Section A: Introduction to Integer Programming

1. What is integer programming primarily used for?
   a) Optimizing decisions in which some or all variables must take integer values
   b) Solving only nonlinear equations
   c) Forecasting demand without constraints
   d) Calculating descriptive statistics
   Answer: a) Optimizing decisions in which some or all variables must take integer values
2. Integer programming is commonly abbreviated as:
   a) IR
   b) IP
   c) IG
   d) LP
   Answer: b) IP
3. An integer-programming model is an optimization model in which:
   a) Every coefficient must be an integer
   b) Every constraint must be an equality
   c) At least some decision variables are restricted to integer values
   d) The objective value must equal zero
   Answer: c) At least some decision variables are restricted to integer values
4. Which decision is best represented by an integer variable?
   a) Liters of fuel consumed
   b) Kilograms of material blended
   c) Hours of machine usage
   d) Number of trucks purchased
   Answer: d) Number of trucks purchased
5. A pure integer-programming problem requires:
   a) All decision variables to be integers
   b) Only one variable to be integer
   c) All coefficients to be binary
   d) Every constraint to be nonlinear
   Answer: a) All decision variables to be integers
6. A mixed-integer programming problem contains:
   a) Only binary variables
   b) Both integer and continuous decision variables
   c) Only continuous variables
   d) No constraints
   Answer: b) Both integer and continuous decision variables
7. Binary integer variables can take the values:
   a) Any positive integers
   b) (-1) and (1)
   c) 0 and 1
   d) Any value between 0 and 1
   Answer: c) 0 and 1
8. A binary variable is also called a:
   a) Slack variable
   b) Surplus variable
   c) Continuous variable
   d) Zero-one variable
   Answer: d) Zero-one variable
9. In a facility-location model, a binary variable may indicate whether a facility is:
   a) Opened or not opened
   b) Producing a fractional quantity
   c) Earning a continuous profit
   d) Using a certain amount of material
   Answer: a) Opened or not opened
10. If (y=1) means a project is selected, then (y=0) means:
    a) The project is selected twice
    b) The project is not selected
    c) The project receives zero profit
    d) The project has no constraints
    Answer: b) The project is not selected
11. Which is a common application of integer programming?
    a) Estimating a population mean
    b) Calculating depreciation only
    c) Selecting projects under a limited budget
    d) Measuring correlation
    Answer: c) Selecting projects under a limited budget
12. Which problem commonly uses binary integer programming?
    a) Determining liters of a chemical blend
    b) Allocating fractional investment amounts only
    c) Calculating average waiting time
    d) Choosing whether to build particular warehouses
    Answer: d) Choosing whether to build particular warehouses
13. Which variable is most naturally modeled as an integer?
    a) Number of employees assigned to a shift
    b) Amount of liquid produced
    c) Percentage of a portfolio invested
    d) Length of a production run in hours
    Answer: a) Number of employees assigned to a shift
14. Integer programming is especially appropriate when decisions involve:
    a) Divisible resources only
    b) Indivisible units or yes-or-no choices
    c) No resource restrictions
    d) Purely descriptive information
    Answer: b) Indivisible units or yes-or-no choices
15. Which statement describes a general integer variable?
    a) It must equal zero or one only
    b) It can take any real value
    c) It may take values such as 0, 1, 2, 3 and so on
    d) It must be negative
    Answer: c) It may take values such as 0, 1, 2, 3 and so on
16. Which restriction represents a binary variable?
    a) (x\geq0)
    b) (x\leq1)
    c) (x) is continuous
    d) (x\in{0,1})
    Answer: d) (x\in{0,1})
