Review F

The shadow price of a nonbinding constraint is: a.) positive b.) zero c.) negative d.) indeterminante

b.) zero

The objective function value for the ILP problem can never be as good as the optimal solution to its LP relaxation. be as poor as the optimal solution to its LP relaxation. be worse than the optimal solution to its LP relaxation. be better than the optimal solution to its LP relaxation.

be better than the optimal solution to its LP relaxation.

How is an LP problem changed into an ILP problem? by adding constraints that the decision variables be non-negative. by adding integrality conditions. by adding discontinuity constraints. by making the RHS values integer.

by adding integrality conditions.

Which of the following is the general format of an objective function? a.) f(X1, X2, ..., Xn) <=b b.) f(X1, X2, ..., Xn) >=b c.) f(X1, X2, ..., Xn) = b d.) f(X1, X2, ..., Xn)

d.) f(X1, X2, ..., Xn)

A change in the right-hand side of a constraint changes: a.) the slope of the objective function. b.) objective function coefficients. c.) other right-hand sides. d.) the feasible region.

d.) the feasible region.

Suppose that X1 equals 4. What are the values for d1+ and d1- in the following constraint? d1- = 4, d1+ = 0 d1- = 0, d1+ = 4 d1- = 4, d1+ = 4 d1- = 8, d1+ = 0

d1- = 4, d1+ = 0

A manager wants to ensure that he does not exceed his budget by more than $1000 in a goal programming problem. If the budget constraint is the third constraint in the goal programming problem which of the following formulas will best ensure that the manager's objective is met?

d3+ <= 1000

An integrality condition indicates that some (or all) of the RHS values for constraints must be integer objective function coefficients must be integer constraint coefficients must be integer decision variables must be integer

decision variables must be integer

The di+, di- variables are referred to as objective variables. goal variables. target variables. deviational variables.

deviational variables.

A production optimization problem has four decision variables and a requirement that at least X1 units of material one are consumed. Which of the following constraints reflects this fact? f( X1, X2, X3, X4) <= b1 f( X1, X2, X3, X4) >= b1 f( X1, X2, X3, X4) = b1 f( X1, X2, X3, X4) /= b1

f( X1, X2, X3, X4) >= b1

The setup cost incurred in preparing a machine to produce a batch of product is an example of a fixed charge random charge sunk cost variable cost

fixed charge

A network flow problem that allows gains or losses along the arcs is called a non-constant network flow model. non-directional, shortest path model. generalized network flow model. transshipment model with linear side constraints.

generalized network flow model.

The absolute value of the shadow price indicates the amount by which the objective function will be: improved if the corresponding constraint is loosened. improved if the corresponding constraint is tightened. made worse if the corresponding constraint is loosened. improved if the corresponding constraint is unchanged.

improved if the corresponding constraint is loosened.

In a transshipment problem, which of the following statements is a correct representation of the balance-of-flow rule if Total Supply < Total Demand? inflow - outflow >= supply or demand inflow + outflow >= supply or demand inflow - outflow <= supply or demand inflow - outflow <= supply or demand

inflow - outflow <= supply or demand

The right hand side value for the starting node in a shortest path problem has a value of -1 0 1 2

-1

A company will be able to obtain a quantity discount on component parts for its three products, X1, X2 and X3 if it produces beyond certain limits. To get the X1 discount it must produce more than 50 X1's. It must produce more than 60 X2's for the X2 discount and 70 X3's for the X3 discount. How many binary variables are required in the formulation of this problem? 3 6 9 12

3

Suppose that the first goal in a GP problem is to make 3 X1 + 4 X2 approximately equal to 36. Using the deviational variables d1- and d1+, what constraint can be used to express this goal?

3 X1 + 4 X2 + d1- - d1+ == 36

An ILP problem has 5 binary decision variables. How many possible integer solutions are there to this problem? 5 10 25 32

32

A company will be able to obtain a quantity discount on component parts for its three products, X1, X2 and X3 if it produces beyond certain limits. To get the X1 discount it must produce more than 50 X1's. It must produce more than 60 X2's for the X2 discount and 70 X3's for the X3 discount. How many decision variables are required in the formulation of this problem? 3 6 9 12

9

Decision-making problems which can be stated as a collection of desired objectives are known as what type of problem? A non-linear programming problem. An unconstrained programming problem. A goal programming problem. An integer programming problem.

