Section A: Concept of Duality

1. In linear programming, the original model is commonly called the:
   a) Primal problem
   b) Dual variable
   c) Shadow-price model
   d) Reduced-cost problem
   Answer: a) Primal problem
2. The mathematical model associated with a primal linear program is called its:
   a) Feasible region
   b) Dual problem
   c) Slack equation
   d) Basis matrix
   Answer: b) Dual problem
3. What is the main relationship between a primal problem and its dual?
   a) They have identical decision variables
   b) They always have identical constraints
   c) One provides an alternative perspective on the other
   d) They must both be maximization problems
   Answer: c) One provides an alternative perspective on the other
4. If the primal problem has (m) constraints and (n) variables, the dual has:
   a) (m) constraints and (n) variables
   b) (m+n) variables
   c) (mn) constraints
   d) (n) constraints and (m) variables
   Answer: d) (n) constraints and (m) variables
5. Each primal constraint generally corresponds to:
   a) One dual variable
   b) One primal slack variable only
   c) One dual constraint
   d) One objective coefficient
   Answer: a) One dual variable
6. Each primal decision variable generally corresponds to:
   a) One dual variable
   b) One dual constraint
   c) One primal constraint
   d) One shadow-price range
   Answer: b) One dual constraint
7. The coefficient matrix of the dual is usually:
   a) The inverse of the primal matrix
   b) The negative of the primal matrix
   c) The transpose of the primal coefficient matrix
   d) The identity matrix
   Answer: c) The transpose of the primal coefficient matrix
8. If a primal model is a standard maximization problem, its dual is generally a:
   a) Maximization problem
   b) Transportation problem
   c) Nonlinear problem
   d) Minimization problem
   Answer: d) Minimization problem
9. If a primal model is a standard minimization problem, its dual is generally a:
   a) Maximization problem
   b) Minimization problem
   c) Queuing problem
   d) Network-flow problem
   Answer: a) Maximization problem
10. In the dual of a standard primal maximization problem, the primal right-hand-side values become:
    a) Dual right-hand-side values
    b) Dual objective coefficients
    c) Dual variable bounds
    d) Dual slack values
    Answer: b) Dual objective coefficients
11. In the dual, the primal objective coefficients become:
    a) Dual objective coefficients
    b) Dual variable values
    c) Dual right-hand-side values
    d) Shadow prices automatically
    Answer: c) Dual right-hand-side values
12. Why is duality useful in linear programming?
    a) It eliminates all model constraints
    b) It guarantees integer solutions
    c) It prevents infeasibility
    d) It provides economic interpretation and alternative solution approaches
    Answer: d) It provides economic interpretation and alternative solution approaches