17. Which restriction represents a nonnegative general integer variable?
    a) (x\in{0,1,2,\ldots})
    b) (0\leq x\leq1)
    c) (x) is unrestricted and continuous
    d) (x<0)
    Answer: a) (x\in{0,1,2,\ldots})
18. A capital-budgeting model may use binary variables to represent:
    a) The amount of continuous cash flow
    b) Whether each investment project is accepted
    c) The interest rate
    d) The total budget value
    Answer: b) Whether each investment project is accepted
19. A fixed-charge problem generally includes:
    a) Only continuous variable costs
    b) No objective function
    c) A fixed cost incurred when an activity is undertaken
    d) Unlimited resources
    Answer: c) A fixed cost incurred when an activity is undertaken
20. In a fixed-charge model, a binary variable is commonly used to indicate:
    a) The exact amount produced
    b) The value of the objective function
    c) The number of constraints
    d) Whether the activity is activated
    Answer: d) Whether the activity is activated
21. The objective function of an integer program may be designed to:
    a) Maximize profit or minimize cost
    b) Eliminate every constraint
    c) Force all coefficients to be equal
    d) Produce only binary objective values
    Answer: a) Maximize profit or minimize cost
22. Constraints in an integer-programming model represent:
    a) Only variable names
    b) Resource limits, requirements and logical relationships
    c) Only the final optimal value
    d) Only integer restrictions
    Answer: b) Resource limits, requirements and logical relationships
23. A feasible integer solution must:
    a) Satisfy only the integer restrictions
    b) Satisfy only the linear constraints
    c) Satisfy all model constraints and integrality requirements
    d) Maximize the objective automatically
    Answer: c) Satisfy all model constraints and integrality requirements
24. An optimal integer solution is:
    a) Any solution containing whole numbers
    b) The continuous LP optimum
    c) A solution with the most variables
    d) The best feasible solution satisfying the integer restrictions
    Answer: d) The best feasible solution satisfying the integer restrictions
25. Which problem may require integer programming rather than ordinary LP?
    a) Selecting the number of aircraft to purchase
    b) Determining the amount of oil to blend
    c) Allocating divisible raw material
    d) Setting a continuous production rate
    Answer: a) Selecting the number of aircraft to purchase
26. Which is an example of a binary decision?
    a) How many units should be produced?
    b) Should a new plant be opened?
    c) How many liters should be shipped?
    d) How many hours should a machine operate?
    Answer: b) Should a new plant be opened?
27. A set-covering model seeks to:
    a) Assign continuous resources to activities
    b) Minimize inventory only
    c) Select a minimum-cost collection of options that covers all requirements
    d) Maximize the number of uncovered requirements
    Answer: c) Select a minimum-cost collection of options that covers all requirements
28. A set-partitioning model generally requires each item to be:
    a) Covered at least twice
    b) Ignored if expensive
    c) Covered by every selected set
    d) Covered exactly once
    Answer: d) Covered exactly once
29. A set-packing model generally limits each item to being:
    a) Included at most once
    b) Included exactly twice
    c) Included in every set
    d) Assigned a continuous quantity
    Answer: a) Included at most once
30. Which integer-programming model is commonly used for crew scheduling?
    a) Queuing model
    b) Set-partitioning model
    c) Inventory model
    d) Replacement model
    Answer: b) Set-partitioning model
31. Logical conditions such as “select A only if B is selected” are modeled using:
    a) Continuous slack variables only
    b) Objective coefficients only
    c) Binary variables and linking constraints
    d) Graphical methods only
    Answer: c) Binary variables and linking constraints
32. If project A can be selected only when project B is selected, the appropriate binary constraint is:
    a) (x_A+x_B=0)
    b) (x_A\geq x_B)
    c) (x_A+x_B\leq1)
    d) (x_A\leq x_B)
    Answer: d) (x_A\leq x_B)
33. If projects A and B are mutually exclusive, an appropriate constraint is:
    a) (x_A+x_B\leq1)
    b) (x_A=x_B=1)
    c) (x_A\geq x_B)
    d) (x_A+x_B\geq2)
    Answer: a) (x_A+x_B\leq1)
34. If exactly one of projects A and B must be selected, the constraint is:
    a) (x_A+x_B\leq1)
    b) (x_A+x_B=1)
    c) (x_A+x_B\geq2)
    d) (x_A=x_B)
    Answer: b) (x_A+x_B=1)
35. If at least one of projects A and B must be selected, the constraint is:
    a) (x_A+x_B=0)
    b) (x_A+x_B\leq1)
    c) (x_A+x_B\geq1)
    d) (x_A-x_B=1)
    Answer: c) (x_A+x_B\geq1)

Section B: Differences Between Integer Programming and Linear Programming

