Terms in this set (83)
If the optimal value of a variable is basic (not zero), then the reduced cost is always:
If the optimal value of a variable is zero and the reduced cost corresponding to the variable is also zero, then:
...there is at least one other corner that is also in the optimal solution.
Reduced cost value is only non-zero when the optimal value of a variable is:
-The reduced cost value indicates:
-In the case of a minimization problem, "improved" means:
-In the case of a maximization problem, "improved" means
-how much the objective function coefficient on the corresponding variable must be improved before the value of the variable will be positive in the optimal solution.
Non basic are?
What is B sub i
the right hand side
What is C sub j
the coefficient for X in the objective function
What is A sub ij
the consumption ratio
What is the shadow price of a redundant constraint?
What are shadow prices in forms of?
How do you change the profit per item?
change C sub j
In the sensitivity report how do you know if you have alternative optima?
• More than one 0 in the allowable inc/allowable dec in the decision variable section
• A row of 0 in the reduced cost and final value of a decision variable
• A row of 0 in the shadow price and final value of a constraint
If the final value is positive then reduced cost is?
How do you know if it's a max/min by just looking at the reduced cost?
max - negative or o
min - positive or 0
What tells you what you are paying above what you need to pay for a resource?
the shadow price
If the final value = RHS what do you know exists?
a binding constraint
E minus some number is? Why?
- virtually zero
- the number is the exponent
You can't have a negative amount of a resource because?
you have a non-neg. constraint on them
How to formulate an LP model (five steps)
1. Understand the problem.
2. Identify the decision variable.
3. State the objective function as a linear combination of decision variables.
4. State the constraints as linear combination of decision variables.
5. Identify any upper or lower bounds on the decision variables.
What is pivoting?
moving from one extreme point to the next
Where does Simplex start? How does it move?
starts us at the origin and mathematically takes us around the feasible region, from extreme point to extreme point, in the direction of the most improvement with respect to the objective function.
Sensitivity analysis does what?
determines when a change in the parameters of a problems will cause a change in the basis.
What are the three parameters?
B sub i, C sub j, and A sub ij
the additional, or
marginal, value of one more unit of a resource.
If a basis changes, what else must change? (and visa versa)
the shadow prices and reduced cost
What is right hand side ranging?
by how much can we change a right-hand-side value without changing the basis.
What is Cj ranging?
by how much can you change the Cj values without changing the basis.
How do you denote it?
marginal value of one more unit of a resource
U sub i
Right hand side value?
amount of a resource
B sub i
the change in our Z value if we make one of our non basic decision variables a basic variable (unit contribution)
fewer basic variable than total constraints or if any of the allowable increase/decrease in the bottom part are zero you will have degeneracy
-means that some of your results in the sensitivity analysis might not be unique (A difference set of shadow prices and ranges might also pertain to the same problem and give you the same Z value for the problem, but not necessarily alternative optima)
the solution to an LP problem is degenerate if the RHS values of the constraints have an allowable increase or allowable decrease of zero
DEGENERACY MAKES THE WAYS TO FIND ALTERNATIVE OPTIMA INACCURATE. IF YOU HAVE DEGENERACY YOU WILL NOT KNOW IF YOU HAVE ALTERNATIVE OPTIMA
How to tell if its min or max on a sensitivity report?
What does it have to be to not know if it is min or max?
look at the reduced cost, if it is negative then it's a max problem and if it is positive then it is a min problem
if it is zero you don't know which one it will be
Objective coefficients need to match what unit?
the unit of the Z value
The C j value, which is the objective coefficient, will always be in the same unit, as the unit of
The z value
The A sub ij and B sub i values, which are parameters, will always be in the unit of
each respective constraints
What are the 3 parameters?
C sub j, A sub ij, B sub i
Any zero in the bottom section, for allowable increase or allowable decrease means what?
What's in the basis?
-the optimal solution
-the basic variables (non-zero) and the Z value
What is another term for the reduced cost?
Another name for the consumption ratio?
A sub ij
RHS is the function of?
the A sub ijs times the amount you make of X1 (X2, X3, X4)
Constraint lower boundary =
Constraint RHS (b sub i) minus the allowable decrease, OR add the allowable increase
What do aij, bi, and Cj stand for?
What are the synonyms of each
Are these values parameters or variables?
- aij = Consumption Ratio/Technical Coefficient; the amount of change in Z value with additional units of corresponding resources
- bi¬ = RHS = Amount of resource required and/or available, depending upon sign of inequality/if it is an equality constraint
- Cj = objective function coefficient; the change in Z value with additional production of corresponding decision variable
- THESE ARE PARAMETERS
What one constraint does every LP problem have?
If an LP problem has decision variables that represent proportions or percentages, what constraint(s) must be included?
• Sum of Decision Variables = 1
What are key characteristics of Mixing/Blending Problems?
• When writing objective function/constraints, percentages representing parameters are generally turned into decimals for purposes of the problem & coming up with an answer that actually makes sense
The decision variables are what you as the decision maker have __________ __________.
• A _____________ variable is a non-zero variable.
• A _____________ variable is a zero variable.
• When a constraint is used to capacity, then it is a _______________ constraint or a ______________ variable.
• When a resource is not used to capacity, then it is a ______________ variable.
• If you have a surplus or a slack constraint, then you have a ____________ variable.
c* variable is a non-zero variable.
c* variable is a zero variable.
• When a constraint is used to capacity, then it is a
constraint or a
• When a resource is not used to capacity, then it is a
• If you have a surplus or a slack constraint, then you have a
• A ___________ occurs when you have used more than the minimum amount required.
