) (2x_1+3x_2\geq60)
    b) (2x_1+3x_2=0)
    c) (2x_1+3x_2\leq60)
    d) (3x_1+2x_2\leq60)
    Answer: c) (2x_1+3x_2\leq60)
68. If at least 20 total units must be produced, the constraint is:
    a) (x_1+x_2\leq20)
    b) (x_1+x_2=0)
    c) (x_1x_2\geq20)
    d) (x_1+x_2\geq20)
    Answer: d) (x_1+x_2\geq20)
69. If no more than 15 units of product A may be produced, the constraint is:
    a) (x_1\leq15)
    b) (x_1\geq15)
    c) (x_2\leq15)
    d) (x_1+x_2\leq15)
    Answer: a) (x_1\leq15)
70. If at least eight units of product B are required, the constraint is:
    a) (x_1\geq8)
    b) (x_2\geq8)
    c) (x_2\leq8)
    d) (x_1+x_2=8)
    Answer: b) (x_2\geq8)
71. If the number of A units must equal twice the number of B units, the constraint is:
    a) (x_1+x_2=2)
    b) (2x_1=x_2)
    c) (x_1=2x_2)
    d) (x_1\leq2x_2)
    Answer: c) (x_1=2x_2)
72. If A production cannot exceed B production, the correct constraint is:
    a) (x_1\geq x_2)
    b) (x_1+x_2\leq0)
    c) (x_1=2x_2)
    d) (x_1\leq x_2)
    Answer: d) (x_1\leq x_2)
73. If product A must represent at least half of total production, an equivalent constraint is:
    a) (x_1\geq x_2)
    b) (x_1\leq x_2)
    c) (x_1+x_2\leq2)
    d) (2x_1+x_2\geq0)
    Answer: a) (x_1\geq x_2)
74. If total production cannot exceed 100 units, the constraint is:
    a) (x_1+x_2\geq100)
    b) (x_1+x_2\leq100)
    c) (x_1x_2\leq100)
    d) (x_1=x_2=100)
    Answer: b) (x_1+x_2\leq100)
75. If a blending model requires at least 30 kilograms of an ingredient, it uses:
    a) A less-than-or-equal-to constraint
    b) A strict inequality
    c) A greater-than-or-equal-to constraint
    d) A nonnegativity restriction only
    Answer: c) A greater-than-or-equal-to constraint
76. If exactly 500 units must be shipped, the relevant constraint is:
    a) Total shipments (\leq500)
    b) Total shipments (\geq500)
    c) Total shipments (<500)
    d) Total shipments (=500)
    Answer: d) Total shipments (=500)
