- 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 forecast demand without constraints
d) To calculate accounting profit only
Answer: a) To optimize a linear objective subject to linear constraints
- In a linear programming problem, the quantities to be determined are called:
a) Objective coefficients
b) Decision variables
c) Shadow prices
d) Slack resources
Answer: b) Decision variables
- The mathematical expression being maximized or minimized is the:
a) Feasible region
b) Resource equation
c) Objective function
d) Nonnegativity condition
Answer: c) Objective function
- Which statement defines a constraint in linear programming?
a) It states the final optimal value
b) It specifies only the decision-variable names
c) It measures unused resources
d) It represents a limitation or requirement on decisions
Answer: d) It represents a limitation or requirement on decisions
- Which is an example of a linear objective function?
a) Maximize (Z=5x_1+8x_2)
b) Maximize (Z=x_1x_2)
c) Minimize (Z=x_1^2+x_2)
d) Maximize (Z=5/x_1+x_2)
Answer: a) Maximize (Z=5x_1+8x_2)
- Which expression is a linear constraint?
a) (x_1x_2\leq20)
b) (3x_1+2x_2\leq40)
c) (x_1^2+x_2\leq30)
d) (\sqrt{x_1}+x_2\leq12)
Answer: b) (3x_1+2x_2\leq40)
- The nonnegativity condition commonly requires:
a) Objective coefficients to be positive
b) Constraint values to be equal
c) Decision variables to be zero or positive
d) Every constraint to be less than or equal to zero
Answer: c) Decision variables to be zero or positive
- Which notation represents nonnegativity for two variables?
a) (x_1+x_2=0)
b) (x_1,x_2<0)
c) (x_1=x_2)
d) (x_1,x_2\geq0)
Answer: d) (x_1,x_2\geq0)
- A feasible solution is one that:
a) Satisfies every constraint, including sign restrictions
b) Always gives the highest objective value
c) Contains no zero-valued variables
d) Uses all available resources
Answer: a) Satisfies every constraint, including sign restrictions
- An optimal solution is:
a) Any point outside the feasible region
b) A feasible solution with the best objective value
c) A solution that violates one constraint
d) A solution containing the most variables
Answer: b) A feasible solution with the best objective value
- The feasible region consists of:
a) All points satisfying the objective function only
b) All infeasible corner points
c) All points satisfying the complete set of constraints
d) Only the origin
Answer: c) All points satisfying the complete set of constraints
- In a two-variable graphical model, the feasible region is determined by:
a) The objective coefficients alone
b) The number of decision variables
c) The largest constraint coefficient
d) The intersection of all constraint regions
Answer: d) The intersection of all constraint regions
- The proportionality assumption means that:
a) Each variable’s contribution is proportional to its value
b) Variables must have equal values
c) All constraints must have the same right-hand side
d) Resources must be used completely
Answer: a) Each variable’s contribution is proportional to its value
- The additivity assumption means that:
a) Variables must be integers
b) Total effects are the sum of individual variable effects
c) All objective coefficients are added to constraints
d) Constraints cannot share resources
Answer: b) Total effects are the sum of individual variable effects
- The divisibility assumption allows decision variables to:
a) Take only zero or one
b) Take only positive integers
c) Take fractional values
d) Take only negative values
Answer: c) Take fractional values
- The certainty assumption means that:
a) Every optimal solution is unique
b) Decision variables are known before formulation
c) Constraints cannot change form
d) Model coefficients are treated as known constants
Answer: d) Model coefficients are treated as known constants
- Which assumption is violated if production must occur in whole units only?
a) Divisibility
b) Additivity
c) Proportionality
d) Certainty
Answer: a) Divisibility
- Which problem is most suitable for ordinary linear programming?
a) A model with quadratic costs
b) A product-mix problem with linear profit and resource limits
c) A model with uncertain coefficients and no estimates
d) A problem requiring logical either-or conditions only
Answer: b) A product-mix problem with linear profit and resource limits
- In a product-mix model, decision variables commonly represent:
a) Available resource quantities
b) Unit contribution margins
c) Quantities of products to produce
d) Constraint relationships
Answer: c) Quantities of products to produce
- The right-hand side of a resource constraint commonly represents:
a) Unit profit
b) Decision-variable value
c) Reduced cost
d) Available resource capacity
Answer: d) Available resource capacity
- In (4x_1+3x_2\leq120), the number 120 represents:
a) The available amount of the constrained resource
b) The profit per unit of (x_1)
c) The number of basic variables
d) The objective-function value
Answer: a) The available amount of the constrained resource
- In (Z=7x_1+5x_2), the number 7 represents:
a) The right-hand-side value
b) The objective contribution per unit of (x_1)
c) The slack in the first constraint
d) The maximum value of (x_1)
Answer: b) The objective contribution per unit of (x_1)
- A binding constraint at a solution is one that:
a) Has no variables
b) Has a negative right-hand side
c) holds as an equality with no slack or surplus
d) Does not affect the feasible region
Answer: c) Holds as an equality with no slack or surplus
- A nonbinding resource constraint generally has:
a) A negative objective coefficient
b) An artificial variable
c) An infeasible right-hand side
d) Positive unused capacity at the solution
Answer: d) Positive unused capacity at the solution
- If (x_1=2) and (x_2=3), what is (Z=4x_1+5x_2)?
