2. Quantitative Analysis & its Steps
Quantitative Analysis
Procedures for finding the “best” or optimal solution/s (Anderson et al., 2019).
The process of developing & solving models because models are the essence of quantitative analysis
It includes four steps:
Model Development
Data Preparation
Model Solution
Report Generation
1. Model Development
Models are representations of real objects or situations presented in various forms.
Generally, experimenting with models (as to the real thing) requires less time, is less expensive, and involves less risk.
The more closely the model represents the real situation, the more accurate the conclusions and predictions will be.
The three forms of models are:
Iconic models
physical replicas (scalar representations) of real objects
car models/plane models
Analog models
Physical in form, but do not physically resemble the modeled object
gas meter in cars/thermometer
Mathematical models
represent real-world problems through a system of mathematical formulas and expressions based on key assumptions, estimates, or statistical analyses.
Objective Function
a mathematical expression that describes the problem's objective, such as maximizing profit or minimizing cost.
e.g. Suppose x denotes the number of units produced and sold each week, and the firm’s objective is to maximize total weekly profit. With a profit of P500 per unit, the objective function is 500x. (P=500x)
Constraints/Non-negativity
a set of restrictions or limitations, such as production capacities
e.g. 6 hours are required to produce each unit and only 48 hours are available per week. The production capacity constraint is given by 6x ≤ 48.
e.g. The value of 6x is the total time required to produce x units; the symbol indicates that the production time required must be less than or equal to the 48 hours available.
Uncontrollable Inputs
environmental factors that are not under the control of the decision-maker.
e.g. the profit per unit (P 500), the production time per unit (6 hours), and the production capacity (48 hours) are environmental factors not under the control of the manager or decision-maker
Controlled Inputs/Decision Variables
factors the decision maker specifies, such as the number of product units to produce.
e.g., the production quantity "x" is the controllable input of the model.
Note: Controlled Inputs (Decision Variables) & Uncontrollable Inputs (Environmental Variables) are placed in a mathematical model to generate Outputs (projected results)
Types of Math Models
Deterministic Model
if all uncontrollable inputs to the model are known and cannot vary
Stochastic/Probabilistic Model
if any uncontrollable inputs are uncertain and subject to variation.
Stochastic models are often more difficult to analyze.
e.g. If the number of hours of production time per unit could vary from 3 to 6 hours depending on the quality of the raw material, the model would be stochastic.
In selecting models, one should consider:
Cost/benefit considerations
These must be made when selecting an appropriate mathematical model.
For example, a less complicated (and perhaps less precise) model is more appropriate than a more complex and accurate one if cost/ease of solution is considered.
2. Data Preparation
Data preparation is not a trivial step due to the time required and the possibility of data collection errors.
A model with 50 decision variables and 25 constraints could have over 1300 data elements.
Often, a fairly large database is needed.
Information systems specialists might be needed.
3. Model Solution
The analyst attempts to identify the alternative (the set of decision variable values) that provides the “best” output for the model.
The “best” output is the optimal solution.
If the alternative does not satisfy all of the model constraints, it is rejected as being infeasible, regardless of the objective function value.
If the alternative satisfies all of the model constraints, it is feasible and a candidate for the “best” solution.
Model Testing and Validation
Often, goodness/accuracy of a model cannot be assessed until solutions are generated.
Small test problems having known, or at least expected, solutions can be used for model testing and validation.
If the model generates expected solutions, use the model on the full-scale problem.
If inaccuracies or potential shortcomings inherent in the model are identified, take corrective action like:
Collection of more accurate input data
Modification of the model
4. Report Generation
A managerial report based on the results of the model should be prepared.
The report should be easily understood by the decision-maker.
The report should include:
the recommended decision
other information about the results (e.g. how sensitive the model solution is to the assumptions and data used in the model)
Outline
Overview
Procedures for finding optimal solutions (Anderson et al., 2019)
Essence of quantitative analysis: model development and solving
Four Steps of Quantitative Analysis
Model Development
Models as representations of real objects/situations
Advantages of modeling: less time, cost, and risk
Accuracy correlates with model realism
Forms of Models:
Iconic Models: Physical replicas (e.g., car models)
Analog Models: Physical but not resembling the object (e.g., gas meters)
Mathematical Models: Use formulas based on assumptions/statistics
Objective Function: Describes the problem's goal (e.g., maximizing profit)
Constraints/Non-negativity: Restrictions on variables (e.g., production capacities)
Uncontrollable Inputs: External factors (e.g., profit per unit)
Controlled Inputs/Decision Variables: Factors specified by the decision-maker (e.g., production quantity)
Types of Math Models:
Deterministic Model: Uncontrollable inputs known
Stochastic/Probabilistic Model: Uncontrollable inputs uncertain/variable
Model Selection Considerations:
Cost/benefit analysis for model complexity vs. accuracy
Data Preparation
Importance of data preparation due to time required
Potential for data collection errors
Large databases may be required
Possible need for information systems specialists
Model Solution
Identifying the alternative providing the best output
Definition of optimal solution
Feasibility vs. infeasibility of alternatives
Model Testing and Validation:
Assessing model accuracy through known solutions
Corrective actions for inaccuracies (e.g., data collection, model modification)
Report Generation
Preparing a managerial report based on model results
Report should be understandable for decision-makers
Key components of the report:
Recommended decision
Sensitivity analysis of model results to assumptions/data used
Conclusion
Importance of each step in the quantitative analysis process for effective decision-making.
Mind Map
Overview
Procedures for finding optimal solutions (Anderson et al., 2019)
Essence of quantitative analysis: Developing & solving models
Steps in Quantitative Analysis
Model Development
Models as representations of real objects/situations
Benefits: Less time, cost, and risk
Accuracy linked to model realism
Forms of Models:
Iconic Models: Physical replicas (e.g., car models)
Analog Models: Physical but not resembling the object (e.g., gas meter)
Mathematical Models: Formulas based on assumptions/estimates
Objective Function: Describes the problem's goal (e.g., maximizing profit)
Constraints/Non-negativity: Restrictions on variables (e.g., production capacity)
Uncontrollable Inputs: External factors (e.g., profit per unit)
Controlled Inputs/Decision Variables: Specified by decision-maker (e.g., production quantity)
Types of Math Models:
Deterministic Model: All inputs known
Stochastic Model: Inputs uncertain/variable
Data Preparation
Critical step; time-consuming and prone to errors
Large databases often required
Potential need for information systems specialists
Model Solution
Identify alternatives providing the "best" output
Optimal solution vs. infeasible solutions
Model Testing and Validation:
Assessing model accuracy through test problems
Corrective actions if inaccuracies are found (e.g., data collection, model modification)
Report Generation
Managerial report based on model results
Must be understandable for decision-makers
Should include:
Recommended decision
Sensitivity analysis of model results
Considerations in Model Selection
Cost/benefit analysis
Simplicity vs. complexity in model choice
Conclusion
Quantitative analysis involves systematic steps to develop, solve, and report on models to find optimal solutions in decision-making contexts.