OR Final

How to find Zj element

(CBi x X-value) + (CBi x X-value)

Matrix form how many different matrices are needed for simplex?

8

Matrix form Columns of N refer to which terms?

A1 and A2

Matrix form Xb (bhat) = (equation)

B^-1(b)

Matrix form ahat equation

B^-1(whichever variable is entering)

A basic solution which also satisfies the condition in which all basic variables are non-negative is called ______________.

Basic feasible solution

Matrix form Z (optimal value)=

Cb(B^-1)(b)

________ are expressed in the form of inequalities or equations

Constraints

A basic feasible solution of a linear programming problem is said to be __________ if at least one of the basic variables is zero.

Degenerate

True or false When an artificial problem is created by introducing artificial variables and using the Big M method, if all artificial variables in an optimal solution for the artificial problem are equal to zero, then the real problem has no feasible solutions

FALSE, if at least ONE of the artificial variables is not zero, then the real problem is feasible

True or false when a linear programming model has an equality contraint, an artificial variable is introduced into this constraint in order to start the simplex method with an obvious basic solution that is feasible for the original model

FALSE, the initial basic solution for the artificial model is not feasible for the original model

True or false the two phase method is commonly used in practice because it usually requires fewer iterations to reach an optimal solution than the Big M does

FALSE, the two methods are basically equivalent, so they should take the same number of iterations

T/F: Every LP with an unbounded feasible region has an unbounded optimal solution.

False, if the feasible region is unbounded, the optimal solution could be maximizing or minimizing in the opposite direction of the unbounded region. The optimal solution constrains the unbounded feasible region.

If the feasible region of a linear programming problem is empty, the solution is _______________.

Infeasible

If an LP's feasible region is bounded you can find the optimal solution to the LP by simply checking the z values at each of the feasible regions extreme points. Does this hold is the LP's feasible region is unbounded?

No, if the feasible region is unbounded there is no way to check the CPF's for an optimal value because there are infinite optimal values.

if all bottom values in simplex tableau contain all negative or zero values....

STOP! the tableau describes an optimal solution

For an LP to be unbounded, the LP's feasible region must be unbounded (T/F)?

True, the feasible region must be unbounded for the LP to be unbounded resulting in no corner points to be analyzed for optimization.

graphic method can be applied to solve a linear programming problem when there are only _____________ variable.

Two

Matrix form Entering variable determination equation (Z1, Z2)

Z1 = Cb(B^-1)(A1)-C1 Z2 = Cb(B^-1)(A2)-C2

in simplex method, we add _________________ variables in the case of "=".

artificial variables

complementary solutions property

at each iteration, the simplex method simultaneously identifies a CFP solution x for the primal and a complementary solution y for the dual if x is not optimal for primal, then y is not feasible for dual

Matrix form Cb=

basic variable coefficients in objective function

Matrix form Xb =

basic variables

B = what matrix in simplex

basis matrix for optimal solution

Matrix form Ratio to determine leaving variable

bhat/ahat (transpose)

Duality Theorem if one problem has feasible solutions and a bounded objective function, then so does the other problem and...

both weak and strong hold true

Cbv(B^-1)(A)-c

coefficients of the slack variables in row 0

B^-1 = what matrix in simplex

coeffiecents of the slack variables in constraint

spanning tree

connected network with no undirected cycles

___________ are expressed in the form of inequalities or equations

constraints and objective functions

cost assumption

cost is directly proportional to number of units distributed

According to the weak and strong duality theories, they imply what?

cx < yb for feasible solutions if one or both of them are NOT optimal for their respective problems equality holds if both are optimal

Matrix form B (must find B^-1) =

decision variable coefficients (slack/surplus)

connected network

every pair of nodes in the network has at least one undirected path between them

until the last iteration in simplex method, the primal solutions are ___________ and the dual is ______ ____________.

feasible, not feasible

requirements assumption each source has a ________ supply entire supply must be __________ to the ___________. each ___________ has a fixed demand entire _________ must be _________ from the sources

fixed distributed; destinations destination demand; recieved

transshipment node

flow in equals flow out

demand node

flow in exceeds flow out

supply node

flow out exceeds flow in

symmetry property

for any primal problem and its dual problem, all relationships between them must be symmetric because the dual of this dual problem is the primal

subtract slack and add artificial

greater than or equal to constraint

if P has an optimal solution then D... and vice versa

has an optimal solution

Weak Duality property

if X is a feasible solution for the primal problem and y is a feasible solution for the dual problem, then cx <= yb

Strong duality property

if x* is an optimal solution for the primal problem and y* is an optimal solution for the dual problem, then cx*=y*b

If D is unbounded then P is (and vice versa)

infeasible

if the feasible region of a linear programming problem is empty, the solution is___________.

infeasible

Pivot column holds the __________ value in the last row

largest

pivot row corresponds to...

leaving variable

add surplus variable

less than or equal to constraint

arc capacity

max amount of flow that can be carried on a directed arc

undirected network

network has only undirected arcs

Matrix form N=

non basic variable coefficients

Matrix form Cn=

non basic variable coefficients in objective function

Matrix form Xn=

non basic variables

Equation for new value in iteration 1 for new value row

old value - ((corr. key column value x corr. key row value)/key element

directed network

only has directed arcs

B^-1(b) = what matrix in simplex

optimal rhs of constraint

z=(Cbv)(B^-1)(b)

optimal value of the objective function

weak duality theory describes what

relationship between any pair of solutions for the primal and dual problems where both solutions are feasible.

Matrix form b =

rhs of constraints

path between two nodes

sequence of distinct arcs connecting the nodes

What is also defined as the non-negative variables which are added in the left hand side of the constraint to convert the inequality "less than or equal too" into an equation?

slack variables

the row with the ___________ ratio is called the _____ _______.

smallest; pivot row

A set of values X1, X2,...Xn which satisfies the constraints of the linear programming problem is called the _____________.

solution

first step in hungarian algorithm

subtract smallest number in each row from every number in the row. enter the results in the new table.

second step in hungarian algorithm

subtract the smallest number in each column of the new table from every number in the column. enter results in another table.

xi-cj plays the role of what in dualality

surplus variable

third step in hungarian method

test whether an optimal set of assignments can be made. to do this, determine the minimum number of lines needed to cross out all zeros.

Cbv(B^-1)

the coefficients of the slack variables in row 0

Duality Theorem If one problem has no feasible solutions, then....

the other problem has either no feasible solutions of an unbounded objective function

Duality Theorem If one problem has feasible solutions and an unbounded objective function (no optimal solution)...

the other problem has no feasible solution

if all entries in the pivot column are zero or negative...

then the problem is unbounded

If p is infeasible then either D is... and vice versa

unbounded or infeasible

The objective functions and constraints are linear relationships between ____________.

variables

the objective functions and constraints are linear relationships between ___________.

variables

Xb = what in simplex

vector of basic variabels