Ch. 2 & 3 Resource Management Flashcards
1 An analytical process for allocating scarce resources among competing alternatives
2 Typically there is a single objective subject to a number of constraints
3 The task is to find the values for the decision variables such that the value of the objective function is optimized (maximized or minimized)
4 Linear programming is part of a larger family of optimization tools called mathematical programming
Principles of Linear Programming (Optimization)
A mathematical technique used to optimize the allocated scarce resources among competing alternatives (decision variables) subject to a series of linear constraints
Linear programming
The quantities that the decision-makers would like to determine that optimizes the objective function
Decision variables
What are the three components used in Linear Programming?
Variables
Constraints
Objective Functions
What are the three linear programming assumptions?
1 Proportionality
2 Deterministic
3 Non-integer
1 Organize the data for the model on the spreadsheet
2 Reserve separate cells in the spreadsheet to represent each decision variable in the algebraic model
3 Create a formula in a cell in the spreadsheet that corresponds to the objective function in the algebraic model
4 For each constraint, create a formula in a separate cell in the spreadsheet that corresponds to the left-hand-side of the constraint
Steps in Implementing an LP Model in a Spreadsheet
First step of LP Model
Start with an objective function that maximizes or minimizes
Second step of LP Model
Right hand side - typically represents resources that are available or requirements
Third step of LP Model
Constraints - can have different types of constraints
ex. Less than or equal to constraints - associated with a maximum level of a resource
Greater than or equal to constraints - associated with a requirement
Fourth step of LP Model
Left hand side - typically rates in which the resources are consumed or meeting requirements
What is the goal of Linear Programming Model?
A model used for determining the “best” allocation
Linear Programming Assumptions
- Linear relationships
- Non-integer solutions
- Independent variables
- Deterministic coefficients
Linear Programming Outputs
- Decision variables values
- Objective function values
- Surplus and slack
- Sensitivity analysis
A limitation on the values of the decision variables
Constraint
The Constraints of the LP Problem define what?
Feasible region
A constraint that does not affect the optimal solution
Redundant constraint
A solution that simultaneously satisfies all the constraints
Feasible solution
A solution with inconsistent constraints
Infeasible solution
The minimum number of constraints required to generate an infeasible solution is how many?
1
A feasible solution that optimizes the value of the objective function
Optimal solution
A linear relationship used to specify the payoff.
identifies some function of the decision variables that the decision maker wants to either MAXimize or MINimize
Objective function
The amount of resources remaining (
Slack
The amount above the minimum requirements (>)
Surplus
The minimum changes in the objective function coefficients required to yield a new optimal solution
Coefficient of sensitivity
A mathematical technique for allocating scarce resources where some of the decision variables are limited to integer or binary values
Integer programming
More than one set of values for the decision variables that optimizes the objective function
Multiple optimal solutions
When does a Multiple optimal solutions in the LP model?
They occur in the LP model when the objective function is parallel to the binding constraint
Maximum willing to pay for one more unit of resource
Shadow prices
A condition where one or more of the decision variables are not constrained
Unbounded solution
Solver uses a special algorithm to solve the problem
Simplex method
LP Table:
Columns =
Rows =
Final Row =
Columns = Decision Variables
Rows = Requirements
Final Row = Profitability
In the LP Table - Its not the number of units to be produced or the unit profit, it’s the product of those two:
Number of units you can make multiplied by the corresponding profitability, summed across the product line.
Determines how efficiently an operating unit (or company) converts inputs to outputs when compared with other units.
Data Envelopment Analysis (DEA)
an area of business analytics that finds the optimal, or most efficient, way of using limited resources to achieve the objectives of an individual business.
Often referred to as OPTIMIZATION
Mathematical Programming -
Determining Product Mix
Manufacturing
Routing Logistics
Financial Planning
Instances you can apply Mathematical Optimization
taking a practical problem and expressing it algebraically in the form of an LP model
Formulating
1 Understand the Problem
2 Identify the decision variables
3 State the objective function as a linear combination of the decision variables
4 State the constraints as linear combinations of the decision variables
5 Identify any upper or lower bounds on the decision variables
Steps to Formulating an LP Model
the set of points or values that the decision variables can assume and simultaneously satisfy all the constraints in the problem
Feasible Region
The optimal solution will always occur at a point in the feasible region where two or more of the boundary lines of the constraints intersect
Corner/Extreme points
the objective functions are sometimes referred to as this
Level Curves
there can be more than one feasible point that maximizes or minimizes the value of the objective function
Alternate optimal solutions
involves increasing the upper limits or reducing the lower limits to expand the range of feasible solutions
Loosening constraints
Every solvable LP problem has:
feasible region and an optimal solution
Solving LP problems graphically is easy when there are
two decision variables
Three Variable Decision graphs are hard to visualize because
they are three-dimensional.