13. The dual of the dual problem is:
    a) The original primal problem
    b) Always infeasible
    c) A different unrelated model
    d) Always unbounded
    Answer: a) The original primal problem
14. Dual variables are commonly associated with:
    a) Decision-variable production quantities
    b) The value of constrained resources
    c) Transportation routes
    d) Objective-function constants
    Answer: b) The value of constrained resources
15. In a resource-allocation maximization problem, dual variables may be interpreted as:
    a) Product quantities
    b) Slack-variable values only
    c) Imputed values of resources
    d) Fixed production costs
    Answer: c) Imputed values of resources
16. The primal and dual are linked mainly through:
    a) Identical variable names
    b) Equal numbers of rows and columns
    c) The graphical method only
    d) Their coefficients, constraints and objective functions
    Answer: d) Their coefficients, constraints and objective functions
17. Which statement about primal and dual feasibility is correct?
    a) They are related through the duality theorems
    b) Both must always be feasible
    c) Both must always be infeasible
    d) Feasibility of one guarantees feasibility of the other
    Answer: a) They are related through the duality theorems
18. A dual variable measures the marginal value of:
    a) A primal decision variable
    b) A primal constraint’s right-hand-side resource
    c) A dual objective coefficient
    d) A reduced-cost column
    Answer: b) A primal constraint’s right-hand-side resource
19. The duality relationship provides a way to place:
    a) Physical limits on all variables
    b) Integer restrictions on the primal
    c) Bounds on objective-function values
    d) Equal values on all resources
    Answer: c) Bounds on objective-function values
20. For a primal maximization problem and a dual minimization problem, any dual feasible solution provides:
    a) A lower bound on the primal objective
    b) The exact primal solution
    c) A primal decision-variable vector
    d) An upper bound on the primal objective
    Answer: d) An upper bound on the primal objective
21. For a primal minimization problem and a dual maximization problem, any dual feasible solution provides:
    a) A lower bound on the primal objective
    b) An upper bound on the primal objective
    c) A guarantee of unboundedness
    d) A basic primal solution
    Answer: a) A lower bound on the primal objective
22. Which concept explains why dual feasible values bound primal feasible values?
    a) Divisibility
    b) Weak duality
    c) Additivity
    d) Proportionality
    Answer: b) Weak duality
23. If feasible primal and dual objective values are equal, the corresponding solutions are:
    a) Degenerate
    b) Infeasible
    c) Optimal
    d) Unbounded
    Answer: c) Optimal
24. Duality is especially useful for interpreting:
    a) Product names
    b) Constraint numbering
    c) Tableau formatting
    d) Resource scarcity and marginal worth
    Answer: d) Resource scarcity and marginal worth
25. A zero-valued dual variable generally indicates that the associated primal constraint is:
    a) Nonbinding under typical nondegenerate conditions
    b) Infeasible
    c) An equality constraint
    d) Artificial
    Answer: a) Nonbinding under typical nondegenerate conditions
26. A positive dual value in a maximization resource problem generally suggests that the resource is:
    a) Unlimited
    b) Economically valuable at the margin
    c) Unused
    d) Irrelevant to the objective
    Answer: b) Economically valuable at the margin
27. Which mathematical operation changes primal rows into dual columns?
    a) Differentiation
    b) Integration
    c) Matrix transposition
    d) Matrix inversion
    Answer: c) Matrix transposition
28. The number of dual objective coefficients equals the number of:
    a) Primal variables
    b) Primal objective coefficients
    c) Dual constraints
    d) Primal constraints
    Answer: d) Primal constraints
29. The number of dual right-hand-side values equals the number of:
    a) Primal decision variables
    b) Primal constraints