a) 23
b) 18
c) 25
d) 30
Answer: a) 23
- If (2x_1+x_2\leq10), is (x_1=3,x_2=2) feasible for this constraint?
a) No, because the left side is 10
b) Yes, because the left side is 8
c) No, because (x_1>x_2)
d) Yes, because all positive points are feasible
Answer: b) Yes, because the left side is 8
- If (3x_1+2x_2\geq12), which point satisfies the constraint?
a) ((1,2))
b) ((2,1))
c) ((2,3))
d) ((0,4))
Answer: c) ((2,3))
- A linear program is infeasible when:
a) Its objective value is zero
b) It has more constraints than variables
c) It contains equality constraints
d) No point satisfies all constraints simultaneously
Answer: d) No point satisfies all constraints simultaneously
- A maximization model is unbounded when:
a) Its objective can increase indefinitely while feasibility is maintained
b) Every constraint is binding
c) It has a finite optimal corner point
d) It includes slack variables
Answer: a) Its objective can increase indefinitely while feasibility is maintained
- Multiple optimal solutions occur when:
a) The feasible region is empty
b) More than one feasible point gives the same best objective value
c) Every decision variable equals zero
d) The model contains an artificial variable
Answer: b) More than one feasible point gives the same best objective value
- In the graphical method, an optimum for a linear program generally occurs at:
a) The center of the feasible region
b) Any point on a constraint line
c) A corner or extreme point of the feasible region
d) The point with the largest coordinate values
Answer: c) A corner or extreme point of the feasible region
- The simplex method moves from:
a) One objective function to another
b) An infeasible point to another infeasible point
c) One constraint equation to another
d) One basic feasible solution to an adjacent one
Answer: d) One basic feasible solution to an adjacent one
- The simplex method is especially useful when a model has:
a) More than two decision variables
b) Exactly one constraint
c) No objective function
d) Only nonlinear relationships
Answer: a) More than two decision variables
- Standard form generally requires constraints to be written as:
a) Strict inequalities
b) Equations with nonnegative variables
c) Nonlinear functions
d) Objective functions
Answer: b) Equations with nonnegative variables
- A basic solution is obtained by:
a) Setting all variables equal
b) Ignoring the constraints
c) Setting selected nonbasic variables to zero and solving for basic variables
d) Maximizing every variable independently
Answer: c) Setting selected nonbasic variables to zero and solving for basic variables
- In a system with (m) independent equality constraints, a basis normally contains:
a) One variable
b) (m-1) variables
c) All variables
d) (m) basic variables
Answer: d) (m) basic variables
- Nonbasic variables in a basic solution are normally assigned:
a) A value of zero
b) A value of one
c) The objective-function value
d) The right-hand-side values
Answer: a) A value of zero
- A basic feasible solution requires the basic-variable values to be:
a) Strictly positive only
b) Nonnegative
c) Equal to one
d) Larger than the nonbasic values
Answer: b) Nonnegative
- A degenerate basic feasible solution has:
a) No constraints
b) Multiple objective functions
c) At least one basic variable equal to zero
d) Every nonbasic variable positive
Answer: c) At least one basic variable equal to zero
- Cycling in the simplex method refers to:
a) Reversing the objective direction
b) Converting constraints repeatedly
c) Changing all coefficient signs
d) Repeating bases without reaching improvement
Answer: d) Repeating bases without reaching improvement
Section B: Slack, Surplus and Artificial Variables
- A slack variable is added to which type of constraint?
a) A less-than-or-equal-to constraint
b) A greater-than-or-equal-to constraint
c) A strict inequality
d) An objective function
Answer: a) A less-than-or-equal-to constraint
- The main purpose of a slack variable is to:
a) Increase available resources
b) convert a (\leq) constraint into an equation
c) Penalize infeasibility
d) Maximize the objective value
Answer: b) Convert a (\leq) constraint into an equation
- The constraint (2x_1+x_2\leq10) becomes:
a) (2x_1+x_2-s_1=10)
b) (2x_1+x_2+A_1=10)
c) (2x_1+x_2+s_1=10)
d) (2x_1+x_2=0)
Answer: c) (2x_1+x_2+s_1=10)
- A positive slack-variable value indicates:
a) A shortage of the resource
b) An infeasible constraint
c) Excess requirement above the minimum
d) Unused resource capacity
Answer: d) Unused resource capacity
- If (x_1+x_2+s_1=20) and (x_1+x_2=14), then (s_1) equals:
a) 6
b) 14
c) 20
d) 34
Answer: a) 6
- A surplus variable is associated with:
a) A (\leq) constraint
b) A (\geq) constraint
c) A nonnegativity condition
d) An objective equation only
Answer: b) A (\geq) constraint
- A surplus variable is:
a) Added to the left side
b) Multiplied by the right-hand side
c) Subtracted from the left side
d) Removed from the model
Answer: c) Subtracted from the left side
- The constraint (3x_1+2x_2\geq18) becomes:
a) (3x_1+2x_2+s_1=18)
b) (3x_1+2x_2=0)
c) (3x_1+2x_2+A_1=18)
d) (3x_1+2x_2-s_1=18), before adding any required artificial variable
Answer: d) (3x_1+2x_2-s_1=18), before adding any required artificial variable
- A positive surplus-variable value measures:
a) The amount by which the left side exceeds the minimum requirement
b) Unused capacity in a maximum constraint
c) The objective-function value
d) The number of simplex iterations
Answer: a) The amount by which the left side exceeds the minimum requirement
- Why is an artificial variable often added to a (\geq) constraint?
