Linear vs Nonlinear Programming: Choosing the Right Model for Business Decisions
100+ Multiple-Choice Questions with Answers
1. What is the primary purpose of linear programming?
A. Predicting outcomes
B. Allocating resources efficiently
C. Evaluating marketing strategies
D. Reducing inventory levels
✅ Answer: B
2. The term “programming” in linear programming originally meant:
A. Coding algorithms
B. Optimization
C. Planning
D. Scheduling computer tasks
✅ Answer: C
3. Linear programming involves identifying:
A. Decision variables, objective function, and constraints
B. Only objective function
C. Resources and demand
D. Cost centers
✅ Answer: A
4. Decision variables represent:
A. Constraints to follow
B. Quantities under managerial control
C. Past performance measures
D. Fixed costs
✅ Answer: B
5. The objective function represents:
A. Random output
B. Limiting condition
C. A measurable goal like profit or cost
D. A constant variable
✅ Answer: C
6. In the insulation example, which variable represents truckloads of Type B insulation?
A. r
B. b
C. t
D. c
✅ Answer: B
7. What is the objective function in the insulation problem?
A. 950b + 1200r
B. 1200b + 950r
C. 3b + r ≤ 65
D. b + r ≤ 30
✅ Answer: A
8. What does the constraint 1.4b + 2.8r ≤ 70 represent?
A. Resource availability
B. Loading dock capacity
C. Machine capacity
D. Chemical limit
✅ Answer: C
9. The constraint 3b + r ≤ 65 represents:
A. Chemical constraint
B. Weight constraint
C. Demand constraint
D. Cost constraint
✅ Answer: A
10. Which of the following is NOT a characteristic of a linear programming model?
A. Linear objective function
B. Nonlinear constraints
C. Linear constraints
D. Divisible decision variables
✅ Answer: B
11. The assumption that fractional choices are allowed is known as:
A. Divisibility
B. Flexibility
C. Constraint satisfaction
D. Continuity
✅ Answer: A
12. If a model requires integer answers, it becomes:
A. Linear programming
B. Integer programming
C. Nonlinear optimization
D. Constraint satisfaction
✅ Answer: B
13. The simplex method was developed by:
A. Jonathan Eckstein
B. George Dantzig
C. John Nash
D. Alan Turing
✅ Answer: B
14. The simplex method is mainly used for:
A. Integer programming
B. Linear programming
C. Nonlinear optimization
D. Simulation
✅ Answer: B
15. What is the shadow price associated with the loading dock in the example?
A. $178.57
B. $700
C. $214.28
D. $1,200
✅ Answer: B
16. What does a shadow price of zero indicate?
A. Excess capacity or non-binding constraint
B. Maximum profit
C. Violation of constraint
D. Feasibility error
✅ Answer: A
17. Slack in a constraint refers to:
A. Extra unused resource
B. Excess demand
C. Negative opportunity cost
D. Marginal profit
✅ Answer: A
18. What is the total contribution in the optimal insulation production?
A. $27,600
B. $33,500
C. $28,900
D. $35,000
✅ Answer: B
19. Shadow prices represent:
A. Actual costs
B. Marginal value of resources
C. Fixed costs
D. Random factors
✅ Answer: B
20. If a shadow price is valid only within certain limits, this range is called:
A. Sensitivity range
B. Feasibility range
C. Opportunity range
D. Decision range
✅ Answer: A
21. What does “pricing out” mean in linear programming?
A. Setting market prices
B. Comparing contribution vs. opportunity cost
C. Measuring inflation
D. Budgeting for resources
✅ Answer: B
22. What is the reduced cost for Type X insulation in the example?
A. $0
B. $214.28
C. $825
D. $1,039.28
✅ Answer: B
23. The contribution per truckload of Type X insulation is:
A. $950
B. $1,075
C. $825
D. $1,200
✅ Answer: C
24. The main reason Type X insulation should not be produced is:
A. It uses too much chemical
B. Its opportunity cost exceeds contribution
C. Lack of demand
D. Insufficient data
✅ Answer: B
25. What does “reduced cost = 0” indicate?
A. Optimal activity is in use
B. Non-optimal variable
C. Negative opportunity cost
D. Infeasibility
✅ Answer: A
26. The concept of “smooth divisibility” in linear programming means:
A. Decisions can only be whole numbers
B. Fractional values of variables are allowed
C. Constraints are nonlinear
D. Costs are fixed
✅ Answer: B
27. Integer programming problems are usually:
A. Easier to solve than linear programming
B. Harder to solve than linear programming
C. The same difficulty level
D. Unsolvable by computers
✅ Answer: B
28. A nonlinear programming problem violates which condition?
A. Linear constraints or objective function
