1) Ch 2 Flashcards
Optimization
allocating limited resources most effectively (maximize profits)
Optimization traits
It is deteminsitic (all input data and parameters are known with certainty)
Simplest type of Optimization is called
Linear Programming
Goals of Optimization Problems
1) max profit
2) min costs
why do we struggle to achieve the goals of optimization problems
supply or demand constraints
How to do LP model formulation
1) determine the mathematical formulation
2) identify the optimal solution
3) interpret the results
Step 1 of LP model formulation
1DETERMINING THE MATHEMATICAL FORMULATION
-> What are the decision variables (x and y)
-> what is the objective function using the decision variables (this is profit or cost formula)
-> what are the constraints
Step 2 of LP Model Formulaiton
Identify the optimal solution
-> Graphically or Spreadsheet
Step 3 of LP Model Formulaiton
Interpret the results
-> what if scenarious to see impact of parameter change
Basic assumptions of an LP model
1) certainty
2) prooportionality and linearity
3) additivity
4) divisibility
Certainty- assumption
all paramenters are known and constant (no probabiilities, 10% chance, 20 % chance etc.)
proportionality and linearity- assumption
the objective function and constraints are linear and proportional
1 unit= 3 hours then 5 units = 15 hours
Additivity- assumption of LP model
the total of all activities is the sum of individual activities
Divisibility:
solutions need not be whole numbers (not always integers)
What can u not have in a LP statement
IF statement
MIN MAX VALUES
SQUARES ^2
SQRTS
MULTIPLY DECISION VBLS
DIVIDE DECISION VBLS
technical steps of LP model formulaiton
1) understand the problem (what are we trying to solve)
2) identify decision vbls (x and y)
3) state the objective function (relating x and y)
4) state the constraints (conditions or limits on the formula)
What are decision varables usually
any amount that can be controlled, like units, litres, people, dollars
How to determine the objective function
Look wherever it says profit or cost
profit per thing(vbl1)+profit per other thing(vbl2)
Making the constraint table
- one column each for each variable
- one row each for the major constraints
- individual rows for each terms individual constraint
-non negativity constraint (use comma and put them in one cell)
MAKE SURE TO WRITE (SUBJECT TO:) AND ALSO NOTE THE UNITS
How to deal with ratio constraints
wheat>= 0.2*(#hectares)
1) Figure out the expression, write it as an inequality
If it is like 20% of land must be wheat then:
# wheat/# hectares total >= 0.2
2) then do math to isolate one vairable
3) simplify the inequlaity further
KEEP IT LIKE IT IS!
Ratio Constraints trait
There are variables in the constraint
DETERMINING ONSTRAINTS??
-what limitations do you have on the situation
- is it percents and ratios? you will have variables in your limits
-Different constraints can be in different units
(hours, minutes, litres, hectares) but each
type of constraint must have the same units(sales volume, labour time)
Costs($) are not included in the constraints, only usage SO…
Costs($) are not included in the constraints, only usage so EXCLUDE THEM FROM THE CONSTRAINTS CHART
UNITS
-Different constraints can be in different units
(hours, minutes, litres, hectares) but each
type of constraint must have the same units(sales volume, labour time)
How to do iso- profit line method for optimization
1) Determine your goal: max profit or min cost
2) Determine decision variables
3) Draw out constraints (linear lines) LABEL THEM
4) Draw out the iso-profit line by solving the equation
5) understand that the iso profi tline can only be parallel to the existing line so move it down until you hit an intersection of constraints
6) solve the system of equaitons form the two constraint lines
7) that is ur max profit
How to use corner point method to find max profit/min cost
1) define the goal
2) find decision variables
3) draw out all the constraints
4) set up systems of equations at all the corners, and solve for the coordinates and also the profit
5) highlight the point with the largest profit!
When do you need to use a spreadsheet to model LP problems
when there are 2+ decision variables
How to Optimize LP models in a spreadsheet
1) Organize the data for the model
-> Set up decision variables
-> Set up the profit
->Set up the OBJECTIVE FUNCTION
(coeffdecision vbl1 + coeffdecisionvl2)
2) Create a constraints section
-> For each constraint have a different row
3)Build the model using formulas, fill out all the constraints, REFER EVERYTHING TO DECISION VBL CELLS
4) USE SOLVER
how to use solver?
1) select the objective function
2) is it max or min
3) select changing cells as vbl cells
4) select the constrians, MIND THE SIGNS
5) solving method = simplex lp
What formulas should you use for constraint and objective cells
ONLY SUM, AVERAGE, SUMPRODUCT
SUMPRODUCT SHOULD BE
SUMPRODUCT (CONSTANT CELLS,CHANGING CELLS)
If a constraint is a ratio and non-linear?
Have a dynamic right hand side constaint w/ variable in the limit
At least 20% of land must be wheat
wheat >= 0.2 *(wheat + canola)
w>= 0.2*(w+c)
leave it
Use Sumproduct when?