77. Which objective is appropriate for a least-cost diet model?
    a) Minimize total food cost
    b) Maximize total food cost
    c) Minimize all nutrient levels
    d) Maximize the number of constraints
    Answer: a) Minimize total food cost
78. Which constraints are common in a diet model?
    a) Maximum-profit constraints
    b) Minimum nutritional requirements
    c) Equality of all food quantities
    d) Negative food quantities
    Answer: b) Minimum nutritional requirements
79. In a transportation formulation, decision variables commonly represent:
    a) Source capacities
    b) Destination demands
    c) Quantities shipped on each route
    d) Unit shipping costs
    Answer: c) Quantities shipped on each route
80. In an advertising-media model, the objective may be to:
    a) Minimize audience exposure
    b) Equalize all media spending
    c) Eliminate the advertising budget
    d) Maximize audience reach within a budget
    Answer: d) Maximize audience reach within a budget
81. Which constraint represents a budget of $10,000 when media costs are $500 and $800?
    a) (500x_1+800x_2\leq10000)
    b) (500x_1+800x_2\geq10000)
    c) (x_1+x_2=10000)
    d) (800x_1+500x_2\leq10)
    Answer: a) (500x_1+800x_2\leq10000)
82. If two products consume 4 and 6 kilograms of material, with 120 kilograms available, the constraint is:
    a) (4x_1+6x_2\geq120)
    b) (4x_1+6x_2\leq120)
    c) (6x_1+4x_2=120)
    d) (x_1+x_2\leq120)
    Answer: b) (4x_1+6x_2\leq120)
83. If demand limits sales of product B to 25 units, the constraint is:
    a) (x_1\leq25)
    b) (x_2\geq25)
    c) (x_2\leq25)
    d) (x_1+x_2=25)
    Answer: c) (x_2\leq25)
84. If contractual requirements demand at least 10 units of A, the constraint is:
    a) (x_2\geq10)
    b) (x_1\leq10)
    c) (x_1+x_2\geq10)
    d) (x_1\geq10)
    Answer: d) (x_1\geq10)
85. If A requires one machine hour and B requires two, with 40 hours available, the constraint is:
    a) (x_1+2x_2\leq40)
    b) (x_1+2x_2\geq40)
    c) (2x_1+x_2\leq40)
    d) (x_1+x_2=40)
    Answer: a) (x_1+2x_2\leq40)
86. Which expression correctly represents total profit when unit profits are 9 and 7?
    a) (Z=x_1+x_2)
    b) (Z=9x_1+7x_2)
    c) (Z=7x_1+9x_2) regardless of product labels
    d) (Z=63x_1x_2)
    Answer: b) (Z=9x_1+7x_2)
87. If management wants to minimize overtime and costs are $12 and $15 per hour, the objective is:
    a) Maximize (12x_1+15x_2)
    b) Minimize (x_1+x_2) only
    c) Minimize (12x_1+15x_2)
    d) Maximize (15x_1-12x_2)
    Answer: c) Minimize (12x_1+15x_2)
88. Which condition should normally be included for quantities produced?
    a) (x_1,x_2\leq0)
    b) (x_1=x_2)
    c) (x_1+x_2=0)
    d) (x_1,x_2\geq0)
    Answer: d) (x_1,x_2\geq0)
89. Which statement best describes good formulation practice?
    a) Define every variable clearly with units
    b) Omit units to simplify the model
    c) Mix decision variables and coefficients
    d) Draw the graph before identifying the objective
    Answer: a) Define every variable clearly with units
90. Dimensional consistency means that:
    a) Every coefficient must equal one
    b) Terms combined in a constraint should use compatible units
    c) Every variable must represent money
    d) All right-hand sides must be equal
    Answer: b) Terms combined in a constraint should use compatible units

Section D: Graphical Solution of Linear Programming Problems

91. The graphical method begins by plotting:
    a) Only the objective function
    b) Only the nonnegativity restrictions
    c) The boundary lines of the constraints
    d) The final optimal solution
    Answer: c) The boundary lines of the constraints
92. To graph (2x_1+x_2\leq10), the boundary line is:
    a) (2x_1+x_2<10)
    b) (2x_1+x_2\geq10)
    c) (2x_1+x_2=0)
    d) (2x_1+x_2=10)
    Answer: d) (2x_1+x_2=10)
93. The (x_1)-intercept of (2x_1+x_2=10) is:
    a) 5
    b) 10
    c) 2
    d) 20
    Answer: a) 5
94. The (x_2)-intercept of (2x_1+x_2=10) is:
    a) 5
    b) 10
    c) 2
    d) 20
    Answer: b) 10
95. The (x_1)-intercept of (3x_1+2x_2=12) is:
    a) 6
    b) 12
    c) 4
    d) 2
    Answer: c) 4
96. The (x_2)-intercept of (3x_1+2x_2=12) is:
    a) 4
    b) 12
    c) 2
    d) 6
    Answer: d) 6
97. A test point commonly used to determine the feasible side of a line is:
    a) The origin, when it is not on the line
    b) The objective-function value
    c) The largest intercept
    d) A random infeasible point
    Answer: a) The origin, when it is not on the line
98. For (2x_1+x_2\leq10), the origin is:
    a) Infeasible because zero is too small
    b) Feasible because (0\leq10)
    c) On the boundary line
    d) Feasible only for maximization
    Answer: b) Feasible because (0\leq10)