A goal programming problem.

For minimization problems, the optimal objective function value to the LP relaxation provides what for the optimal objective function value of the ILP problem? An upper bound A lower bound An alternative optimal solution An additional constraint for the ILP problem.

A lower bound

For maximization problems, the optimal objective function value to the LP relaxation provides what for the optimal objective function value of the ILP problem? An upper bound. A lower bound. An alternative optimal solution. An additional constraint for the ILP problem.

An upper bound.

signs of shadow prices

Binding MIN <= - shadow price MIN >= + shadow price MAX <= + shadow price MAX >= - shadow price

Which of the following are potential pitfalls of using a non-zero optimality gap? No assurance the returned solution is optimal. No assurance the returned solution is integer. The true optimal solution may be worse than the returned solution. There are no pitfalls to consider since the Solver will obtain solutions quicker.

No assurance the returned solution is optimal.

Consider the constraint X3+X4+X5+X6+X7≥27 representing Air Express' Monday minimum worker requirement. Why was a "≥" used versus an "="? The "≥" is needed to accommodate workers held over from Sunday. Gurobi only accepts "≥" constraints. The "≥" is less restrictive. The "=" will always produce an infeasible constraint.

The "≥" is less restrictive.

When performing sensitivity analysis, which of the following assumptions must apply? a.) All other coefficients remain constant. b.) Only right hand side changes really mean anything. c.) The X1 variable change is the most important. d.) The non-negativity assumption can be relaxed.

a.) All other coefficients remain constant.

One approach to solving integer programming problems is to ignore the integrality conditions and solve the problem with continuous decision variables. This is referred to as quickest solution method. LP satisficing. LP relaxation. LP approximation.

LP relaxation.

Which of the following formulas is a deviation-minimizing objective function for a goal programming problem?

MIN SUM(d1- + d1+)

What is the meaning of the ti term in this objective function for a goal programming problem? The time required for each decision variable. The percent of goal i met. The coefficient for the ith decision variable The target value for goal i.

The target value for goal i.

What are binary integer variables? Variables with any two values, a and b. Variables with values 0 and 1. Variables whose sum of digits is 2. Variables with values between 0 and 1.

Variables with values 0 and 1.

Which of the following is not a benefit of using binary variables? With only 2 values, Gurobi Solver can work faster. Binary variables are useful in selection problems. Binary variables can replace some IF() conditions. Binary variables can enforce logical conditions.

With only 2 values, Gurobi Solver can work faster.

A company wants to select no more than 2 projects from a set of 4 possible projects. Which of the following constraints ensures that no more than 2 will be selected? X1 + X2 + X3 + X4 = 2 X1 + X2 + X3 + X4 <= 2 X1 + X2 + X3 + X4 >= 2 X1 + X2 + X3 + X4 = 0

X1 + X2 + X3 + X4 <= 2

A company wants to select 1 project from a set of 4 possible projects. Which of the following constraints ensures that only 1 will be selected? X1 + X2 + X3 + X4 = 1 X1 + X2 + X3 + X4 <= 1 X1 + X2 + X3 + X4 >= 1 X1 + X2 + X3 + X4 = 4

X1 + X2 + X3 + X4 = 1

If a company selects either of Project 1 or Project 2 (or both), then either Project 3 or Project 4 (or both) must also be selected. Which of the following constraints enforce this condition? X1 + X2 <= 2(X3 + X4) X1 + X2 <= X3 + X4 X1 - X3 = X2 - X4 X1 + X2 + X3 + X4 <= 2

X1 + X2 <= 2(X3 + X4)

If a company produces Product 1, then it must produce at least 150 units of Product 1. Which of the following constraints enforces this condition? X1 <= 150Y1 X1 - 150Y1 >= 0 X1Y1 <= 150 X1 >= 150 + Y1

X1 - 150Y1 >= 0

If a company selects Project 1 then it must also select either Project 2 or Project 3. Which of the following constraints enforces this condition? X1 - X2 - X3 >= 0 X1 + (X2 - X3) <= 0 X1 - X2 - X3 <= 2 X1 - X2 - X3 <= 0

X1 - X2 - X3 <= 0

A company must invest in project 1 in order to invest in project 2. Which of the following constraints ensures that project 1 will be chosen if project 2 is invested in? X1 + X2 = 0 X1 + X2 = 1 X1 - X2 >= 0 X1 - X2 <= 0

X1 - X2 >= 0

A production company wants to ensure that if Product 1 is produced, production of Product 1 not exceed production of Product 2. Which of the following constraints enforce this condition?

