Man. Sci. Exam 2

conditional constraint

-0-1 model -completion of one project is dependent upon the completion of another -x2</x1

Objective function

-a linear mathematical relationship describing an objective of the g/f in terms of decision variables -can be maximized or minimized

Assumptions - Transportation

-a product is transferred from sources to a number of destinations at the minimum possible cost -each source is able to supply a fixed number of units, each destination has a fit demand for the product

characteristics of LPP

-choice between alternative actions -defining the decision variables -objective the decision maker wants to achieve -restrictions exist, limited resources -restrictions & objective must be defined in linear mathematical relationships

feasible solution

-does not violate any constraints, so the solution is possible

changes in objective function coefficients

-may be reactions to anticipated uncertainties in the parameters or to new or changed information concerning the model -changes optimal solution point -increasing profit of x1 makes the obj. function line steeper -increasing profit of x2 makes the obj. function line flatter

the solution to the LP relaxation of a minimization integer linear program provides

-relaxed program has a value less than or equal to that of the original program -optimistic bound on the integer solution

Shadow prices in excel

-value is change in Z if constant increases/decreases by 1 unit

Assume that X2, X7 and X8 are the dollars invested in three different common stocks from New York Stock Exchange. In order to diversify the investments, the investing company requires that no more than 60% of the dollars invested can be in "stock 2". The constraint for this requirement can be written as:

.4X2- .6X7 -.6X8 </ 0

At least one or both deviational variables in a goal constraint must equal:

0

binary variables are

0 or 1 only

Steps in Model formulation

1. clearly define the decision variables 2. construct the objective function 3. Formulate constraints

Which of the following assumptions is not an assumption of the transportation model?

Actual total supply and actual total demand must be equal

The region that satisfies all of the constraints in a graphical linear programming problem is the:

Feasible solution space

Solve the system of 3x+15y>21 and x+4y=6 for y.

None of the answer choices is correct

_____ is used to analyze changes in model parameters.

Sensitivity analysis

Which of the following is not a typical characteristic of a linear programming problem?

The problem has an objective

The assignment problem constraint x41 + x42 + x43 + x44 ≤ 3 means:

agent 4 can be assigned to three tasks

total integer model

all decision variables have integer solution values

Which of the following statements is not true?

an infeasible solution violates all constraints

In a transportation problem, items are allocated from sources to destinations

at a minimum cost

The objective function min P1d1- , P2d2+:

attempts to avoid being below target for the priority 2 goal

Unbalanced types - Transportation

constraints contain inequalities where supply does not equal demand

Which of the following is not an integer linear programming problem?

continious

Two or more goals at the same priority level can be assigned weights to indicate their relative:

importance

the optimal solution to a linear programming model that has been solved using the graphical approach

is typically located at the origin

decision variables

mathematical symbols (under your control) representing levels of activity of a public/government problem or a firm/business

multiple optimal solutions

occurs when the objective function is parallel to a constraint line

A decision with more than one objective

requires the decision maker to put the objectives in some order of importance

Unbalanced mode - assignment

supply does not equal demand

Balanced mode - assignment

supply equals demand

slack variables

-adding an inequality to a constraint to convert it to an equation - represents unused resources, does not affect obj. function

the solution to the LP relaxation of a maximization integer linear program provides

-an upper bound for the value of the objective function -relaxed program has a value greater than or equal to that of the original program

optimal solution

-best feasible solution -the extreme point closest to the origin

problem with rounding of non-integer solutions in the integer programming

-can result in an infeasible solution -feasible solutions are ensured by rounding down non-integer solution values -rounding could affect the profit or cost by millions -rounding down can result in a less-than optimal solution -can result in a higher profit

LP formulation of a transportation problem

-constraints for supply and demand at each source and demand at each destination

sensitivity analysis for integer linear programming

-does not have the same interpretation as LPP and should be disregarded

multiple choice constraint

-either one project MUST be built but not both -forces a choice between two options, x1+x2 = 1, individually, they would not -if two facilities out of 4 must be builts: x1+x2+x3+x4=2

mutually exclusive constraint

-either one project or the other CAN be built but not both -action not selected will have a value of zero, if selected will have a value of one

irregular types of LPP

-for some models, general rules don't apply -special types: multiple optimal, infeasible, unbounded

corequisite constraints

-if one project is completed, then the other one will also be and vice versa -x2=x1

graphical solutions

-limited to linear programming problems with only two decision variables -provides an idea how a solution is obtained for a LP problem

Integer programming as the violation of the assumption of linear programming

-linear programming models have the assumption that solutions could be fractional or real numbers (non-integer). -non integer solutions are not always practical

Parameters

-numerical coefficients and constants used in the objective function and constraints -uncontrollable, known w/ certainty

deviational variables

-positive: d+ is the amount a goal level is exceeded -negative: d- is the amount a goal level is underachieved

sensitivity range

-range of values over which the current optimal solution point will remain optimal

Constraints

-requirements or restrictions placed on the g/f by the operating environment (e.g. budget, resources, etc.) -stated in linear relationships of the decision variables

LP formulation of an assignment problem

-similar to transportation -except all supply values for each source are equal to one -demand values at each destination equal one

proportionality (LPP)