    c) Dual variables
    d) Primal slack variables
    Answer: a) Primal decision variables
30. In standard duality, a primal maximization model with (\leq) constraints usually has dual variables that are:
    a) Unrestricted
    b) Nonnegative
    c) Nonpositive
    d) Binary
    Answer: b) Nonnegative

Section B: Formulation of the Dual Problem

31. Consider the primal objective Maximize (Z=4x_1+6x_2). In the dual, 4 and 6 become:
    a) Dual objective coefficients
    b) Dual-variable values
    c) Dual constraint right-hand sides
    d) Primal shadow prices
    Answer: c) Dual constraint right-hand sides
32. If the primal right-hand side is (b=(20,30)), the dual objective is based on:
    a) (4y_1+6y_2)
    b) (20x_1+30x_2)
    c) (20+30) only
    d) (20y_1+30y_2)
    Answer: d) (20y_1+30y_2)
33. For a standard primal maximization problem, the dual objective is to:
    a) Minimize the total imputed value of resources
    b) Maximize total product contribution
    c) Minimize the number of constraints
    d) Maximize unused capacity
    Answer: a) Minimize the total imputed value of resources
34. For a standard primal minimization problem, the dual objective is to:
    a) Minimize resource value
    b) Maximize the dual objective
    c) Minimize slack variables
    d) Maximize artificial variables
    Answer: b) Maximize the dual objective
35. If the primal is Maximize (c^Tx) subject to (Ax\leq b), (x\geq0), the dual is:
    a) Maximize (b^Ty), (A^Ty\leq c)
    b) Minimize (c^Ty), (Ay\geq b)
    c) Minimize (b^Ty), (A^Ty\geq c), (y\geq0)
    d) Maximize (b^Ty), (A^Ty=c)
    Answer: c) Minimize (b^Ty), (A^Ty\geq c), (y\geq0)
36. If the primal is Minimize (c^Tx) subject to (Ax\geq b), (x\geq0), the dual is:
    a) Minimize (b^Ty), (A^Ty\leq c)
    b) Maximize (c^Ty), (Ay\geq b)
    c) Minimize (c^Tx), (Ax=b)
    d) Maximize (b^Ty), (A^Ty\leq c), (y\geq0)
    Answer: d) Maximize (b^Ty), (A^Ty\leq c), (y\geq0)
37. A primal (\leq) constraint in a maximization problem generally produces a dual variable that is:
    a) Nonnegative
    b) Nonpositive
    c) Unrestricted
    d) Integer
    Answer: a) Nonnegative
38. A primal (\geq) constraint in a maximization problem generally produces a dual variable that is:
    a) Nonnegative
    b) Nonpositive
    c) Unrestricted
    d) Binary
    Answer: b) Nonpositive
39. A primal equality constraint generally produces a dual variable that is:
    a) Nonnegative
    b) Nonpositive
    c) Unrestricted in sign
    d) Equal to zero
    Answer: c) Unrestricted in sign
40. A nonnegative primal variable in a maximization problem generally corresponds to a dual constraint of type:
    a) Equality
    b) Less than or equal to
    c) Strict inequality
    d) Greater than or equal to
    Answer: d) Greater than or equal to
41. A nonpositive primal variable in a maximization problem generally corresponds to a dual constraint of type:
    a) Less than or equal to
    b) Greater than or equal to
    c) Equality
    d) No constraint
    Answer: a) Less than or equal to
42. An unrestricted primal variable corresponds to a dual constraint that is:
    a) Less than or equal to
    b) An equality
    c) Greater than or equal to
    d) Nonnegative
    Answer: b) An equality
43. In a minimization problem, a nonnegative primal variable generally corresponds to a dual constraint of type:
    a) Greater than or equal to
    b) Equality
    c) Less than or equal to
    d) Strictly positive
    Answer: c) Less than or equal to
44. In a minimization problem, a nonpositive primal variable generally corresponds to a dual constraint of type:
    a) Less than or equal to
    b) Equality
    c) Nonnegative
    d) Greater than or equal to
    Answer: d) Greater than or equal to
45. In a minimization problem, an unrestricted primal variable corresponds to:
    a) An equality dual constraint
    b) A nonnegative dual variable
    c) A less-than dual constraint
    d) A surplus variable
    Answer: a) An equality dual constraint
46. A primal (\geq) constraint in a standard minimization problem generally produces a dual variable that is:
    a) Nonpositive
    b) Nonnegative
