IS 380 final exam
What is the constraint for node 2 in the following shortest path problem?
- Xl2 + X24 = 0
Values of the binary integer variable are restricted to
0 or 1
The right-hand side value for the ending node in a shortest path problem has a value of
1
the objective function for an LP model is 3 XI + 2 X2. If XI = 20 and X2 = 30, what is the value of the objective function?
120
An ILP problem has 4 binary decision variables. How many possible integer solutions are there to this problem?
16
The following linear programming problem has been written to plan the production of two products. The company wants to maximize its profits.
2
a diet is being developed that 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?
20 A+ 30 B:::,: 100
The constraint for resource 1 is 5 Xl + 4 X2 200. If X2 = 20, what is the minimum value for X1?
24
The constraint for resource 1 is 5 Xl + 4 X2 :S 200. If Xl = 20, what is the maximum value for X2?
25
a company will be able to obtain a quantity discount on component parts for its three products, Xl, X2, and X3 if it produces beyond certain limits. To get the Xl discount, it must produce more than 5'0 units of product XL It must produce more than 60 X2 units for the X2 discount and 70 X3 units for the X3 discount. How many binary variables are required in the formulation of this problem?
3
The constraint for resource 1 is 5 Xl + 4 X2::; 200. If Xl = 20 and X2 = 5, how much of resource 1 is unused?
80
a company is developing its weekly production plan. The company produces two products, A and B, which are processed in two departments. Setting up each batch of A requires $60 of labor, while setting up a batch of B costs $80. Each unit of A generates a profit of $17, while a unit of B earns a profit of $21. The company can sell all the units it produces. The data for the problem are summarized as follows: What is the appropriate formula to use in cell E8 of the following Excel implementation of the ILP model for this problem?
=SUMPRODUCT(B5:C5,B7:C7) SUMPRODUCT(B8:C8,Bl4:Cl4)
how is an LP problem changed into an ILP problem?
By adding integrality conditions
In a decision-making framework, bad luck refers to a situation when which of the following occur?
Good decision quality and bad outcome quality
In a decision-making framework, deserved success refers to a situation when which of the following occur?
Good decision quality and good outcome quality
Which of the following special conditions in an LP model represents potential errors in the mathematical formulation?
Infeasibility and unbounded solutions
which of the following statements is true of using models in problem solving and decision analysis?
It is something virtually everyone has done before.
What does the Analytic Solver Platform integer tolerance factor of 0 accomplish?
It stops B & B when the true optimal integer solution has been found.
one approach to solving integer programming problems is to ignore the integrality conditions and solve the problem with continuous decision variables. this is referred to
LP relaxation
What is the objective function in the following maximal flow problem?
MAXX41
A sub-problem in a B & B is solved and found infeasible. Should the B & B algorithm continue further analysis on this candidate problem?
No, adding more constraints will not restore problem feasibility.
Solutions to which of the following categories of modeling techniques indicate a course of action to the decision maker?
Prescriptive models
which formula should be used to determine the net flow values in cell K6 in the following spreadsheet model?
SUMIF($E$6:$E$16,I6,$B$6:$B$16)-SUMIF($C$6:$C$16,I6,$B$6 :$B$16)
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 the inflows or the sum of the outflows to 100?
Separate the warehouse node into two nodes, connected by a single arc, with a capacity of 100.
Consider the constraint X3 + X4 + XS + X6 + X7 :::: 27
The "?:" is less restrictive.
what is the goal in optimization?
To find the decision variable values that result in the best objective function and satisfy all constraints.
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?
Xl + X2 + X3 + X4 :S: 2
If a company selects either 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 enforces this condition?
Xl + X2 :S 2(X3 + X4)
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?
Xl-X2-X3:S0
The B & B algorithm solves ILP problems by solving
a series of LP problems
binary variables are
a subset of integer variables
recall that in the network model formulation of this problem a node represents a year when the equipment was purchased. an arc from node i to node j indicates that the equipment purchased in year i can be replaced at the beginning of year j. how could the following network model be modified to depict an equipment purchase in year 4 and operating costs only through the remainder of the planning window
add a dummy node, 6, so that the 4-6 represents just the operating costs
how could a network be modified if demand exceeds supply?
add a dummy supply
a minimum or maximum flow restriction in the network flow problem can be modeled by all of the following except
adding dummy constraints
the minimal spanning tree solution algorithm works by defining a subnetwork and
adding the least expensive arc which connects any node in the current subnetwork to any node not in the current subnetwork.
ILP formulation requires that
all decision variables must be integers
Errors in human judgment often arise because of what type of effects associated with decision problems?
anchoring and framing
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 units of product XL It must produce more than 60 units of X2 for the X2 discount and 70 units of X3 for the X3 discount. Which of the following pairs of constraints enforces the quantity discount relationship on X3?
b. X32 :'.S M3Y3, X31 ::: 50 Y3
all of the following are reasons why someone would wish to use a spreadsheet model except
b. ecause spreadsheets lead to increased opportunities.
he objective function value for the ILP problem can never be
better than the optimal solution to its LP relaxation.
In the B & B algorithm, B & B stands for
branch and bound
which of the following fields of study uses computers, statistics, and mathematics to solve business problems?
business analytics
a purely rational decision maker should
consistently select the same alternative, regardless of how the problem is framed.