Spreadsheets are best used for:
multiple variable decisions
The main challenge in linear programing problems is ensuring that you:
formulate the LP problem correctly and communicate this formulation to the computer accurately.
built-in spreadsheet optimization tool that excel offers?
Solvers
“1. Organize the data for the model on the spreadsheet.
“(Note that some or all of the coefficients and values for an LP model might be calculated from other data, often referred to as the primary data.
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“2. Reserve separate cells in the spreadsheet to represent each decision variable in the algebraic model.
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3. Create a formula in a cell in the spreadsheet that corresponds to the objective function.
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4. For each constraint, create a formula in a separate cell in the spreadsheet that corresponds to the left-hand side (LHS) of the constraint.
What are the tips for Solving LP Problems in a Spreadsheet?
Three components of our spreadsheet model for Solver
Objective Cell (Target Cell) Variable Cells (Changing Cell) Constraint Cells
the cell in the spreadsheet that represents the objective function
Objective Cell (Target Cell)
the cells in the spreadsheet representing the decision variables
Variable Cells (Changing Cell)
the cells in the spreadsheet representing the LHS formulas on the constraints
Constraint Cells
indicate that the decision variables can assume only nonnegative values
occurs when there are simple lower bounds on our decision variables represented by X1 >= 0 and X2 >= 0
Nonnegativity conditions
provides an efficient way of solving LP problems and, therefore, requires less solution time
Simplex method
Communication, Reliability, Auditability, Modifiability
How can one achieve the end goal of a logical spreadsheet design?
A spreadsheet’s primary business purpose is communicating information to managers.
Communication
This term shows The output a spreadsheet generates should be correct and consistent.
Reliability
A manager should be able to retrace the steps followed to generate the different outputs from the model in order to understand and verify results.
Auditability
A well-designed spreadsheet should be easy to change or enhance in order to meet dynamic user requirements.
Modifiability
- Organize the data, then build the model around the data.
- Do not embed numeric constants in formulas.
- Things which are logically related should be physically related.
- Use formulas that can be copied.
- Column/rows totals should be close to the columns/rows being totaled.
- The English-reading eye scans left to right, top to bottom.
- Use color, shading, borders and protection to distinguish changeable parameters from other model elements.
- Use text boxes and cell notes to document various elements of the model.
Spreadsheet Design Guidelines
will not change if the formula containing the reference is copied to another location
absolute cell reference
rule-of-thumb for making decisions that might work well in some instances, but is not guaranteed to produce optimal solutions or decisions
heuristic
- Notice the coefficient for X2 in the ‘corn’ constraint is 0.05/8000 = 0.00000625
- As Solver runs, intermediate calculations are made that make coefficients larger or smaller.
- Storage problems may force the computer to use approximations of the actual numbers.
- Such ‘scaling’ problems sometimes prevents Solver from being able to solve the problem accurately.
- Most problems can be formulated in a way to minimize scaling errors…
A Comment About Scaling
When using DEA
output variables should be expressed on a scale where “more is better” and input variables should be expressed on a scale where “less is better”.
Risk Solver Platform includes a number of custom functions that all begin with the letters PSI (short for polymorphic spreadsheet interpreter)
Psi Functions
When Running Multiple Optimizations:
PsiCurrentOpt( )
PsiOptValue( )
How to Formulate a LP Model
Identify the decision variables
Construct the objective function
Identify the right-hand-side (RHS) values
Specify the types of constraint relations ()
Develop the coefficients for the left-hand-side (LHS)
Make sure that both the RHS and LHS have the same units
Three Special Conditions in LP
Unbounded solution:
Infeasible solutions:
Multiple optimal solutions:
One or more of the decision variables are unconstrained
an unacceptable situation. And fortunately the Excel software will give us a heads up. In those cases, we need to investigate what variables are not constrained.
Unbounded solution:
No feasible solution can be found since one or more of the constraints are inconsistent
when we cannot find a solution that matches all of the constraints.
Infeasible solutions:
There exists more than one solution that optimizes the objective function
There may be more than one way in which we can allocate scarce resources to achieve the same value of the objective function.
Multiple optimal solutions:
The inputted or internal worth of resources owned by the firm
Shadow Prices
Is an indication of the maximum the firm would be willing to pay for an additional unit of a specific resource
Shadow Prices
Are only viable over the range between the lower and upper limits given by the constraint sensitivity analysis
Shadow Prices
Service center staff (1,2,3,…) Snow removal trucks (1,2,3,…) Nuclear power plants (1,2,3 …) Funding multiple projects (0/1) Facility location (0/1) Job offers (0/1)
Integer Variable Problems examples
Resource allocation is a primary function of management.
Linear programming provides an analytical approach to resource allocation.
Shadow prices indicate the worth of a resource.
Blending involves combining resources.
Transportation model optimizes the production and shipment schedule.
LP RECAP POINTS