• A ___________ occurs when you have used less than the maximum possible amount available.
• A ___________ occurs when you have used < the amount of resource available.
• A ___________ occurs when you have used > the amount required of a resource.
s* occurs when you have used more than the minimum amount required.
k* occurs when you have used less than the maximum possible amount available.
k* occurs when you have used < the amount of resource available.
s* occurs when you have used > the amount required of a resource.
• What is a shadow price?
• When will the shadow price ALWAYS be zero?
• When the shadow price is
t* zero, what type of constraint do you have?
• When maximizing profit, shadow prices are above and beyond what you are already paying for a_______________.
• Marginal Value (in terms of contribution to Z-value) of a Resource
___________ is the process of starting at the origin and moving from point to point around the feasible region in order to find the optimal solution.
___________ is the action of moving from one extreme (/________) point to another.
Sensitivity analysis determines when a change in a ____________ will cause a change in the basis.
For a maximization problem, we will only produce a new item if the item's unit contribution is __________________.
• The basis changes when an objective function coefficient is changed to a value outside of the ______ range or a constraint is changed to a value outside of the _____/_____ range. When the basis changes, so do the decision variables' ________ ___________ and the resource's ___________ ______________.
The basis changes when an objective function coefficient is changed to a value outside of the
range or a constraint is changed to a value outside of the
range. When the basis changes, so do the decision variables'
and the resource's
• If we are considering changing the amount of a constraint, we need to look at the _____ range.
• If we are considering changing the price of our product (decision variable), we need to look at the ________ range.
C sub j
• If the coefficient of the objective function is changed to a value inside the allowable range, will the basis change? ________ What if the value is on the border of the allowable range? ________
i. Will the final Z value change? If so, by how much?
- No, except there would be alternative optima indicating that more than one combination of basic variables would result in the same Z value corresponding to that particular Cj value
-Yes - (new Cj - old Cj) * amount of Xi produced
1. Reduced cost represents how much each __________ ___________ ____________ would have to improve before the corresponding __________ ___________ could assume a positive (__________) value in the optimal solution.
• It also represents how much the Z value will increase/decrease if we decide to produce a product we _____ _____ currently ____________.
• In a minimization problem, improvement refers to a ___________ in the objective function coefficient.
• In a maximization problem, improvement refers to an ___________ in the objective function coefficient.
• In a minimization problem, the range for a reduced cost is: ____________
• In a maximization problem, the range for a reduced cost is: ____________
1. Reduced cost represents how much each
objective function coefficient
would have to improve before the corresponding
could assume a positive
value in the optimal solution.
• It also represents how much the Z value will increase/decrease if we decide to produce a product we
• In a minimization problem, improvement refers to a
e* in the objective function coefficient.
• In a maximization problem, improvement refers to an
e* in the objective function coefficient.
• In a minimization problem, the range for a reduced cost is:
• In a maximization problem, the range for a reduced cost is:
(neg infinity, 0)
Unit contribution is another name for __________ ___________, and is calculated from (choose one: aij, bi, and Cj) _____ - _____________ _______________.
Unit contribution is another name for
, and is calculated from (choose one: aij, bi, and Cj)
Cj - opportunity cost
The opportunity cost is calculated by....
take A sub ij and multiply it by U (shadow price) for all constraints. Sum all of these.
For a minimization problem, we will only produce a new item if the item's unit contribution is __________________.
If final value is basic, reduced cost is?
How do you find the reduced cost?
1. Look at final value, if it is non-zero, RC is zero
2. If final value = 0 check if it is a max or a min problem
3. Max prob. the reduced cost is the opposite of allowable increase
3. Min prob. the reduced cost is the
In a sub ij, the i represents the ________, value, while the j represents the ______ position
List the two rates of change
Cj and Aij
What determines when a change in the parameters of a problem will cause a change in the basis?
What are the four facts about a reduced cost?
- represents a unit contribution
- when considering to produce a new decision variable, is a function of the new variable's objective value coefficient and the opportunity cost associated with producing a unit of the new variable (definition)
- Is always negative when the final value is 0
- represents the increase or decrease to the Z value from producing another unit of its corresponding decision variable
If there is a row of zeros in the decision variables final value and reduced cost then....?
Alt. optima exists
When can you be sure that a constraint is non binding and the resource has no marginal value to the optimal value of the solution?
When the shadow price of a given constraint has a value of zero
If the basis changes what else changes?
The reduced cost
Three ways to know alt. optima exist?
- final and shadow price are zeros
- final and reduced are zero
- one or more zero(s) in the allowable increase/decrease column of the decision variable portion
The slope of the level curve for the objective function value can be changed by
changing a coefficient in the objective function
The solution to an LP problem is degenerate if
the right hand sides of any of the
have an allowable increase or allowable decrease of zero
With respect to LP problems, a better name for the reduced cost is
When would you have more basic variables than constraint variables?
You won't - it is mathematically impossible
The amount used of a resource is represented by...
left hand side (Aij times X)
Which of the following is not true of redundant constraints?
A. always have shadow price of 0
B. always have marginal value of 0
C. always have a non-zero U sub i
D. they are non-binding
E. all of the above are true
T/F If the opportunity cost of producing a new product is equal to the potential product's Cj value, then alternative optima would result from its production
Which of the following is true about the basis?
A. it reports optimal value
B. it reports the basic variables
C. it provides Cj and RHS ranges
D. A and B
E. all of the above
How do you find the Cj value when you have RHS range, allowable increase, and allowable decrease
Lower side of the range plus allowable decrease. Upper side of the range minus the allowable increase.