36. The main difference between IP and LP is that IP:
    a) Has no objective function
    b) Uses only nonlinear constraints
    c) Cannot be solved by computers
    d) Restricts some or all variables to integer values
    Answer: d) Restricts some or all variables to integer values
37. In ordinary linear programming, decision variables are generally allowed to take:
    a) Continuous values
    b) Binary values only
    c) Positive integers only
    d) Negative integers only
    Answer: a) Continuous values
38. Which solution may be acceptable in LP but invalid in an IP model?
    a) (x=4)
    b) (x=4.6) when (x) represents the number of trucks
    c) (x=0)
    d) (x=10)
    Answer: b) (x=4.6) when (x) represents the number of trucks
39. The LP relaxation of an integer program is formed by:
    a) Removing the objective function
    b) Deleting all constraints
    c) Dropping the integer restrictions
    d) Making all variables binary
    Answer: c) Dropping the integer restrictions
40. The feasible region of an LP relaxation contains:
    a) Only integer points
    b) No feasible points
    c) Fewer points than the IP
    d) All IP-feasible points plus possible fractional points
    Answer: d) All IP-feasible points plus possible fractional points
41. For a maximization problem, the LP-relaxation optimum generally provides:
    a) An upper bound on the integer optimum
    b) A lower bound on the integer optimum
    c) The exact integer optimum in every case
    d) No useful information
    Answer: a) An upper bound on the integer optimum
42. For a minimization problem, the LP-relaxation optimum generally provides:
    a) An upper bound on the integer optimum
    b) A lower bound on the integer optimum
    c) A binary solution automatically
    d) The number of branches required
    Answer: b) A lower bound on the integer optimum
43. Why is rounding an LP solution generally unreliable for solving IP?
    a) Rounding always increases the objective
    b) Rounding always creates binary values
    c) The rounded solution may be infeasible or nonoptimal
    d) LP solutions can never be fractional
    Answer: c) The rounded solution may be infeasible or nonoptimal
44. Suppose an LP solution is (x_1=2.6), (x_2=3.4). Rounding both values may:
    a) Always produce the integer optimum
    b) Preserve every constraint automatically
    c) Improve the objective in every case
    d) Violate one or more model constraints
    Answer: d) Violate one or more model constraints
45. Compared with LP, integer programming is generally:
    a) Computationally more difficult
    b) Easier because it has fewer feasible points
    c) Solved only graphically
    d) Free from combinatorial complexity
    Answer: a) Computationally more difficult
46. The feasible set of an integer program is often:
    a) A continuous convex region only
    b) A collection of discrete feasible points
    c) Always empty
    d) Always a straight line
    Answer: b) A collection of discrete feasible points
47. Which property of an LP feasible region supports simplex methods?
    a) It contains only isolated points
    b) It is always circular
    c) It is convex
    d) It contains no corner points
    Answer: c) It is convex
48. The feasible integer points within an LP region are generally:
    a) Every point in the region
    b) Only the corner points
    c) All points on the axes
    d) Discrete points satisfying integrality
    Answer: d) Discrete points satisfying integrality
49. Which technique is commonly used to solve integer programs?
    a) Branch and bound
    b) Northwest Corner Method
    c) Exponential smoothing
    d) Queuing analysis
    Answer: a) Branch and bound
50. Branch and bound solves an IP by:
    a) Ignoring integrality
    b) Dividing the feasible region into smaller subproblems and using bounds
    c) Enumerating only objective coefficients
    d) Replacing all variables with slack variables