-slope of a constraint or objective function line is constant -changes in the value of a decision variable will result in the same change in a functional variable

surplus variable

-subtracting from an inequality constraint to convert to an equation -represents the excess above a minimum resource requirement level

Assignment problem

-supply at each source and demand at each destination OFTEN limited to one unit -all supply and demand values equal one

changes in constraint quantity values

-the sensitivity range for a right-hand side value is the range of values over which the quantity's value can change without changing the solution variable mix, including slack variables

infeasible solution

-violates at least one constraint, solution is not possible

Quickbrush Paint Company is developing a linear program to determine the optimal quantities of ingredient A and ingredient B to blend together to make oil-base and water-base paint. The oil-base paint contains 90 percent A and 10 percent B, whereas the water-base paint contains 30 percent A and 70 percent B. Quickbrush currently has 10,000 gallons of ingredient A and 5,000 gallons of ingredient B in inventory and cannot obtain more at this time. Assuming that x represents the number of gallons of oil-base paint, and y represents the gallons of water-base paint, which constraint is correctly represents the constraint on ingredient A?

.9x+.3y </ 10,000

In an assignment problem all supply and demand values equal are

1

Applications of LP

1. identify a problem as solvable by LP 2. create a mathematical model 3. solve 4. implementation

The feasible region based upon certain constraints is shown in the graph for the objective function Z=11x+7y. What is the minimum value of the objective function?

21

The corner points of a feasible region based upon certain constraints are (0,6), (1,5), (2,4), and (10,0) for the objective function z=3x+5y. What is the minimum value of the objective function?

26

In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, and 3, which have selling prices of $15, $47.25, and $110, respectively. The investor has up to $50,000 to invest. The stockbroker suggests limiting the investments so that no more than $10,000 is invested in stock 2 or the total number of shares of stocks 2 and 3 does not exceed 350, whichever is more restrictive. How would this be formulated as a linear programming constraint?

47.25X2</10,000 and X2+X3</350

Which of these statements is best?

An unbounded problem has feasible solutions

If the feasible region for a linear programming problem is unbounded, then the solution to the corresponding linear programming problem is _____ unbounded.

Sometimes

In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, and 3, which have selling prices of $15, $47.25, and $110, respectively. The investor has up to $50,000 to invest. The investor stipulates that stock 1 must not account for more than 35% of the number of shares purchased. Which constraint is correct?

X1 ≤ 0.35(X1 + X2 + X3)

branch and bound method

a mathematical solution approach for solving integer programming problems total set of feasible solutions can be divided into smaller subsets of solutions smaller solutions evaluated until best solution is found

Linear Programming

a model that consists of linear relationships representing a firm's decision(s), given an objective and resource constraints

Balanced types - Transportation

all constraints are equalities where supply equals demand

0-1 integer model

all the decision variables have integer values of zero or one

which of the following special cases does not require reformulation of the problem in order to obtain a solution?

alternate optimality

optimal solution in a minimization problem

boundary of the feasible solution area, boundary contains the point CLOSEST to the origin

optimal solution in a maximization problem

boundary of the feasible solution area, contains the points FURTHEST from the origin

slope

computed as the "rise" over "run, objective function into an equation for the straight line

If we are solving a 0-1 integer programming problem, the constraint x1=x2 is a ____ constraint.

corequisite

Extreme points

corner points on the boundary of the feasible solution area

goal constraints

equalities that include deviational variables, redoing equation, one or both deviational variables must equal 0

formulation of goal programming models

includes multiple goals instead of a single objective 1. transform the linear programming model constraints into goals 2. represent the first goal by creating a new objective function 3.Solve 4. move on to the next goal and repeat

In a _______ integer model, some solution values for decision variables are integers and others can be non-integer

mixed

If we are solving a 0-1 integer programming problem, the constraint x1+x2 </ 1 is a _____ constraint

multiple choice

the ______ property of linear programming models indicates that the rate of change or slope of the objective function or a constraint is constant

proportionality

standard form

requires all variables in the constraint equations to appear on the left of the inequality (or equality) and all numeric values to be on the right hand side

Non-negativity constraints

restrict the decision variables to zero or positive values

objective function in goal programming

seeks to minimize the deviation from goals in order of the goal priorities deviations from the goal may be minimized individually in order of priority

mixed integer model

some of the decision variables, but not all are required to have integer solutions

The linear programming model for a transportation problem has constraints for supply at each _____ and ______ at each destination

source, destination

In the formulation of a >/ constraint

surplus variable is subtracted

additive (LPP)

terms of the objective function or constraints must be able to be combined

sensitivity analysis

the analysis of parameter changes and their effects on the optimal solution

Shadow prices

the price one would be willing to pay to obtain one more unit of a resource in a linear programming problem

Compared to product mix problems, transportation problems are unique because

the solution values are always integers

Types of integer programming models are:

total, 0-1, and mixed

unbounded solution

value of the objective function increases indefinitely

certainty (LPP)

values of all model parameters are assumed to be constant and known

divisible (LPP)

values of decision variables are continuous, solution methods may not be whole numbers