Prescriptive Analytics

What is the interpretation of units "shipped" along arcs from dummy supply nodes to demand nodes? a. Indicates unmet supply at supply nodes b. Indicates unmet demand at supply nodes c. Indicates unmet demand at demand nodes d. Indicates unmet supply at demand nodes

c. Indicates unmet demand at demand nodes

Which balance of flow rule should be applied at each node in a network flow problem when Total Supply > Total Demand? a. Inflow − Outflow ≤ Supply or Demand b. Inflow − Outflow = Supply or Demand c. Inflow − Outflow ≥ Supply or Demand d. Inflow − Supply ≥ Outflow or Demand

c. Inflow − Outflow ≥ Supply or Demand

The following linear programming problem has been written to plan the production of two products. The company wants to maximize its profits. X 1 = number of product 1 produced in each batch X 2 = number of product 2 produced in each batch MAX:150 X1 + 250 X2 Subject to: 2 X1 + 5 X2 ≤ 200 3 X1 + 7 X2 ≤ 175 X1, X2 ≥ 0

250

What function is used to add the contents of cells A1, A2 and A3?

=A1+A2+A3

Models which are setup in an intuitively appealing, logical layout tend to be the most

Auditable

The built-in Solver in Excel is found under which tab on the ribbon?

Data

If constraints are added to an LP model the feasible solution space will generally

Decrease

Examining the effect of changes ​in the RHS values of constraints is part of the answer report.

False

In mathematical programming formulations the objective function may ​contain cubic terms.

False

Values of the binary integer variable are ​restricted to -1 or 1.

False

For an infeasible problem, an infeasible region

Is an empty set

Objective cell, variable cells and constraint cells are terms used in Excel solver to describe the purpose of the cells. ​

True

When the lines connecting the nodes in a network are arrows that indicate a direction, the arcs in the network are called directed arcs.

True

​One of the benefits of using the simplex method to solve LP problems is its speed.

True

When the objective function can increase without ever contacting a constraint the LP model is said to be

Unbounded

A variable with a final value equal to its simple lower or upper bound and a reduced cost of zero indicates that a. an alternate optimal solution exists. b. the right hand sides should be increased. c. an error in formulation has been made. d. the objective function needs new coefficients.

a. an alternate optimal solution exists

In the B & B algorithm, B & B stands for a. Branch and Bound b. Best Bound c. Brooks and Baker d. Best Branch

a. Branch and Bound

How are general integrality requirements indicated in the Analytic Solver Platform? a. Specifying the INT option for the appropriate changing cells. b. Adding additional RHS values to constraints. c. Choosing the BIN setting in the Value field in the Solver Parameters dialog box. d. Specifying the INT option for the constraint rows.

a. Specifying the INT option for the appropriate changing cells.

A company is planning next month's production. It has to pay a setup cost to produce a batch of X 4'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 Y 4 and the production quantity variable X 4? a. X4 ≤ M4Y4, X4 ≥ 100 Y4 b. X4 ≤ M4Y4, X4 ≤ 100 Y4 c. X4 ≤ M4Y4, X4 ≥ 100 d. X4 ≤ M4Y4, X4 = 100 Y4

a. X4 ≤ M4Y4, X4 ≥ 100 Y4

Binary variables are: a. a subset of integer variables b. negative c. continuous on an interval (0,+∞) d. continuous on an interval (0,1)

a. a subset of integer variables

The Simplex method works by first a. identifying any basic feasible solution. b. setting X1 at one-half of the its maximum value. c. choosing the largest value for X1. d. going directly to the optimal solution.

a. identifying any basic feasible solution

Which property of network flow models guarantees integer solutions? a. integer constraint RHS values and balance of flow equation format b. linear objective function coefficients c. integer objective function coefficients d. linear constraints and balance of flow equation format

a. integer constraint RHS values and balance of flow equation format

When a solution is degenerate the shadow prices and their ranges a. may be interpreted in the usual way but they may not be unique. b. are always understated c. are always valid and unique. d. must be disregarded

a. may be interpreted in the usual way but they may not be unique

The branch-and-bound algorithm starts by a. relaxing all the integrality conditions in an ILP and solving the resulting LP problem. b. determining the most likely RHS values and solving for them. c. solving two LP problems in which X1 is set at 0 and 1 respectively. d. relaxing all the RHS values in an ILP and solving the resulting LP problem.

a. relaxing all the integrality conditions in an ILP and solving the resulting LP problem..