99. For (x_1+x_2\geq6), the origin is:
    a) Feasible
    b) On the boundary
    c) Infeasible
    d) Optimal
    Answer: c) Infeasible
100. The feasible region is obtained by:
     a) Selecting the largest half-plane
     b) Using only nonnegativity
     c) Evaluating the objective first
     d) Intersecting all feasible half-planes
     Answer: d) Intersecting all feasible half-planes
101. In a standard two-variable nonnegative model, the feasible region lies in:
     a) The first quadrant
     b) The second quadrant
     c) The third quadrant
     d) All quadrants equally
     Answer: a) The first quadrant
102. The corner-point method evaluates the objective function at:
     a) Every point in the plane
     b) Each extreme point of the feasible region
     c) Only the origin
     d) Only line intercepts
     Answer: b) Each extreme point of the feasible region
103. Why are corner points important in linear programming?
     a) Every corner is optimal
     b) They always use all resources
     c) A finite optimum occurs at at least one extreme point
     d) They eliminate the need for constraints
     Answer: c) A finite optimum occurs at at least one extreme point
104. The intersection of two boundary lines is commonly found by:
     a) Estimating from the graph only
     b) Multiplying their slopes
     c) Adding their intercepts
     d) Solving the two equations simultaneously
     Answer: d) Solving the two equations simultaneously
105. Solve (x_1+x_2=8) and (x_1-x_2=2). The intersection is:
     a) ((5,3))
     b) ((3,5))
     c) ((4,4))
     d) ((6,2))
     Answer: a) ((5,3))
106. Solve (x_1+x_2=7) and (x_1=3). The intersection is:
     a) ((4,3))
     b) ((3,4))
     c) ((3,7))
     d) ((7,3))
     Answer: b) ((3,4))
107. At ((x_1,x_2)=(4,3)), what is (Z=5x_1+2x_2)?
     a) 20
     b) 22
     c) 26
     d) 30
     Answer: c) 26
108. At ((x_1,x_2)=(2,5)), what is (Z=3x_1+4x_2)?
     a) 20
     b) 22
     c) 24
     d) 26
     Answer: d) 26
109. In a maximization problem, the optimal corner point has:
     a) The largest feasible objective value
     b) The smallest coordinate values
     c) The greatest number of binding constraints
     d) The largest (x_1) value only
     Answer: a) The largest feasible objective value
110. In a minimization problem, the optimal corner point has:
     a) The largest objective value
     b) The smallest feasible objective value
     c) The smallest (x_1) value only
     d) The most constraints
     Answer: b) The smallest feasible objective value
111. An iso-profit line represents:
     a) A resource constraint
     b) The nonnegativity boundary
     c) Combinations giving the same profit
     d) All feasible solutions
     Answer: c) Combinations giving the same profit
112. For (Z=4x_1+2x_2), the slope of an iso-profit line is:
     a) 2
     b) (-1/2)
     c) 1/2
     d) (-2)
     Answer: d) (-2)
113. For (Z=3x_1+6x_2), the slope of the objective line is:
     a) (-1/2)
     b) (-2)
     c) (1/2)
     d) 2
     Answer: a) (-1/2)
114. In the iso-profit method, the objective line is moved:
     a) Perpendicular to itself