B. Decision variables
C. Feasibility
D. Slackness
✅ Answer: A
29. In the insulation example, which constraint ensures that total trucks per day don’t exceed capacity?
A. b + r ≤ 30
B. 3b + r ≤ 65
C. 1.4b + 2.8r ≤ 70
D. b ≥ 0
✅ Answer: A
30. Which constraint prevents negative production levels?
A. 3b + r ≤ 65
B. b ≥ 0 and r ≥ 0
C. b + r ≤ 30
D. None of the above
✅ Answer: B
31. In business, linear programming helps managers primarily in:
A. Advertising
B. Optimization and resource allocation
C. Accounting reconciliation
D. Customer relationship management
✅ Answer: B
32. The simplex method was first developed in which decade?
A. 1930s
B. 1940s
C. 1950s
D. 1960s
✅ Answer: C
33. A shadow price of $700 on the loading dock implies:
A. Each extra truck adds $700 contribution
B. Each extra truck costs $700
C. Each extra truck reduces contribution
D. Shadow price is negative
✅ Answer: A
34. A constraint with zero slack is said to be:
A. Non-binding
B. Redundant
C. Binding
D. Infeasible
✅ Answer: C
35. In the insulation example, how many canisters of flame retardant are used optimally per day?
A. 70
B. 50
C. 65
D. 30
✅ Answer: B
36. If a constraint is non-binding, its shadow price is:
A. Positive
B. Negative
C. Zero
D. Infinite
✅ Answer: C
37. What happens when the availability of a non-binding resource increases?
A. Objective value increases
B. Objective value decreases
C. Objective value remains unchanged
D. Feasibility decreases
✅ Answer: C
38. The “lower” and “upper” range of shadow prices indicate:
A. Feasibility of constraints
B. Range of resource change for which shadow price holds true
C. Cost per resource
D. Optimal variable limits
✅ Answer: B
39. When the shadow price range is exceeded, predictions become:
A. Exact
B. Optimistic
C. Pessimistic
D. Unchanged
✅ Answer: B
40. The shadow price for the flame retardant is zero because:
A. It is a limiting constraint
B. The resource is abundant
C. It is nonlinear
D. It’s not used in production
✅ Answer: B
41. Opportunity cost in linear programming reflects:
A. Historical cost
B. The value of resources used elsewhere
C. Direct cost
D. Average cost
✅ Answer: B
42. In the “Type X” insulation example, opportunity cost was:
A. $825
B. $1,039.28
C. $700
D. $214.28
✅ Answer: B
43. The “reduced cost” represents:
A. Contribution minus opportunity cost
B. Total contribution
C. Fixed overhead
D. Slack variable
✅ Answer: A
44. If reduced cost is positive for a non-produced product, it means:
A. It should be produced
B. It is unprofitable
C. It’s in the optimal plan
D. It has no contribution
✅ Answer: B
45. In the modified insulation example, what is the reduced cost of product X?
A. 0
B. $825
C. $214.28
D. $1,039.28
✅ Answer: C
46. Why do optimal products have a reduced cost of zero?
A. They are non-binding
B. They already satisfy optimality
C. They violate feasibility
D. They are fixed variables
✅ Answer: B
47. Shadow prices can also be viewed as:
A. Resource rental rates
B. Machine downtime costs
C. Depreciation values
D. Average variable costs
✅ Answer: A
48. When production and shadow prices are balanced so that no change is desired, the system is:
A. Unstable
B. Redundant
C. In equilibrium
D. Infeasible
✅ Answer: C
49. Linear programming equilibrium implies:
A. Supply exceeds demand
B. Demand equals supply
C. Negative profit
D. No feasible region
✅ Answer: B
50. Which of the following steps is not part of applying linear programming in practice?
A. Formulating the model
B. Collecting data
C. Marketing execution
D. Running the model
✅ Answer: C
51. Which step in LP is usually the most time-consuming?
A. Data collection
B. Model formulation
C. Software selection
D. Result analysis
✅ Answer: A
52. The first step in applying LP to a business problem is:
A. Running the model
B. Formulating the model
C. Conducting sensitivity analysis
D. Hiring an analyst
✅ Answer: B
53. Sensitivity analysis in LP helps:
A. Evaluate robustness under uncertain data
B. Remove all constraints
C. Change objective functions
D. Replace shadow prices
✅ Answer: A
54. Stochastic programming deals with:
A. Deterministic problems
B. Uncertainty in parameters
C. Static models
D. Graphical solutions
✅ Answer: B
55. The use of stochastic programming is expected to grow due to:
A. Manual data entry
B. Increasing computer power
C. Human intuition
D. Budget constraints
✅ Answer: B
56. In linear programming, “binding constraint” refers to:
A. A redundant restriction
B. A limitation fully used in the optimal solution
C. A flexible resource
D. An infeasible limit
✅ Answer: B
57. Which of these represents a nonlinear constraint?
A. 7x + xy ≤ 8
B. 2x + 3y = 20
C. x − y ≥ 0
D. 3x + y ≤ 50
✅ Answer: A
58. Why might a manager ignore minor nonlinearities when modeling with LP?
A. To simplify analysis while keeping it useful
B. To increase model complexity
C. To make it unsolvable
D. To remove all constraints
✅ Answer: A
59. Sensitivity analysis is especially important when:
A. All data are known with certainty
B. Uncertainty exists in model parameters
C. Objective function is fixed
D. Only one variable is present
✅ Answer: B
60. In real-world practice, linear programming success depends heavily on:
A. Mathematical formulas
B. Managerial understanding and trust
C. Hardware specifications
D. Random experimentation
✅ Answer: B
61. The term “binding constraint” is synonymous with:
A. Non-binding constraint
B. Fully utilized resource
C. Unused resource
D. Redundant condition
✅ Answer: B
62. When a constraint has positive slack, it indicates:
A. Resource is fully used
B. Some of the resource remains unused
C. Constraint is violated
D. Objective is maximized
✅ Answer: B
63. In LP terminology, the term “objective value” refers to:
A. Value of a decision variable
B. Maximum or minimum result of the objective function
C. Slack variable output
D. Range of feasibility
✅ Answer: B
64. In the insulation problem, the optimal objective value is:
A. $27,600
B. $33,500
C. $29,500
D. $31,200
✅ Answer: B
65. If the capacity of the loading dock decreases below the lower range, the shadow price becomes:
A. Invalid
B. Constant
C. Zero
D. Infinite
✅ Answer: A
66. The shadow price of a constraint is best described as:
A. The marginal worth of an additional unit of a resource
B. The total value of a resource
C. The contribution margin
D. A fixed cost
✅ Answer: A
67. If a constraint’s shadow price is $0, what does that imply?
A. The resource is scarce
B. The constraint is binding
C. The resource is non-binding
D. The problem is infeasible
✅ Answer: C
68. A manager can use shadow prices to:
A. Measure how much to pay for additional resources
B. Replace fixed costs
C. Eliminate constraints
D. Forecast sales
✅ Answer: A
69. In the equilibrium interpretation of LP, shadow prices act as:
A. Discounts
B. Transfer prices between product and resource managers
C. Fixed costs
D. Residual income
✅ Answer: B
70. The product managers in the equilibrium model “price out” their products by comparing:
A. Contribution vs. overhead
B. Contribution vs. transfer cost
C. Fixed vs. variable costs
D. Revenue vs. depreciation
✅ Answer: B
71. If the transfer cost equals the contribution, the manager will:
A. Increase production
B. Decrease production
C. Maintain the same level
D. Stop production
✅ Answer: C
72. In the equilibrium model, the flame-retardant manager’s price is zero because:
A. He overcharges
B. There is unused capacity
C. The resource is scarce
D. The system is unstable
✅ Answer: B
73. In LP, what ensures that the system reaches equilibrium?
A. Random decisions
B. Market demand changes
C. Matching resource prices and production levels
D. Arbitrary costs
✅ Answer: C
74. The LP system achieves stability when:
A. No manager has an incentive to change behavior
B. All constraints are ignored
C. Prices fluctuate indefinitely
D. One constraint dominates
✅ Answer: A
75. Which of the following is not a step in linear programming analysis?
A. Formulation
B. Sensitivity analysis
C. Model implementation
D. Market segmentation
✅ Answer: D
76. A linear function must have:
A. Variables multiplied by each other
B. Only addition and scalar multiplication
C. Exponential relationships
D. Logarithmic terms
✅ Answer: B
77. The function ( xy + 7x = 10 ) is:
A. Linear
B. Nonlinear
C. Integer
D. Polynomial
✅ Answer: B
78. If a constraint reads ( 2x + 3y = 20 ), it represents:
A. A linear equality constraint
B. A nonlinear constraint
C. A stochastic model
D. An infeasible condition
✅ Answer: A
79. A nonlinear optimization problem typically involves:
A. Linear objective function only
B. Nonlinear relationships between variables
C. Integer variables
D. No constraints
✅ Answer: B
80. The main advantage of linear programming is:
A. Simplicity and solvability
B. Requirement of integer data
C. Ignoring constraints