Data Envelopment Analysis (DEA) is an LP-based methodology in which weighted sums of inputs and outputs are calculated and a. the objective for each unit is to maximize the weighted sum of its outputs. b. the objective is to maximize every units output. c. the constraints capture the maximum effectiveness of each unit. d. the constraints ensure the sum of the weighted outputs is one.

a. the objective for each unit is to maximize the weighted sum of its outputs

The right hand side value for the starting node in a shortest path problem has a value of a. −1 b. 1 c. 2 d. 0

a. −1

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

b. A lower bound

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

b. The "≥" is less restrictive.

What happens to the solution of a network flow model if side constraints are added that do not obey the balance of flow rules? a. The model solution is not guaranteed to be feasible. b. The model solution is not guaranteed to be integer. c. The model solution will be integer but more accurate. d. The model solution will more accurately reflect reality.

b. The model solution is not guaranteed to be integer

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

b. Variables with values 0 and 1.

When might a network flow model for a transportation/assignment problem be preferable to a matrix form for the problem? a. When supply exceeds demand. b. When the problem is large and not fully connected. c. When the problem is large and fully connected. d. When an integer solution is required.

b. When the problem is large and not fully connected.

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

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

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

b. improved if the corresponding constraint is loosened

Which method is preferred for solving fully connected transportation problems? a. simulation b. linear programming c. trial and error d. network flow methods

b. linear programming

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

b.continous

The street intersections in a city road network represent a. arcs. b. nodes. c. expenses. d. resources.

b.nodes

Consider the formulation below. The standard form of the second constraint is: MAX:8 X1 + 4 X2 Subject to: 5 X1 + 5 X2 ≤ 20 6 X1 + 2 X2 ≤ 18 X1, X2 ≥ 0 a. 6 X1 + 2 X2 = 18 b. 6 X1 + 2 X2 - S2 = 18 c. 6 X1 + 2 X2 + S2 = 18 d. 5 X1 + 5 X2 = 20

c. 6 X1 + 2 X2 + S2 = 18

Consider the formulation below. The standard form of the second constraint is: MAX: 8 X1 + 4 X2 Subject to:5 X1 + 5 X2 ≤ 20 6 X1 + 2 X2 ≥ 18 X1, X2 ≥ 0 a. 6 X1 + 2 X2 = 18 b. 6 X1 + 2 X2 + S2 = 18 c. 6 X1 + 2 X2 - S2 = 18 d. 5 X1 + 5 X2 = 2

c. 6 X1 + 2 X2 - S2 = 18

What does the Excel "=SUMPRODUCT(A1:A5,C6;C10)" function do? a. Sums each range and multiplies the sums. b. Sum each pair of cells and multiples each sum. c. Multiplies each pair of cells in two arrays matched by position and sums the products. d. Multiplies the contents of cells containing the =SUM() command.

c. Multiplies each pair of cells in two arrays matched by position and sums the products

Consider modeling a warehouse with three in-flow arcs and three outflow arcs. The warehouse node is a transshipment node but has a capacity of 100. How would one modify the network model to avoid adding a side constraint that limits either the sum of in-flows or the sum of the out-flows to 100? a. Place a limit of 34 on each in-flow arc. b. Add a side constraint limiting the out-flow arcs sum to 100. c. Separate the warehouse node into two nodes, connected by a single arc, with capacity of 100. d. It cannot be accomplished, a side constraint must be added.

c. Separate the warehouse node into two nodes, connected by a single arc, with capacity of 100.

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

c. With only 2 values, Solver can work faster

ILP formulations can be used to model: a. investment allocation problems b. personnel scheduling problems c. all of the above d. production problems

c. all of the above

The Cell Value column in the Solver Answer Report shows a. Right hand sides of constraints. b. which constraints are binding. c. final (optimal) value assumed by each constraint cell. d. objective function values.