     b) Parallel to itself
     c) Along one constraint only
     d) Toward the origin in every maximization problem
     Answer: b) Parallel to itself
115. For maximization, the objective line is moved in the direction of:
     a) Decreasing objective value
     b) The origin only
     c) Increasing objective value
     d) The steepest constraint
     Answer: c) Increasing objective value
116. For minimization, the objective line is moved toward:
     a) Increasing costs
     b) The farthest infeasible point
     c) The largest intercept
     d) Lower objective values while touching the feasible region
     Answer: d) Lower objective values while touching the feasible region
117. A constraint is binding at a solution when:
     a) It holds as an equality
     b) It has positive slack
     c) It does not affect the solution
     d) Its boundary does not touch the feasible region
     Answer: a) It holds as an equality
118. A nonbinding (\leq) constraint at a solution has:
     a) Negative slack
     b) Positive slack
     c) Zero right-hand side
     d) An artificial variable
     Answer: b) Positive slack
119. For (x_1+x_2\leq10), if (x_1=4) and (x_2=3), the slack is:
     a) 7
     b) 4
     c) 3
     d) 10
     Answer: c) 3
120. For (2x_1+x_2\leq12), if (x_1=5) and (x_2=2), the slack is:
     a) 2
     b) 5
     c) 10
     d) 0
     Answer: d) 0
121. If the objective line is parallel to a binding edge at the optimum, the model may have:
     a) Multiple optimal solutions
     b) No feasible solution
     c) An unbounded solution only
     d) A redundant objective
     Answer: a) Multiple optimal solutions
122. Multiple optimal solutions occur when:
     a) The feasible region is empty
     b) An entire feasible edge gives the same optimal value
     c) Every constraint is nonbinding
     d) The objective has no coefficients
     Answer: b) An entire feasible edge gives the same optimal value
123. An unbounded maximization problem occurs when:
     a) The feasible region contains one point
     b) All constraints are equalities
     c) The objective can increase indefinitely within the feasible region
     d) The origin is infeasible
     Answer: c) The objective can increase indefinitely within the feasible region
124. An infeasible graphical problem has:
     a) A very large feasible region
     b) Multiple optimal edges
     c) No objective function
     d) No common intersection satisfying every constraint
     Answer: d) No common intersection satisfying every constraint
125. A redundant constraint is one that:
     a) Does not change the feasible region
     b) Makes the model infeasible
     c) Determines the unique optimum
     d) Must be binding
     Answer: a) Does not change the feasible region
126. Which graphical feature indicates a redundant constraint?
     a) It removes all feasible points
     b) Its feasible half-plane already contains the region defined by other constraints
     c) It is parallel to the objective function
     d) It passes through the origin
     Answer: b) Its feasible half-plane already contains the region defined by other constraints