    c) Unrestricted
    d) Binary
    Answer: b) Nonnegative
47. A primal (\leq) constraint in a minimization problem generally produces a dual variable that is:
    a) Nonnegative
    b) Unrestricted
    c) Nonpositive
    d) Equal to one
    Answer: c) Nonpositive
48. A primal equality constraint in a minimization problem produces a dual variable that is:
    a) Nonnegative
    b) Nonpositive
    c) Zero
    d) Unrestricted in sign
    Answer: d) Unrestricted in sign
49. Suppose the primal has two constraints and three decision variables. The dual has:
    a) Two variables and three constraints
    b) Three variables and two constraints
    c) Two variables and two constraints
    d) Three variables and three constraints
    Answer: a) Two variables and three constraints
50. Suppose the primal has four constraints and two variables. The dual has:
    a) Two variables and four constraints
    b) Four variables and two constraints
    c) Four variables and four constraints
    d) Two variables and two constraints
    Answer: b) Four variables and two constraints
51. The first dual constraint is formed using:
    a) The first primal row
    b) The primal right-hand sides only
    c) The coefficients of the first primal variable across all constraints
    d) The first dual-variable value
    Answer: c) The coefficients of the first primal variable across all constraints
52. The second dual variable corresponds to:
    a) The second primal variable
    b) The second dual constraint
    c) The second objective coefficient
    d) The second primal constraint
    Answer: d) The second primal constraint
53. Consider Maximize (Z=3x_1+5x_2), subject to (2x_1+x_2\leq8) and (x_1+3x_2\leq9). The dual objective is:
    a) Minimize (8y_1+9y_2)
    b) Maximize (8y_1+9y_2)
    c) Minimize (3y_1+5y_2)
    d) Maximize (3y_1+5y_2)
    Answer: a) Minimize (8y_1+9y_2)
54. For the model in Question 53, the dual constraint corresponding to (x_1) is:
    a) (3y_1+5y_2\geq8)
    b) (2y_1+y_2\geq3)
    c) (2y_1+y_2\leq3)
    d) (y_1+3y_2\geq5)
    Answer: b) (2y_1+y_2\geq3)
55. For the model in Question 53, the dual constraint corresponding to (x_2) is:
    a) (2y_1+y_2\geq3)
    b) (y_1+3y_2\leq5)
    c) (y_1+3y_2\geq5)
    d) (3y_1+5y_2\geq9)
    Answer: c) (y_1+3y_2\geq5)
56. In the dual of Question 53, the sign restrictions are:
    a) (y_1,y_2\leq0)
    b) (y_1) unrestricted and (y_2\geq0)
    c) (y_1,y_2) unrestricted
    d) (y_1,y_2\geq0)
    Answer: d) (y_1,y_2\geq0)
57. Consider Minimize (W=6x_1+4x_2), subject to (x_1+2x_2\geq10) and (3x_1+x_2\geq12). The dual objective is:
    a) Maximize (10y_1+12y_2)
    b) Minimize (10y_1+12y_2)
    c) Maximize (6y_1+4y_2)
    d) Minimize (6y_1+4y_2)
    Answer: a) Maximize (10y_1+12y_2)
58. For Question 57, the dual constraint corresponding to (x_1) is:
    a) (y_1+2y_2\leq6)
    b) (y_1+3y_2\leq6)
    c) (y_1+3y_2\geq6)
    d) (3y_1+y_2\leq4)
    Answer: b) (y_1+3y_2\leq6)
59. For Question 57, the dual constraint corresponding to (x_2) is:
    a) (y_1+3y_2\leq6)
    b) (2y_1+y_2\geq4)
    c) (2y_1+y_2\leq4)
    d) (6y_1+4y_2\leq12)
    Answer: c) (2y_1+y_2\leq4)
60. The dual variables in Question 57 satisfy:
    a) (y_1,y_2\leq0)
    b) (y_1,y_2) unrestricted
    c) (y_1\geq0,y_2\leq0)
    d) (y_1,y_2\geq0)
    Answer: d) (y_1,y_2\geq0)

Section C: Duality Theorems

61. Weak duality states that for a primal maximization and dual minimization pair:
    a) Every primal feasible value is no greater than every dual feasible value
    b) Every primal feasible value equals every dual feasible value
    c) The primal value always exceeds the dual value
    d) Both models must be optimal
    Answer: a) Every primal feasible value is no greater than every dual feasible value
62. Strong duality states that if both primal and dual have optimal solutions:
    a) Their variables are identical
    b) Their optimal objective values are equal
    c) Their constraints are identical
    d) Their slack values are equal
    Answer: b) Their optimal objective values are equal
63. If a primal maximization feasible solution has value 100 and a dual feasible solution has value 120, weak duality implies:
    a) The primal is unbounded
    b) The dual is infeasible
    c) 100 is a lower bound and 120 is an upper bound on the optimum
    d) Both solutions are optimal
    Answer: c) 100 is a lower bound and 120 is an upper bound on the optimum
64. If a primal feasible solution and a dual feasible solution both have objective value 85, then:
    a) Both are infeasible
    b) The primal is unbounded
    c) The dual must be degenerate
    d) Both solutions are optimal
    Answer: d) Both solutions are optimal
65. If the primal is unbounded, the dual must be:
    a) Infeasible
    b) Optimal
    c) Degenerate
    d) Feasible and bounded
    Answer: a) Infeasible
66. If the dual is unbounded, the primal must be:
    a) Optimal
    b) Infeasible
    c) Bounded
    d) Degenerate
    Answer: b) Infeasible
67. If the primal is feasible and has a finite optimum, the dual:
    a) Must be infeasible
    b) Must be unbounded
    c) Also has a finite optimum
    d) Must have zero objective value
    Answer: c) Also has a finite optimum
68. If the primal is infeasible, the dual may be:
    a) Only feasible and bounded
    b) Only infeasible
    c) Only unbounded
    d) Infeasible or unbounded
    Answer: d) Infeasible or unbounded
69. Which theorem establishes equality of optimal primal and dual objective values?
    a) Strong duality theorem
    b) Weak duality theorem
    c) Fundamental counting theorem
    d) Additivity theorem
    Answer: a) Strong duality theorem
70. Which theorem provides objective-value bounds before optimality is established?
    a) Complementary slackness
    b) Weak duality
    c) Strong duality
    d) Divisibility
    Answer: b) Weak duality
71. Complementary slackness links:
    a) Primal and dual objective coefficients only
    b) Two primal feasible solutions
    c) Primal slacks with dual variables and dual slacks with primal variables
    d) Artificial and surplus variables
    Answer: c) Primal slacks with dual variables and dual slacks with primal variables
72. If a primal constraint has positive slack at optimality, complementary slackness implies that its corresponding dual variable is:
    a) Positive
    b) Negative
    c) Unrestricted
    d) Zero
    Answer: d) Zero
73. If a dual variable is positive at optimality, the corresponding primal constraint must be:
    a) Binding
    b) Nonbinding
    c) Infeasible
    d) Redundant in every case
    Answer: a) Binding
74. If a primal variable is positive at optimality, the corresponding dual constraint must be:
    a) Nonbinding
    b) Binding
    c) Infeasible
    d) Unbounded
    Answer: b) Binding
75. If a dual constraint has positive slack at optimality, the corresponding primal variable must be:
    a) Positive
    b) Unrestricted
    c) Zero
    d) Artificial
    Answer: c) Zero
76. Complementary slackness conditions are especially useful for:
    a) Adding artificial variables
    b) Identifying all corner points
    c) Converting nonlinear constraints
    d) Recovering one optimal solution from the other
    Answer: d) Recovering one optimal solution from the other
77. For a primal maximization and dual minimization pair, the duality gap is:
    a) Dual objective value minus primal objective value
    b) Primal value minus dual value only
    c) The sum of both objective values
    d) The number of nonbinding constraints
    Answer: a) Dual objective value minus primal objective value
78. At optimality under strong duality, the duality gap equals:
    a) One
    b) Zero
    c) The primal slack
    d) The shadow price
    Answer: b) Zero
79. A positive duality gap between feasible primal and dual solutions indicates that:
    a) Both are optimal
    b) Both are infeasible
    c) At least one solution is not yet optimal
    d) The model is unbounded
    Answer: c) At least one solution is not yet optimal
80. If a feasible primal value exceeds a feasible dual value in a max-primal/min-dual pair, this suggests:
    a) Strong duality has been proved
    b) Both solutions are optimal
    c) The dual is degenerate
    d) At least one solution or formulation is incorrect
    Answer: d) At least one solution or formulation is incorrect
81. The complementary slackness product for a primal slack and its dual variable must equal:
    a) Zero
    b) One
    c) The optimal objective value
    d) The resource quantity
    Answer: a) Zero
82. If primal slack (s_i>0), then complementary slackness requires:
    a) (y_i>0)
    b) (y_i=0)
    c) (y_i<0)
    d) (y_i=s_i)
    Answer: b) (y_i=0)
83. If (x_j>0), complementary slackness requires the associated dual constraint slack to be:
    a) Positive
    b) Negative
    c) Zero
    d) Equal to (x_j)
    Answer: c) Zero
84. If both primal and dual are feasible, weak duality guarantees that:
    a) Both are optimal
    b) The primal is unbounded
    c) The duality gap is negative
    d) The primal objective cannot exceed the dual objective in a max-min pair
    Answer: d) The primal objective cannot exceed the dual objective in a max-min pair
85. Which situation is impossible under weak duality for a max-primal/min-dual pair?
    a) A primal feasible value of 150 and a dual feasible value of 140
    b) A primal value of 100 and dual value of 120
    c) Equal primal and dual values
    d) A positive duality gap
    Answer: a) A primal feasible value of 150 and a dual feasible value of 140
86. If the primal optimum is 250, strong duality implies that the dual optimum is:
    a) Less than 250
    b) 250
    c) Greater than 250
    d) Zero
    Answer: b) 250
87. Which theorem can certify optimality without further simplex iterations when feasible primal and dual values match?
    a) Divisibility theorem
    b) Additivity theorem
    c) Strong duality theorem
    d) Proportionality theorem
    Answer: c) Strong duality theorem
88. The existence of alternate optimal primal solutions does not prevent:
    a) Feasibility
    b) Complementary slackness