a) To create unused capacity
b) To provide an initial basic variable
c) To increase the right-hand side
d) To represent profit
Answer: b) To provide an initial basic variable
- An equality constraint may require:
a) A slack variable only
b) A surplus variable only
c) An artificial variable to establish an initial basis
d) No variable under any circumstances
Answer: c) An artificial variable to establish an initial basis
- Artificial variables are introduced primarily as:
a) Permanent production decisions
b) Measures of unused resources
c) Final optimal variables
d) Temporary computational devices
Answer: d) Temporary computational devices
- In a valid final solution, artificial variables should normally be:
a) Zero and removed from the basis when possible
b) Positive and large
c) Equal to the right-hand side
d) Larger than slack variables
Answer: a) Zero and removed from the basis when possible
- The equation for (x_1+x_2\geq8) using a surplus and artificial variable is:
a) (x_1+x_2+s_1=8)
b) (x_1+x_2-s_1+A_1=8)
c) (x_1+x_2+A_1=0)
d) (x_1+x_2-s_1=0)
Answer: b) (x_1+x_2-s_1+A_1=8)
- The equality (2x_1+x_2=12) may be written for simplex initialization as:
a) (2x_1+x_2+s_1=12)
b) (2x_1+x_2-s_1=12)
c) (2x_1+x_2+A_1=12)
d) (2x_1+x_2=0)
Answer: c) (2x_1+x_2+A_1=12)
- Which variable has a coefficient of (+1) in a converted (\leq) constraint?
a) Surplus variable
b) Artificial penalty
c) Decision variable only
d) Slack variable
Answer: d) Slack variable
- Which variable typically has a coefficient of (-1) in a converted (\geq) constraint?
a) Surplus variable
b) Slack variable
c) Artificial variable
d) Basic decision variable
Answer: a) Surplus variable
- Which variable usually enters with a coefficient of (+1) after subtracting surplus?
a) A second surplus variable
b) An artificial variable
c) A shadow-price variable
d) A reduced-cost variable
Answer: b) An artificial variable
- In a resource constraint, a slack value of zero means the resource is:
a) Unlimited
b) Unused
c) Fully utilized at that solution
d) Infeasible
Answer: c) Fully utilized at that solution
- If a minimum requirement constraint has zero surplus, the achieved level is:
a) Below the minimum
b) Unbounded
c) Infeasible
d) Exactly equal to the minimum
Answer: d) Exactly equal to the minimum
- Which conversion is correct for (4x_1+x_2\leq16)?
a) (4x_1+x_2+s_1=16)
b) (4x_1+x_2-s_1=16)
c) (4x_1+x_2+A_1=16)
d) (4x_1+x_2-s_1+A_1=16)
Answer: a) (4x_1+x_2+s_1=16)
- Which conversion is correct for (x_1+5x_2\geq25)?
a) (x_1+5x_2+s_1=25)
b) (x_1+5x_2-s_1+A_1=25)
c) (x_1+5x_2+s_1+A_1=25)
d) (x_1+5x_2-A_1=25)
Answer: b) (x_1+5x_2-s_1+A_1=25)
- Which conversion is normally used for (3x_1+4x_2=24)?