X1 <= X2

A wedding caterer has several wine shops from which it can order champagne. The caterer needs 100 bottles of champagne on a particular weekend for 2 weddings. The first supplier can supply either 40 bottles or 90 bottles. The relevant decision variable is defined as X1 = the number of bottles supplied by supplier 1 Which set of constraints reflects the fact that supplier 1 can supply only 40 or 90 bottles?

X1 = 40Y11 + 90Y12 , Y11 + Y12 <= 1

A company will be able to obtain a quantity discount on component parts for its three products, X1, X2 and X3 if it produces beyond certain limits. To get the X1 discount it must produce more than 50 X1's. It must produce more than 60 X2's for the X2 discount and 70 X3's for the X3 discount. Which of the following pair of constraints enforces the quantity discount relationship on X3?

X32 <= M3Y3 , X31 >= 50Y3

A company is planning next month's production. It has to pay a setup cost to produce a batch of X4's so if it does produce a batch it wants to produce at least 100 units. Which of the following pairs of constraints show the relationship(s) between the setup variable Y4 and the production quantity variable X4?

X4 <= M4Y4 , X4 >= 100Y4

A diet is being developed which must contain at least 100 mg of vitamin C. Two fruits are used in this diet. Bananas contain 30 mg of vitamin C and apples contain 20 mg of vitamin C. The diet must contain at least 100 mg of vitamin C. Which of the following constraints reflects the relationship between bananas, apples, and vitamin C? a.) 20A + 30B >= 100 b.) 20A = 100 c.) 20A + 30B <= 100 d.) 20A + 30b = 100

a.) 20A + 30B >= 100

Which of the following actions would expand the feasible region of a linear programming (LP) model? a.) Loosening the constraints b.) Tightening the constraints c.) Multiplying each constraint by two d.) Adding an additional constraint

a.) Loosening the constraints

The slope of the level curve for the objective function value can be changed by: a.) changing a coefficient in the objective function. b.) increasing the right-hand sides of constraints. c.) increasing the value of the decision variables. d.) doubling all the coefficients in the objective function.

a.) changing a coefficient in the objective function.

The number of units to ship from Chicago to Memphis is an example of a(n) a.) decision b.) objective c.) constraint d.) parameter

a.) decision

If constraints are added to an LP model the feasible solution space will generally: a.) decrease b.) increase c.) remain the same d.) become for feasible

a.) decrease

Binding constraints have: a.) zero slack b.) negative slack c.) positive slack d.) surplus resources

a.) zero slack

In generalized network flow problems solutions may not be integer values. flows along arcs may increase or decrease. it can be difficult to tell if total supply is adequate to meet total demand. all of these

all of these

The LP relaxation of an ILP problem always encompasses all the feasible integer solutions to the original ILP problem. encompasses at least 90% of the feasible integer solutions to the original ILP problem. encompasses different set of feasible integer solutions to the original ILP problem. will not contain the feasible integer solutions to the original ILP problem.

always encompasses all the feasible integer solutions to the original ILP problem.

Decision variables in network flow problems are represented by nodes arcs demands supplies

arcs

The constraint for resource 1 is 5X1 + 4X2 ≥ 200. If X2=20, what is the minimum value for X1? a.) 20 b.) 24 c.) 40 d.) 50

b.) 24

What are the three common elements of an optimization problem? a.) Objectives, resources, goals b.) Decisions, constraints, an objective c.) Decision variables, profit levels, costs d.) Decisions, resource requirements, a profit function

b.) Decisions, constraints, an objective

In which of the following categories of modeling techniques do the independent variables have unknown or uncertain values or coefficients? a.) Probabilistic models b.) Descriptive models c.) Prescriptive models d.) Predictive models

b.) Descriptive models

Limited resources are modeled in optimization problems as: a.) An objective function b.) constraints c.) decision variables d.) alternatives

b.) constraints

The allowable increase for a decision variable is: a.) how many more units to produce to maximize profits. b.) the amount by which the objective function coefficient can increase without changing the optimal solution. c.) how much to charge to get the optimal solution. d.) the amount by which constraint coefficient can increase without changing the optimal solution.

b.) the amount by which the objective function coefficient can increase without changing the optimal solution.