    c) Zero reduced costs
    d) Equality of primal and dual optimal objective values
    Answer: d) Equality of primal and dual optimal objective values
89. If a primal constraint is binding, its dual variable:
    a) May be positive or zero
    b) Must always be positive
    c) Must always be zero
    d) Must be unrestricted
    Answer: a) May be positive or zero
90. If a dual variable equals zero, the related primal constraint:
    a) Must be binding
    b) May be binding or nonbinding
    c) Must be infeasible
    d) Must be redundant
    Answer: b) May be binding or nonbinding

Section D: Economic Interpretation of Duality

91. In a production maximization problem, the primal variables typically represent:
    a) Resource prices
    b) Constraint slacks
    c) Quantities of products to produce
    d) Opportunity costs of unused resources
    Answer: c) Quantities of products to produce
92. In the corresponding dual, the variables typically represent:
    a) Product quantities
    b) Total demand
    c) Production capacities
    d) Imputed prices of resources
    Answer: d) Imputed prices of resources
93. The dual objective in a resource-pricing interpretation seeks to:
    a) Minimize the total imputed value of available resources
    b) Maximize unused capacity
    c) Minimize production quantities
    d) Maximize the number of products
    Answer: a) Minimize the total imputed value of available resources
94. A dual constraint ensures that the imputed resource cost of producing one unit of a product is:
    a) Less than zero
    b) At least as large as the product’s unit contribution in a maximization problem
    c) Equal to total available capacity
    d) No greater than the product contribution
    Answer: b) At least as large as the product’s unit contribution in a maximization problem
95. If the imputed resource cost of a product is below its unit profit, then the dual constraint is:
    a) Binding
    b) Redundant
    c) Violated
    d) Nonbinding but feasible
    Answer: c) Violated
96. At optimality, the total maximum contribution from the primal equals:
    a) Total unused resources
    b) The sum of all primal variables
    c) Total fixed cost
    d) The minimum imputed value of resources in the dual
    Answer: d) The minimum imputed value of resources in the dual
97. A dual value can be interpreted as the maximum amount a firm should be willing to pay for:
    a) One additional unit of the associated resource, within the allowable range
    b) All available resources
    c) One unit of every product
    d) A nonbasic decision variable
    Answer: a) One additional unit of the associated resource, within the allowable range
98. If a machine-hour constraint has a shadow price of $12, one extra machine hour may increase profit by:
    a) $1
    b) $12
    c) The total machine cost
    d) An unlimited amount
    Answer: b) $12
99. The interpretation in Question 98 is valid only:
    a) For all possible capacity changes
    b) If every variable remains positive
    c) Within the allowable right-hand-side range
    d) When the resource constraint is nonbinding
    Answer: c) Within the allowable right-hand-side range
100. If a resource has a shadow price of zero, an additional unit of that resource generally provides:
     a) A guaranteed profit increase
     b) A negative benefit
     c) An unbounded benefit
     d) No immediate improvement in the objective
     Answer: d) No immediate improvement in the objective
101. A scarce resource is more likely to have:
     a) A positive shadow price
     b) A negative slack value
     c) An artificial variable
     d) An unrestricted primal variable
     Answer: a) A positive shadow price
102. A resource with unused capacity is more likely to have:
     a) A high positive shadow price
     b) A zero shadow price
     c) An infinite shadow price
     d) A negative production quantity
     Answer: b) A zero shadow price
103. In economic terms, dual constraints compare product contribution with:
     a) Product demand only
     b) Fixed operating cost
     c) The value of resources consumed by the product
     d) The number of constraints
     Answer: c) The value of resources consumed by the product
104. If a product is produced at a positive level, complementary slackness implies that:
     a) Its selling price is zero
     b) Its resource use is zero
     c) All resources are unused
     d) Its imputed resource cost equals its objective contribution
     Answer: d) Its imputed resource cost equals its objective contribution
105. If the imputed resource cost of a product exceeds its unit contribution, the product will generally be:
     a) Nonbasic at zero in the optimal primal solution
     b) Produced without limit
     c) Assigned an artificial variable
     d) Required by complementary slackness
     Answer: a) Nonbasic at zero in the optimal primal solution
106. The economic interpretation of duality helps managers evaluate:
     a) Only current production quantities
     b) Resource acquisition and capacity-expansion decisions
     c) Employee performance
     d) Accounting depreciation
     Answer: b) Resource acquisition and capacity-expansion decisions
107. If a resource shadow price is $20 and an additional unit costs $15, acquiring that unit may be:
     a) Unprofitable in every case
     b) Irrelevant
     c) Economically attractive within the valid range
     d) Infeasible by definition
     Answer: c) Economically attractive within the valid range
108. If a resource shadow price is $8 and an extra unit costs $12, the firm should generally:
     a) Buy unlimited units
     b) Ignore the cost difference
     c) Change the objective function
     d) Not buy the unit based solely on this marginal comparison
     Answer: d) Not buy the unit based solely on this marginal comparison
109. Dual values help identify which constraints are:
     a) Economically important
     b) Graphically parallel
