Modeling workflow
Modeling Workflow
Modeling is the process of translating a real-world decision problem into mathematics.
In operational research, the goal is to build a model that can be solved by an optimization method.
Core Idea
A model must answer three questions:
- What do we choose?
- What do we want?
- What limits us?
Mathematically:
| Question | Model Part |
|---|---|
| What do we choose? | decision variables |
| What do we want? | objective function |
| What limits us? | constraints |
Standard Workflow
A typical OR workflow is:
- identify the problem
- identify the variables
- formulate the mathematical model
- solve the model
- validate the model
- analyze the result
Step 1: Understand the Problem
Read the problem carefully.
Identify:
- the decision-maker
- the available choices
- the goal
- the limitations
- the units
- the time period
Example questions:
- Are we deciding production per day, week, or year?
- Are variables measured in units, kilograms, euros, or hours?
- Is the goal profit, cost, revenue, or return?
Step 2: Define Decision Variables
Decision variables are the unknowns.
They should be precise.
Bad:
\[x_1 = \text{lettuce}\]Good:
\[x_1 = \text{hectares allocated to lettuce}\]Bad:
\[x_2 = \text{fund B}\]Good:
\[x_2 = \text{number of units purchased of fund B}\]Step 3: Write the Objective Function
The objective function expresses the goal.
Use maximization for benefits:
\[\max f(x)\]Examples:
- maximize profit
- maximize return
- maximize production
- maximize number of hires
Use minimization for costs:
\[\min f(x)\]Examples:
- minimize cost
- minimize time
- minimize waste
- minimize distance
Step 4: Write the Constraints
Constraints express the rules and limitations.
Common phrases:
| Phrase | Mathematical Meaning |
|---|---|
| at most | $\le$ |
| no more than | $\le$ |
| maximum | $\le$ |
| at least | $\ge$ |
| no less than | $\ge$ |
| minimum | $\ge$ |
| exactly | $=$ |
Step 5: Add Sign Restrictions
Most quantity variables are nonnegative:
\[x_j\ge 0\]Examples:
- production cannot be negative
- investment cannot be negative
- transported goods cannot be negative
- hectares cannot be negative
- food quantity cannot be negative
If variables must be whole numbers, add:
\[x_j\in\mathbb{Z}\]If variables are yes/no, add:
\[x_j\in\{0,1\}\]Step 6: Check Units
Every coefficient must match the unit of the variable.
Example:
If:
\[x_1 = \text{hectares of lettuce}\]and each hectare requires $18$ labor hours, then labor use is:
\[18x_1\]If:
\[x_2 = \text{hectares of tomatoes}\]and each hectare requires $24$ labor hours, then total labor is:
\[18x_1+24x_2\]Step 7: Check the Objective Units
All terms in the objective must have the same unit.
Example:
If lettuce revenue is euros and tomato revenue is euros, then:
\[\max 2000x_1+4500x_2\]is meaningful.
Do not add incompatible quantities such as:
\[\text{euros}+\text{hours}\]unless they have been converted into a common scale.
Step 8: Write the Final Model Clearly
A complete LP model should have:
- objective
- constraints
- sign restrictions
- variable definitions
Example structure:
\[\max c_1x_1+c_2x_2\]subject to:
\[a_{11}x_1+a_{12}x_2\le b_1\] \[a_{21}x_1+a_{22}x_2\ge b_2\] \[x_1,x_2\ge 0\]where:
\[x_1=\ldots,\qquad x_2=\ldots\]Example: Production Problem
A factory produces two products.
Let:
\[x_1 = \text{units of product A produced}\] \[x_2 = \text{units of product B produced}\]If profit per unit is $5$ and $12$, then:
\[\max 5x_1+12x_2\]If resource constraints are:
\[20x_1+10x_2\le 200\] \[10x_1+20x_2\le 120\] \[10x_1+30x_2\le 150\]and:
\[x_1,x_2\ge 0\]then the model is complete.
Example: Diet Problem
Let:
\[x_j = \text{quantity of food }j\]If food $j$ costs $c_j$ per unit, the objective is:
\[\min \sum_{j=1}^n c_jx_j\]If nutrient $i$ contained in food $j$ is $a_{ij}$ and the required amount is $b_i$, then:
\[\sum_{j=1}^n a_{ij}x_j\ge b_i\]for each nutrient $i$.
Also:
\[x_j\ge 0\]Example: Transportation Problem
Let:
\[x_{ij}=\text{quantity shipped from source }i\text{ to destination }j\]If shipping cost per unit is $c_{ij}$, then:
\[\min \sum_i\sum_j c_{ij}x_{ij}\]Supply constraints:
\[\sum_j x_{ij}\le s_i\]Demand constraints:
\[\sum_i x_{ij}\ge d_j\]Nonnegativity:
\[x_{ij}\ge 0\]Common Modeling Mistakes
Missing Variable Definitions
Do not write equations before defining variables.
The reader must know what every variable means.
Wrong Inequality Direction
“At most” means $\le$.
“At least” means $\ge$.
“Exactly” means $=$.
Missing Nonnegativity
If a variable represents a physical quantity, include:
\[x_j\ge 0\]Mixing Units
Do not mix kilograms, grams, euros, and units without converting.
Optimizing the Wrong Quantity
If the problem asks for maximum gain, do not accidentally minimize cost unless cost is the intended objective.
Validation
After writing the model, check whether it makes sense.
Ask:
- Can the variables take realistic values?
- Do all constraints match the problem statement?
- Are all units consistent?
- Does the objective represent the stated goal?
- Are nonnegativity or integer constraints needed?
- Is the model too restrictive or too loose?
Analyzing Results
After solving the model, interpret the result in words.
A complete answer should include:
- the optimal solution $x^*$
- the optimal value $z^*$
- the real-world meaning of each variable
- whether constraints are binding or nonbinding, if relevant
Example:
\[x^*=(6,3),\qquad z^*=66\]means:
- produce 6 units of product A
- produce 3 units of product B
- maximum profit is 66
Modeling Template
Use this template for exam problems:
Let:
x1 = ...
x2 = ...
Objective:
max/min ...
Subject to:
constraint 1
constraint 2
constraint 3
Sign restrictions:
x1, x2, ... >= 0
Checklist
You understand the modeling workflow if you can:
- define decision variables clearly
- write the objective function
- translate words into constraints
- choose the correct inequality direction
- add sign restrictions
- check units
- write the final model cleanly
- interpret the solution in words
See Also
- Optimization Foundations
- Decision Variables
- Objective Functions
- Constraints
- Feasible Sets
- LP Formulation
- Production Problems
- Transportation Problems
- Diet Problems
Exam checkpoint
In exam problems, translate the story into variables, objective, constraints, feasible set, and optimal value. Always state whether the problem is a maximization or minimization and what the units are.