The allowable increase for a constraint is: a.) how many more units of resource to purchase to maximize profits b.) the amount by which the resource can increase given shadow price. c.) how much resource to use to get the optimal solution. d.) the amount by which the constraint coefficient can increase without changing the final optimal value.

b.) the amount by which the resource can increase given shadow price.

A binding greater than or equal to () constraint in a minimization problem means that: a.) the variable is up against an upper limit. b.) the minimum requirement for the constraint has just been met. c.) another constraint is limiting the solution d.) the shadow price for the constraint will be positive.

b.) the minimum requirement for the constraint has just been met.

The constraint for resource 1 is 5X1 + 4X2 ≤ 200. If X1=20, what is the maximum value for X2? a.) 50 b.) 20 c.) 25 d.) 40

c.) 25

The best models a.) Replicate the characteristics of a component in isolation from the rest of the system. b.) Are mathematical models. c.) Accurately reflect relevant characteristics of the real-world object or decision. d.) Replicate all aspects of the real-world object or decision.

c.) Accurately reflect relevant characteristics of the real-world object or decision.

What is the goal in optimization? a.) Find the values of the decision variables that use all available resources. b.) None of the options. c.) Find the decision variable values that result in the best objective function and satisfy all constraints. d.) Find the values of the decision variables that satisfy all constraints.

c.) Find the decision variable values that result in the best objective function and satisfy all constraints.

A production optimization problem has 4 decision variables and resource 1 limits how many of the 4 products can be produced. Which of the following constraints (assuming that b1 > 0) reflect this fact? a.) f(X1, X2, X3, X4) =/ b1 b.) f(X1, X2, X3, X4) = b1 c.) f(X1, X2, X3, X4) <= b1 d.) f(X1, X2, X3, X4) >= b1

c.) f(X1, X2, X3, X4) <= b1

The desire to maximize profits is an example of a(n): a.) decision b.) constraint c.) objective d.) parameter

c.) objective

Variables, which are not required to assume strictly integer values are referred to as strictly non-integer. continuous. discrete. infinite.

continuous.

The MINIMAX objective yields the smallest possible deviations. minimizes the maximum deviation from any goal. chooses the deviation which has the largest value. maximizes the minimum value of goal attainment.

minimizes the maximum deviation from any goal.

The following linear programming problem has been written to plan the production of two products. The company wants to maximize its profits. X1 = number of product 1 produced in each batch X2 = number of product 2 produced in each batch MAX 150 X1 + 250X2 ST 2X1 + 5X2 <= 200 3X1 + 7X2 <= 175 X1, X2>=0 How many units of resource 1 are consumed by each unit of product 1 produced? a.) 1 b.) 3 c.) 5 d.) 2

d.) 2

The constraint for resource 1 is 5X1 + 4X2 ≤ 200. If X1 = 20 and X2 = 5, how much of resource 1 is unused? a.) 0 b.) 100 c.) 200 d.) 80

d.) 80

Linear programming problems have: a.) Linear objective functions, non-linear constraints b.) Non-linear objective functions, non-linear constraints c.) Non-linear objective functions, linear constraints d.) Linear objective functions, linear constraints

d.) Linear objective functions, linear constraints

The constraints X1≥0 and X2≥0 are referred to as: a.) Positivity constraints b.) Optimality conditions c.) Left hand sides d.) Nonnegativity conditions

d.) Nonnegativity conditions

An optimization technique useful for solving problems with more than one objective function is dual programming. sensitivity analysis. multi-objective linear programming. goal programming.

multi-objective linear programming.

The di+ variable indicates the amount by which each goal's target value is missed. underachieved. overachieved. overstated.

overachieved

A node which can both send to and receive from other nodes is a demand node supply node random node transhippment node

transhippment node

A manufacturing company has costs associated with production preparation and with per unit production. The per unit production costs are referred to as decision variables. production cost constraint coefficients. variable costs. marginal costs.

variable costs