How much money an individual should withdraw each year from various retirement accounts is an example of a
decision
the number of units to ship from Chicago to Memphis is an example of a(n)
decision
In an optimization problem, the mathematical symbols Xl, X2, ... , Xn often represent the
decision variables
An integrality condition indicates that some (or all) of the
decision variables must be integers.
what are the three common elements of an optimization problem?
decisions, constraints, and an objective
in the model Y = f(Xl, X2), Y is called a(n)
dependent variable
Which of the following categories of modeling techniques addresses uncertainty in the values of the independent variables?
descriptive models
A situation when decision quality is bad and the resulting outcome quality is good is referred to as
dumb luck
Suppose that the left side of the constraint cannot take a specific value, b. This can be expressed mathematically as
f(Xl, X2, ..., Xn) t b.
A greater than or equal to constraint can be expressed mathematically as
f(Xl, X2, ..., Xn):::: b.
Some resources (e.g., meat and dairy products, pharmaceuticals, a can of paint) are perishable. This means that once a package (e.g., a can or a bag) is open, the content should be used in its entirety. Which of the following constraints reflects this fact?
f(Xl, X2, X3, X4) = bl
nary variables are useful for modeling the
fixed-charge problem.
The specification or description of the relationship between the dependent and independent variables is generally called a
function
A network flow problem that allows gains or losses along the arcs is called a
generalized network flow model
In which step of the problem-solving process is the concept of "probortunity" introduced?
identify problem
which step of the problem-solving process is considered the most important?
identify problem
Business analytics focuses on
identifying and leveraging business opportunities.
Which of the following problem-solving steps is often considered the most difficult?
implement the solution
because they simplify reality, models are generally helpful in examining things that would be
impossible to do in reality
Independent variables are represented as which type of cells in a spreadsheet model?
input cells
if an LP problem has an optimal solution with a finite objective function value, this solution will always occur at a point in the feasible region where two or more of the boundary lines of the constraints
intersect
A mathematical model is considered to be valid when
it accurately represents the relevant characteristics of the object or decision.
why is the graphical method of solving LP problems studied?
it helps develop an understanding of the linear programming strategy
retail companies try to find the
least costly method of transferring goods from warehouses to stores
When building a model, it is often to analyze decision problems using a model than carrying out the decision in reality.
less expensive
Which method is preferred for solving fully connected transportation problems?
linear programming
which of the following is the type of model used throughout this textbook?
mathematical model
Rounding the LP relaxed problem solution up or down to the nearest integer
may produce a suboptimal solution
A facility produces two products. The labor constraint (in hours) is formulated as 350 Xl + 300 X2:::; 10,000. The number 10,000
means there are 10,000 hours of labor available for use.
A common objective when manufacturing printed circuit boards (PCBs) is
minimizing the total distance the drill bit must be moved.
all of the following are benefits of modeling except that
modeling finds the right answers to incorrect or flawed problem statements.
ILP problems are computationally
more demanding than their LP relaxations
How many constraints are there in a transshipment problem that has n nodes and m arcs?
n
A facility produces two products. The labor constraint (in hours) is formulated as 350 Xl + 300 X2 :=:: 10,000. The number 350 means that
one unit of product 1 uses 350 hours of labor.
Mathematical programming is referred to as
optimization
In which of the following categories of modeling techniques are the specifications of the relationships between dependent and independent variables unknown or poorly defined?
predictive models
Which of the following categories of modeling techniques involves determining the value of a dependent variable based on specific values of independent variables?
predictive models
If the constraint 2 Xl + 3 X2 2: 900 is binding, then the constraint 4 Xl + 6 X2 2: 600 is
redundant
the branch-and-bound algorithm starts by
relaxing all the integrality conditions in an ILP and solving the resulting LP problem.
All of the following are CEO alternatives to start the OR/MS collaboration process except
require the OR/MS group to save their yearly salary in every study.
Identifying the real problems faced by the decision maker
requires insight, some imagination, time and a good bit of detective work.
the equipment replacement problem is a common type of business problem that can be modeled as a
shortest path problem
Suppose you want to minimize an objective function Z = 2 XI + 3 X2. Both decision variables must be integers. The optimal solution to the LP relaxation will be
smaller than the optimal IP solution
The third step in formulating a linear programming problem is to
state the objective function as a linear combination of the decision variables.
"Probortunity" is
the first step in the problem-solving process.
what happens to the solution of a network flow model if side constraints are added that do not obey the balance-of-flow rules?
the model solution is not guaranteed to be integral
A linear formulation means that
the objective function and all constraints must be linear.
The intuitive approach to solving LP problems cannot be trusted because
there actually may be a better solution to the problem being modeled.
The supply nodes in the graphical representation of the transshipment problem have
total available quantities expressed as negative numbers
the demand nodes in the graphical representation of the transshipment problem have
total demanded quantities expressed as positive numbers
a node that can both send to and receive from other nodes is a
transshipment node
Any shortest path problem can be modeled as a(n)
transshipment problem
To be effective, a modeler must
understand how modeling fits into the entire problem-solving process.
The first step in formulating an LP model is
understanding the problem to be modeled.
For maximization problems, the optimal objective function value to the LP relaxation provides a(n) for the optimal objective function value of the ILP problem.
upper bound
a manufacturing company has costs associated with production preparation and with per-unit production. The per-unit production costs are referred to as
variable costs
to illustrate how a complex system will be built, an engineer will likely use a
visual model