a) Add a slack variable
b) Subtract a surplus variable only
c) Add an artificial variable when an initial basis is needed
d) Reverse the objective function
Answer: c) Add an artificial variable when an initial basis is needed
- If the right-hand side of a constraint is negative, a common first step is to:
a) Delete the constraint
b) Add two artificial variables
c) Set every variable to zero
d) Multiply the entire constraint by (-1) and reverse the inequality
Answer: d) Multiply the entire constraint by (-1) and reverse the inequality
- Multiplying (-2x_1+x_2\leq-6) by (-1) gives:
a) (2x_1-x_2\geq6)
b) (2x_1-x_2\leq6)
c) (-2x_1+x_2\geq6)
d) (2x_1+x_2=6)
Answer: a) (2x_1-x_2\geq6)
- In the Big M method for a maximization problem, artificial variables receive:
a) A large positive reward
b) A very large negative objective coefficient
c) A zero objective coefficient
d) The same coefficient as slack variables
Answer: b) A very large negative objective coefficient
- In a minimization Big M model, an artificial variable commonly receives:
a) A large negative coefficient
b) A zero coefficient
c) A large positive penalty coefficient
d) A coefficient of one only
Answer: c) A large positive penalty coefficient
- The purpose of the Big M penalty is to:
a) Encourage artificial variables to remain positive
b) measure unused resources
c) create alternate optima
d) force artificial variables out of the optimal solution
Answer: d) Force artificial variables out of the optimal solution
- Phase I of the Two-Phase method seeks to:
a) Minimize the sum of artificial variables
b) Maximize the original profit
c) Calculate shadow prices
d) Determine allowable objective ranges
Answer: a) Minimize the sum of artificial variables
- Phase II begins after:
a) Every slack variable is positive
b) Phase I finds a feasible basis with artificial variables at zero
c) The objective function becomes unbounded
d) All constraints become nonbinding
Answer: b) Phase I finds a feasible basis with artificial variables at zero
- If the minimum Phase I objective value is positive, the original model is:
a) Unbounded
b) Degenerate only
c) Infeasible
d) Guaranteed optimal
Answer: c) Infeasible
- If Phase I ends at zero, this indicates that:
a) The original model is automatically optimal
b) The original objective value is zero
c) No constraints are binding
d) A feasible solution to the original constraints has been found
Answer: d) A feasible solution to the original constraints has been found
- Which variable is not part of the original real-world decision problem?
a) Artificial variable
b) Production variable
c) Shipment variable
d) Investment variable
Answer: a) Artificial variable
- Slack variables usually have what coefficient in the original objective function?
a) One
b) Zero
c) A large positive number
d) A large negative number
Answer: b) Zero
- Surplus variables normally have what coefficient in the original objective function?
a) The largest profit coefficient
b) One
c) Zero
d) The right-hand-side value
Answer: c) Zero
- The identity columns needed for an obvious initial basis are commonly supplied by:
a) Objective coefficients
b) Right-hand-side values
c) Reduced costs
d) Slack or artificial variables
Answer: d) Slack or artificial variables
- If all constraints are (\leq) with nonnegative right-hand sides, the initial basis can often consist of:
a) Slack variables
b) Surplus variables
c) Decision variables only
d) Artificial variables only
Answer: a) Slack variables
- A surplus variable alone cannot provide the usual initial basis because its column contains:
a) A zero coefficient
b) A (-1) rather than the required (+1) identity entry
c) A nonlinear term
d) An unknown right-hand side
Answer: b) A (-1) rather than the required (+1) identity entry
- An artificial variable remaining positive at the end of optimization signals:
a) Multiple optimal solutions
b) An unused resource
c) Infeasibility of the original problem
d) A nonbinding constraint
Answer: c) Infeasibility of the original problem
- Which method avoids using an unspecified numerical value (M)?
a) Graphical method
b) Primal simplex only
c) Big M method
d) Two-Phase method
Answer: d) Two-Phase method
Section C: Simplex Tableau and Maximization Problems
- The initial simplex tableau contains:
a) Constraint coefficients, objective information and right-hand-side values
b) Only the objective coefficients
c) Only the decision-variable values
d) Only slack-variable columns
Answer: a) Constraint coefficients, objective information and right-hand-side values
- In the (C_j-Z_j) maximization convention, the entering variable is commonly chosen from the column with:
a) The most negative right-hand side
b) The largest positive (C_j-Z_j) value
c) The smallest objective coefficient
d) The greatest slack value
Answer: b) The largest positive (C_j-Z_j) value
- In the (Z_j-C_j) maximization convention, optimality is reached when all values are:
a) Negative only
b) Equal to one
c) Zero or positive
d) Larger than the right-hand side
Answer: c) Zero or positive
- The entering column is also called the:
a) Slack column
b) identity column
c) feasibility column
d) Pivot column
Answer: d) Pivot column
- The leaving variable is normally determined using:
a) The minimum positive-ratio test
b) The largest objective coefficient
c) The smallest row coefficient
d) The maximum negative ratio
Answer: a) The minimum positive-ratio test
- In the ratio test, each eligible ratio is calculated as:
a) Pivot-column entry divided by right-hand side
b) Right-hand side divided by a positive pivot-column entry
c) Objective coefficient divided by right-hand side
d) Basic cost divided by pivot entry
Answer: b) Right-hand side divided by a positive pivot-column entry
- Rows with zero or negative pivot-column entries are generally:
a) Always selected
b) Converted into objective rows
c) Excluded from the standard positive-ratio test
d) Assigned a ratio of zero automatically
Answer: c) Excluded from the standard positive-ratio test
- The intersection of the pivot row and pivot column is the:
a) Reduced cost
b) shadow price
c) basic coefficient
d) Pivot element
Answer: d) Pivot element
- The first row operation in pivoting usually makes the pivot element:
a) Equal to one
b) Equal to zero
c) Negative
d) Equal to the right-hand side
Answer: a) Equal to one
- After normalizing the pivot row, other row operations make the remaining pivot-column entries:
a) Equal to one
b) Equal to zero
c) Positive
d) Identical to the objective coefficient
Answer: b) Equal to zero
- A variable entering the basis becomes:
a) A surplus variable
b) An objective coefficient
c) A basic variable
d) A right-hand-side value
Answer: c) A basic variable
- A variable leaving the basis normally becomes:
a) Artificial
b) unrestricted
c) binding
d) Nonbasic with value zero
Answer: d) Nonbasic with value zero
- The basis column of a basic variable should look like:
a) A unit or identity column
b) A column of negative ones
c) The objective-function column
d) The right-hand-side column
Answer: a) A unit or identity column
- The current values of basic variables are read from:
a) The objective-coefficient row
b) The right-hand-side column
c) The (C_j) row
d) The variable-name column only
Answer: b) The right-hand-side column
- A nonbasic variable’s value in a standard tableau solution is:
a) Its objective coefficient
b) Its reduced cost
c) Zero
d) Its shadow price
Answer: c) Zero
- The objective-function value is commonly found in:
a) The slack-variable column
b) The pivot column
c) The first constraint row
d) The objective row’s right-hand-side position
Answer: d) The objective row’s right-hand-side position
- Consider maximize (Z=3x_1+5x_2). Which variable has the larger initial profit coefficient?