    Answer: b) Dividing the feasible region into smaller subproblems and using bounds
51. In branch and bound, branching usually occurs on:
    a) An integer-valued variable only
    b) The objective function
    c) A variable with a fractional LP-relaxation value
    d) A slack variable with value zero
    Answer: c) A variable with a fractional LP-relaxation value
52. If an LP relaxation gives (x=4.7), a branch may create the subproblems:
    a) (x=4.7) and (x=5)
    b) (x\leq4.7) and (x\geq4.7)
    c) (x=0) and (x=1) only
    d) (x\leq4) and (x\geq5)
    Answer: d) (x\leq4) and (x\geq5)
53. A branch-and-bound node can be fathomed when its relaxation is:
    a) Infeasible
    b) Fractional
    c) Larger than the root node
    d) Based on continuous variables
    Answer: a) Infeasible
54. A node may also be fathomed when its LP-relaxation solution is:
    a) Fractional and promising
    b) Integer feasible
    c) Unbounded only
    d) Equal to the root solution only
    Answer: b) Integer feasible
55. For a maximization IP, a node may be pruned when its upper bound is:
    a) Greater than every possible objective
    b) Fractional
    c) No better than the current best integer solution
    d) Equal to the number of variables
    Answer: c) No better than the current best integer solution
56. The best integer-feasible solution found so far is called the:
    a) Relaxation
    b) Branching variable
    c) Root solution
    d) Incumbent solution
    Answer: d) Incumbent solution
57. A cutting-plane method improves an IP relaxation by:
    a) Adding valid constraints that eliminate fractional solutions
    b) Removing the objective function
    c) Making all variables continuous
    d) Adding infeasible integer points
    Answer: a) Adding valid constraints that eliminate fractional solutions
58. A valid inequality in integer programming:
    a) Eliminates all integer solutions
    b) Preserves feasible integer solutions while tightening the relaxation
    c) Removes every constraint
    d) Guarantees a binary solution immediately
    Answer: b) Preserves feasible integer solutions while tightening the relaxation
59. A branch-and-cut algorithm combines:
    a) Dynamic programming and simulation
    b) Queuing and inventory models
    c) Branch and bound with cutting planes
    d) Transportation and assignment methods
    Answer: c) Branch and bound with cutting planes
60. The integrality gap measures the difference between:
    a) Two integer solutions only
    b) The primal and dual LP objectives only
    c) Two branch-and-bound nodes
    d) The LP-relaxation bound and the best integer objective
    Answer: d) The LP-relaxation bound and the best integer objective
61. A small integrality gap generally indicates that:
    a) The LP relaxation closely approximates the integer problem
    b) The model is infeasible
    c) No integer solution exists
    d) The objective function is nonlinear
    Answer: a) The LP relaxation closely approximates the integer problem
62. If the LP-relaxation optimum is already integer feasible, then it is:
    a) Only a lower bound
    b) Also optimal for the integer program
    c) Necessarily infeasible for IP
    d) Required to be rounded
    Answer: b) Also optimal for the integer program
63. Which problem can sometimes yield integer LP solutions without explicit integer restrictions due to model structure?
    a) Every nonlinear program
    b) Every fixed-charge problem
    c) Certain assignment and transportation models
    d) Every capital-budgeting problem
    Answer: c) Certain assignment and transportation models
64. Which statement about IP and LP is correct?
    a) Every IP solution is fractional
    b) Every LP requires binary variables
    c) IP cannot include continuous variables
    d) Mixed-integer models may contain both discrete and continuous decisions
    Answer: d) Mixed-integer models may contain both discrete and continuous decisions