     c) Artificial
     d) Nonlinear
     Answer: a) Economically important
110. In a cost-minimization problem, a dual variable may represent:
     a) Product profit
     b) The value of meeting an additional unit of a requirement
     c) Unused production capacity only
     d) A fixed cost
     Answer: b) The value of meeting an additional unit of a requirement
111. In a diet minimization model, dual variables may represent the marginal value of:
     a) Food quantities
     b) Ingredient prices only
     c) Nutritional requirements
     d) Total calories consumed
     Answer: c) Nutritional requirements
112. In a workforce minimization model, a dual value may indicate the marginal cost of:
     a) Reducing all wages
     b) Hiring one manager
     c) Eliminating one shift
     d) Increasing a staffing requirement by one unit
     Answer: d) Increasing a staffing requirement by one unit
113. The dual objective provides an economic valuation of:
     a) The resources or requirements represented by primal constraints
     b) Only primal decision variables
     c) Only slack variables
     d) The number of simplex iterations
     Answer: a) The resources or requirements represented by primal constraints
114. Duality helps explain why limited resources should be assigned:
     a) Equally to all products
     b) To uses that create the greatest economic value
     c) Only to the highest-volume product
     d) Without considering opportunity cost
     Answer: b) To uses that create the greatest economic value
115. A positive reduced cost for a nonbasic variable in a minimization problem often indicates:
     a) The variable must enter immediately
     b) The constraint is infeasible
     c) Its objective coefficient must improve before it becomes attractive
     d) The dual is unbounded
     Answer: c) Its objective coefficient must improve before it becomes attractive
116. Reduced cost and shadow price are related because both arise from:
     a) The graphical method only
     b) Transportation balancing
     c) Integer restrictions
     d) Marginal information in primal-dual optimality
     Answer: d) Marginal information in primal-dual optimality
117. If the dual price of a resource rises, this generally signals that the resource has become:
     a) More scarce or valuable
     b) More abundant
     c) Nonbinding
     d) Irrelevant
     Answer: a) More scarce or valuable
118. A zero dual price does not necessarily mean that a resource:
     a) Is nonbinding
     b) Has no physical value
     c) Has unused capacity
     d) Is outside its allowable range
     Answer: b) Has no physical value
119. A shadow price measures marginal value rather than:
     a) Objective improvement
     b) Resource scarcity
     c) The full market price of the entire resource stock
     d) Sensitivity to right-hand-side changes
     Answer: c) The full market price of the entire resource stock
120. The dual model can be viewed as finding resource prices that:
     a) Maximize production quantities
     b) Ignore product profitability
     c) Make every product profitable
     d) Support the optimal primal production plan
     Answer: d) Support the optimal primal production plan

Section E: Shadow Prices and Sensitivity Concepts

121. A shadow price is the change in optimal objective value caused by:
     a) A one-unit change in a constraint right-hand side, within its allowable range
     b) A one-unit change in every objective coefficient
     c) Adding one decision variable
     d) Removing one constraint
     Answer: a) A one-unit change in a constraint right-hand side, within its allowable range
122. Shadow prices are also called:
     a) Slack values
     b) Dual values
     c) Artificial costs
     d) Pivot ratios
     Answer: b) Dual values
123. If the shadow price of labor is $15, increasing labor availability by two hours may increase profit by:
     a) $15
     b) $17
     c) $30, provided the change stays within the allowable range
     d) An unlimited amount
     Answer: c) $30, provided the change stays within the allowable range
124. If the shadow price is $6 and resource availability falls by three units, the objective may decrease by:
     a) $2
     b) $6
     c) $9
     d) $18, within the allowable range
     Answer: d) $18, within the allowable range
125. A shadow price remains valid as long as:
     a) The current optimal basis remains unchanged
     b) Every constraint remains binding
     c) The objective value remains zero
     d) No decision variable is positive
     Answer: a) The current optimal basis remains unchanged
126. The allowable increase for a right-hand side tells how much it may rise before:
     a) The objective coefficient changes
     b) The current shadow price or basis may change
     c) The model becomes linear
     d) A dual variable is created
     Answer: b) The current shadow price or basis may change
127. The allowable decrease for a right-hand side identifies:
     a) The minimum objective coefficient
     b) The number of constraints removable
     c) How much the resource may decline while the current basis remains valid
     d) The maximum dual objective
     Answer: c) How much the resource may decline while the current basis remains valid
128. Applying a shadow price beyond its allowable range may be misleading because:
     a) The model becomes nonlinear automatically
     b) The primal becomes infeasible in every case
     c) The resource becomes free
     d) The optimal basis and marginal value may change
     Answer: d) The optimal basis and marginal value may change
129. A binding constraint usually has:
     a) Zero slack
     b) Positive slack
     c) Negative resource use
     d) No dual variable
     Answer: a) Zero slack
130. A nonbinding (\leq) resource constraint usually has:
     a) Zero slack
     b) Positive slack
     c) Negative shadow price in every case
     d) An artificial variable
     Answer: b) Positive slack