a) (x_2)
b) (x_1)
c) Both have equal coefficients
d) Neither variable
Answer: a) (x_2)
- For (2x_1+x_2+s_1=8), if (x_1=x_2=0), the initial value of (s_1) is:
a) 0
b) 8
c) 2
d) 1
Answer: b) 8
- For (x_1+3x_2+s_2=12), the slack variable’s initial value is:
a) 3
b) 1
c) 12
d) 0
Answer: c) 12
- In a maximization tableau using (C_j-Z_j), no positive values in the evaluation row indicate:
a) Infeasibility
b) Degeneracy
c) Unboundedness
d) Optimality
Answer: d) Optimality
- If the entering column has no positive constraint coefficient, the maximization problem is:
a) Unbounded in the entering direction
b) Infeasible in all cases
c) Degenerate only
d) Already optimal
Answer: a) Unbounded in the entering direction
- A tie in the minimum-ratio test may indicate the possibility of:
a) Unboundedness only
b) Degeneracy
c) A nonlinear objective
d) No feasible basis
Answer: b) Degeneracy
- A zero-valued basic variable indicates:
a) An alternate optimum automatically
b) An unbounded model
c) A degenerate basic feasible solution
d) A negative slack value
Answer: c) A degenerate basic feasible solution
- Bland’s rule is designed primarily to prevent:
a) Artificial-variable penalties
b) Multiple objective functions
c) infeasibility
d) Cycling
Answer: d) Cycling
- Multiple optimal solutions may be detected when:
a) A nonbasic variable has zero reduced cost at optimality
b) Every slack variable is positive
c) An artificial variable is positive
d) The model has one constraint
Answer: a) A nonbasic variable has zero reduced cost at optimality
- In a maximization problem, a positive (C_j-Z_j) for a nonbasic variable means:
a) The model is infeasible
b) Introducing it may improve the objective
c) The current basis is necessarily degenerate
d) The variable must remain zero
Answer: b) Introducing it may improve the objective
- Reduced cost for a nonbasic variable describes:
a) Its current production quantity
b) Its resource consumption
c) The objective improvement or deterioration associated with entering the basis
d) The constraint’s right-hand side
Answer: c) The objective improvement or deterioration associated with entering the basis
- The simplex method stops when:
a) Every variable is basic
b) Every constraint is nonbinding
c) Every slack variable equals zero
d) The optimality criterion is satisfied
Answer: d) The optimality criterion is satisfied
- For maximize (Z=4x_1+2x_2), subject to (x_1+x_2\leq5), which corner gives the higher value between ((5,0)) and ((0,5))?
a) ((5,0))
b) ((0,5))
c) Both give the same value
d) Neither point is feasible
Answer: a) ((5,0))
- At ((x_1,x_2)=(2,3)), the value of (Z=6x_1+4x_2) is:
a) 20
b) 24
c) 30
d) 36
Answer: b) 24
- If a constraint (x_1+x_2\leq10) has solution (x_1=4,x_2=6), its slack is:
a) 10
b) 4
c) 0
d) 6
Answer: c) 0
- If (2x_1+x_2\leq15) and the solution is (x_1=4,x_2=3), slack equals:
a) 11
b) 8
c) 3
d) 4
Answer: d) 4
- If a maximization model has all (\leq) constraints and positive right-hand sides, the initial solution normally sets:
a) Decision variables to zero and slack variables to resource amounts
b) Slack variables to zero and decisions to maximum values
c) Every variable to one
d) Artificial variables to positive values
Answer: a) Decision variables to zero and slack variables to resource amounts
- Which condition guarantees that the all-zero decision solution satisfies a (\leq) resource constraint with positive RHS?