D. Random outcomes
✅ Answer: A
81. The graphical method of LP can be used when:
A. There are only two variables
B. Constraints are nonlinear
C. More than five variables exist
D. Decision variables are integers only
✅ Answer: A
82. Sensitivity analysis examines:
A. Impact of changes in parameters on optimal solution
B. Random errors
C. Fixed resource cost
D. Managerial turnover
✅ Answer: A
83. In LP, the concept of “pricing out” helps determine:
A. Resource profitability
B. Whether a new activity should be added
C. Slack value
D. Shadow price
✅ Answer: B
84. What happens if contribution < opportunity cost in LP?
A. Product should be produced
B. Product should be eliminated
C. Slack increases
D. Shadow price decreases
✅ Answer: B
85. When reduced cost is negative for a maximization problem, it implies:
A. Potential increase in profit
B. No change
C. Violation of constraint
D. Infeasibility
✅ Answer: A
86. Which of these industries is a major user of LP techniques?
A. Airlines
B. Entertainment
C. Fashion
D. Education
✅ Answer: A
87. In the insulation example, which resources were fully utilized?
A. Machine and loading dock
B. Flame retardant and trucks
C. All three resources
D. None
✅ Answer: A
88. The “reduced cost” for optimal variables (b and r) equals:
A. Zero
B. Positive value
C. Negative value
D. Undefined
✅ Answer: A
89. Linear programming assumes which type of relationship between variables?
A. Linear and proportional
B. Exponential
C. Irregular
D. Random
✅ Answer: A
90. The function ( 3b + r ≤ 65 ) limits which factor?
A. Machine capacity
B. Flame retardant availability
C. Truck capacity
D. Product price
✅ Answer: B
91. “Constraint slack” can be interpreted as:
A. The difference between LHS and RHS in a constraint
B. Total contribution
C. Shadow price range
D. Reduced cost value
✅ Answer: A
92. In LP, a “feasible region” represents:
A. Area of all possible variable combinations that satisfy constraints
B. Area of negative outcomes
C. Infeasible set
D. Objective boundary
✅ Answer: A
93. What happens if there is no feasible solution?
A. LP model cannot be solved
B. Shadow prices are infinite
C. Slack variables become zero
D. All constraints are redundant
✅ Answer: A
94. The Simplex Method operates by moving from:
A. One infeasible point to another
B. One feasible corner point to another
C. One random solution to another
D. Midpoints of constraints
✅ Answer: B
95. The simplex method continues iterations until:
A. No further improvement in objective function
B. Slack is negative
C. Shadow price increases
D. Constraints double
✅ Answer: A
96. Professional LP modeling tools are mostly used for:
A. Large and complex problems
B. Small, ad-hoc problems
C. Classroom exercises only
D. Manual computation
✅ Answer: A
97. Spreadsheet LP add-ins (like Excel Solver) are best for:
A. Large-scale optimization
B. Small managerial problems
C. AI modeling
D. Machine learning
✅ Answer: B
98. A key managerial challenge in LP implementation is:
A. Lack of understanding or trust in models
B. Unavailability of software
C. High data accuracy
D. Lack of constraints
✅ Answer: A
99. One way to increase trust in LP models is to:
A. Involve managers in model development
B. Hide assumptions
C. Avoid sensitivity analysis
D. Simplify excessively
✅ Answer: A
100. The ultimate managerial goal of linear programming is to:
A. Solve equations
B. Gain insight into resource allocation and decision making
C. Replace all human judgment
D. Eliminate uncertainty
✅ Answer: B
101. The term “stochastic programming” is used when:
A. All parameters are known exactly
B. Some parameters are uncertain
C. Constraints are fixed
D. Objectives are constant
✅ Answer: B
102. What is the expected long-term impact of successful LP projects?
A. Reduced analytical use
B. Increased trust and adoption of quantitative tools
C. Rejection of optimization models
D. Elimination of human roles
✅ Answer: B
103. A linear programming solution is meaningful only if:
A. The model accurately represents reality
B. The shadow prices are zero
C. Constraints are ignored
D. Data are arbitrary
✅ Answer: A
104. The final step of LP application is:
A. Running the model
B. Implementation of results
C. Defining variables
D. Reviewing formulas
✅ Answer: B
105. When over-detailing an LP model, it may become:
A. Unwieldy and less practical
B. More accurate
C. Easier to solve
D. More realistic
✅ Answer: A
xxxxxxx