127. If the feasible region is closed and bounded, a continuous linear objective:
     a) Is always infeasible
     b) Must have multiple optima
     c) Has a finite maximum and minimum
     d) Must be zero
     Answer: c) Has a finite maximum and minimum
128. Which statement about an unbounded feasible region is correct?
     a) It always produces an unbounded objective
     b) It cannot contain an optimum
     c) It is always infeasible
     d) It may still have a finite optimum depending on objective direction
     Answer: d) It may still have a finite optimum depending on objective direction
129. The graphical method can verify feasibility by:
     a) Inspecting whether the constraint half-planes overlap
     b) Ignoring nonnegativity
     c) Evaluating only one corner
     d) Comparing objective coefficients only
     Answer: a) Inspecting whether the constraint half-planes overlap
130. Which point should be excluded if (x_1,x_2\geq0)?
     a) ((3,4))
     b) ((-2,5))
     c) ((0,6))
     d) ((5,0))
     Answer: b) ((-2,5))

Section E: Applications of Linear Programming

131. A product-mix model determines:
     a) Employee salaries
     b) Machine-maintenance dates
     c) Quantities of products that optimize an objective
     d) The location of every customer
     Answer: c) Quantities of products that optimize an objective
132. In a production-planning problem, a common objective is to:
     a) Minimize all output
     b) Equalize every product quantity
     c) Maximize unused resources
     d) Maximize contribution or minimize production cost
     Answer: d) Maximize contribution or minimize production cost
133. In a diet problem, the decision variables commonly represent:
     a) Quantities of foods selected
     b) Nutritional minimums
     c) Unit nutrient coefficients
     d) Total budget only
     Answer: a) Quantities of foods selected
134. The objective of a standard diet model is often to:
     a) Maximize calories
     b) Minimize cost while meeting nutritional needs
     c) Eliminate all nutrients
     d) Equalize food quantities
     Answer: b) Minimize cost while meeting nutritional needs
135. In a blending problem, constraints often ensure:
     a) Equal costs for all ingredients
     b) Maximum workforce
     c) Required composition or quality levels
     d) Unlimited material use
     Answer: c) Required composition or quality levels
136. In a media-selection problem, decision variables may represent:
     a) Audience members
     b) Product prices
     c) Employee working hours
     d) Numbers of advertisements placed in each medium
     Answer: d) Numbers of advertisements placed in each medium
137. A media-planning objective may be to:
     a) Maximize audience exposure within a budget
     b) Minimize all advertising activity
     c) Maximize cost without restrictions
     d) Eliminate audience requirements
     Answer: a) Maximize audience exposure within a budget
138. In a workforce-scheduling model, decision variables may represent:
     a) Daily customer demand
     b) Numbers of employees assigned to shifts
     c) Wage rates only
     d) Total overtime cost only
     Answer: b) Numbers of employees assigned to shifts
139. A workforce-scheduling constraint commonly ensures:
     a) Maximum advertising exposure
     b) Equal production quantities
     c) Minimum staffing coverage
     d) Unlimited overtime
     Answer: c) Minimum staffing coverage
140. In an investment-allocation model, the objective may be to:
     a) Minimize every return
     b) Equalize all investments
     c) Remove risk limits
     d) Maximize expected return subject to budget and risk constraints
     Answer: d) Maximize expected return subject to budget and risk constraints
141. A capital-budgeting constraint usually limits:
     a) Total amount invested
     b) Number of constraints
     c) Objective-function coefficients
     d) Shadow-price values
     Answer: a) Total amount invested
142. In an agricultural-planning model, decision variables may represent:
     a) Crop prices
     b) Acres allocated to different crops
     c) Rainfall levels
     d) Available land only
     Answer: b) Acres allocated to different crops
143. An agricultural LPP may include constraints for:
     a) Only crop profits
     b) Only selling prices
     c) Land, labor, water and budget
     d) Graphical slopes only
     Answer: c) Land, labor, water and budget
144. In a transportation model, the objective usually is to:
     a) Maximize unused supply
     b) Equalize all route shipments
     c) Increase the number of routes
     d) Minimize total shipping cost
     Answer: d) Minimize total shipping cost
145. In an assignment problem, decision variables indicate:
     a) Whether a particular resource is assigned to a particular task
     b) Total supply at each source
     c) Unit transportation cost
     d) Available machine time
     Answer: a) Whether a particular resource is assigned to a particular task
146. A cutting-stock application seeks to:
     a) Maximize waste
     b) Minimize material waste or the number of stock pieces used
     c) Equalize all cutting patterns
     d) Eliminate demand constraints
     Answer: b) Minimize material waste or the number of stock pieces used
147. In a portfolio model, diversification constraints may:
     a) Require all money in one asset
     b) eliminate the budget
     c) Limit the amount invested in individual asset categories
     d) Make all returns equal
     Answer: c) Limit the amount invested in individual asset categories
148. A warehouse-distribution model may use LPP to:
     a) Select employee benefits
     b) Forecast weather
     c) calculate depreciation
     d) allocate products among locations at minimum cost
     Answer: d) Allocate products among locations at minimum cost
149. Which statement best describes the value of LPP applications?
     a) They support structured allocation of scarce resources
     b) They eliminate the need for managerial judgment
     c) They guarantee that all data are certain
     d) They apply only to manufacturing
     Answer: a) They support structured allocation of scarce resources
150. Which statement best summarizes formulation and graphical analysis?
     a) The graph alone defines the business problem
     b) A sound LPP requires clear variables, a linear objective, valid constraints and evaluation of feasible corner points
     c) Every LPP has exactly one optimal solution
     d) Graphical analysis works equally well for hundreds of variables
     Answer: b) A sound LPP requires clear variables, a linear objective, valid constraints and evaluation of feasible corner points