a) Objective coefficients are positive
b) The left side becomes zero, which does not exceed the RHS
c) Every slack variable is nonbasic
d) The model is unbounded
Answer: b) The left side becomes zero, which does not exceed the RHS
- The number of basic variables in a tableau normally equals:
a) The number of decision variables
b) The total number of variables
c) The number of equality constraints
d) The number of objective coefficients
Answer: c) The number of equality constraints
- Each simplex pivot operation changes:
a) The mathematical meaning of the model
b) The number of constraints
c) The feasible region
d) The current basis
Answer: d) The current basis
- The simplex algorithm improves the objective while maintaining:
a) Feasibility of the basic solution
b) Positive reduced costs only
c) Equal variable values
d) A fixed pivot column
Answer: a) Feasibility of the basic solution
- Which variable leaves when ratios are 8, 5 and 12?
a) The row with ratio 8
b) The row with ratio 5
c) The row with ratio 12
d) No row leaves
Answer: b) The row with ratio 5
- If a pivot element is 4, the pivot row is normalized by:
a) Multiplying the row by 4
b) Adding 4 to each entry
c) Dividing every pivot-row entry by 4
d) Subtracting 4 from the RHS
Answer: c) Dividing every pivot-row entry by 4
- To eliminate a coefficient of 3 in another pivot-column row, one may:
a) Divide the row by 3 only
b) Replace the objective function
c) Add the unnormalized pivot row
d) Subtract three times the normalized pivot row
Answer: d) Subtract three times the normalized pivot row
- If (C_j-Z_j=0) for a basic variable, this is:
a) Expected because a basic variable’s reduced cost is zero
b) Proof of infeasibility
c) Evidence of unboundedness
d) A reason to remove the constraint
Answer: a) Expected because a basic variable’s reduced cost is zero
- If a tableau’s RHS contains a negative value during primal simplex, the current basis is generally:
a) Optimal
b) Primal infeasible
c) Unbounded
d) Nonlinear
Answer: b) Primal infeasible
- Which method is often useful when the tableau is dual feasible but primal infeasible?
a) Graphical method
b) Big M method only
c) Dual simplex method
d) Transportation method
Answer: c) Dual simplex method
- The revised simplex method differs mainly by:
a) Solving nonlinear constraints
b) Eliminating the need for a basis
c) Checking every feasible point
d) Updating basis-related matrices rather than the full tableau
Answer: d) Updating basis-related matrices rather than the full tableau
- The geometric interpretation of a simplex pivot is movement to:
a) An adjacent extreme point
b) The center of the feasible region
c) An infeasible interior point
d) A random point
Answer: a) An adjacent extreme point
Section D: Minimization, Big M and Two-Phase Methods
- A minimization objective seeks:
a) The largest feasible objective value
b) The smallest feasible objective value
c) The largest slack value
d) The most decision variables
Answer: b) The smallest feasible objective value
- Which is a common minimization application?
a) Maximizing advertising reach
b) Maximizing contribution margin
c) Minimizing production and distribution cost
d) Maximizing investment return
Answer: c) Minimizing production and distribution cost
- Minimization models often contain:
a) Only (\leq) resource constraints
b) No constraints
c) Nonlinear variables
d) Minimum-requirement constraints of the (\geq) type
Answer: d) Minimum-requirement constraints of the (\geq) type
- A diet problem commonly minimizes:
a) Cost while meeting nutritional minimums
b) Nutritional content
c) Number of food types only
d) Available supply
Answer: a) Cost while meeting nutritional minimums
- A blending model may minimize cost subject to:
a) Only maximum-profit conditions
b) Minimum quality or composition requirements
c) No resource restrictions
d) Integer requirements only
Answer: b) Minimum quality or composition requirements
- To convert a minimization objective to a maximization objective algebraically, one may:
a) Reverse every constraint
b) Add slack to the objective
c) Maximize the negative of the original objective
d) Set the objective equal to zero
Answer: c) Maximize the negative of the original objective
- The dual of a minimization problem may sometimes be solved because it becomes:
a) A nonlinear model
b) An assignment model
c) A transportation table
d) A maximization problem with a convenient simplex form
Answer: d) A maximization problem with a convenient simplex form
- The Big M method includes artificial variables in:
a) The objective function with severe penalties
b) The nonnegativity restrictions only
c) The software installation
d) The final report only
Answer: a) The objective function with severe penalties
- In minimizing (Z=4x_1+6x_2), an artificial variable may be assigned:
a) A coefficient of zero
b) A coefficient of (+M)
c) A coefficient of (-M)
d) The coefficient 4
Answer: b) A coefficient of (+M)
- In maximizing (Z=5x_1+3x_2), an artificial variable may be assigned:
a) (+M)
b) Zero
c) (-M)
d) (+1)
Answer: c) (-M)
- The symbol (M) represents:
a) The number of constraints
b) The number of basic variables
c) The objective value
d) A conceptually very large positive number
Answer: d) A conceptually very large positive number
- A practical difficulty with Big M in numerical software is:
a) Very large coefficients may create numerical instability
b) It cannot represent equality constraints
c) It always returns integer answers
d) It removes the objective function
Answer: a) Very large coefficients may create numerical instability
- The Two-Phase method handles artificial variables by:
a) Ignoring them
b) Solving a separate feasibility objective first
c) Giving them random coefficients
d) Treating them as slack resources
Answer: b) Solving a separate feasibility objective first
- In Phase I, the auxiliary objective is commonly:
a) Maximize the original profit
b) Maximize the artificial-variable sum
c) Minimize the sum of artificial variables
d) Minimize the slack-variable sum
Answer: c) Minimize the sum of artificial variables
- If Phase I produces a positive minimum, the conclusion is:
a) The original problem is optimal
b) The original model has alternate optima
c) The model is unbounded
d) The original constraint system is infeasible
Answer: d) The original constraint system is infeasible
- At the start of Phase II, the original objective function is:
a) Restored and optimized from the feasible basis
b) Permanently discarded
c) Replaced by the artificial-variable sum
d) Set equal to the Phase I value
Answer: a) Restored and optimized from the feasible basis
- Artificial-variable columns are generally removed before Phase II because:
a) They represent real decisions
b) They are no longer needed in the original model
c) They measure unused resources
d) They are always nonnegative
Answer: b) They are no longer needed in the original model
- If an artificial variable is basic at zero after Phase I, it may indicate:
a) Unboundedness
b) A positive Phase I objective
c) A redundant constraint or degenerate basis
d) A negative right-hand side
Answer: c) A redundant constraint or degenerate basis
- A redundant constraint is one that:
a) Makes the model infeasible
b) Contains no variables
c) Must always be binding
d) Does not further restrict the feasible region
Answer: d) Does not further restrict the feasible region
- Which model requires an artificial variable most directly?