When you have 2+ variables in objective function
Different revenues per type
$7.50vase + $4.40cup + $6.70*bowl
integer programming
when DECISION VBLS you are only allowed to have integers, you must use integer linear programming (ILP)
binary programming
when DECISION VBLS only allowed 2 integers (0 or 1) use BLP
0= NO ACTIVITY
1= ACTIVITY
Drawback of ILP
No sensitivity report in solver
If you need your decision variables too be integers what do you do? 3 steps
1) When you do integer constraints add the int
2) AND ALSO GO INTO OPTIONS AND UNCLICK IGNORE INTEGER CONSTRAINT
3) Type 0 in the integer optimality
Logical Conditions:
At least 2 of the projects 1,3,5,6 must be undertaken
X1 + X3 + X5 +X6 >= 2
Whatever is 1 is true
Whatever is 0 is true
Projects 3 and 5 can only be taken simultuaneously
X3 =X5
Either project 1 or 4 must be undertaken not both
X1 + X4 =1
Project 4 cant be done unless 2 and 3 are
2X4 < = X2 + X3
or
X4<= X2 and X4 <= X3
Sensitivity Analysis is what
basically measuring how much the values change if the decision vairable increase by different amounts
what are the two ways we measure changes in sensitivty analysis?
1) Changing the value of OFC (objective function coefficeint) [Represented by the slope of the profit line[
2) changing the value of the right hand side constraints (The resources) [ Changes the shape of the feasible region of graph]
THIS IS IMPORTANT
what are the two ways we measure changes in sensitivty analysis?
1) Changing the value of OFC (objective function coefficeint) [Represented by the slope of the profit line[
2) changing the value of the right hand side constraints (The resources) [ Changes the shape of the feasible region of graph]
Sophisticated method of sensitivty analusis?
use the excel solver sensitivty report
How to clean sensitvity report?
set everything to decimal places!!!
Undestanding excel senstivity report> what are the two sections for?
TOP SECTION: relates to objective function
BOTTOME SECTION: relates to constsraints
wwhat does 1E+30 represent?
infintiy
OFC section of sensitivty report:
Maximize 200W+ 150C
1) what does allowable increase/decrease mean
1) This is how much you can change the coefficients 200 and 150, and still have the same optimal solution (Same values in decision variable cells)
What is the product mix?
the values in the decision variables cellsW
What if there is an increase or decrease outside of the allowables increase and decrease in excel sensitivty analysis?
the problem needs to be re-solved!!!
what is a binding constraint
constriant that has 0 slack in the optimal solution
why do binding constarints matter
we can not make any improvements to the original optimal solution- so in senstivity analysis all we can do is change the RHS value
Constraints section of Sensitivty Analysis:
What is the shadow price?
The change in the OFC value if the RHS value of the specific cosntraint increase by 1 unit
EX: How uch the profit of canola would go up by if the demand for canola went up by 1
Why is the shadow price useful?
allows us to determine how valuable additional resources (increase the rhs constraint) can be comparing to the costs
Do all constraint s have shadow price?
no !!! only the binding constraints do cuz the non-binding ones can still be increased (no extra analysis needed)
How to determine the impact on profit by changing rhs constratins?
USE SHADOW PRICE
Impact on profit = change in resource * shadow price
Process: Say you are given a chance to increase your RHS constraint, how do you decide if yes or no?
1) Check if this is allowable or not (increase/decrease)
2) If yes: then
Impact on profit= shadow price *change in resource
3) Compare the impact on profit, with the cost that will be found (use critical thinking)
4) should you take the deal?
what is reduced cost
amount the objective function coeffs would have to change to make the respective decision variable important in the solution
how to use the reduced cost formula?
Profit= objective coefficient- reduced cost
**keep the sign
this is what the profit must be to make the resepective variable important
When do you have a negative shadow price
when the constraint is binding and is a >= constraint (maximizing)
still use
impact on profit =change in resource *shadow price
SPECIAL CASE!!!!!!!!!!1
ALL OR NONE DEALS: HOW DO YOU SOLVE
1) These questions will force u to go beyond allowable increase
2) its okay- go through w the process
3) multiply increase by shadow price
4) if profit > the cost , they should make the deal and throw away the rest since it is still beneficial
If the resource is a sunk or fixed cost:
→ the cost is not included in the objective cell.
→ Since sunk / fixed costs stay the same regardless of
quantity, we want Excel Solver Optimization to ignore
them when it tries different combinations
If the resource is a variable cost:
→ The cost is included in the objective cell
→ Shadow price is what you would gain net of the
variable cost for one more unit of the resource.
→ This is the situation in the Wilton Pottery case (but not
in the assignment questions nor the exam question).
90
Fixed Cost Versus Variable Cost
try processing this baby
greater
has to have less than or equal to sign
smaller
has to have equal sign
In assignment problesm
set the solve constraints to =
in transportation problems and assignment problems
you do NOT need the binary rule
the conditions on slide 128 might be on digitial exam
YES
LOOK AT SLIDE 142 TO FIGURE OUT HOW TO FORMULATE RATIO CONSTRAINTS!!!!
DO THIS