65. Which model is most appropriate when production quantities are divisible?
    a) Linear programming
    b) Pure binary programming
    c) Zero-one programming only
    d) Set-partitioning programming
    Answer: a) Linear programming
66. Which model is most appropriate when machines must be purchased in whole numbers?
    a) Continuous LP
    b) Integer programming
    c) Queuing theory
    d) Simulation only
    Answer: b) Integer programming
67. Compared with LP sensitivity reports, IP sensitivity analysis is often:
    a) Identical in interpretation
    b) Unnecessary
    c) More complex because small data changes can alter discrete decisions
    d) Based only on shadow prices
    Answer: c) More complex because small data changes can alter discrete decisions
68. Shadow prices from an LP relaxation should be applied cautiously to an IP because:
    a) IP has no constraints
    b) All IP variables are continuous
    c) LP relaxations have no objective
    d) Discrete changes may invalidate marginal interpretations
    Answer: d) Discrete changes may invalidate marginal interpretations
69. Which statement best distinguishes an integer solution from a continuous solution?
    a) Integer solutions restrict selected variables to whole-number values
    b) Integer solutions contain no objective function
    c) Continuous solutions must be negative
    d) Continuous models have no constraints
    Answer: a) Integer solutions restrict selected variables to whole-number values
70. The computational effort required for an IP is strongly influenced by:
    a) The font used in the model
    b) The number of integer variables and strength of the formulation
    c) The number of comments in the code
    d) The order in which constraints are written only
    Answer: b) The number of integer variables and strength of the formulation

Section C: Solving Integer-Programming Problems Using LINGO

71. LINGO is primarily used for:
    a) Word processing
    b) Statistical chart design
    c) Mathematical optimization modeling and solution
    d) Database administration only
    Answer: c) Mathematical optimization modeling and solution
72. A LINGO model commonly begins and ends with:
    a) START and STOP
    b) BEGIN and FINISH
    c) OPEN and CLOSE
    d) MODEL: and END
    Answer: d) MODEL: and END
73. Which keyword is used to specify a maximization objective in LINGO?
    a) MAX
    b) MAXIMIZE ONLY
    c) HIGH
    d) OPTMAX
    Answer: a) MAX
74. Which keyword is used to specify a minimization objective in LINGO?
    a) LOW
    b) MIN
    c) MINIMUMVALUE
    d) OPTMIN
    Answer: b) MIN
75. Which symbol normally terminates a LINGO statement?
    a) Colon
    b) Comma
    c) Semicolon
    d) Full stop
    Answer: c) Semicolon
76. Which declaration makes a variable integer in LINGO?
    a) @BIN(variable)
    b) @FREE(variable)
    c) @BND(0,variable,1)
    d) @GIN(variable)
    Answer: d) @GIN(variable)
77. Which declaration makes a variable binary in LINGO?
    a) @BIN(variable)
    b) @GIN(variable)
    c) @INT(variable)
    d) @ZEROONE(variable)
    Answer: a) @BIN(variable)
78. In LINGO, @GIN(X) means that (X) is:
    a) Binary only
    b) A general integer variable
    c) Unrestricted and continuous
    d) A parameter
    Answer: b) A general integer variable
79. In LINGO, @BIN(Y) restricts (Y) to:
    a) Any integer
    b) Any nonnegative real value
    c) Zero or one
    d) Values between (-1) and (1)
    Answer: c) Zero or one
80. Which function can be used to define lower and upper bounds in LINGO?
    a) @SUM
    b) @FOR
    c) @GIN
    d) @BND
    Answer: d) @BND
81. Which LINGO expression correctly declares (X) as a general integer?
    a) @GIN(X);
    b) GIN = X;
    c) X := INTEGER;
    d) INTEGER(X) = 1;
    Answer: a) @GIN(X);
82. Which LINGO expression correctly declares (Y) as binary?
    a) Y = BINARY;
    b) @BIN(Y);
    c) @GIN(Y) = 1;
    d) BIN := Y;
    Answer: b) @BIN(Y);
83. A LINGO objective to maximize (5X+8Y) may be written as:
    a) OBJECTIVE = 5X + 8Y;
    b) MAXIMUM 5X + 8Y;
    c) MAX = 5X + 8Y;
    d) MAX(5X,8Y);
    Answer: c) MAX = 5X + 8Y;