a) (x_1+x_2=10)
b) (x_1+x_2\leq10)
c) (x_1,x_2\geq0)
d) (Z=3x_1+2x_2)
Answer: a) (x_1+x_2=10)
- Which converted constraint includes both surplus and artificial variables?
a) (x_1+x_2\leq8)
b) (x_1+x_2\geq8)
c) (x_1+x_2=8) with an existing identity column
d) (x_1,x_2\geq0)
Answer: b) (x_1+x_2\geq8)
- If (2x_1+x_2\geq10), the correct standard equation is:
a) (2x_1+x_2+s_1=10)
b) (2x_1+x_2+A_1=10)
c) (2x_1+x_2-s_1+A_1=10)
d) (2x_1+x_2-s_1=0)
Answer: c) (2x_1+x_2-s_1+A_1=10)
- If (x_1+4x_2=20), a common initial-basis equation is:
a) (x_1+4x_2+s_1=20)
b) (x_1+4x_2-s_1=20)
c) (x_1+4x_2=0)
d) (x_1+4x_2+A_1=20)
Answer: d) (x_1+4x_2+A_1=20)
- The simplex optimality criterion for minimization depends on:
a) The tableau convention used for reduced costs
b) Whether the model has two variables
c) The graphical slope only
d) The number of slack variables
Answer: a) The tableau convention used for reduced costs
- Under a (C_j-Z_j) minimization convention, optimality commonly requires all values to be:
a) Positive only
b) Zero or positive, depending on the stated convention
c) Strictly negative only
d) Equal to the RHS
Answer: b) Zero or positive, depending on the stated convention
- Why must the reduced-cost sign convention be stated clearly?
a) It determines the number of constraints
b) It changes the feasible region
c) Different tableau formats reverse the apparent optimality signs
d) It changes minimization into nonlinear programming
Answer: c) Different tableau formats reverse the apparent optimality signs
- A minimization model is unbounded below when:
a) No feasible point exists
b) All constraints bind
c) Artificial variables remain positive
d) The objective can decrease indefinitely while remaining feasible
Answer: d) The objective can decrease indefinitely while remaining feasible
- If a minimization problem has no common feasible point, it is:
a) Infeasible
b) Degenerate
c) Alternate optimal
d) Redundant
Answer: a) Infeasible
- If several feasible solutions have the same minimum cost, the model has:
a) An unbounded solution
b) Multiple optimal solutions
c) No basic variables
d) A positive Phase I value
Answer: b) Multiple optimal solutions
- A zero reduced cost for a nonbasic variable at a minimization optimum suggests:
a) Infeasibility
b) Unboundedness
c) An alternate optimal solution may exist
d) A negative right-hand side
Answer: c) An alternate optimal solution may exist
- Complementary slackness connects:
a) Two primal constraints only
b) Slack and surplus in the same equation only
c) Two graphical corner points
d) Optimal primal and dual solutions
Answer: d) Optimal primal and dual solutions
- The dual of a primal maximization model with (\leq) constraints is commonly a:
a) Minimization model with (\geq) constraints
b) Maximization model with (\leq) constraints
c) Nonlinear model
d) Queuing model
Answer: a) Minimization model with (\geq) constraints
- The number of dual variables equals the number of:
a) Primal variables
b) Primal constraints
c) Slack variables only
d) Artificial variables
Answer: b) Primal constraints
- The number of dual constraints equals the number of:
a) Primal constraints
b) Basic variables
c) Primal decision variables
d) Tableau rows plus one
Answer: c) Primal decision variables
- Under strong duality, when both models have optimal solutions:
a) The primal value is always larger
b) The dual value is always larger
c) Both variable vectors are identical
d) The primal and dual objective values are equal
Answer: d) The primal and dual objective values are equal
- A shadow price measures:
a) The change in optimal objective value from a one-unit RHS increase within an allowable range
b) The market price of a finished product
c) The value of a slack variable
d) The Big M penalty
Answer: a) The change in optimal objective value from a one-unit RHS increase within an allowable range
- A zero shadow price commonly indicates that a resource constraint is:
a) Infeasible
b) Nonbinding at the optimum
c) Unbounded
d) Artificial
Answer: b) Nonbinding at the optimum
- Sensitivity analysis examines:
a) Only the current decision-variable values
b) Only the number of iterations
c) How changes in coefficients affect the optimal solution
d) How to install solver software
Answer: c) How changes in coefficients affect the optimal solution
- The allowable increase for an objective coefficient identifies:
a) How much the RHS can increase
b) The maximum decision-variable value
c) The number of additional constraints