c. final (optimal) value assumed by each constraint cell

The demand nodes in the graphical representation of the transshipment problem: a. have some unidirectional arcs b. have all directed arcs originating at them c. have total demanded quantities expressed as positive numbers d. have total available quantities expressed as negative numbers

c. have total demanded quantities expressed as positive numbers

If the allowable increase for a constraint is 100 and we add 110 units of the resource what happens to the objective function value? a. decrease of 100 b. increase of 110 c. increases but by unknown amount d. increase of 100

c. increases but by unknown amount

Mixed IP formulation requires that: a. the optimal objective function value must be integer b. all decision variables must be integer c. some decision variables must be integer d. some decision variables and the optimal value of the objective function must be integer

c. some decision variables must be integer

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

c. the amount by which objective function coefficient can decrease without changing the final optimal solution

In the generalized network flow problem solver could not find a feasible solution. This means that: a. dummy demand is needed in the formulation b. the total supply is capable of meeting the total demand c. the total supply is not capable of meeting the total demand d. the total demand is not capable of meeting the total supply

c. the total supply is not capable of meeting the total demand

What is the value of the slack variable in the following constraint when X 1 and X 2 are nonbasic and only non-negativity is used as simple bounds? X1 + X2 + S1 = 100 a. 0 b. 50 c. 100 d. can't be determined from the given information

c.100

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

c.be better than the optimal solution to its LP relaxation

The difference between the right-hand side (RHS) values of the constraints and the final (optimal) value assumed by the left-hand side (LHS) formula for each constraint is called the slack and is found in the a. Slack report b. Status report c. Cell Value report d. Answer report

d. Answer report

Given an objective function value of 150 and a shadow price for resource 1 of 5, if 10 more units of resource 1 are added (assuming the allowable increase is greater than 10), what is the impact on the objective function value? a. decrease of 50 b. increase of 10 c. increase of unknown amount d. increase of 50

d. Increase of 50

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? a. X1 + X2 ≤ X3 + X4 b. X1 − X3 = X2 − X4 c. X1 + X2 + X3 + X4 ≤ 2 d. X1 + X2 ≤ 2(X3 + X4)

d. X1 + X2 ≤ 2(X3 + X4)

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? a. X1 + X2 = 1 b. X1 − X2 ≤ 0 c. X1 + X2 = 0 d. X1 − X2 ≥ 0

d. 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? a. X1 ≥ M2Y2 b. X1 ≤ M1Y1, X1 ≤ Y1X2 c. X1 ≤ M2X2 d. X1 ≤ X2

d. X1 ≤ X2

How could a network be modified if demand exceeds supply? a. remove the extra demand arcs b. add a dummy demand c. add extra supply arcs d. add a dummy supply

d. add a dummy supply

Suppose you want to minimize an objective function z=2x 1+3x 2. Both decision variables must be integer. The optimal solution to the LP relaxation will: a. be larger than the optimal IP solution b. will be within 5% of the optimal IP solution value c. can be either smaller or larger than the optimal IP solution d. be smaller than the optimal IP solution

d. be smaller than the optimal IP solution

The supply nodes in the graphical representation of the transshipment problem: a. have some unidirectional arcs b. have all directed arcs terminating at them c. have total demanded quantities expressed as positive numbers d. have total available quantities expressed as negative numbers

d. have total available quantities expressed as negative numbers

The transshipment nodes in the graphical representation of the transshipment problem: a. have all directed arcs terminating at them b. have all directed arcs originating at them c. have some unidirectional arcs d. have total demanded quantities expressed as positive numbers

d. have total demanded quantities expressed as positive numbers

The reduced cost for a changing cell (decision variable) is a. how many more units to product to maximize profits. b. equal to zero for variables at their optimal values. c. the amount by which the objective function value changes if the variable is increased by one unit. d. the per unit profits minus the per unit costs for that variable.

d. the per unit profits minus the per unit costs for that variable

Almost all network problems can be viewed as special cases of the a. shortest path problem. b. minimal spanning tree problem. c. maximal flow problem. d. transshipment problem.

d. transshipment problem.

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

d. variable costs

Which of the following is the general format of an objective function?

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

Suppose that the left side of the constraint cannot take a specific value, b. This can be expressed mathematically as

f(X1, X2, ..., Xn) ≠ b.

The constraints X 1 ≥ 0 and X 2 ≥ 0 are referred to as

nonnegativity conditions.

Microsoft Excel contains a built-in optimizer called

solvers

A number of practical decision problems in business fall into a category known as network flow problems. These problems can be described or displayed in a graphical form known as a network.

true

A solvable problem must have:

​a feasible region that is not an empty set.