84. Which LINGO statement represents the constraint (2X+3Y\leq20)?
    a) 2X + 3Y =< 20; b) 2X + 3Y => 20;
    c) 2X + 3Y = 20;
    d) 2X + 3Y <= 20;
    Answer: d) 2X + 3Y <= 20;
85. Which LINGO statement represents (X+Y\geq6)?
    a) X + Y >= 6;
    b) X + Y <= 6; c) X + Y = 6; d) X + Y <> 6;
    Answer: a) X + Y >= 6;
86. If (X) and (Y) must both be general integers, the declarations are:
    a) @BIN(X); @BIN(Y);
    b) @GIN(X); @GIN(Y);
    c) @FREE(X); @FREE(Y);
    d) @SUM(X,Y);
    Answer: b) @GIN(X); @GIN(Y);
87. If (X) is continuous and (Y) is integer, the model is a:
    a) Pure linear program
    b) Pure integer program
    c) Mixed-integer program
    d) Binary-only program
    Answer: c) Mixed-integer program
88. To model a yes-or-no plant-opening decision in LINGO, the most appropriate declaration is:
    a) @GIN(OPEN); only
    b) @FREE(OPEN);
    c) @BND(0,OPEN,10);
    d) @BIN(OPEN);
    Answer: d) @BIN(OPEN);
89. The SETS section in LINGO is used to:
    a) Define collections of related objects and attributes
    b) Display the final solution only
    c) Declare the objective value
    d) Stop the solver
    Answer: a) Define collections of related objects and attributes
90. The DATA section in LINGO is commonly used to:
    a) Add branching constraints
    b) Supply numerical parameter values
    c) Define only binary variables
    d) Display reduced costs
    Answer: b) Supply numerical parameter values
91. The @FOR function in LINGO is used to:
    a) Define a single scalar objective
    b) Declare one binary variable
    c) Generate constraints over members of a set
    d) Calculate only averages
    Answer: c) Generate constraints over members of a set
92. The @SUM function in LINGO is used to:
    a) Stop the solution process
    b) Define a set
    c) Declare integer restrictions
    d) Sum an expression over a set
    Answer: d) Sum an expression over a set
93. Which LINGO construct is useful for creating one capacity constraint for every factory?
    a) @FOR
    b) @BIN
    c) @BND
    d) @FREE
    Answer: a) @FOR
94. Which LINGO construct is useful for calculating total cost across all products?
    a) @GIN
    b) @SUM
    c) @FOR only
    d) @BND
    Answer: b) @SUM
95. In a set-based LINGO model, attributes typically represent:
    a) Only comments
    b) Only constraint names
    c) Data or decision variables associated with set members
    d) Solver-status messages
    Answer: c) Data or decision variables associated with set members
96. If LINGO reports the model as infeasible, this means:
    a) The objective is zero
    b) The solver found a fractional solution
    c) The model has multiple optima
    d) No solution satisfies all stated constraints
    Answer: d) No solution satisfies all stated constraints
97. If LINGO finds a global optimum for an integer model, the reported solution:
    a) Satisfies the integer restrictions and has the best proven objective value
    b) Must contain fractional integer variables
    c) Is only an LP-relaxation solution
    d) Requires manual rounding
    Answer: a) Satisfies the integer restrictions and has the best proven objective value
98. Before accepting a LINGO result, the analyst should:
    a) Ignore model status
    b) Check solver status, variable values and constraint satisfaction
    c) Round all continuous variables
    d) Delete integer declarations
    Answer: b) Check solver status, variable values and constraint satisfaction
99. Which is a common modeling error in LINGO integer programs?
    a) Using semicolons
    b) Defining an objective
    c) Forgetting to declare required variables as integer or binary
    d) Including constraints
    Answer: c) Forgetting to declare required variables as integer or binary
100. Which statement best describes effective use of LINGO for integer programming?
     a) LINGO automatically determines the correct business model without user input
     b) Every variable should be declared binary
     c) Solver output eliminates the need for validation
     d) Correct formulation, proper integer declarations and careful result review are essential
     Answer: d) Correct formulation, proper integer declarations and careful result review are essential