d) How much the coefficient may rise without changing the current optimal basis
Answer: d) How much the coefficient may rise without changing the current optimal basis
- Reduced cost for a nonbasic maximization variable can indicate:
a) How much its objective coefficient must improve before it may enter the basis
b) Its current slack
c) The available resource quantity
d) The Phase I objective value
Answer: a) How much its objective coefficient must improve before it may enter the basis
- The allowable range for a right-hand side preserves:
a) The objective coefficients only
b) The current basis and associated shadow-price validity
c) The number of decision variables
d) The use of artificial variables
Answer: b) The current basis and associated shadow-price validity
- Sensitivity results are valid under the usual assumption that:
a) Every coefficient changes simultaneously without limit
b) The model becomes nonlinear
c) Other data remain fixed when one parameter range is interpreted
d) Decision variables become integer
Answer: c) Other data remain fixed when one parameter range is interpreted
- A shadow price should not be applied beyond its allowable RHS range because:
a) The objective becomes zero
b) The resource disappears
c) Slack variables become negative automatically
d) The optimal basis may change
Answer: d) The optimal basis may change
- If a binding resource has a positive shadow price in a maximization model, one more unit may:
a) Increase the optimal objective value within the valid range
b) Always reduce profit
c) Have no effect
d) Make the model infeasible in every case
Answer: a) Increase the optimal objective value within the valid range
- If a nonbinding constraint has unused capacity, adding more of that resource will commonly:
a) Change every decision variable
b) Have no immediate objective benefit
c) Make the model unbounded
d) require an artificial variable
Answer: b) Have no immediate objective benefit
Section E: Software for Solving Linear Programs
- Which Microsoft Excel feature is commonly used to solve linear programs?
a) PivotTable
b) Goal Seek only
c) Solver
d) Conditional Formatting
Answer: c) Solver
- Which Excel Solver method should be selected for a purely linear model?
a) GRG Nonlinear
b) Evolutionary
c) Automatic scaling only
d) Simplex LP
Answer: d) Simplex LP
- In Excel Solver, the objective cell should contain:
a) A formula calculating the objective-function value
b) A text description of the model
c) Only a decision-variable name
d) A constraint label
Answer: a) A formula calculating the objective-function value
- Excel Solver’s changing variable cells correspond to:
a) Shadow prices
b) Decision variables
c) Slack-variable reports only
d) Constraint labels
Answer: b) Decision variables
- Solver constraints are used to represent:
a) Worksheet colors
b) Chart titles
c) Resource limits and model requirements
d) Only nonnegativity
Answer: c) Resource limits and model requirements
- To enforce nonnegative variables in Excel Solver, users may:
a) Delete the objective formula
b) Select GRG Nonlinear
c) Remove all constraints
d) Use the nonnegative-variable option or explicit lower bounds
Answer: d) Use the nonnegative-variable option or explicit lower bounds
- Which Excel function is commonly used to calculate a linear objective from coefficients and variables?
a) SUMPRODUCT
b) VLOOKUP
c) COUNTIF
d) CONCAT
Answer: a) SUMPRODUCT
- A Solver Answer Report commonly shows:
a) Only worksheet formatting
b) Final variable values and constraint status
c) Python source code
d) The simplex tableau for every iteration automatically
Answer: b) Final variable values and constraint status
- A Solver Sensitivity Report may provide:
a) Only the optimal objective value
b) A list of worksheet errors
c) Shadow prices, reduced costs and allowable ranges
d) Artificial-variable formulas only
Answer: c) Shadow prices, reduced costs and allowable ranges
- The Excel Solver add-in must often be:
a) Rewritten in VBA
b) Purchased separately in every version
c) Used only online
d) Enabled before it appears on the Data tab
Answer: d) Enabled before it appears on the Data tab
- In Python, scipy.optimize.linprog is used for:
a) Linear optimization
b) Image processing only
c) Database management
d) Symbolic integration only
Answer: a) Linear optimization
- SciPy’s linprog interface is formulated primarily as a:
a) Profit maximization problem only
b) Minimization problem
c) Nonlinear least-squares problem
d) Simulation model
Answer: b) Minimization problem