131. Under standard nondegenerate conditions, a nonbinding resource constraint has a shadow price of:
     a) One
     b) Infinity
     c) Zero
     d) The right-hand-side value
     Answer: c) Zero
132. A binding constraint may have a zero shadow price when:
     a) The model is always infeasible
     b) The primal is unbounded
     c) The constraint has positive slack
     d) Degeneracy or redundancy is present
     Answer: d) Degeneracy or redundancy is present
133. If an additional resource unit increases optimal profit by $9, the resource’s shadow price is:
     a) $9
     b) $1
     c) The total profit
     d) The resource cost
     Answer: a) $9
134. If a one-unit increase in a requirement increases minimum cost by $11, the shadow price is:
     a) $1
     b) $11
     c) Zero
     d) Negative $11
     Answer: b) $11
135. A negative shadow price may occur when increasing a constraint right-hand side:
     a) Always increases profit
     b) Has no effect
     c) Worsens the objective under the constraint’s sign convention
     d) Makes the dual infeasible automatically
     Answer: c) Worsens the objective under the constraint’s sign convention
136. In a standard maximization model with (\leq) resource constraints, shadow prices are generally:
     a) Unrestricted
     b) Negative
     c) Binary
     d) Nonnegative
     Answer: d) Nonnegative
137. The shadow price of a resource should be compared with:
     a) The marginal cost of obtaining more of that resource
     b) The total fixed cost only
     c) The number of primal variables
     d) The slack-variable coefficient
     Answer: a) The marginal cost of obtaining more of that resource
138. If the market cost of one more unit is below its positive shadow price, purchasing the unit may:
     a) Reduce profit automatically
     b) Improve the objective within the allowable range
     c) Cause infeasibility
     d) Eliminate the dual model
     Answer: b) Improve the objective within the allowable range
139. If a resource has a zero shadow price, buying additional units is generally:
     a) Always profitable
     b) Required for feasibility
     c) Not beneficial at the margin under current conditions
     d) Guaranteed to change the basis
     Answer: c) Not beneficial at the margin under current conditions
140. Shadow-price information is normally found in:
     a) The objective-function statement only
     b) The initial simplex tableau only
     c) The primal constraint labels
     d) A sensitivity or dual-value report
     Answer: d) A sensitivity or dual-value report
141. In Excel Solver, the shadow price is commonly reported for:
     a) Each constraint
     b) Each worksheet cell
     c) Each objective coefficient only
     d) Each formula label
     Answer: a) Each constraint
142. A shadow price is valid for simultaneous changes in several right-hand sides only when:
     a) No limits apply
     b) appropriate sensitivity rules and ranges are respected
     c) Every shadow price is zero
     d) The primal has two variables
     Answer: b) Appropriate sensitivity rules and ranges are respected
143. The 100% rule is sometimes used to assess:
     a) A single unrestricted change
     b) Integer feasibility
     c) Simultaneous changes in objective coefficients or right-hand sides
     d) Whether a constraint is linear
     Answer: c) Simultaneous changes in objective coefficients or right-hand sides
144. If a right-hand-side change exceeds its allowable increase, the analyst should:
     a) Continue using the same shadow price indefinitely
     b) Delete the constraint
     c) Change the problem to nonlinear programming
     d) Re-solve the linear program
     Answer: d) Re-solve the linear program
145. Which statement best describes a shadow price?
     a) It is a local marginal value, not necessarily a permanent market price
     b) It is always the purchase price of a resource
     c) It is the value of a primal decision variable
     d) It is always positive
     Answer: a) It is a local marginal value, not necessarily a permanent market price
146. If a constraint’s allowable increase is 10 units, the current shadow price is guaranteed to remain valid for:
     a) Any increase
     b) An increase of up to 10 units, assuming other conditions remain unchanged
     c) A decrease of exactly 10 units only
     d) No change at all
     Answer: b) An increase of up to 10 units, assuming other conditions remain unchanged
147. If the allowable decrease is 5 units, reducing the right-hand side by 7 units means:
     a) The shadow price must remain valid
     b) The model is automatically infeasible
     c) The current sensitivity result may no longer apply
     d) The objective value remains unchanged
     Answer: c) The current sensitivity result may no longer apply
148. Shadow prices are most useful for:
     a) Naming decision variables
     b) Creating artificial variables
     c) Counting simplex iterations
     d) Evaluating marginal resource and requirement changes
     Answer: d) Evaluating marginal resource and requirement changes
149. Which statement best summarizes duality?
     a) It connects optimal decisions with the marginal value of constraints and resources
     b) It replaces every primal variable with a slack variable
     c) It guarantees that all constraints bind
     d) It applies only to minimization problems
     Answer: a) It connects optimal decisions with the marginal value of constraints and resources
150. Which statement best summarizes the role of shadow prices in management decisions?
     a) They provide exact values for unlimited resource changes
     b) They help assess whether small changes in resource availability are economically worthwhile
     c) They replace the need to solve the primal problem
     d) They measure total accounting cost
     Answer: b) They help assess whether small changes in resource availability are economically worthwhile