Ch. 2 & 3 Resource Management Flashcards by Anthony LoBasso

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)

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A mathematical technique used to optimize the allocated scarce resources among competing alternatives (decision variables) subject to a series of linear constraints

Linear programming

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The quantities that the decision-makers would like to determine that optimizes the objective function

Decision variables

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What are the three components used in Linear Programming?

Variables
Constraints
Objective Functions

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What are the three linear programming assumptions?

1 Proportionality
2 Deterministic
3 Non-integer

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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

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First step of LP Model

Start with an objective function that maximizes or minimizes

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Second step of LP Model

Right hand side - typically represents resources that are available or requirements

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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

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Fourth step of LP Model

Left hand side - typically rates in which the resources are consumed or meeting requirements

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What is the goal of Linear Programming Model?

A model used for determining the “best” allocation

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Linear Programming Assumptions

Linear relationships
Non-integer solutions
Independent variables
Deterministic coefficients

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Linear Programming Outputs

Decision variables values
Objective function values
Surplus and slack
Sensitivity analysis

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A limitation on the values of the decision variables

Constraint

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The Constraints of the LP Problem define what?

Feasible region

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A constraint that does not affect the optimal solution

Redundant constraint

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A solution that simultaneously satisfies all the constraints

Feasible solution

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A solution with inconsistent constraints

Infeasible solution

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The minimum number of constraints required to generate an infeasible solution is how many?

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A feasible solution that optimizes the value of the objective function

Optimal solution

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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

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The amount of resources remaining (

Slack

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The amount above the minimum requirements (>)

Surplus

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The minimum changes in the objective function coefficients required to yield a new optimal solution

Coefficient of sensitivity

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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. - "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. - 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